diff --git a/-NFIT4oBgHgl3EQf8ysQ/content/tmp_files/load_file.txt b/-NFIT4oBgHgl3EQf8ysQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..690f4e677ada45a878acb677d8763b647990fce1 --- /dev/null +++ b/-NFIT4oBgHgl3EQf8ysQ/content/tmp_files/load_file.txt @@ -0,0 +1,713 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf,len=712 +page_content='Detecting Pump&Dump Stock Market Manipulation from Online Forums D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Nam D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Skillicorn School of Computing Queen’s University Kingston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Canada skill@queensu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='ca Abstract The intersection of social media, low-cost trading platforms, and naive investors has created an ideal situation for information-based market manipulations, especially pump&dumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Manipulators accumulate small-cap stocks, disseminate false information on social media to inflate their price, and sell at the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' We collect a dataset of stocks whose price and volume profiles have the characteristic shape of a pump&dump, and social media posts for those same stocks that match the timing of the initial price rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' From these we build predictive models for pump&dump events based on the language used in the social media posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' There are multiple difficulties: not every post will cause the intended market reaction, some pump&dump events may be triggered by posts in other forums, and there may be accidental con- fluences of post timing and market movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Nevertheless, our best model achieves a prediction accuracy of 85% and an F1-score of 62%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Such a tool can provide early warning to investors and regulators that a pump&dump may be underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' 1 Introduction New financial products and technologies have allowed naive investors to easily enter financial mar- kets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' This has increased the risk of manipulation, and detecting and investigating fraudulent activities has become much more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Many go undetected [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Social media has created new methods for manipulating markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' A scheme known as Pump and Dump (P&D) is one popular mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Fraudsters buy quantities of a stock, disseminate false information about it to artificially raise its price, and then sell their purchased shares at the higher price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Social media provides a channel for rapid dissemination and a pool of investors with little knowledge or experience who may not detect that the information is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Conventional approaches to detecting manipulation look for known patterns, and for anomalous activity such as exceeded thresholds for prices and trading volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Suspicious activities can be detected using sets of rules and triggers that cause notifications of potential manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' However, those methods struggle in the presence of behaviours that deviate from historical patterns [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Previous work has also focused on detecting manipulations so that regulators can penalise those who carry them out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' This does little to help investors, either to prevent their being deceived or recovering their investments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='11403v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='SI] 26 Jan 2023 Data-analytic techniques have the potential to detect false information as it being disseminated [11, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Natural language analytics can detect the posts in social media that are intended to pump particular stocks, providing a real-time warning to potential investors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' We investigate how well P&D schemes can be detected in posts on social media, by matching the language patterns in the posts to the pattern of stock price corresponding to a P&D manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' A penny stock is a stock that is traded by a small public company for less than $5 per share [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Many of these companies are known for their volatility due to their limited coverage by analysts and interest from institutional buyers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Because of their low price, retail investors can buy large quantities of these stocks without having to invest much money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' This, however, makes their prices volatile and so creates the potential for large returns on investments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' but also leaves them vulnerable to manipulation by malicious actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' One study found that 50% of manipulated stocks are those with a small market capitalization [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' It might be supposed that the connection between a social media post and a P&D event is too tenuous to be detected – after all, not every post will have the desired effect, and a P&D might be triggered by some less visible social media activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' We show that, at least for penny stocks, the connection is reasonably detectable, and we achieve prediction accuracies (that a post is intended to cause a P&D event) of 85%, with an F1 score of 67% (± 12 percentage points) from posts alone, and 62% (± 3 percentage points) from posts and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' 2 Tools Stance detection is a technique to determine the attitude or viewpoint of a text towards a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' It aims to detect whether the author of the text is in support of or against a given entity [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Some applications of stance detection have been in political debates, fake news, and social media [15, 26, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Empath is a tool that was developed by Fast et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' [13] for researchers to generate and validate new lexical categories on demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' It uses deep learning to establish connections between words and phrases used in modern fiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Given a small set of seed words that represents a category, Empath can provide new related terms using its neural embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' It also employs the use of crowd-sourcing to validate the terms that it considers are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Along with the ability to create new categories, Empath comes with 200 built-in, pre-validated categories for common topics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=', neglect, government, social media).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' SHAP (SHapley Additive exPlanation) is a tool that was developed by Lundberg and Lee [22] to determine the impact of each attribute on the output of a predictive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' It is based on Shapley values, a concept from game theory that determines a fair way to distribute the payoff for players that have worked in coalition towards an outcome [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Extreme Gradient Boosting is a decision-tree based ensemble algorithm that has become known for its speed and performance [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Decision trees are built sequentially so that each one reduces the errors of the previous one [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Random Forests is a decision-tree based ensemble algorithm with each tree built from a subset of the rows and columns of the dataset [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' This allows for variation among the trees and results in lower correlation among their predictions [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Support Vector Machines are a supervised learning algorithm that finds a hyperplane that best separates the data points from two classes [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Artificial Neural Networks are computational networks that are inspired by the biological ner- vous system [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' ANNs excel at prediction for data where the amount of information in each 2 Figure 1: Stages of Pump and Dump attribute is small and there are non-linear interactions among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Deep learning models are a class of extensions to ANNs that have solved long standing prediction problems in image recogni- tion and natural language [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Convolutional Neural Networks (CNNs) are a class of deep learning networks that were designed initially to work with images but work surprisingly well with sequence data such as texts as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Long Short-Term Memory (LSTM) deep learning networks are a type of recurrent neural network designed to handle the long-term dependencies present in sequence pre- diction problems [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Understanding text often requires looking ahead (think of verbs in German) and so processing text in both directions, using a biLSTM, provides better results for language [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' 3 Experiments Within a typical online forum, there are two different categories of texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The first is a post, which initiates a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The second is a set of comments responding to the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' For example, an individual may post saying that, in their opinion, a stock’s price is about to rise, with others respond by sharing their opinions in the same thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Responders may agree with the original post, or disagree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' P&D is an information-based manipulation, artificially raising the price of a stock through the dissemination of false information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' As shown in Figure 1, this manipulation strategy involves three different stages [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The operators of the scheme first purchase the stock that they are planning to manipulate (Accumulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Once they have acquired enough shares, they will disseminate false information to make it appear more desirable, driving up the price (Pump).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Once the price has risen to the desired level of profit, the operators sell off their shares before anyone uncovers that the information has no basis or the hype dies down (Dump).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' To identify P&Ds within the market, patterns associated with the scheme must be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' 3 Price Accumulation Pump Dump TimeWhile the method of conducting a P&D may vary, two indicators that can identify them are sharp changes in price and volume [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' A P&D will cause a significant price increase within a short amount of time, larger than the fluctuations that the stock typically experiences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' followed by a decrease once the dump phase has begun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The volume also increases as the stock gains interest among investors during and after the dissemination phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' However, the volume will typically not immediately experience as sharp a decline as the price when the operators begin to dump their shares because of the reluctance of investors to believe that the price is illusory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' If the profile of a P&D manipulation can be detected in the market, then the post that putatively caused it can be straightforwardly labelled and its language patterns investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' (Of course, it is possible that some of the apparent connections are spurious, but it is relatively unlikely that a post touting a particular stock will be disseminated exactly when the stock’s price and volume begin a sharp rise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Labelling comments is more complex, since the comments may agree with the original post, or disagree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Only the language of those that agree can contribute to predicting a P&D event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='1 Data Sources Two different data sources were utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The first is the popular online website Reddit, where users discuss the stock market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The second is Yahoo Finance, a financial market website that provides historical data about companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Reddit contains forums referred to as subreddits, each dedicated to the discussion of a specific topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Popular forums for the discussions of stocks are r/pennystocks, r/wallstreetbets, r/stocks, r/RobinHoodPennyStocks, r/TheWallStreet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' We use r/pennystocks and r/RobinHoodPennyStocks, Yahoo Finance is a website provided by Yahoo for investors to access financial news, market data, and basic financial tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Given a stock symbol or company name, it provides the relevant market data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Classification techniques such as Extreme Gradient Boosting (XGBoost), Random Forests, Sup- port Vector Machine (SVM), and Artificial Neural Networks (ANNs) were used to learn predictive models, and then to identify which attributes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' words) are most predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Figure 2 shows the experimental workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Figure 2: Experiment workflow Data from Reddit and Yahoo Finance were collected daily for the period October 1, 2019, to June 28, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' A breakdown of the data is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The majority of the data is retrieved 4 Redldit Yahoo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Finance Anomaly Detection Text Labelling Model Model Data Preprocessing Training Testing Historical Data Agreement Model Data Retriever Data Preparation Dataset Modelling Model Comparison/ EvaluationSubreddit Number of Posts Number of Comments Total r/pennystocks 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='049 234,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='149 246,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='198 r/RobinHoodPennyStocks 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='506 78,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='429 84,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='935 Total 18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='555 312,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='578 331,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='133 Table 1: Breakdown of records collected from subreddits Figure 3: Data Collection Volumes from r/pennystocks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' with about a third from r/RobinHoodPennyStocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The number of comments is much larger than the number of posts, with posts making up only about 5% of the texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' As shown in Figure 3, there was a sharp increase in the number of submissions over the period of data collection: r/pennystocks - 139,000 Members ⇒ 257,000 Members r/RobinHoodPennyStocks - 52,000 Members ⇒ 133,0000 Members This seems to reflect an increase in amateur stock market investing because of the covid-19 pan- demic, and a corresponding increase in manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='e, as manipulators look to take advantage of new, naive investors during the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Alerts and press releases by the SEC and the Canadian Securities Administrators warned new investors to be vigilant about the increasing number of P&D schemes that have occurred around that time [9, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The median number of words per post or comment was 22, and the total number of distinct words was 4,862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Replacing stock symbols by the market sector to which each business belongs allows us to see which sectors are discussed the most, and which are the targets of P&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Figure 4 shows that healthcare stocks are the most mentioned, followed by technology stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The pandemic clearly had an effect on both attention to markets and manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Temporal trends in the healthcare 5 10000 8000 Number of Records 6000 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='00 2000 DatesFigure 4: Histogram of market sectors discussed within subreddits sector, Figure 5 , show an increase in online activity at the beginning of the pandemic, and then a further increase in the middle of 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Figure 6 shows that P&D manipulations also increased in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Table 2 shows the information collected for each post and comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Data from Yahoo Finance was scraped using the yfinance tool [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Stock symbols were extracted from Reddit posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' This step is non-trivial and required regular expression extraction, and look ups against the publicly traded exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Posts which mentioned more than one stock were discarded, partly because of the complexity of deciding which stock may be being touted, and partly because P&D posts typically focus on one particular stock they are pumping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' If a stock symbol was found, yfinance was used to collect the financial information described in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' As shown in Figure 7, the daily Open, High, Low, Close, and Volume (OHLCV) data was collected over nine business days surrounding an event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Data was collected over five days before each post event to establish a baseline for price and volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Penny stocks almost always shows minor variation in price and volume so this baseline is typically quite flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The remaining four days contain the pump event (sharp increase) followed by a decrease in price and a slower decrease in volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' 6 00008 70000 60000 of Records 50000 40000 30000 20000 10000 SectorConglomerates SectorServices SectorUtilities SectorConsumerDefensive SectorBasicMaterials SectorRealEstate SectorFinancialServices SectorEnergy Sectorlndustrials SectorCommunicationServices SectorUnknown SectorTechnology SectorHealthcare SectorsFigure 5: Trend of posts and comments that discussed healthcare stocks Feature Description Post Title Title of the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Post ID Unique identification code for post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Post Author Author of the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Post Created Unix Timestamp of when post was submitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Post Body Text of the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Comment ID Unique identification code for comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Comment Author Author of the comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Comment Created Unix Timestamp of when comment was submit- ted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Comment Body Text of the comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Table 2: Features of collected Reddit data Sabherwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' [27] studied the effects of online message boards on market manipulation and found that dumps typically occur within four days and this is plausible because the manipulators want to sell as soon as the price reaches a peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Texts from subreddits were preprocessed using the following steps: remove URLs, expand con- tractions, remove HTML Tags, remove punctuation, remove extra whitespaces, remove numbers, lemmatization, and remove stopwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Stock symbols within the text were replaced by dummy stock names representing the market sector associated with each business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' This is required because the name of the particular stock being pumped and dumped in one case has nothing to do with the name of the stock being used in another case – but there might be correspondences within sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Here is an example: 7 4000 3500 3000 of Records 25:00 Yumber 2000 15:00 1000 500 DatesFigure 6: Trend of posts that have been labelled as P&D Feature Description Open Opening price of the stock for the given period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' High Highest price for the stock within the given pe- riod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Low Lowest price for the stock within the given pe- riod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Close Closing price of the stock for the given period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Volume Total number of shares traded within the given period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Market Sector Associated industry that the company is in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Market Capitalization Total market value of the company’s outstand- ing shares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Table 3: Features of Yahoo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Finance data “AYTU perfect time to buy” ⇒ “SectorHealthcare perfect time to buy” 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='2 Data Labelling To label each post, stock data surrounding the day in which the post was submitted to Reddit were analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' If the market data exhibited that pattern associated with P&D (a notable rise from the time of the post, followed by a sharp drop) then the post was labelled accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' A rise was detected by calculating the average price and volume in the five-day window before the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The 8 120 100 Posts Number of i 60 40 20 DatesFigure 7: Time window used to collect market data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' daily average price (DAP) of the values was first calculated for each of the five days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' DAP(Xt) = 1 4(Xtopen + Xthigh + Xtlow + Xtclose) (1) and then the baseline average price (BAP) was calculated by BAP(Xest) = 1 5 · T1 � t=T0 DAP(Xt) (2) The baseline average volume (BAV) was calculated by taking the average of the volume values over the estimation window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' BAV (Xest) = 1 5 · T1 � t=T0 Xtvolume (3) A threshold was set at two standard deviations above the average price within the five-day estimation window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Price increases above this threshold were considered to be pump events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' A similar threshold was used to define a volume anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Events were considered to be the result of P&D if they exceeded the threshold for both price and volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Figure 8 shows a comparison of the stock behaviours labelled using this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' A sudden price rise or volume increase might coincide with a post, but is not necessarily caused by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The rising region of each stock trend of a potential P&D event was min-max normalised, 9 Reddit Post Date Price Event To T2 Volume 27 29 May 5 11 Estimation Event Window (5 Days) Window (4 Days)Figure 8: Comparison of stock behaviours that have been labelled using anomaly detection and its slope calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Steep price increases are more likely to arise from genuine information and less likely to have resulted from a single manipulation post, so the median slope across the entire dataset was calculated, and only slopes below the median were considered as potential P&D events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Figure 9 shows the distribution of stock price trend slopes from the entire the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The median value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='3 Agreement Model The comments associated with the P&D post cannot all be labelled as examples of P&D language, since not all of them will be supportive of the post they are responding to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Manipulators, of course, will post comments in support of the post, either from the same identity or from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' We developed an agreement model, using ideas from stance detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' This was done using Empath to generate a lexicon of agreement, seeding it with the words: bought, agree, positive, increasing, good, and now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Empath returned the words listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Posts touting stocks also use a specialised vocabulary, shown in these examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='“probably go to shoot up tomorrow” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='TRNX2020-04-16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='CCO2020-03-31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='USWS2020-06-07 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='GNUS2020-06-09 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Stockbehaviours ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Stockbehaviours ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='labelledasP&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='labelledasnotP&DFigure 9: Distribution of stock price trend slopes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='only ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='done ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='better ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='true ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='knew ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='besides ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='like ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='good ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='understood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='needed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='work ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='because ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='successful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='knowing ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='course ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='glad ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='well ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='considering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='anyway ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='agree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='meaning ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='positive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='thankful ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='actually ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='agreed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='special ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='doubt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='guess ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='though ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='bet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='buy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='surpass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='worth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='suppose ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='although ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='especially ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='definitely ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='certain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='figured ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='given ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='means ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Table 4: List of generated agreement words from Empath ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='“this bad boy just rocket” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='“i will see you on the moon” ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='An extended lexicon was determined manually by inspecting posts associated with manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Table 5 contains the list of words that were chosen using this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Comments were labelled as associated with pumping if they contained two or more of the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Number of Posts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='200moon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='fast ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='massive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='rich ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='surprise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='rocket ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='profit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='top ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='easy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='move ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='pump ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='rally ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='peak ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='early ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='load ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='soar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='climb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='worth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='shoot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='quick ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='jump ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='rise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='sale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='money ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='burst ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='pop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='high ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='gain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='breakout ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='drive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='hype ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='spike ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='run ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='cash ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='nice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='fly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='go ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='up ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='hit ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='bank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='awesome ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='confident ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='surpass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='more ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='zoom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='big ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='great ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='potential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='advantage ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Table 5: List of custom words used in the Agreement Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='agreement words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' or if they were (visibly) authored by the original poster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The following are some examples of comments that were labelled as not P&D related based on the agreement model: “it be the american dream to fall for snake oil salesman and then lose everything it be a story as old as humanity” “clearly a pump and dump scheme” “do not touch it if the chart look like a hockey stick” This labelling of comments is limited by the completeness of the agreement lexicon, and also does not account for negations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' P&D posts and comments are relatively rare and so the dataset is naturally imbalanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Tech- niques such as SMOTE [3] and ADASYN [17] were tried but proved ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Instead, where predictors allowed it, class weight parameters were set to penalise mistakes in the minority class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='4 Modelling The following predictors were used: Extreme Gradient Boosting (XGBoost) Random Forest (RF) Support Vector Machine (SVM) Artificial Neural Networks – Multilayer Perceptron (MLP) – Convolutional Neural Network (CNN) – Bidirectional Long Short Term Memory (BiLSTM) In each case the standard performance measures (accuracy, precision, recall, F1-Score, confusion matrix) were calculated, as well as the Shapley values which rank words by their importance to the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' 12 Record Type P&D Not P&D Total Posts 3,006 15,549 18,555 Comment 26,727 285,851 312,578 Total 29,733 312,142 331,133 Table 6: Dataset class distribution Model TP FP TN FN Accuracy Precision Recall F1-Score XGBoost Posts 1728 6615 8934 1278 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='46 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='73) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='71 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='48) 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='49 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='68) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='45 (±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='25) XGBoost Posts and Comments 2007 7646 7903 999 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='41 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='42) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='79 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='85) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='77 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='58) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='71 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='96) RF Posts 271 646 14903 2735 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='78 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='51) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='55 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='40) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='01 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='52) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='81 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='78) RF Posts and Comments 414 211 15338 2592 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='89 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='69) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='24 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='69) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='77 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='47) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='80 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='75) SVM Posts 1752 5263 10286 1254 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='88 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='14) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='98 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='76) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='28 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='05) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='97 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='16) SVM Posts and Comments 2125 4559 10990 881 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='6 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='49) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='79 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='43) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='69 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='56) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='86 (±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='57) MLP Posts 2382 1718 13831 624 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='38 (±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='66) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='10 (±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='65) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='24 (±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='76) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='04 (±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='12) MLP Posts and Comments 2103 2602 12947 903 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='11 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='71) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='70 (±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='28) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='96 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='80) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='55 (±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='36) CNN Posts 2373 1709 13840 633 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='38 (±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='04) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='13 (±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='02) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='94 (±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='76) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='96 (±12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='37) CNN Posts and Comments 2304 2068 13481 702 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='07 (±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='25) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='70 (±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='33) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='65 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='45) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='46 (±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='64) biLSTM Posts 2297 2495 13054 709 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='73 (±8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='11) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='93 (±9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='92) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='41 (±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='94) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='91 (±10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='82) biLSTM Posts and Comments 2288 2370 13179 718 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='36 (±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='27) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='12 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='25) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='11 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='86) 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='71 (±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='54) Table 7: Summary of model performance 4 Results Table 6 shows the class distribution for the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Less than 9% of the records are labelled as being P&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' This is typical of datasets where fraud is present;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' indeed it is striking that the rate of fraud is this high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The results of each of the predictive model are reported in Table 7 using 5-fold cross validation and upweighting the fraud class when the model permits it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The neural network models perform well as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Models such as XGBoost, Random Forests, and SVM had disappointing performance, and a heterogeneous stacked classifier combining their predictions did not improve on the performance of the individual predictors, suggesting that they make their errors on the same records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' At first glance, the ANN models using posts perform better than those using posts and com- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' However, the standard deviations of the performance numbers show that the inclusion of comments provides stability for correctly identifying P&D posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The best performing model over- all is CNN, especially with comments included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Its precision is relatively low;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' of all the records that the model predicts to be P&D, only 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='7% are actually correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' If we look at the rate at which each class is predicted to be positive, a better outlook of the model is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Given a positive P&D text, the model has a 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='65% chance of classifying it correctly, whereas, if it is given a negative text, it has a 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='3% chance of classifying it incorrectly as positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' It is perhaps a little surprising that biLSTM did not perform best since they are typically strong predictors for natural language problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The SHAP Explainers produce diagrams that rank the attributes by their impact on outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Figure 10 shows the diagram for the CNN predictor for posts and comments and the 30 most impactful words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Although the influence of any single word is inevitably weak, there are visible red dots to the right for many of these words, indicating that higher frequencies of these words are associated with P&D events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' The names of the popular sectors are indicator of P&Ds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' as are words ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Predicted Label ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Actual Label ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Misclassified Post ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Not P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='sectorunknown about to soar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Not P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='sectorunknown fitness equipment maker owner ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='of bow flex completely sell out of most retail ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='store how be this look just buy in share ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Not P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='quick all in sectorcommunicationservices pump ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='my first time actually do something right the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='lambos go to be green for gain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Not P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='blast off look like gold and oil will be big player ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='this i also suggest look at sectortechnology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Not P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='sectorenergy drop time to buy it be drop below ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='which be its day low be it a good time to buy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Not P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='sectortechnology release patent news on thermal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='tech could be a mark sympathy play bust out ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='over ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Not P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='sectorhealthcare do anyone understand why sec- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='torhealthcare shoot up soo much i be not able ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='to find any real catalyst ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Not P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='sectorhealthcare on the move this have potential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='reach today ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Not P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='sectorhealthcare to the moon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Not P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='P&D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='any thought on when to sell sectorenergy bought ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='in late i be up after hour should i wait til tomor- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='row or sell as soon as possible in the am ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='Table 8: Examples of misclassified posts from CNN model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='from the agreement model such as “buy” and “go”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Across the best performing models, the same set of words emerge as the most impactful features (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Misclassifications by the model have different impacts depending on how and where it is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' For an ordinary investor, a false positive (a post predicted to be a P&D when it isn’t) means a missed opportunity for profit, but a false negative means a financial loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' For a regulatory body, a false positive is problematic, but a false negative less so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Table 8 shows some of the examples of misclassifications by the CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Some false positives, predicted to be P&D from the text, but without a corresponding market movement may be instances where the post failed to attract enough attention to cause a measurable market movement, or was so blatant that it was not credible to typical investors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Some false negatives may be because the posts were too short to contain the required two words, because the pumping took place on another platform or because a market movement happened to match the timing of the post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' 14 5 Related work The application of data analytics for detecting market manipulation is a relatively new in the field of finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Most research has focused on detecting trade-based manipulation because it is most common [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Huang and Chang found that of the manipulation cases prosecuted in Taiwan from 1991 to 2010, 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='61% were trade-based, and only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='39% were information-based [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Some examples detecting trade-based manipulation are: Ogut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' [38] in the emerging Istanbul Stock Exchange, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' [32] for prosecuted manipulation cases reported by the China Securities Regulatory Commission, Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' [7] using real trading data from four popular NASDAQ stocks with synthetic cases of manipulation (spoofing and quote stuffing), Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' [36] using seven popular NASDAQ and LSE stocks data injecting ten simulated stock price manipulations, Diaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' [12] using manipulation cases pursued by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Securities and Exchange Commission (SEC) in 2003, and Golomohammadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' [16] trying to detect three groups of manipulation schemes: marking the close, wash trades, and cornering the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' For information-based manipulation, Victor and Hagemann [31] looked at 149 confirmed P&D schemes coordinated through Telegram chats and pumped via Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Using XGBoost, they built a model that achieved a sensitivity of 85% and specificity of 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' They concluded that P&Ds were frequent among cryptocurrencies that had a market capitalization of $50 million or below and often involved trading volumes of several hundred thousand dollars within a short time-frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Mirtaheri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' [23] looked specifically at forecasting P&Ds by combining the information from Twitter and Telegram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' They manually labelled known P&D operation messages on Telegram, and then used SVMs with a stochastic gradient descent optimizer to label the remaining messages as P&D or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' They used Random Forests to detect whether a manipulation event was going to take place within the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Their results showed that they were able to detect, with reasonable accu- racy, whether there is an unfolding manipulation scheme occurring on Telegram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Their proposed model was able to achieve an accuracy of 87% and an F1-Score of 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Some partially automated tools have also been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' These flag suspicious activities that can then by investigated by regulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Delort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' [11] used Naive Bayes classifiers to examine collected messages from HotCopper, an Australian stock message board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' They successfully identified messages of concern, but the number of false positives was too high to use the model in an automated way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Owda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' [25] compared messages to lexicon templates of known illegal financial activities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Pump and Dump, Insider Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' They found that, of the 3000 comments that were collected on a daily basis, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='2% were deemed suspicious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' 6 Conclusion The intersection of social media with low-cost trading platforms and naive investors has made market manipulation an attractive strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Pump&dump is particularly simple to implement since it requires only the dissemination of fictional information about the future prospects for a stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' This is particular easy for penny stocks where validating information is difficult for ordinary investors, and where relatively small purchase volumes can cause large price movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' We investigate protecting investors, and assisting regulators, by building predictive models that label social media posts (and the responses they elicit) as potential drivers of P&D events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' We do this by collecting posts and comments, developing a model for a P&D event based on patterns of price and volume changes, using the match between posts and P&D events to label posts, and 15 extending this labelling to comments using an agreement model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Natural language predictors then learn the language patterns associated with P&D manipulations, so that new manipulations can be detected before they affect the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Data is imbalanced, since manipulations are rare, but our best predictive model achieves an F1-score of 62% and an accuracy of 85%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' Improvements in performance are limited by potential coincidences between a post and a price and volume change that mimics a P&D, posts that fail to reach a sufficient audience to cause the desired buying behaviour, and natural language issues that arise from informal and short texts, and a specialised vocabulary used in stock discussion forums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content=' References 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} +page_content='5 LOW SHAP value (impact on model output)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFIT4oBgHgl3EQf8ysQ/content/2301.11403v1.pdf'} diff --git a/-dAyT4oBgHgl3EQfdfdm/content/tmp_files/2301.00303v1.pdf.txt b/-dAyT4oBgHgl3EQfdfdm/content/tmp_files/2301.00303v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..083cdef9da7e7917d3f83174d4bc24ad72518c16 --- /dev/null +++ b/-dAyT4oBgHgl3EQfdfdm/content/tmp_files/2301.00303v1.pdf.txt @@ -0,0 +1,1483 @@ +Rethinking with Retrieval: Faithful Large Language Model Inference +Hangfeng He†∗ +Hongming Zhang‡ +Dan Roth§ +†University of Rochester +‡Tencent AI Lab, Seattle +§University of Pennsylvania +hanfeng.he@rochester.edu, hongmzhang@global.tencent.com +danroth@seas.upenn.edu +Abstract +Despite the success of large language mod- +els (LLMs) in various natural language pro- +cessing (NLP) tasks, the stored knowledge +in these models may inevitably be incom- +plete, out-of-date, or incorrect. +This mo- +tivates the need to utilize external knowl- +edge to assist LLMs. Unfortunately, current +methods for incorporating external knowl- +edge often require additional training or +fine-tuning, which can be costly and may +not be feasible for LLMs. To address this +issue, we propose a novel post-processing +approach, rethinking with retrieval (RR), +which retrieves relevant external knowledge +based on the decomposed reasoning steps +obtained from the chain-of-thought (CoT) +prompting. This lightweight approach does +not require additional training or fine-tuning +and is not limited by the input length of +LLMs. We evaluate the effectiveness of RR +through extensive experiments with GPT-3 +on three complex reasoning tasks: common- +sense reasoning, temporal reasoning, and +tabular reasoning. Our results show that RR +can produce more faithful explanations and +improve the performance of LLMs.1 +1 +Introduction +Large language models (LLMs) have shown +exceptional performance across various tasks +through in-context learning without task-specific +training or fine-tuning (Brown et al., 2020; +Chowdhery et al., 2022; Zhang et al., 2022; +Ouyang et al., 2022). Recent progress in prompt- +ing (Wei et al., 2022; Zhou et al., 2022; Kojima +et al., 2022) and decoding (Wang et al., 2022) has +made it feasible for LLMs to tackle tasks that de- +mand complex reasoning. +∗Part of this work was done while the author was at the +University of Pennsylvania. +1Our code is publicly available at https://github. +com/HornHehhf/RR. +Query +Prediction +LLM +Query +Explanation + Prediction +LLM +Query +Explanation + Prediction +LLM +(a) +(b) +(c) +Knowledge +Chain of thought +Chain of thought +Retrieval +Rethinking +Figure 1: An overview of three approaches for using +LLMs: (a) Standard prompting for generating a pre- +diction in response to a query. (b) Chain-of-thought +prompting for generating both an explanation and a +prediction in response to a query. (c) Rethinking with +retrieval, our proposed approach for using the decom- +posed reasoning steps obtained from chain-of-thought +prompting to retrieve relevant external knowledge for +LLMs, leading to more faithful explanations and im- +proved predictions in response to a query. +However, the knowledge stored in LLMs might +inevitably be incomplete, out-of-date, or incorrect. +As a result, external sources of knowledge, such +as Wikipedia, may be essential for the success- +ful deployment of LLMs for real-world applica- +tions. Previously, people tried to utilize knowl- +edge for smaller language models (LMs), such +as T5 (Raffel et al., 2020), BERT (Devlin et al., +2019), and RoBERTa (Liu et al., 2019). However, +these methods often require additional training or +fine-tuning, which can be costly and thus imprac- +tical for LLMs. +In this paper, we present a post-processing +approach called rethinking with retrieval (RR) +for utilizing external knowledge in LLMs. Our +method begins by using the chain-of-thought +(CoT) prompting method (Wei et al., 2022) to gen- +erate a diverse set of reasoning paths, as described +in Wang et al. (2022). +We then use each rea- +soning step in those paths to retrieve relevant ex- +ternal knowledge, which enables RR to provide +arXiv:2301.00303v1 [cs.CL] 31 Dec 2022 + +more faithful explanations and more accurate pre- +dictions, as illustrated in Figure 1. +We evaluate the effectiveness of our proposed +method, RR, on three complex reasoning tasks: +commonsense reasoning, temporal reasoning, and +tabular reasoning, using GPT-3 175B (Brown +et al., 2020) and different external knowledge +sources: +Wikipedia, Wikidata (Vrandeˇci´c and +Krötzsch, 2014), WordNet (Miller, 1995), and +Conceptnet (Speer et al., 2017). +The results +demonstrate that RR consistently outperforms all +baselines on all three tasks without requiring ad- +ditional training or fine-tuning, indicating the su- +periority of our approach in leveraging external +knowledge to enhance the performance of LLMs. +2 +Related Work +Enhancing LMs through retrieval. +Retrieval- +enhanced LMs have received significant attention +as a means of improving performance through the +incorporation of external knowledge. For exam- +ple, the k-most similar training contexts can be re- +trieved to improve the estimation of the next word +distribution in both the training stage (Borgeaud +et al., 2021) and the inference stage (Khandelwal +et al., 2020). Furthermore, search query genera- +tors have been adopted to generate search queries +for search engines to retrieve relevant documents +(Komeili et al., 2022; Shuster et al., 2022; Thop- +pilan et al., 2022). +Other approaches have uti- +lized retrieved documents as the additional con- +text in generation tasks (Joshi et al., 2020; Guu +et al., 2020; Lewis et al., 2020). Nakano et al. +(2021) instead use human feedback in a text-based +web-browsing environment. +Among these pre- +vious works, Khandelwal et al. (2020) is most +closely related to our approach. +However, they +focus on improving local inference by using the +nearest neighbor datastore constructed from train- +ing data, whereas we focus on conducting faith- +ful inference using external knowledge. In con- +trast to other aforementioned approaches, which +require training or fine-tuning to incorporate re- +trieved knowledge, we propose a post-processing +method for leveraging retrieved knowledge with- +out additional training or fine-tuning. +Incorporating external knowledge into LMs. +Significant effort has been devoted to leveraging +external knowledge to improve the reasoning abil- +ity of LMs. Previous work has incorporated exter- +nal knowledge sources such as WordNet (Miller, +1995) and ConceptNet (Speer et al., 2017) to en- +hance LMs for tabular reasoning tasks (Neeraja +et al., 2021; Varun et al., 2022). +Explicit rules +have also been added to inputs to improve rea- +soning ability over implicit knowledge (Talmor +et al., 2020). In addition, explicit knowledge from +Wikidata (Vrandeˇci´c and Krötzsch, 2014) and im- +plicit knowledge in LLMs have been integrated +into a transformer (Vaswani et al., 2017) for vi- +sual question answering (Gui et al., 2021). Nye +et al. (2021) instead introduces a symbolic reason- +ing module to improve coherence and consistency +in LLMs. Among these previous works, Nye et al. +(2021) is the most relevant to our approach. Still, +they focus on incorporating logical constraints to +improve coherence and consistency, whereas we +aim to improve the faithfulness of explanations +through the use of external knowledge. In con- +trast to other aforementioned approaches that in- +corporate external knowledge before generation +and require additional training or fine-tuning, our +proposal leverages external knowledge in a post- +processing manner to enhance LMs without addi- +tional training or fine-tuning. +Uncovering latent Knowledge in LLMs. +There +has been a line of work exploring the knowledge +hidden within LLMs for reasoning. This has in- +cluded the use of careful prompting to encourage +LLMs to generate explanations in the reasoning +process, such as through chain of thought prompt- +ing in few-shot (Wei et al., 2022) or zero-shot +(Kojima et al., 2022) learning, or through the use +of scratchpads for intermediate computation (Nye +et al., 2022). In addition, various methods based +on sampling a diverse set of reasoning paths in +LLMs have been proposed, including training ver- +ifiers to judge the correctness of model comple- +tions (Cobbe et al., 2021), calibrating model pre- +dictions based on the reliability of the explana- +tions (Ye and Durrett, 2022), and promoting self- +consistency over diverse reasoning paths (Wang +et al., 2022). Zelikman et al. (2022) instead it- +eratively bootstrap the ability of LLMs to gener- +ate high-quality rationales from a few initial ex- +amples. Liu et al. (2022) further propose generat- +ing knowledge from LLMs, which is then used as +additional input to improve commonsense reason- +ing. In contrast to this line of work, our proposal +focuses on leveraging external knowledge to en- +hance LLMs, while they aim to explore the knowl- +edge hidden within LLMs. + +3 +Rethinking with Retrieval +LLMs have been shown to generate incorrect sup- +porting facts from time to time, even when they ac- +curately capture the perspective needed to answer +a question. This phenomenon highlights intrinsic +issues in the way LLMs store and retrieve knowl- +edge, including (1) the presence of out-of-date, +incorrect, or missing relevant knowledge in the +pre-training corpus; (2) incorrect memorization of +relevant knowledge during pre-training; and (3) +incorrect retrieval of relevant knowledge during +the inference stage. To address these issues, we +propose the use of RR, which leverages external +knowledge through the retrieval of relevant infor- +mation based on decomposed reasoning steps. +Overview. +Given a query Q, we utilize chain-of- +thought prompting to generate a diverse set of rea- +soning paths R1, R2, · · · RN, where each reason- +ing path Ri consists of an explanation Ei followed +by a prediction Pi. After that, we retrieve relevant +knowledge K1, · · · KM from a suitable knowledge +base KB to support the explanation in each reason- +ing path, and select the prediction ˆP that is most +faithful to this knowledge. To better illustrate our +proposal, we use “Did Aristotle use a laptop?” as +a running example in this work. +Chain-of-thought prompting. +In contrast to +standard prompting, CoT prompting (Wei et al., +2022) includes demonstrations of step-by-step rea- +soning examples in the prompt to produce a series +of short sentences that capture the reasoning pro- +cess. For instance, given the question “Did Aris- +totle use a laptop?”, CoT prompting aims to gen- +erate the complete reasoning path “Aristotle died +in 322 BC. The first laptop was invented in 1980. +Thus, Aristotle did not use a laptop. So the answer +is no.” rather than simply outputs “No.” Empirical +results show that CoT prompting significantly im- +proves the performance of LLMs on many multi- +step reasoning tasks. Therefore, we adopt CoT +prompting to obtain both explanation E and pre- +diction P for the query Q. +Sampling diverse reasoning paths. +Similar to +Wang et al. (2022), we sample a diverse set of rea- +soning paths R1, R2, · · · RN rather than only con- +sidering the greedy path as in Wei et al. (2022). +For the question “Did Aristotle use a laptop?”, the +potential reasoning paths can be as follows: +(R1) Aristotle died in 2000. The first laptop was +invented in 1980. Thus, Aristotle used a lap- +top. So the answer is yes. +(R2) Aristotle died in 322BC. The first laptop was +invented in 2000. Thus, Aristotle did not use +a laptop. So the answer is no. +(R3) Aristotle died in 322BC. The first laptop was +invented in 1980. Thus, Aristotle did not use +a laptop. So the answer is no. +Knowledge +retrieval. +Different +knowledge +bases can be used to address different tasks. For +example, to address the question “Did Aristotle +use a laptop?”, we can use Wikipedia as the ex- +ternal knowledge base KB. Information retrieval +techniques can be applied to retrieve the relevant +knowledge K1, · · · KM from Wikipedia based +on the decomposed reasoning steps. Ideally, we +would obtain the following two paragraphs from +Wikipedia for this question: +(K1) Aristotle (384–322 BC) was a Greek philoso- +pher and polymath during the Classical pe- +riod in Ancient Greece. ... +(K2) The Epson HX-20, the first laptop computer, +was invented in 1980. ... +Faithful inference. +The faithfulness of each rea- +soning path Ri can be estimated using a function +fKB(Ri), which is based on relevant knowledge +K1, · · · , KM retrieved from the knowledge base +KB. The final prediction is obtained through the +application of the following inference procedure2: +ˆP = +arg max +Pi∈{P1,··· ,PN} +N +� +i=1 +1(Pi = P)fKB(Ri), (1) +where Pi denotes the corresponding prediction in +the reasoning path Ri. This inference procedure +is designed to identify the most faithful prediction +ˆP to the knowledge base among all predictions in +the N reasoning paths. For instance, in the run- +ning example, given reasoning paths R1, R2, R3 +and the retrieved knowledge K1, K2, the above in- +ference procedure would output the prediction “So +the answer is no.”, as it is supported by both R2 +and R3 and has a higher faithfulness score com- +pared to the prediction “So the answer is yes.”, +which is only supported by R1. +2Note that this is the basic version of faithful inference, +and further variations can be found in Section 5.3. + +4 +Experiments +In this section, we present the evaluation of our +proposed method, RR, on three complex reason- +ing tasks: commonsense reasoning, temporal rea- +soning, and tabular reasoning. +4.1 +Baselines +We compare with the following baselines. +Zero-shot/few-shot prompting. +In our experi- +ments, we consider GPT-3 with standard zero- +shot/few-shot prompting as baselines, following +the approach described in Brown et al. (2020), in +which zero or few in-context exemplars of input- +output pairs are provided in the prompt. +Chain-of-thought prompting. +In addition to +the standard zero-shot/few-shot prompting, we +also consider GPT-3 with the CoT prompting pro- +posed in (Wei et al., 2022) as a baseline in our ex- +periments. This approach involves feeding LLMs +step-by-step reasoning examples instead of stan- +dard input-output examples. +Self-consistency. +In addition, we also consider +self-consistency (Wang et al., 2022) as a baseline +in our experiments. This approach, proposed as an +alternative to the naive greedy decoding used in +CoT prompting (Wei et al., 2022), involves sam- +pling a diverse set of reasoning paths and select- +ing the most consistent answer by marginalizing +the sampled paths. +4.2 +Commonsense Reasoning +Dataset description. +For commonsense reason- +ing, we consider the StrategyQA dataset (Geva +et al., 2021), which includes questions that require +implicit reasoning strategies. +For example, the +question “Did Aristotle use a laptop?” requires +implicit decomposition into reasoning steps, while +the question “Was Aristotle alive when the laptop +was invented?” explicitly specifies the reasoning +process. The StrategyQA dataset includes 2, 290 +training examples, each consisting of a question +(Q), a yes/no answer (A), a decomposition (D), +evidence paragraphs (E), and supporting facts (F). +On average, each question requires about 2.93 rea- +soning steps and 2.33 evidence paragraphs. In ad- +dition, a development set is constructed by ran- +domly sampling 10% of the training examples +(i.e., 229 examples). The answer distribution is +roughly balanced, with approximately 47% "yes" +questions in both the training and development +sets. Unless otherwise specified, the models are +evaluated on the development set3 for StrategyQA. +Implementation details. +In this part, we uti- +lize Wikipedia as the external knowledge base +KB. For each sentence in the explanation of ev- +ery reasoning path, we first apply BM25 (Robert- +son et al., 2009) to retrieve the top 10 most rele- +vant paragraphs from Wikipedia. In particular, we +use the re-implementation of the sparse retrieval +BM254 in Karpukhin et al. (2020) from Pyserini +(Lin et al., 2021). Subsequently, we use the pre- +trained MPNet model (Song et al., 2020) to se- +lect the most similar paragraph based on the cosine +similarity between the sentence embeddings of the +retrieved paragraph and the sentence. +We then +employ a pre-trained natural language inference +(NLI) model (Nie et al., 2020) to obtain the en- +tailment and contradiction scores for the sentence, +treating the most similar paragraph as the premise. +The faithfulness of each reasoning path is then +calculated using fKB(·) based on the entailment +scores, contradiction scores, and MPNet similari- +ties of all sentences in the explanation of the rea- +soning path. The final prediction for each ques- +tion is obtained through faithful inference (Equa- +tion 1). More details about fKB(·) can be found in +Appendix A.2. +4.3 +Temporal Reasoning +Dataset description. +In this experiment, we use +the TempQuestions dataset (Jia et al., 2018) to +investigate temporal reasoning. This dataset in- +cludes 1, 271 temporal questions that are divided +into four classes: explicit temporal, implicit tem- +poral, temporal answer, and ordinal constraints. +The questions are paired with their answers from +Freebase (Bollacker et al., 2008). To examine the +most challenging aspect of temporal reasoning, we +focus on the set of implicit temporal questions, +which contain implicit temporal expressions, in- +cluding free-text temporal expressions. +For ex- +ample, the question “who was governor of oregon +when shanghai noon was released?” is an implicit +temporal question. To facilitate our analysis, we +only consider questions with a single answer, re- +sulting in a total of 175 examples. Of these ex- +3As the annotations for the test set are not publicly avail- +able, we use the development set for evaluation. This allows +us to perform a more comprehensive analysis. +4We also experimented with DPR and BM25+DPR, and +found that BM25 outperformed these methods in our experi- +ments. More details can be found in Appendix A.3. + +Methods +Commonsense +Temporal +Tabular +GPT-3 +Zero-shot prompting +58.08 +28.40 +82.00 +Few-shot prompting +63.32 +29.59 +83.08 +Chain-of-thought prompting +65.94 +33.14 +83.33 +Self-consistency +73.36 +37.28 +84.00 +Rethinking with retrieval +77.73 +39.05 +84.83 +Table 1: Performance of different methods using GPT-3 on three reasoning tasks. +amples, the first 6 are used for prompting, and the +remaining 169 are used for evaluation. +Implementation details. +In this part, we utilize +Wikidata (Vrandeˇci´c and Krötzsch, 2014) as the +external knowledge base KB, as it is the largest +publicly available knowledge graph, and the data +from Freebase has been migrated to Wikidata. To +incorporate this knowledge into our system, we +apply an entity linking system5 to each sentence +in the explanation of each reasoning path to iden- +tify the corresponding Wikidata pages for all enti- +ties in the sentence. Next, we extract all temporal +relations from these relevant Wikidata pages and +use templates to convert these temporal relations +into sentences. This step generates a set of rele- +vant knowledge sentences for each sentence in the +explanation of each reasoning path. The final pre- +diction is then obtained by applying the procedure +described in Section 4.2, in which the retrieved +paragraphs are replaced with the relevant knowl- +edge sentences from the current part. +4.4 +Tabular Reasoning +Dataset +description. +We +consider +the +IN- +FOTABS dataset (Gupta et al., 2020) for tabu- +lar reasoning, which consists of 23, 738 human- +written textual hypotheses based on premises in +the form of tables extracted from 2, 540 unique +Wikipedia info-boxes. We focus on the develop- +ment set, which includes 1, 800 hypotheses based +on 200 tables, and only consider entailed and con- +tradictory hypotheses as it is tricky to write CoT +demonstrations for neutral hypotheses. This re- +sults in a total of 1, 200 hypotheses based on 200 +tables for evaluation, with an equal number of en- +tailed and contradictory hypotheses. +Implementation details. +In this part, we utilize +WordNet (Miller, 1995) and ConceptNet (Speer +5We use the spacy entity linker: https://pypi.org/ +project/spacy-entity-linker/. +et al., 2017) as external knowledge bases. To con- +vert tables into textual premises, we follow the +same technique as in Varun et al. (2022). For each +premise-hypothesis pair, we follow the procedure +outlined in Varun et al. (2022) to retrieve rele- +vant word relation triples that connect the premise +and hypothesis words, such as “married” RelatedTo +←−−−−→ +“spouse”. +These triples are then converted into +sentences using some simple templates. The re- +sulting sentences, along with the textual premises +from the tables, serve as relevant knowledge for +each sentence in the explanation of each reasoning +path. To obtain the final prediction, the procedure +described in Section 4.2 is applied, whereby the +retrieved paragraphs in Section 4.2 are replaced +with the relevant knowledge from the current part. +4.5 +Evaluation +Experimental settings. +In all experiments, we +utilize GPT-3 text-davinci-002 unless oth- +erwise stated. The maximum number of tokens for +generation during completion is set to 256. For +zero-shot, few-shot, and chain-of-thought prompt- +ing, the temperature is fixed at 0. +For self- +consistency and rethinking with retrieval, we ran- +domly sample 10 outputs6 with temperature 0.7. +Detailed prompts can be found in Appendix A.1. +We evaluate the performance of different methods +on commonsense and tabular reasoning using ac- +curacy, and on temporal reasoning using the exact +match metric as defined in Rajpurkar et al. (2016). +Results. +As shown in Table 1, our proposed +method, rethinking with retrieval, consistently +outperforms all baselines on all three reasoning +tasks without requiring additional training or fine- +tuning. The results highlight the effectiveness of +our approach in leveraging external knowledge to +improve the performance of LLMs. +6For commonsense reasoning, we sample 9 outputs, as we +have found that odd numbers of outputs tend to yield better +voting performance for self-consistency on StrategyQA. + +Wrong supporting facts +Wrong inference +Question +Was Lil Jon’s top ranked Billboard song a +collaboration with a member of The Lox? +Would the top of Mount Fuji stick out of +the Sea of Japan? +Answer +No +Yes +Facts +Lil Jon’s highest ranked billboard song +was Yeah; Yeah was a collaboration be- +tween Lil Jon, Usher, and Ludacris; The +Lox is a rap trio consisting of: Styles P, +Sheek Louch, and Jadakiss. +The average depth of the Sea of Japan is +5,748 feet (1,752 metres) and its maxi- +mum depth is 12,276 feet (3,742 metres); +Mount Fuji is 3,776.24 metres (12,389.2 +ft) tall. +GPT-3 +Lil Jon’s top ranked Billboard song was +"Get Low" with the East Side Boyz. The +Lox is not a member of the East Side +Boyz. +Thus, Lil Jon’s top ranked Bill- +board song was not a collaboration with +a member of The Lox. So the answer is +no. +Mount Fuji is 3,776 meters tall. +The +Sea of Japan is about 3,741 meters deep. +Thus, the top of Mount Fuji would not +stick out of the Sea of Japan. So the an- +swer is no. +Table 2: Examples of incorrect outputs from GPT-3 with CoT prompting. +5 +Analysis +In this section, we perform a thorough analysis to +gain a deeper understanding of RR. +5.1 +Limitations of LLMs in Reasoning +In this subsection, we present an analysis of GPT- +3 with CoT prompting on the StrategyQA dataset. +Upon closer examination of the outputs of GPT- +3, we observed that it can provide reasonable ex- +planations and correct predictions for a number +of questions. For example, when given the ques- +tion “Will the Albany in Georgia reach a hundred +thousand occupants before the one in New York?”, +GPT-3 produced the following output: +The Albany in New York has a pop- +ulation of about 98,000. +The Albany +in Georgia has a population of about +77,000. Thus, the Albany in New York +is more populous than the Albany in +Georgia. So the answer is no. +The above output consists of three components: +(1) supporting facts (in cyan) that are based on a +particular perspective, (2) chaining arguments (in +orange), and (3) a prediction (in green). +Com- +ponents (1) and (2) contribute to the explanation. +Overall, the output exhibits a high level of quality. +However, we also observed that GPT-3 may occa- +sionally produce incorrect supporting facts for its +explanations or make incorrect inferences for its +Retrieval +Commonsense +Tabular +Query-based +73.36 +36.69 +Decomposition-based +77.73 +39.05 +Table +3: +Comparison +of +query-based +and +decomposition-based +retrieval +on +commonsense +and tabular reasoning. +predictions, despite generally being able to iden- +tify suitable perspectives. +Wrong supporting facts. +As shown in Table 2, +GPT-3 provides the incorrect supporting fact for +Lil Jon’s top-ranked Billboard song, stating that +it was “Get Low” instead of the correct answer, +“Yeah”. However, it does have the correct per- +spective on how to answer the question, “Was Lil +Jon’s top ranked Billboard song a collaboration +with a member of The Lox?”. +Wrong inference. +As shown in Table 2, GPT-3 +makes an incorrect inference, stating that the top +of Mount Fuji “would not stick out” of the Sea of +Japan, rather than the correct answer, “would stick +out”. However, it does provide correct supporting +facts based on the appropriate perspective for the +question, “Would the top of Mount Fuji stick out of +the Sea of Japan?”. +5.2 +Ablation Study +Importance of decomposition-based retrieval. +In our proposed method, we retrieve relevant ex- + +Knowledge +Tabular +External +79.92 +Background +84.75 +Background + External +84.83 +Table 4: Performance of RR with different types of +knowledge on tabular reasoning: external only, back- +ground only, and a combination of both. +External +knowledge refers to WordNet and ConceptNet, while +background knowledge refers to the tables. +ternal knowledge based on the decomposed rea- +soning steps rather than the original query. To fur- +ther investigate the impact of this choice, we con- +ducted additional experiments in which we used +the original query for knowledge retrieval while +keeping other aspects of our method unchanged. +As shown in Table 3, the results for these experi- +ments are poor for both commonsense and tempo- +ral reasoning, indicating the importance of using +decomposition-based retrieval in our approach. +The impact of different types of knowledge. +For tabular reasoning, we use both external knowl- +edge (WordNet and ConceptNet) and background +knowledge (tables) in our experiments. +In this +section, we further examine the effect of differ- +ent types of knowledge on the performance of our +proposed method. As shown in Table 4, the addi- +tional improvement gained by incorporating Wiki- +data and ConceptNet in addition to tables is lim- +ited, indicating that GPT-3 already captures many +word-level relations in these external knowledge +sources. In addition, the observed significant im- +provement in tabular reasoning from using tables +alone suggests that our proposed method can also +effectively leverage background knowledge. +5.3 +Variations of the Proposed Approach +Basic approach: Weighting outputs. +In Sec- +tion 3, we present a basic version of our proposal +for taking advantage of external knowledge. Our +basic approach involves weighting outputs as indi- +vidual units and using a voting mechanism to se- +lect the best-supported prediction. We can also di- +rectly choose the best-supported output, which in- +cludes both an explanation and a prediction, with- +out using voting. +For example, in the running +example of “Did Aristotle use a laptop?” +(see +more in Section 3), the third reasoning path R3 is +the output most supported by the knowledge para- +graphs K1 and K2. +Variant I: Fact selection. +The first variant of +our approach involves selecting facts from the out- +puts of LLMs based on external knowledge. For +example, consider the running example of “Did +Aristotle use a laptop?”, where we only have ac- +cess to the first two reasoning paths, R1 and R2. +In this case, the first sentence in R2 and the sec- +ond sentence in R1 are supported by knowledge +K1 and K2, respectively. Therefore, the first vari- +ant would output the first sentence in R2 and the +second sentence in R1 as the supporting facts. +Variant II: Fact generation. +The second vari- +ant of our approach involves generating facts +based on both the outputs of LLMs and external +knowledge. For example, consider the running ex- +ample of “Did Aristotle use a laptop?”, where we +only have access to the first reasoning path R1. +The second sentence in R1 is supported by the sec- +ond knowledge paragraph K2. However, the first +sentence is not supported by any evidence para- +graphs. We can generate questions about the first +sentence, such as “When did Aristotle die?” and +use the first knowledge paragraph K1 to generate +a new fact: “Aristotle died in 322BC.”. As a result, +the second variant would output the generated fact +“Aristotle died in 322 BC.” and the second sen- +tence in R1 as the supporting facts. +Inference with supporting facts. +For the two +variants of our approach, we only have the sup- +porting facts and need to perform a final inference +step to obtain the corresponding prediction. One +option for this inference is to use LLMs, but they +can be costly (Brown et al., 2020) or difficult to +use (Zhang et al., 2022). An alternative is to use an +off-the-shelf model for inference with supporting +facts, such as UnifiedQA (Khashabi et al., 2020, +2022). As discussed in Appendix A.5, UnifiedQA +is more robust to noisy supporting facts than GPT- +3. We thus use the second version of UnifiedQA, +UnifiedQA-v2 (Khashabi et al., 2022), for the final +step of inference. +Experimental settings. +In this part, we focus +on commonsense reasoning and use the evidence +paragraphs provided in StrategyQA as the rele- +vant knowledge, rather than the retrieved para- +graphs discussed in Section 4.2. To evaluate the +quality of the explanations, we adopt the best met- +ric for factual consistency evaluation in Honovich + +1.3B +2.7B +6.7B +13B +30B +175B +Model Size +0 +20 +40 +60 +80 +Accuracy (%) +Chain-of-thought prompting +Rethinking with retrieval +(a) Accuracy of predictions +1.3B +2.7B +6.7B +13B +30B +175B +Model Size +20 +25 +30 +35 +40 +45 +50 +55 +Factuality (%) +Chain-of-thought prompting +Rethinking with retrieval +(b) Faithfulness of explanations +Figure 2: The effect of LM size on the performance of our proposed method (Variant II) and CoT prompting. We +use various sizes of OPT models, with the exception of the 175B model, which is GPT-3. +Methods +Accuracy (%) +Faithfulness (%) +CoT prompting +65.94 +38.73 +Basic (w/o voting) +76.86 +50.02 +Variant I +78.60 +54.11 +Variant II +78.60 +54.54 +Table 5: Comparison of various variations of RR and +the CoT prompting baseline on StrategyQA using evi- +dence paragraphs. +et al. (2022). For simplicity, we use the pre-trained +NLI model released by Nie et al. (2020) to com- +pute the NLI-based metric, rather than fine-tuning +T5-11B (Raffel et al., 2020) ourselves. The imple- +mentation details of the two variants can be found +in Appendix A.4. +Results. +Table 5 illustrates that the fact selec- +tion and fact generation variants of our proposal +improve the faithfulness of the supporting facts in +explanations, leading to increased prediction ac- +curacy compared to the basic approach without +voting. Across all variations of our proposal, we +observe significant improvements in both predic- +tion accuracy and the faithfulness of explanations +when compared to the CoT prompting baseline. +The incorporation of a voting mechanism leads +to an increased prediction accuracy of 79.91% for +the basic approach. Comparison with the perfor- +mance (i.e., 77.73%) of the same approach us- +ing retrieved paragraphs rather than evidence para- +graphs in Table 1 demonstrates that retrieved para- +graphs are also effective for our proposal, as both +significantly outperform the voting baseline, self- +consistency (i.e., 73.36%), as shown in Table 1. +It is noteworthy that UnifiedQA performs +poorly on StrategyQA, achieving an accuracy of +only 58.95%. +However, when provided with +gold supporting facts in StrategyQA, UnifiedQA +demonstrates excellent performance with an accu- +racy of 90.83%. This suggests that UnifiedQA is +suitable for last-step inference, but not effective +for answering questions in StrategyQA. +5.4 +Impact of the Size of LMs +In this subsection, we examine the effect of the +size of LMs on the performance of our proposed +method, specifically in the context of the fact gen- +eration variant. We compare the performance of +our method using various sizes of OPT models +(Zhang et al., 2022) in addition to GPT-3 (175B) +using the same experimental setup as in Sec- +tion 5.3. +As shown in Figure 2, our proposed +method (Variant II) consistently outperforms CoT +prompting in terms of both prediction accuracy +and the faithfulness of explanations, even when +using smaller LMs. +6 +Conclusion +In conclusion, the proposed approach is a promis- +ing solution for utilizing external knowledge to as- +sist LLMs. Unlike traditional methods, RR does +not require additional training or fine-tuning, mak- +ing it a lightweight and feasible option for LLMs. +Through extensive experiments on three reason- +ing tasks using GPT-3, we have shown that RR is +able to produce more faithful explanations and im- +prove the performance of LLMs. In the future, we +plan to investigate various variations of RR to en- +hance its effectiveness and efficiency in augment- +ing LLMs with external knowledge. + +References +Kurt Bollacker, Colin Evans, Praveen Paritosh, +Tim Sturge, and Jamie Taylor. 2008. 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The few-shot prompt utilizes +the same exemplars as the CoT prompt and does +not involve CoT reasoning processes. +A.2 +Description of Faithfulness Functions +For a sentence s, we denote its MPNet similarity, +entailment score, and contradiction score as M(s), +E(s), and C(s), respectively. In our experiments, +the corresponding thresholds for these scores are +Tm = 0.5, Te = 0.6, and Tc = 0.99. Given the +entailment scores, contradiction scores, and MP- +Net similarities of all supporting facts (denoted as +S) in the explanation of a reasoning path R, differ- +ent faithfulness functions fKB(·) can be adopted in +different settings as follows: +(1) fKB(R) = � +s∈S[M(s)×(M(s) >= Tm)+ +E(s) × (M(s) < Tm) − C(s)] +(2) fKB(R) = � +s∈S[M(s) + E(s)] +(3) fKB(R) = � +s∈S[E(s) × (E(s) >= Te) − +C(s) × (C(s) >= Tc)] +In Section 4, we employ function (1) for com- +monsense and tabular reasoning. For temporal rea- +soning, we use function (2) as the distinct nature of +sentences converted from temporal relations leads +to unreliable contradiction scores. In Sections 5.3- +5.4, we use function (3) for commonsense reason- +ing with evidence paragraphs, as the high quality +of the relevant knowledge negates the need for the +complementary use of the MPNet similarity to im- +prove the entailment score. +A.3 +Comparison of Retrieval Systems +For commonsense reasoning, we utilized different +retrieval systems in Karpukhin et al. (2020) to re- +trieve relevant paragraphs from Wikipedia. The +performance of BM25, DPR, and BM25+DPR +were 77.73%, 58.52%, and 77.29%, respectively, +indicating that BM25 is the best choice in our case. +A.4 +Implementation Details for the Two +Variants of RR +Fact selection implementation details. +In this +work, we utilize the information present in the top- +ranked output produced by our basic approach as +a guide. To this end, we apply a greedy clustering +algorithm to group the sentences from all outputs +into distinct topic categories based on the cosine +similarity of their MPNet sentence embeddings. +For each fact in the top-ranked output of our ba- +sic approach, we identify the fact with the highest +faithfulness within the same topic group and re- +place it in the output. The faithfulness of a fact is +calculated using the fKB function by replacing the +supporting facts with a single fact. +Fact generation implementation details. +In +this part, we generate questions for the named en- +tities present in each fact of the top-ranked output +produced by our basic approach, and retrieve the +corresponding answers from the evidence para- +graphs using UnifiedQA. We employ the ques- +tion generation model described in Deutsch et al. +(2021), which has been shown to be more ex- +tractive compared to other models as demon- +strated in Fabbri et al. (2021). +We adopt the +question filtering approach proposed in Honovich +et al. (2021) using an off-the-shelf extractive +QA model (ktrapeznikov/albert-xlarge-v2-squad- +v2 from Hugging Face (Wolf et al., 2020)). We +then use an off-the-shelf model (MarkS/bart-base- +qa2d from Hugging Face) to convert the generated +QA pairs into declarative sentences. +We apply +simple rules based on the entailment and contra- +diction scores of the selected facts from the fact se- +lection variant and the generated declarative sen- +tences to obtain the final generated facts. +A.5 +Comparison of Different Inference +Methods with Supporting Facts +In our experiments, we utilize UnifiedQA for the +final step of inference in both variants. However, +it is worth noting that GPT-3 could also be used +for this purpose. As shown in Table 7, we observe +that UnifiedQA performs better at inference with +generated facts, while GPT-3 with CoT prompt- +ing performs better with empty or gold facts. This +suggests that UnifiedQA is more robust to noisy + +Q: who was governor of minnesota when maathaad maathaadu mallige was released? +A: Maathaad Maathaadu Mallige was released on 24 August 2007. Tim Pawlenty served as the 39th gov- +ernor of Minnesota from 2003 to 2011. Thus, Tim Pawlenty was governor of minnesota when maathaad +maathaadu mallige was released. So the answer is Tim Pawlenty. +Q: who was us president during the costa rican civil war? +A: The Costa Rican civil war was a civil war in Costa Rica from 12 March to 24 April 1948. Harry S. +Truman was the 33rd president of the United States, serving from 1945 to 1953. Thus, Harry S. Truman +was us president during the costa rican civil war. So the answer is Harry S. Truman. +Q: who was governor of oregon when the collector was released? +A: The Collector premiered at the Cannes Film Festival on May 20, 1965. Mark Hatfield served as the +29th governor of Oregon from 1959 to 1967. Thus, Mark Hatfield was governor of oregon when the +collector was released. So the answer is Mark Hatfield. +Q: who was governor of oregon when shanghai noon was released? +A: Shanghai Noon was released on May 26, 2000. John Kitzhaber served as the 35th governor of Oregon +from 1995 to 2003. Thus, John Kitzhaber was governor of oregon when shanghai noon was released. So +the answer is John Kitzhaber. +Q: who was us president when john andrew shulze was a teenager? +A: John Andrew Shulze was born on July 19, 1775. A teenager is someone who is between 13 and 19 +years old. George Washington served as the first president of the United States from 1789 to 1797. Thus, +George Washington was us president when john andrew shulze was a teenager. So the answer is George +Washington. +Q: who was us president during the seventh coalition? +A: The War of the Seventh Coalition was from 20 March to 8 July 1815. James Madison served as the +fourth president of the United States from 1809 to 1817. Thus, James Madison was us president during +the seventh coalition. So the answer is James Madison. +Table 6: The CoT prompt for temporal reasoning. +Methods +Accuracy (%) +Empty facts +GPT-3 (zero-shot) +58.08 +GPT-3 (CoT) +65.94 +UnifiedQA +58.95 +Gold facts +GPT-3 (zero-shot) +81.66 +GPT-3 (CoT) +91.70 +UnifiedQA +90.83 +Generated facts +GPT-3 (zero-shot) +69.87 +GPT-3 (CoT) +76.42 +UnifiedQA +78.60 +Table 7: Comparison of different inference methods on +empty, gold, and generated facts. +inputs compared to GPT-3. +Additionally, both +UnifiedQA and GPT-3 with CoT prompting signif- +icantly outperform GPT-3 with zero-shot prompt- +ing, indicating that the CoT prompting is also ben- +eficial for the final step of inference. + +Charles Sumner Tainter was Born on April 25, 1854 ( 1854-04-25 ) Watertown, Massachusetts, U.S.. +Charles Sumner Tainter was Died on April 20, 1940 ( 1940-04-21 ) (aged 85) San Diego, California, +U.S.. The Nationality of Charles Sumner Tainter are American. The Known for of Charles Sumner +Tainter are Photophone, phonograph Father Of The Speaking Machine. +Question: Charles Sumner Tainter never left the state of Massachusetts. True or False? +Answer: Charles Sumner Tainter was died in San Diego, California, U.S.. California is a state. Thus, +Charles Sumner Tainter has left the state of Massachusetts. So the answer is false. +The Region of Curitiba are South. The Elevation of Curitiba are 934.6 m (3,066.3 ft). The Density of +Curitiba are 4,062/km 2 (10,523/sq mi). The Metro density of Curitiba are 210.9/km 2 (546.2/sq mi). +Question: Curitiba is above sea level. True or False? +Answer: The elevation of Curitiba are 934.6 m (3,066.3 ft). Elevation is a hypernym of level. Thus, +Curitiba is above sea level. So the answer is true. +Charles (Prince of Wales) was Born on 14 November 1948 ( 1948-11-14 ) (age 70) Buckingham Palace, +London, England. The Spouse of Charles (Prince of Wales) are Lady Diana Spencer ( m. 1981 ; div. +1996 ) , and Camilla Parker Bowles ( m. 2005 ). The Issue of Charles (Prince of Wales) are Prince +William, Duke of Cambridge , and Prince Harry, Duke of Sussex. +Question: Charles was born in 1948 and has been married twice. True or False? +Answer: Charles (Prince of Wales) was Born on 14 November 1948. The Spouse of Charles (Prince of +Wales) are Lady Diana Spencer ( m. 1981 ; div. 1996 ) , and Camilla Parker Bowles ( m. 2005 ). Married +is related to spouse. Thus, Charles was born in 1948 and has been married twice. So the answer is true. +The Born of Idris Elba are 6 September 1972 (age 46) Hackney, London, England. The Residence of +Idris Elba are London. The Other names of Idris Elba are DJ Big Driis, Big Driis the Londoner, Big +Driis, and 7 Dub. The Occupation of Idris Elba are Actor, producer, director, musician, and DJ. +Question: Idris Elba is an English entertainer. True or False? +Answer: The residence of Idris Elba is London. English is related to London. The occupation of Idris +Elba are actor, producer, director, musician, and DJ. Actor is a hyponym of entertainer. Musician is a +hyponym of entertainer. DJ is an entertainer. Thus, Idris Elba is an English entertainer. So the answer +is true. +The Breed of Jean, the Vitagraph Dog are Scotch Collie. The Sex of Jean, the Vitagraph Dog are Female. +The Born of Jean, the Vitagraph Dog are 1902 Eastport, Maine. The Years active of Jean, the Vitagraph +Dog are 1909 - 1916. +Question: Jean, the Vitagraph Dog was a Golden Retriever which perform in circus. True or False? +Answer: The Breed of Jean, the Vitagraph Dog are Scotch Collie. Collie is a hyponym of dog. Retriever +is a hyponym of dog. Thus, Jean, the Vitagraph Dog was not a Golden Retriever which perform in circus. +So the answer is false. +The Studio of Hydrograd are Sphere Studios, North Hollywood, Los Angeles. The Genre of Hydrograd +are Hard rock. The Label of Hydrograd are Roadrunner. The Producer of Hydrograd are Jay Ruston. +Question: Hydrograd is in the rap genre. True or False? +Answer: The Genre of Hydrograd are Hard rock. Rap is distinct from rock. Thus, Hydrograd is not in +the rap genre. So the answer is false. +Table 8: The CoT prompt for tabular reasoning. + diff --git a/-dAyT4oBgHgl3EQfdfdm/content/tmp_files/load_file.txt b/-dAyT4oBgHgl3EQfdfdm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..41439dfe4e75ed1edfe3cf176488f21dec1c5407 --- /dev/null +++ b/-dAyT4oBgHgl3EQfdfdm/content/tmp_files/load_file.txt @@ -0,0 +1,840 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf,len=839 +page_content='Rethinking with Retrieval: Faithful Large Language Model Inference Hangfeng He†∗ Hongming Zhang‡ Dan Roth§ †University of Rochester ‡Tencent AI Lab, Seattle §University of Pennsylvania hanfeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='he@rochester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='edu, hongmzhang@global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='com danroth@seas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='edu Abstract Despite the success of large language mod- els (LLMs) in various natural language pro- cessing (NLP) tasks, the stored knowledge in these models may inevitably be incom- plete, out-of-date, or incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' This mo- tivates the need to utilize external knowl- edge to assist LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Unfortunately, current methods for incorporating external knowl- edge often require additional training or fine-tuning, which can be costly and may not be feasible for LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' To address this issue, we propose a novel post-processing approach, rethinking with retrieval (RR), which retrieves relevant external knowledge based on the decomposed reasoning steps obtained from the chain-of-thought (CoT) prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' This lightweight approach does not require additional training or fine-tuning and is not limited by the input length of LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We evaluate the effectiveness of RR through extensive experiments with GPT-3 on three complex reasoning tasks: common- sense reasoning, temporal reasoning, and tabular reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Our results show that RR can produce more faithful explanations and improve the performance of LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='1 1 Introduction Large language models (LLMs) have shown exceptional performance across various tasks through in-context learning without task-specific training or fine-tuning (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Chowdhery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Ouyang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Recent progress in prompt- ing (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Kojima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022) and decoding (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022) has made it feasible for LLMs to tackle tasks that de- mand complex reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' ∗Part of this work was done while the author was at the University of Pennsylvania.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 1Our code is publicly available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' com/HornHehhf/RR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Query Prediction LLM Query Explanation + Prediction LLM Query Explanation + Prediction LLM (a) (b) (c) Knowledge Chain of thought Chain of thought Retrieval Rethinking Figure 1: An overview of three approaches for using LLMs: (a) Standard prompting for generating a pre- diction in response to a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (b) Chain-of-thought prompting for generating both an explanation and a prediction in response to a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (c) Rethinking with retrieval, our proposed approach for using the decom- posed reasoning steps obtained from chain-of-thought prompting to retrieve relevant external knowledge for LLMs, leading to more faithful explanations and im- proved predictions in response to a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' However, the knowledge stored in LLMs might inevitably be incomplete, out-of-date, or incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' As a result, external sources of knowledge, such as Wikipedia, may be essential for the success- ful deployment of LLMs for real-world applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Previously, people tried to utilize knowl- edge for smaller language models (LMs), such as T5 (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020), BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2019), and RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' However, these methods often require additional training or fine-tuning, which can be costly and thus imprac- tical for LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In this paper, we present a post-processing approach called rethinking with retrieval (RR) for utilizing external knowledge in LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Our method begins by using the chain-of-thought (CoT) prompting method (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022) to gen- erate a diverse set of reasoning paths, as described in Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We then use each rea- soning step in those paths to retrieve relevant ex- ternal knowledge, which enables RR to provide arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='00303v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='CL] 31 Dec 2022 more faithful explanations and more accurate pre- dictions, as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We evaluate the effectiveness of our proposed method, RR, on three complex reasoning tasks: commonsense reasoning, temporal reasoning, and tabular reasoning, using GPT-3 175B (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020) and different external knowledge sources: Wikipedia, Wikidata (Vrandeˇci´c and Krötzsch, 2014), WordNet (Miller, 1995), and Conceptnet (Speer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The results demonstrate that RR consistently outperforms all baselines on all three tasks without requiring ad- ditional training or fine-tuning, indicating the su- periority of our approach in leveraging external knowledge to enhance the performance of LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 2 Related Work Enhancing LMs through retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Retrieval- enhanced LMs have received significant attention as a means of improving performance through the incorporation of external knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For exam- ple, the k-most similar training contexts can be re- trieved to improve the estimation of the next word distribution in both the training stage (Borgeaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2021) and the inference stage (Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Furthermore, search query genera- tors have been adopted to generate search queries for search engines to retrieve relevant documents (Komeili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Shuster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thop- pilan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Other approaches have uti- lized retrieved documents as the additional con- text in generation tasks (Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Guu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Nakano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2021) instead use human feedback in a text-based web-browsing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Among these pre- vious works, Khandelwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2020) is most closely related to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' However, they focus on improving local inference by using the nearest neighbor datastore constructed from train- ing data, whereas we focus on conducting faith- ful inference using external knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In con- trast to other aforementioned approaches, which require training or fine-tuning to incorporate re- trieved knowledge, we propose a post-processing method for leveraging retrieved knowledge with- out additional training or fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Incorporating external knowledge into LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Significant effort has been devoted to leveraging external knowledge to improve the reasoning abil- ity of LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Previous work has incorporated exter- nal knowledge sources such as WordNet (Miller, 1995) and ConceptNet (Speer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2017) to en- hance LMs for tabular reasoning tasks (Neeraja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Varun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Explicit rules have also been added to inputs to improve rea- soning ability over implicit knowledge (Talmor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In addition, explicit knowledge from Wikidata (Vrandeˇci´c and Krötzsch, 2014) and im- plicit knowledge in LLMs have been integrated into a transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2017) for vi- sual question answering (Gui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Nye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2021) instead introduces a symbolic reason- ing module to improve coherence and consistency in LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Among these previous works, Nye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2021) is the most relevant to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Still, they focus on incorporating logical constraints to improve coherence and consistency, whereas we aim to improve the faithfulness of explanations through the use of external knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In con- trast to other aforementioned approaches that in- corporate external knowledge before generation and require additional training or fine-tuning, our proposal leverages external knowledge in a post- processing manner to enhance LMs without addi- tional training or fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Uncovering latent Knowledge in LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' There has been a line of work exploring the knowledge hidden within LLMs for reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' This has in- cluded the use of careful prompting to encourage LLMs to generate explanations in the reasoning process, such as through chain of thought prompt- ing in few-shot (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022) or zero-shot (Kojima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022) learning, or through the use of scratchpads for intermediate computation (Nye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In addition, various methods based on sampling a diverse set of reasoning paths in LLMs have been proposed, including training ver- ifiers to judge the correctness of model comple- tions (Cobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2021), calibrating model pre- dictions based on the reliability of the explana- tions (Ye and Durrett, 2022), and promoting self- consistency over diverse reasoning paths (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Zelikman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2022) instead it- eratively bootstrap the ability of LLMs to gener- ate high-quality rationales from a few initial ex- amples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2022) further propose generat- ing knowledge from LLMs, which is then used as additional input to improve commonsense reason- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In contrast to this line of work, our proposal focuses on leveraging external knowledge to en- hance LLMs, while they aim to explore the knowl- edge hidden within LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 3 Rethinking with Retrieval LLMs have been shown to generate incorrect sup- porting facts from time to time, even when they ac- curately capture the perspective needed to answer a question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' This phenomenon highlights intrinsic issues in the way LLMs store and retrieve knowl- edge, including (1) the presence of out-of-date, incorrect, or missing relevant knowledge in the pre-training corpus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2) incorrect memorization of relevant knowledge during pre-training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' and (3) incorrect retrieval of relevant knowledge during the inference stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' To address these issues, we propose the use of RR, which leverages external knowledge through the retrieval of relevant infor- mation based on decomposed reasoning steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Given a query Q, we utilize chain-of- thought prompting to generate a diverse set of rea- soning paths R1, R2, · · · RN, where each reason- ing path Ri consists of an explanation Ei followed by a prediction Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' After that, we retrieve relevant knowledge K1, · · · KM from a suitable knowledge base KB to support the explanation in each reason- ing path, and select the prediction ˆP that is most faithful to this knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' To better illustrate our proposal, we use “Did Aristotle use a laptop?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' as a running example in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Chain-of-thought prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In contrast to standard prompting, CoT prompting (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022) includes demonstrations of step-by-step rea- soning examples in the prompt to produce a series of short sentences that capture the reasoning pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For instance, given the question “Did Aris- totle use a laptop?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', CoT prompting aims to gen- erate the complete reasoning path “Aristotle died in 322 BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The first laptop was invented in 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, Aristotle did not use a laptop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is no.” rather than simply outputs “No.” Empirical results show that CoT prompting significantly im- proves the performance of LLMs on many multi- step reasoning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Therefore, we adopt CoT prompting to obtain both explanation E and pre- diction P for the query Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Sampling diverse reasoning paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Similar to Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2022), we sample a diverse set of rea- soning paths R1, R2, · · · RN rather than only con- sidering the greedy path as in Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For the question “Did Aristotle use a laptop?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', the potential reasoning paths can be as follows: (R1) Aristotle died in 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The first laptop was invented in 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, Aristotle used a lap- top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (R2) Aristotle died in 322BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The first laptop was invented in 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, Aristotle did not use a laptop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (R3) Aristotle died in 322BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The first laptop was invented in 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, Aristotle did not use a laptop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Knowledge retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Different knowledge bases can be used to address different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For example, to address the question “Did Aristotle use a laptop?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', we can use Wikipedia as the ex- ternal knowledge base KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Information retrieval techniques can be applied to retrieve the relevant knowledge K1, · · · KM from Wikipedia based on the decomposed reasoning steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Ideally, we would obtain the following two paragraphs from Wikipedia for this question: (K1) Aristotle (384–322 BC) was a Greek philoso- pher and polymath during the Classical pe- riod in Ancient Greece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (K2) The Epson HX-20, the first laptop computer, was invented in 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Faithful inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The faithfulness of each rea- soning path Ri can be estimated using a function fKB(Ri), which is based on relevant knowledge K1, · · · , KM retrieved from the knowledge base KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The final prediction is obtained through the application of the following inference procedure2: ˆP = arg max Pi∈{P1,··· ,PN} N � i=1 1(Pi = P)fKB(Ri), (1) where Pi denotes the corresponding prediction in the reasoning path Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' This inference procedure is designed to identify the most faithful prediction ˆP to the knowledge base among all predictions in the N reasoning paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For instance, in the run- ning example, given reasoning paths R1, R2, R3 and the retrieved knowledge K1, K2, the above in- ference procedure would output the prediction “So the answer is no.”, as it is supported by both R2 and R3 and has a higher faithfulness score com- pared to the prediction “So the answer is yes.”, which is only supported by R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 2Note that this is the basic version of faithful inference, and further variations can be found in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 4 Experiments In this section, we present the evaluation of our proposed method, RR, on three complex reason- ing tasks: commonsense reasoning, temporal rea- soning, and tabular reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='1 Baselines We compare with the following baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Zero-shot/few-shot prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In our experi- ments, we consider GPT-3 with standard zero- shot/few-shot prompting as baselines, following the approach described in Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2020), in which zero or few in-context exemplars of input- output pairs are provided in the prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Chain-of-thought prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In addition to the standard zero-shot/few-shot prompting, we also consider GPT-3 with the CoT prompting pro- posed in (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022) as a baseline in our ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' This approach involves feeding LLMs step-by-step reasoning examples instead of stan- dard input-output examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In addition, we also consider self-consistency (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022) as a baseline in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' This approach, proposed as an alternative to the naive greedy decoding used in CoT prompting (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022), involves sam- pling a diverse set of reasoning paths and select- ing the most consistent answer by marginalizing the sampled paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='2 Commonsense Reasoning Dataset description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For commonsense reason- ing, we consider the StrategyQA dataset (Geva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2021), which includes questions that require implicit reasoning strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For example, the question “Did Aristotle use a laptop?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' requires implicit decomposition into reasoning steps, while the question “Was Aristotle alive when the laptop was invented?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' explicitly specifies the reasoning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The StrategyQA dataset includes 2, 290 training examples, each consisting of a question (Q), a yes/no answer (A), a decomposition (D), evidence paragraphs (E), and supporting facts (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' On average, each question requires about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='93 rea- soning steps and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='33 evidence paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In ad- dition, a development set is constructed by ran- domly sampling 10% of the training examples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 229 examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The answer distribution is roughly balanced, with approximately 47% "yes" questions in both the training and development sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Unless otherwise specified, the models are evaluated on the development set3 for StrategyQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In this part, we uti- lize Wikipedia as the external knowledge base KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For each sentence in the explanation of ev- ery reasoning path, we first apply BM25 (Robert- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2009) to retrieve the top 10 most rele- vant paragraphs from Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In particular, we use the re-implementation of the sparse retrieval BM254 in Karpukhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2020) from Pyserini (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Subsequently, we use the pre- trained MPNet model (Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020) to se- lect the most similar paragraph based on the cosine similarity between the sentence embeddings of the retrieved paragraph and the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We then employ a pre-trained natural language inference (NLI) model (Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020) to obtain the en- tailment and contradiction scores for the sentence, treating the most similar paragraph as the premise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The faithfulness of each reasoning path is then calculated using fKB(·) based on the entailment scores, contradiction scores, and MPNet similari- ties of all sentences in the explanation of the rea- soning path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The final prediction for each ques- tion is obtained through faithful inference (Equa- tion 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' More details about fKB(·) can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='3 Temporal Reasoning Dataset description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In this experiment, we use the TempQuestions dataset (Jia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2018) to investigate temporal reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' This dataset in- cludes 1, 271 temporal questions that are divided into four classes: explicit temporal, implicit tem- poral, temporal answer, and ordinal constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The questions are paired with their answers from Freebase (Bollacker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' To examine the most challenging aspect of temporal reasoning, we focus on the set of implicit temporal questions, which contain implicit temporal expressions, in- cluding free-text temporal expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For ex- ample, the question “who was governor of oregon when shanghai noon was released?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' is an implicit temporal question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' To facilitate our analysis, we only consider questions with a single answer, re- sulting in a total of 175 examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Of these ex- 3As the annotations for the test set are not publicly avail- able, we use the development set for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' This allows us to perform a more comprehensive analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 4We also experimented with DPR and BM25+DPR, and found that BM25 outperformed these methods in our experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' More details can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Methods Commonsense Temporal Tabular GPT-3 Zero-shot prompting 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='08 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='40 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='00 Few-shot prompting 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='32 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='59 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='08 Chain-of-thought prompting 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='94 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='14 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='33 Self-consistency 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='36 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='28 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='00 Rethinking with retrieval 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='73 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='05 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='83 Table 1: Performance of different methods using GPT-3 on three reasoning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' amples, the first 6 are used for prompting, and the remaining 169 are used for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In this part, we utilize Wikidata (Vrandeˇci´c and Krötzsch, 2014) as the external knowledge base KB, as it is the largest publicly available knowledge graph, and the data from Freebase has been migrated to Wikidata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' To incorporate this knowledge into our system, we apply an entity linking system5 to each sentence in the explanation of each reasoning path to iden- tify the corresponding Wikidata pages for all enti- ties in the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Next, we extract all temporal relations from these relevant Wikidata pages and use templates to convert these temporal relations into sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' This step generates a set of rele- vant knowledge sentences for each sentence in the explanation of each reasoning path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The final pre- diction is then obtained by applying the procedure described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='2, in which the retrieved paragraphs are replaced with the relevant knowl- edge sentences from the current part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='4 Tabular Reasoning Dataset description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We consider the IN- FOTABS dataset (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020) for tabu- lar reasoning, which consists of 23, 738 human- written textual hypotheses based on premises in the form of tables extracted from 2, 540 unique Wikipedia info-boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We focus on the develop- ment set, which includes 1, 800 hypotheses based on 200 tables, and only consider entailed and con- tradictory hypotheses as it is tricky to write CoT demonstrations for neutral hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' This re- sults in a total of 1, 200 hypotheses based on 200 tables for evaluation, with an equal number of en- tailed and contradictory hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In this part, we utilize WordNet (Miller, 1995) and ConceptNet (Speer 5We use the spacy entity linker: https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='org/ project/spacy-entity-linker/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2017) as external knowledge bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' To con- vert tables into textual premises, we follow the same technique as in Varun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For each premise-hypothesis pair, we follow the procedure outlined in Varun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2022) to retrieve rele- vant word relation triples that connect the premise and hypothesis words, such as “married” RelatedTo ←−−−−→ “spouse”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' These triples are then converted into sentences using some simple templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The re- sulting sentences, along with the textual premises from the tables, serve as relevant knowledge for each sentence in the explanation of each reasoning path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' To obtain the final prediction, the procedure described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='2 is applied, whereby the retrieved paragraphs in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='2 are replaced with the relevant knowledge from the current part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='5 Evaluation Experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In all experiments, we utilize GPT-3 text-davinci-002 unless oth- erwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The maximum number of tokens for generation during completion is set to 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For zero-shot, few-shot, and chain-of-thought prompt- ing, the temperature is fixed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For self- consistency and rethinking with retrieval, we ran- domly sample 10 outputs6 with temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Detailed prompts can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We evaluate the performance of different methods on commonsense and tabular reasoning using ac- curacy, and on temporal reasoning using the exact match metric as defined in Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' As shown in Table 1, our proposed method, rethinking with retrieval, consistently outperforms all baselines on all three reasoning tasks without requiring additional training or fine- tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The results highlight the effectiveness of our approach in leveraging external knowledge to improve the performance of LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 6For commonsense reasoning, we sample 9 outputs, as we have found that odd numbers of outputs tend to yield better voting performance for self-consistency on StrategyQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Wrong supporting facts Wrong inference Question Was Lil Jon’s top ranked Billboard song a collaboration with a member of The Lox?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Would the top of Mount Fuji stick out of the Sea of Japan?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Answer No Yes Facts Lil Jon’s highest ranked billboard song was Yeah;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Yeah was a collaboration be- tween Lil Jon, Usher, and Ludacris;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Lox is a rap trio consisting of: Styles P, Sheek Louch, and Jadakiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The average depth of the Sea of Japan is 5,748 feet (1,752 metres) and its maxi- mum depth is 12,276 feet (3,742 metres);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Mount Fuji is 3,776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='24 metres (12,389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='2 ft) tall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' GPT-3 Lil Jon’s top ranked Billboard song was "Get Low" with the East Side Boyz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Lox is not a member of the East Side Boyz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, Lil Jon’s top ranked Bill- board song was not a collaboration with a member of The Lox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Mount Fuji is 3,776 meters tall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Sea of Japan is about 3,741 meters deep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, the top of Mount Fuji would not stick out of the Sea of Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the an- swer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Table 2: Examples of incorrect outputs from GPT-3 with CoT prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 5 Analysis In this section, we perform a thorough analysis to gain a deeper understanding of RR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='1 Limitations of LLMs in Reasoning In this subsection, we present an analysis of GPT- 3 with CoT prompting on the StrategyQA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Upon closer examination of the outputs of GPT- 3, we observed that it can provide reasonable ex- planations and correct predictions for a number of questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For example, when given the ques- tion “Will the Albany in Georgia reach a hundred thousand occupants before the one in New York?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', GPT-3 produced the following output: The Albany in New York has a pop- ulation of about 98,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Albany in Georgia has a population of about 77,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, the Albany in New York is more populous than the Albany in Georgia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The above output consists of three components: (1) supporting facts (in cyan) that are based on a particular perspective, (2) chaining arguments (in orange), and (3) a prediction (in green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Com- ponents (1) and (2) contribute to the explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Overall, the output exhibits a high level of quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' However, we also observed that GPT-3 may occa- sionally produce incorrect supporting facts for its explanations or make incorrect inferences for its Retrieval Commonsense Tabular Query-based 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='36 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='69 Decomposition-based 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='73 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='05 Table 3: Comparison of query-based and decomposition-based retrieval on commonsense and tabular reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' predictions, despite generally being able to iden- tify suitable perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Wrong supporting facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' As shown in Table 2, GPT-3 provides the incorrect supporting fact for Lil Jon’s top-ranked Billboard song, stating that it was “Get Low” instead of the correct answer, “Yeah”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' However, it does have the correct per- spective on how to answer the question, “Was Lil Jon’s top ranked Billboard song a collaboration with a member of The Lox?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Wrong inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' As shown in Table 2, GPT-3 makes an incorrect inference, stating that the top of Mount Fuji “would not stick out” of the Sea of Japan, rather than the correct answer, “would stick out”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' However, it does provide correct supporting facts based on the appropriate perspective for the question, “Would the top of Mount Fuji stick out of the Sea of Japan?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='2 Ablation Study Importance of decomposition-based retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In our proposed method, we retrieve relevant ex- Knowledge Tabular External 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='92 Background 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='75 Background + External 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='83 Table 4: Performance of RR with different types of knowledge on tabular reasoning: external only, back- ground only, and a combination of both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' External knowledge refers to WordNet and ConceptNet, while background knowledge refers to the tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' ternal knowledge based on the decomposed rea- soning steps rather than the original query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' To fur- ther investigate the impact of this choice, we con- ducted additional experiments in which we used the original query for knowledge retrieval while keeping other aspects of our method unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' As shown in Table 3, the results for these experi- ments are poor for both commonsense and tempo- ral reasoning, indicating the importance of using decomposition-based retrieval in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The impact of different types of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For tabular reasoning, we use both external knowl- edge (WordNet and ConceptNet) and background knowledge (tables) in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In this section, we further examine the effect of differ- ent types of knowledge on the performance of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' As shown in Table 4, the addi- tional improvement gained by incorporating Wiki- data and ConceptNet in addition to tables is lim- ited, indicating that GPT-3 already captures many word-level relations in these external knowledge sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In addition, the observed significant im- provement in tabular reasoning from using tables alone suggests that our proposed method can also effectively leverage background knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='3 Variations of the Proposed Approach Basic approach: Weighting outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In Sec- tion 3, we present a basic version of our proposal for taking advantage of external knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Our basic approach involves weighting outputs as indi- vidual units and using a voting mechanism to se- lect the best-supported prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We can also di- rectly choose the best-supported output, which in- cludes both an explanation and a prediction, with- out using voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For example, in the running example of “Did Aristotle use a laptop?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (see more in Section 3), the third reasoning path R3 is the output most supported by the knowledge para- graphs K1 and K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Variant I: Fact selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The first variant of our approach involves selecting facts from the out- puts of LLMs based on external knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For example, consider the running example of “Did Aristotle use a laptop?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', where we only have ac- cess to the first two reasoning paths, R1 and R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In this case, the first sentence in R2 and the sec- ond sentence in R1 are supported by knowledge K1 and K2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Therefore, the first vari- ant would output the first sentence in R2 and the second sentence in R1 as the supporting facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Variant II: Fact generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The second vari- ant of our approach involves generating facts based on both the outputs of LLMs and external knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For example, consider the running ex- ample of “Did Aristotle use a laptop?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', where we only have access to the first reasoning path R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The second sentence in R1 is supported by the sec- ond knowledge paragraph K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' However, the first sentence is not supported by any evidence para- graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We can generate questions about the first sentence, such as “When did Aristotle die?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' and use the first knowledge paragraph K1 to generate a new fact: “Aristotle died in 322BC.”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' As a result, the second variant would output the generated fact “Aristotle died in 322 BC.” and the second sen- tence in R1 as the supporting facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Inference with supporting facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For the two variants of our approach, we only have the sup- porting facts and need to perform a final inference step to obtain the corresponding prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' One option for this inference is to use LLMs, but they can be costly (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020) or difficult to use (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' An alternative is to use an off-the-shelf model for inference with supporting facts, such as UnifiedQA (Khashabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' As discussed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='5, UnifiedQA is more robust to noisy supporting facts than GPT- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We thus use the second version of UnifiedQA, UnifiedQA-v2 (Khashabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022), for the final step of inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Experimental settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In this part, we focus on commonsense reasoning and use the evidence paragraphs provided in StrategyQA as the rele- vant knowledge, rather than the retrieved para- graphs discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' To evaluate the quality of the explanations, we adopt the best met- ric for factual consistency evaluation in Honovich 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='3B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='7B 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='7B 13B 30B 175B Model Size 0 20 40 60 80 Accuracy (%) Chain-of-thought prompting Rethinking with retrieval (a) Accuracy of predictions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='3B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='7B 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='7B 13B 30B 175B Model Size 20 25 30 35 40 45 50 55 Factuality (%) Chain-of-thought prompting Rethinking with retrieval (b) Faithfulness of explanations Figure 2: The effect of LM size on the performance of our proposed method (Variant II) and CoT prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We use various sizes of OPT models, with the exception of the 175B model, which is GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Methods Accuracy (%) Faithfulness (%) CoT prompting 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='94 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='73 Basic (w/o voting) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='86 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='02 Variant I 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='60 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='11 Variant II 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='60 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='54 Table 5: Comparison of various variations of RR and the CoT prompting baseline on StrategyQA using evi- dence paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For simplicity, we use the pre-trained NLI model released by Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2020) to com- pute the NLI-based metric, rather than fine-tuning T5-11B (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020) ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The imple- mentation details of the two variants can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Table 5 illustrates that the fact selec- tion and fact generation variants of our proposal improve the faithfulness of the supporting facts in explanations, leading to increased prediction ac- curacy compared to the basic approach without voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Across all variations of our proposal, we observe significant improvements in both predic- tion accuracy and the faithfulness of explanations when compared to the CoT prompting baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The incorporation of a voting mechanism leads to an increased prediction accuracy of 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='91% for the basic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Comparison with the perfor- mance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='73%) of the same approach us- ing retrieved paragraphs rather than evidence para- graphs in Table 1 demonstrates that retrieved para- graphs are also effective for our proposal, as both significantly outperform the voting baseline, self- consistency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='36%), as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' It is noteworthy that UnifiedQA performs poorly on StrategyQA, achieving an accuracy of only 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' However, when provided with gold supporting facts in StrategyQA, UnifiedQA demonstrates excellent performance with an accu- racy of 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='83%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' This suggests that UnifiedQA is suitable for last-step inference, but not effective for answering questions in StrategyQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='4 Impact of the Size of LMs In this subsection, we examine the effect of the size of LMs on the performance of our proposed method, specifically in the context of the fact gen- eration variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We compare the performance of our method using various sizes of OPT models (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2022) in addition to GPT-3 (175B) using the same experimental setup as in Sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' As shown in Figure 2, our proposed method (Variant II) consistently outperforms CoT prompting in terms of both prediction accuracy and the faithfulness of explanations, even when using smaller LMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 6 Conclusion In conclusion, the proposed approach is a promis- ing solution for utilizing external knowledge to as- sist LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Unlike traditional methods, RR does not require additional training or fine-tuning, mak- ing it a lightweight and feasible option for LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Through extensive experiments on three reason- ing tasks using GPT-3, we have shown that RR is able to produce more faithful explanations and im- prove the performance of LLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In the future, we plan to investigate 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Chain of thought prompting elic- its reasoning in large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='11903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Transformers: State-of-the-art natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In Proceedings of the 2020 conference on empir- ical methods in natural language processing: system demonstrations, pages 38–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Xi Ye and Greg Durrett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The unreliability of explanations in few-shot in-context learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='03401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Eric Zelikman, Yuhuai Wu, and Noah D Good- man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Star: Bootstrapping reasoning with reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='14465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Susan Zhang, Stephen Roller, Naman Goyal, Mikel Artetxe, Moya Chen, Shuohui Chen, Christopher Dewan, Mona Diab, Xian Li, Xi Victoria Lin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Opt: Open pre- trained transformer language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='01068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Denny Zhou, Nathanael Schärli, Le Hou, Ja- son Wei, Nathan Scales, Xuezhi Wang, Dale Schuurmans, Olivier Bousquet, Quoc Le, and Ed Chi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Least-to-most prompting enables complex reasoning in large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='10625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' A Appendix In this section, we provide additional details on our experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Further information can be found in our code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='1 Detailed Prompts We adopt the same CoT prompt for commonsense reasoning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', StrategyQA) as those presented in Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The CoT prompt for tempo- ral reasoning is provided in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For tabular reasoning, we adopt the method of Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2020) for converting NLI into QA for RTE (Da- gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2005), and randomly sample 6 examples from the training data to construct the prompt, as shown in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The few-shot prompt utilizes the same exemplars as the CoT prompt and does not involve CoT reasoning processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='2 Description of Faithfulness Functions For a sentence s, we denote its MPNet similarity, entailment score, and contradiction score as M(s), E(s), and C(s), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In our experiments, the corresponding thresholds for these scores are Tm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='5, Te = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='6, and Tc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Given the entailment scores, contradiction scores, and MP- Net similarities of all supporting facts (denoted as S) in the explanation of a reasoning path R, differ- ent faithfulness functions fKB(·) can be adopted in different settings as follows: (1) fKB(R) = � s∈S[M(s)×(M(s) >= Tm)+ E(s) × (M(s) < Tm) − C(s)] (2) fKB(R) = � s∈S[M(s) + E(s)] (3) fKB(R) = � s∈S[E(s) × (E(s) >= Te) − C(s) × (C(s) >= Tc)] In Section 4, we employ function (1) for com- monsense and tabular reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For temporal rea- soning, we use function (2) as the distinct nature of sentences converted from temporal relations leads to unreliable contradiction scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='3- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='4, we use function (3) for commonsense reason- ing with evidence paragraphs, as the high quality of the relevant knowledge negates the need for the complementary use of the MPNet similarity to im- prove the entailment score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='3 Comparison of Retrieval Systems For commonsense reasoning, we utilized different retrieval systems in Karpukhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2020) to re- trieve relevant paragraphs from Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The performance of BM25, DPR, and BM25+DPR were 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='73%, 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='52%, and 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='29%, respectively, indicating that BM25 is the best choice in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='4 Implementation Details for the Two Variants of RR Fact selection implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In this work, we utilize the information present in the top- ranked output produced by our basic approach as a guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' To this end, we apply a greedy clustering algorithm to group the sentences from all outputs into distinct topic categories based on the cosine similarity of their MPNet sentence embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' For each fact in the top-ranked output of our ba- sic approach, we identify the fact with the highest faithfulness within the same topic group and re- place it in the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The faithfulness of a fact is calculated using the fKB function by replacing the supporting facts with a single fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Fact generation implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' In this part, we generate questions for the named en- tities present in each fact of the top-ranked output produced by our basic approach, and retrieve the corresponding answers from the evidence para- graphs using UnifiedQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We employ the ques- tion generation model described in Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2021), which has been shown to be more ex- tractive compared to other models as demon- strated in Fabbri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We adopt the question filtering approach proposed in Honovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' (2021) using an off-the-shelf extractive QA model (ktrapeznikov/albert-xlarge-v2-squad- v2 from Hugging Face (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=', 2020)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We then use an off-the-shelf model (MarkS/bart-base- qa2d from Hugging Face) to convert the generated QA pairs into declarative sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' We apply simple rules based on the entailment and contra- diction scores of the selected facts from the fact se- lection variant and the generated declarative sen- tences to obtain the final generated facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='5 Comparison of Different Inference Methods with Supporting Facts In our experiments, we utilize UnifiedQA for the final step of inference in both variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' However, it is worth noting that GPT-3 could also be used for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' As shown in Table 7, we observe that UnifiedQA performs better at inference with generated facts, while GPT-3 with CoT prompt- ing performs better with empty or gold facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' This suggests that UnifiedQA is more robust to noisy Q: who was governor of minnesota when maathaad maathaadu mallige was released?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' A: Maathaad Maathaadu Mallige was released on 24 August 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Tim Pawlenty served as the 39th gov- ernor of Minnesota from 2003 to 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, Tim Pawlenty was governor of minnesota when maathaad maathaadu mallige was released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is Tim Pawlenty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Q: who was us president during the costa rican civil war?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' A: The Costa Rican civil war was a civil war in Costa Rica from 12 March to 24 April 1948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Harry S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Truman was the 33rd president of the United States, serving from 1945 to 1953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, Harry S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Truman was us president during the costa rican civil war.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is Harry S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Truman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Q: who was governor of oregon when the collector was released?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' A: The Collector premiered at the Cannes Film Festival on May 20, 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Mark Hatfield served as the 29th governor of Oregon from 1959 to 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, Mark Hatfield was governor of oregon when the collector was released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is Mark Hatfield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Q: who was governor of oregon when shanghai noon was released?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' A: Shanghai Noon was released on May 26, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' John Kitzhaber served as the 35th governor of Oregon from 1995 to 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, John Kitzhaber was governor of oregon when shanghai noon was released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is John Kitzhaber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Q: who was us president when john andrew shulze was a teenager?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' A: John Andrew Shulze was born on July 19, 1775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' A teenager is someone who is between 13 and 19 years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' George Washington served as the first president of the United States from 1789 to 1797.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, George Washington was us president when john andrew shulze was a teenager.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is George Washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Q: who was us president during the seventh coalition?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' A: The War of the Seventh Coalition was from 20 March to 8 July 1815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' James Madison served as the fourth president of the United States from 1809 to 1817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, James Madison was us president during the seventh coalition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is James Madison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Table 6: The CoT prompt for temporal reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Methods Accuracy (%) Empty facts GPT-3 (zero-shot) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='08 GPT-3 (CoT) 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='94 UnifiedQA 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='95 Gold facts GPT-3 (zero-shot) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='66 GPT-3 (CoT) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='70 UnifiedQA 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='83 Generated facts GPT-3 (zero-shot) 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='87 GPT-3 (CoT) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='42 UnifiedQA 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='60 Table 7: Comparison of different inference methods on empty, gold, and generated facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' inputs compared to GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Additionally, both UnifiedQA and GPT-3 with CoT prompting signif- icantly outperform GPT-3 with zero-shot prompt- ing, indicating that the CoT prompting is also ben- eficial for the final step of inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Charles Sumner Tainter was Born on April 25, 1854 ( 1854-04-25 ) Watertown, Massachusetts, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='. Charles Sumner Tainter was Died on April 20, 1940 ( 1940-04-21 ) (aged 85) San Diego, California, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='. The Nationality of Charles Sumner Tainter are American.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Known for of Charles Sumner Tainter are Photophone, phonograph Father Of The Speaking Machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Question: Charles Sumner Tainter never left the state of Massachusetts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' True or False?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Answer: Charles Sumner Tainter was died in San Diego, California, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='. California is a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, Charles Sumner Tainter has left the state of Massachusetts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Region of Curitiba are South.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Elevation of Curitiba are 934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='6 m (3,066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='3 ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Density of Curitiba are 4,062/km 2 (10,523/sq mi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Metro density of Curitiba are 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='9/km 2 (546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='2/sq mi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Question: Curitiba is above sea level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' True or False?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Answer: The elevation of Curitiba are 934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='6 m (3,066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content='3 ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Elevation is a hypernym of level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, Curitiba is above sea level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Charles (Prince of Wales) was Born on 14 November 1948 ( 1948-11-14 ) (age 70) Buckingham Palace, London, England.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Spouse of Charles (Prince of Wales) are Lady Diana Spencer ( m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 1981 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 1996 ) , and Camilla Parker Bowles ( m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 2005 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Issue of Charles (Prince of Wales) are Prince William, Duke of Cambridge , and Prince Harry, Duke of Sussex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Question: Charles was born in 1948 and has been married twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' True or False?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Answer: Charles (Prince of Wales) was Born on 14 November 1948.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Spouse of Charles (Prince of Wales) are Lady Diana Spencer ( m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 1981 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' div.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 1996 ) , and Camilla Parker Bowles ( m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' 2005 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Married is related to spouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, Charles was born in 1948 and has been married twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Born of Idris Elba are 6 September 1972 (age 46) Hackney, London, England.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Residence of Idris Elba are London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Other names of Idris Elba are DJ Big Driis, Big Driis the Londoner, Big Driis, and 7 Dub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Occupation of Idris Elba are Actor, producer, director, musician, and DJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Question: Idris Elba is an English entertainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' True or False?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Answer: The residence of Idris Elba is London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' English is related to London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The occupation of Idris Elba are actor, producer, director, musician, and DJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Actor is a hyponym of entertainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Musician is a hyponym of entertainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' DJ is an entertainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, Idris Elba is an English entertainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Breed of Jean, the Vitagraph Dog are Scotch Collie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Sex of Jean, the Vitagraph Dog are Female.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Born of Jean, the Vitagraph Dog are 1902 Eastport, Maine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Years active of Jean, the Vitagraph Dog are 1909 - 1916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Question: Jean, the Vitagraph Dog was a Golden Retriever which perform in circus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' True or False?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Answer: The Breed of Jean, the Vitagraph Dog are Scotch Collie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Collie is a hyponym of dog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Retriever is a hyponym of dog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, Jean, the Vitagraph Dog was not a Golden Retriever which perform in circus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Studio of Hydrograd are Sphere Studios, North Hollywood, Los Angeles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Genre of Hydrograd are Hard rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Label of Hydrograd are Roadrunner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' The Producer of Hydrograd are Jay Ruston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Question: Hydrograd is in the rap genre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' True or False?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Answer: The Genre of Hydrograd are Hard rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Rap is distinct from rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Thus, Hydrograd is not in the rap genre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' So the answer is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} +page_content=' Table 8: The CoT prompt for tabular reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAyT4oBgHgl3EQfdfdm/content/2301.00303v1.pdf'} diff --git a/-tE3T4oBgHgl3EQfrgrp/content/tmp_files/2301.04661v1.pdf.txt b/-tE3T4oBgHgl3EQfrgrp/content/tmp_files/2301.04661v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b58d9911f30530c1662b9b89a570a6f532eb6f11 --- /dev/null +++ b/-tE3T4oBgHgl3EQfrgrp/content/tmp_files/2301.04661v1.pdf.txt @@ -0,0 +1,4184 @@ +Kondo Resonance, Pomeranchuk Effect, and Heavy Fermi Liquid in Twisted Bilayer +Graphene - A Numerical Renormalization Group Study +Geng-Dong Zhou1 and Zhi-Da Song1, ∗ +1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China +(Dated: January 13, 2023) +Low energy electron Hamiltonian in the magic-angle twisted bilayer graphene can be equivalently +reformulated as a topological heavy fermion model [Phys. Rev. Lett. 129, 047601 (2022)]. It consists +of effective localized f-electrons at AA-stacking regions and itinerant Dirac c-electrons. In this work, +we applied systematic analytical and numerical renormalization group analyses to a single-impurity +version of this model. +We obtained a phase diagram consisting of a Fermi liquid phase in the +Kondo regime, a Fermi liquid phase in the frozen impurity regime, and various local moment phases +with different spin momenta. Remarkably, this single-impurity phase diagram explains a series of +experimental discoveries reported recently: (i) the zero-energy peak at fillings 1 ≲ |ν| < 2 observed +in STM at low temperatures (T < 1K) [Nature 588, 610 (2020), Nature Physics 17, 1375 (2021), +Nature 600, 240 (2021), Nature 589, 536 (2021)], (ii) the cascade of transitions observed in STM at +higher temperatures [Nature 582, 198 (2020), Nature Physics 17, 1375 (2021)], (iii) the Pomeranchuk +effect at ν ≈ ±1 observed in transport and compressibility measurements [Nature 592, 214 (2021), +Nature 592, 220 (2021)], which show that the Fermi liquid ground state develops local moments +upon heating, and (iv) various transport experiments showing resistance peaks but no gaps around +ν ≈ ±1. For the first time, we point out that all these phenomena result from a simple unified +mechanism - the Kondo effect. The Fermi liquid state at ν ≈ ±1 exhibiting the zero-energy peak +is stabilized by the Kondo screening with a Kondo temperature TK ≈ 1.5K. A higher temperature +will suppress the Kondo screening and favor a local moment phase that obeys Curie’s law and +contributes to an entropy of the order of Boltzmann’s constant (per moir´e cell). +We computed +the spectral densities, entropies, and spin susceptibilities at various fillings and temperatures, and +obtained results quantitatively comparable to experiments. We also predict the heavy Fermi liquid +as the ground state in a wide range of fractional fillings and conjecture that it is the parent state +for the observed unconventional superconductivity. +I. +INTRODUCTION +Since the first discovery of the superconductivity [1] +and correlated insulators [2] in magic-angle twisted bi- +layer graphene (MATBG) [3], MATBG has become a new +platform to study novel correlation effects in flat-band +systems and has attracted extensive attentions. Remark- +ably rich physics, including interplay between supercon- +ductivity [4–11] and strong correlation [4, 6–8, 12–19], in- +teraction driven Chern insulators [20–26], strange metal +behaviors [27–29], and the Pomeranchuk effect [30, 31], +etc., have been observed in MATBG. Several theoretical +understandings of the correlated states have also been +achieved recently. The strong correlation arises from the +two topological flat bands [32–37], each of which is four- +fold degenerate due to the spin and valley d.o.f. A large +U(4) symmetry group [38–42] emerges in the flat-band +limit, where the actual bandwidth is counted as negligi- +ble. Then the observed correlated states at integer fillings +ν = 0, ±1, ±2, ±3 can be understood as flavor polarized +states [38–40, 42–58] that spontaneously break the U(4) +symmetry. Here |ν| is the number of electrons (ν > 0) +or holes (ν < 0) per moir´e cell counted from the charge +neutrality point (CNP). The continuous U(4) degeneracy +leads to Goldstone mode fluctuations [59, 60] that may +∗ songzd@pku.edu.cn +destroy the long-range order due to the Mermin-Wagner +theorem. +Less theoretical understandings have been achieved for +the gapless states, which are observed at both fractional +and integer fillings. For example, at ν ≈ ±1, depend- +ing on the experimental setup, both gapped correlated +insulators [4, 6, 7] and gapless Fermi liquid states [6, 25– +27, 29–31] have been observed in transport experiments, +suggesting that they are competing ground states with +close energies. Interestingly, the observed gapless Fermi +liquid states around ν = ±1 usually exhibit resistivity +peaks [25–27, 29, 31] above a few kelvins, and the peaks +could become increasingly pronounced as temperature +rises. +Scanning tunneling microscope (STM) measure- +ments [11, 19, 21, 22] have constantly seen that, at low +temperatures of about a few hundred millikelvins, the +conduction (valence) band will be pinned at the Fermi +level and form a sharp zero-energy peak for the fillings +1 ≲ ν < 2 (−2 < ν ≲ −1). The sharp peak does not fit +the intuition of Stoner instability given that the interac- +tion is indeed strong. When the temperature increases to +a few or ten kelvins, these peaks develop into a cascade +of transitions like a quantum dot model [17, 19]. +In this work, based on the recently developed topolog- +ical heavy fermion (THF) model [61, 62], we find that +the zero-energy peak, as well as the gapless Fermi liq- +uid states at ν ≈ ±1, are results of the Kondo resonance +[63–80]. At a higher temperature exceeding the Kondo +arXiv:2301.04661v1 [cond-mat.str-el] 11 Jan 2023 + +2 +energy scale, the local moments (LMs) formed by local- +ized electrons give rise to the transition cascades, and +the resistance peaks around ±1. We have numerically +reproduced the temperature-dependent features of the +spectral density. Our theory is also fully consistent with +the Pomeranchuk effect observed around fillings ν = ±1 +[30, 31], which show that local moments appear upon +heating. We have calculated LM entropies as functions +of the temperature, filling, and an external field, and +obtained curves comparable to the experimentally mea- +sured data in Ref. [30, 31]. +Our theory shows that the observed gapless states at +the fillings 1 ≲ |ν| < 2 are the strongly correlated heavy +Fermi liquid state. Since this filling range overlaps with +the superconductivity [1, 4–11] and the strange metal +[27–29] around ν = −2+δ (for small δ), the heavy Fermi +liquid state could be the parent state for the unconven- +tional superconductivity, as it is in the heavy fermion +materials with 4f or 5f electrons. +This opens a new +perspective - with a solid theoretical and experimental +basis at the same time - to study the superconductivity +in MATBG. +This work is organized as the followings. For this work +to be self-contained, in Sec. II we will review the THF +model and its symmetry shortly. In Sec. III, based on a +poor man’s scaling analysis and experimental facts, we +argue that the Kondo screening effect is irrelevant at +CNP, and hence the ground state at CNP is the pre- +viously identified symmetry-broken correlated insulator +[38–40, 42–44]. Then we derive a simpler effective peri- +odic Anderson model describing active excitations upon +the correlated ground state. +We further simplify the +model to a single-impurity version. In Sec. IV, by ap- +plying poor man’s scaling and Wilson’s numerical renor- +malization group (NRG) method [81–83] to the single- +impurity problem, we obtain a phase diagram charac- +terized by strong coupling fixed points and various LM +fixed points. The strong coupling phase is divided into +a Kondo regime and a frozen impurity (FI) regime. We +also present a detailed analysis of the RG flows at these +fixed points. The gapless 1 ≲ |ν| < 2 states are found +to be in the Kondo regime. In Secs. V and VI we cal- +culate the spectral densities, spin susceptibilities, and +entropies as functions of the filling ν and the temper- +ature T. The spectral densities feature sharp Kondo res- +onances at low temperatures smaller than the Kondo en- +ergy scale. Whereas at higher temperatures, the Kondo +resonances are suppressed and the Hubbard bands be- +come clearer, which periodically cross the Fermi level as +ν changes from 0 to 4 as that of a quantum dot model, +matching the STM experiments. The spin susceptibili- +ties obey Curie’s law at high temperatures, suggesting +the existence of LM, and approach constants at lower +temperatures, suggesting the Fermi liquid behavior. The +Kondo temperature TK can be estimated as the turning +temperature of the two behaviors of the spin susceptibil- +ity. For 1 ≲ |ν| < 2, we find kBTK ranges from 0.129meV +to 0.675meV. (Here kB is Boltzmann’s constant.) +At +c +c +Energy (meV) +0 +40 +-20 +(b) +ΓM +KM +MM +2|M| +(a) +-40 +-60 +20 +60 +KM +ΓM +MM +c +-80 +80 +f +f +(c) +G +ΓM +KM +MM +c +L=0,0 +L=1,-1 +(d) +G +15 +10 +0 +∆(ω) (meV) +ω (meV) +5 +FIG. 1. +The THF model. +(a) Top: red spheres represent +the effective f-electrons located at AA-stacking regions of +MATBG, and blue spheres represent the itinerant c-electrons. +Bottom: the moir´e Brillouin zone. (b) Black bands are given +by the THF model. The red and blue bands are the decou- +pled f- and c-bands, respectively. +M is a parameter that +determines the bandwidth of the flat bands. We focus on the +M → 0 limit in this work. (c) Excitation spectrum of the ac- +tive c-electrons upon the symmetry-breaking parent state for +ν > 0, where M and µc are set to zero. (d) The hybridization +function ∆(ω) contributed by the c-bands in (c). A nonzero +M will not change the asymptotic behavior of ∆(ω) around +the gap. +|ν| = 1 and kBTK ≈ 1meV, the entropy contributed by +the LM is around 2 ln 2 · kB ≈ 1.39kB. In the presence of +a strong in-plane magnetic field, the entropy is quenched +to about ln 2·kB ≈ 0.69kB. These values will be appreci- +ated if one notices that the measured entropies at ν ≈ 1 +and T ≈ 10K in the absence and presence of a strong in- +plane field are about 1.2kB and 0.6kB, respectively [30]. +In the Sec. VII, we discuss the heavy Fermi liquid states +at 1 ≲ |ν| < 2 and propose experiments to confirm them. +We estimate the heavy fermion bands and their quasi- +particle weights using spectral information provided by +the NRG calculation. The possible effects of the RKKY +interactions are also discussed. +II. +THE THF MODEL +One theoretical challenge in studying correlation +physics in MATBG is the lack of a fully symmetric lat- +tice model for low energy physics, which is forbidden +by the band topology protected by a C2zT symmetry +[32–34] and an emergent particle-hole symmetry P [37] +- even though extended Hubbard models [84–88] can be +constructed at the sacrifice of either symmetry or local- +ity. The band topology was thought as fragile [32–34] +but was later shown to be a stable symmetry anomaly +jointly protected by C2zT and P [37]. The THF model +[61, 62] resolved this problem by ascribing the strong +correlation to effective f-orbitals at the AA-stacking re- +gions, which form a triangular lattice, and leaving the +remaining low energy states to continuous c-bands de- +scribed by a topological Dirac Hamiltonian (Fig. 1). The +THF model faithfully reproduces the symmetry, topol- +ogy, dispersion, and Coulomb interaction of the continu- +ous Bistritzer-MacDonald model [3]. Its free part is given + +3 +by +ˆH0 = −µ ˆ +N + +� +ηs +� +aa′ +� +|k|<Λc +H(c,η) +aa′ (k)c† +kaηsckaηs ++ +� +ηsαa +� +|k|<Λc +� +e− |k|2λ2 +2 +H(cf,η) +aα +(k)c† +kaηsfkαηs + h.c. +� +. +(1) +Here µ is the chemical potential, ˆN is the particle-number +operator, ckaηs is the fermion operator for the c-electron +of the momentum k, orbital a (= 1, 2, 3, 4), valley η (= +±), and spin s (=↑, ↓), fkαηs is the corresponding fermion +operator for the f-electron of the orbital α (=1,2). The +summation over k for the c-bands is in principle limited +within the cutoff Λc. But the theory is well-defined and +yields the same low energy physics if we take the Λc → ∞ +limit. H(c,η)(k) = v⋆(ησx⊗σ0kx−σy⊗σzky)+02×2⊕Mσx +is the Dirac Hamiltonian of the c-bands. When M ̸= 0, +c-bands have a quadratic band touching at the zero en- +ergy, whereas when M = 0, c-bands become linear. The +two-by-two block of H(cf,η) +aα +(k) for a = 1, 2 is given by +γσ0 + v′ +⋆(ησxkx + σyky), and the two-by-two block of +H(cf,η) +aα +(k) for a = 3, 4 vanishes. +The parameter λ in +the second line of Eq. (1) is the spread of the Wan- +nier functions of f-electrons, and it truncates the hy- +bridization at |k| ≫ λ−1. +In this work we adopt the +w0/w1 = 0.8 parameters of Ref. [61]: γ = −24.75meV, +v⋆ = −4.303eV · ˚A, v′ +⋆ = 1.623eV · ˚A, λ = 1.4131/kθ, +kθ = 1.703˚A−1 · 2 sin θm +2 with θm = 1.05◦ being the first +magic angle. (w0 and w1 are the interlayer couplings of +MATBG at the AA-stacking and AB-stacking regions, +respectively. Due to the corrugation effect [89–92], w0 +is usually smaller than w1. Ref. [85] estimates w0/w1 as +0.817.) The resulting band structure with a nonzero M +(3.697meV) is shown in Fig. 1(b). One can see that the +topological flat bands result from the hybridization be- +tween c- and f-bands. As explained in detail in Ref. [61] +and shown in Fig. 1(b), the parameter M determines the +bandwidth of the flat-bands. +In each valley, the Hamiltonian ˆH0 respects a magnetic +space group P6′2′2 [32] (#177.151 in the BNS setting +[93]), generated by C3z, C2x, C2zT, and translation sym- +metries. The two f-orbitals and the four c-bands have +effective angular momenta L = −η, η and L = −η, η, +0, 0, respectively. +As shown in detail in [61], the six- +by-six representations of C3z, C2x, C2zT symmetries on +these orbitals are eiη 2π +3 σz ⊕ eiη 2π +3 σz ⊕ σ0, I3×3 ⊗ σx, and +I3×3 ⊗ σxK, respectively, where σ0,x,y,z are Pauli matri- +ces and K the complex conjugation. One can verify that +the Hamiltonian matrices H(c,η) and H(cf,η) given in the +last paragraph respect these crystalline symmetries. The +interaction Hamiltonian given in the following paragraph +also respects these crystalline symmetries. +The interaction Hamiltonian is given by +ˆHI =U1 +2 +� +R +δnf +Rδnf +R + U2 +2 +� +⟨RR′⟩ +δnf +Rδnf +R′ + +1 +2NM +� +qaa′ +V (q)δnc +−qa′δnc +qa + +1 +NM +� +Rqa +Wae−iq·Rδnf +Rδnc +qa − +J +2NM +� +ηη′αα′ +ss′ +� +|k|,|k′|<Λc +R +� +(ηη′ + (−1)α+α′)e−i(k−k′)·R− λ2(k2+k′2) +2 +(f † +Rα′η′s′fRαηs − 1 +2δηη′δαα′δss′)(c† +k,α+2ηsck′,α′+2η′s′ − 1 +2δkk′δηη′δαα′δss′) +� +, +(2) +where NM is the number of moir´e cells, fRαηs is the real +space fermion operator for the f-electrons, R are the tri- +angular lattice shown in Fig. 1(a), ⟨RR′⟩ represents near- +est neighbor pairs (ordered), δnf +R = � +αηs(f † +RαηsfRαηs − +1 +2) is the density operator of f-electrons, +δnc +qa += +� +ηsk(c† +k+qaηsckaηs − 1 +2δq0) is the density operator for c- +electrons of the orbital a with k and k + q being limited +within the cutoff Λc. U1,2, V (q), Wa are the density- +density interaction between ff, cc, cf electrons, respec- +tively, and J is an exchange interaction between cf elec- +trons. We adopt the w0/w1 = 0.8 parameters of Ref. [61]: +U1 = 57.95meV, U2 = 2.329meV, W1 = W2 = 44.03meV, +W3 = W4 = 50.20meV, J = 16.38meV. We choose +V (q) as the double-gate-screened Coulomb interaction, +πξ2Uξ +Ω0 +tanh(ξ|q|/2) +ξ|q|/2 +, with ξ = 10nm being distance between +MATBG and the gates, Uξ = 24meV the Coulomb inter- +action at the distance ξ, and Ω0 ≈ 156nm2 the area of +the moir´e cell. +Hereafter, we mainly focus on the flat-band limit, i.e., +M = 0. In this limit, an exact U(4) symmetry of ˆH0+ ˆHI +between the spin, valley, and orbital flavors emerge, as +previously recognized in the projected Coulomb Hamil- +tonian of the continuous model [38–42]. We emphasize +that this U(4) symmetry is not related to the so-called +chiral limit [35, 94], i.e., w0 = 0, which leads a distinct +U(4) symmetry [41, 39]. The U(4) symmetry in the flat- +band limit has been shown as a good approximation for +realistic parameters such as w0/w1 = 0.8 [41, 43]. The +sixteen U(4) generators acting on fRαηs, ckaηs (a = 1, 2), +and ckaηs (a = 3, 4) are +Σf +µν = {σ0τ0ςν, σyτxςν, σyτyςν, σ0τzςν} , +(3) +Σc12 +µν = {σ0τ0ςν, σyτxςν, σyτyςν, σ0τzςν} , +(4) +and +Σc34 +µν = {σ0τ0ςν, −σyτxςν, −σyτyςν, σ0τzςν} , +(5) + +4 +respectively, where ςν (ν = 0, x, y, z) are Pauli matrices +acting in the spin subspace, τµ (µ = 0, x, y, z) are Pauli +matrices acting in the valley subspace, and σ0,x,y,z are +Pauli matrices acting in the orbital subspace. With the +help of U(4) symmetry, the J term in Eq. (2) can be writ- +ten as a ferromagnetic coupling between the U(4) LM of +f-electrons and the U(4) LM of c-electrons [61]. When +M ̸= 0, only the µ = 0, z U(4) generators commute with +the Hamiltonian, leading to a lower U(2)×U(2) symme- +try group. The rotation generated by µ = z, ν = 0 is +referred to as the valley-U(1) symmetry. +Consistent with previous results [38–40, 42–44], a +Hartree-Fock treatment of the THF model has predicted +the ground state at CNP as a U(4) LM state that re- +spects a U(2)×U(2) subgroup [61]. The LM forms a 20- +fold multiplet belonging to the [2, 2]4 representation [43] +of the U(4) group. These states can be approximately +written as +|Ψ0⟩ = e−iθµν ˆΣµν � +R +f † +R1+↑f † +R1+↓f † +R2+↑f † +R2+↓|FS⟩ , +(6) +where the |FS⟩ is the Fermi sea state with the half-filled +c-bands, ˆΣµν are the U(4) generator operators defined +by the matrices in Eqs. (3) to (5), and θµν are the ro- +tation parameters, and an implicit summation over re- +peated µ, ν indices is assumed. When θµν’s are zero, |Ψ0⟩ +is the valley-polarized state because all the occupied f- +electrons are in the η = + valley, and the U(2)×U(2) +subgroup is generated by Σ0,ν and Σz,ν (ν = 0, x, y, z). +For nonzero θµν’s, |Ψ0⟩ respects an equivalent U(2)×U(2) +subgroup. The Kramers inter-valley coherent states can +be obtained by choosing nonzero θx0 and θy0 satisfying +θ2 +x0 + θ2 +y0 = (π/4)2. When M ̸= 0, the Kramers inter- +valley coherent states are the ground states, while the +valley polarized states have higher energy (∼ 0.1meV) +[43, 61]. +III. +EFFECTIVE ANDERSON MODEL FOR +ν > 0 STATES +A. +Irrelevance of Kondo screening at CNP +Here we argue that the Kondo screening effect is ir- +relevant at CNP; hence, the U(4) LM state in Eq. (6) +is valid as an approximate ground state. We first exam- +ine the energy scale of a fully symmetric Kondo state +at CNP. Since the f-sites are almost decoupled from +each other, a reasonable approximation is to view each +f-site as a single Anderson impurity coupled to a bath +of c-electrons. If we only consider the on-site U1 inter- +action and the single-particle hybridization between f- +and c-electrons (H(cf,η)(k) in Eq. (1)), then it is almost +a standard Anderson model with eight flavors. The ef- +fect of c-bath is described by the hybridization function +∆(ω), defined as the imaginary part of the self-energy of +a free f-electron (in the absence of U1) coupled to the c- +bath, i.e., Im[Σ(f) +αηs,α′η′s′(ω)] = δα,α′δηη′δss′sgn(ω)∆(ω). +The identity matrix form of Im[Σαηs,α′η′s′(ω)] is guar- +anteed by the spin-SU(2) (δss′), the valley-U(1) and the +time-reversal (δηη′), and crystalline (δαα′) symmetries. +Low energy c-bands (k → 0) are coupled to the impu- +rity through the constant coupling γ in H(cf,η)(k). Then +∆(ω) would be proportional to the density of states ρ(ω) +of c-bands. In the flat-band limit (M = 0), c-bands have +a linear dispersion (Fig. 1(b)) and hence the density of +states, as well as the hybridization function, linear in en- +ergy, i.e., ρ(ω) ∼ |ω|, ∆(ω) ∼ |ω|. As a consequence, +low-lying states of the impurity will see vanishing bath +electrons when the energy scale is small enough. Both nu- +merical [95–97] and analytical [98] RG studies have shown +that Anderson impurity models with such a ∆(ω) ∼ |ω| +hybridization function do not have the strong coupling +fixed point that exhibits Kondo screening. Instead, the +only stable fixed point is the LM phase. +With a finite M, the c-bands given by H(c,η)(k) in +Eq. (1) have a quadratic touching at the zero energy, +i.e., ±(−M/2 + +� +M 2/4 + v2⋆k2), leading to a finite den- +sity of states at the zero energy and hence a finite ∆(0). +Nevertheless, as explained in the following, the Kondo +energy scale resulting from realistic parameters is neg- +ligible. +In Appendix B 2 we derived an analytical ex- +pression of ∆(ω) for the symmetric state at CNP. For +|ω| > M, ∆(ω) is almost linear in |ω|, i.e., ∆(ω) ≈ b·|ω|. +For |ω| < M, ∆(ω) is a constant plus a linear term: +∆(ω) ≈ ∆(0)(1 + |ω|/M). Using the parameters given +in Sec. II and M = 3.697meV, we have b ≈ 0.129, +∆0 ≈ 0.239meV. A rough estimation of the Kondo en- +ergy scale can be made by replacing the ω-dependent +∆(ω) with the constant ∆(0) at ω = 0. Then naively +applying the large-N formula at second order [99], i.e., +kBTK ≈ De− +πU1 +4N ∆(0) with N = 8 being the number of +flavors and D = U1/2 the energy scale up to which the +perturbation theory applies, leads to an extremely small +kBTK ≈ U1 · 2 × 10−11. A better estimation is given by +a poor man’s scaling that considers the ω-dependence of +∆(ω). Readers may refer to Appendix B 2 for more de- +tails. Here we only present the main results. There are +two stages in the RG process: (i) a stage with energy +scale from U1/2 - scale up to which the perturbation the- +ory applies - to M. +(ii) a stage with an energy scale +below M. RG in the first stage effectively enhances ∆(0) +to g1∆(0) with g1 ≈ 2.34. Then, RG in the second stage +gives +kBTK ≈Meye− +πU1 +4N g1∆(0) +≈3.8 × 10−4meV +(2M = 7.39meV). +(7) +where y ≈ 1 is a factor contributed by the ω-dependence +of ∆(ω) in the second stage. This value is still much lower +than the energy gain of the symmetry-broken correlated +state [43, 59]. The bandwidth of the Goldstone modes +at CNP from ΓM to MM is about 8meV. (See Fig. 2 of +Ref. [59]). +If we understand this spectrum as a tight- +binding band of the Holstein–Primakoff bosons on the +f-sites, which form a triangular lattice, then the nearest + +5 +neighbor hopping is about 8meV/8=1meV. This hopping +indicates an RKKY interaction much larger than kBTK +evaluated in Eq. (7). +The actual TK can even be much smaller than the value +in Eq. (7). First, as ∆(0) → 0 when M → 0, TK decays +exponentially when M decreases. For example, a band- +width 2M = 5meV corresponds to +kBTK ≈ 5.1 × 10−6meV +(2M = 5meV) . +(8) +Second, because we only considered the U1 interaction +and the cf hybridization that gives all flavors of f- +electrons the same ∆(ω) (guaranteed by symmetries), the +single-impurity model has a U(8) symmetry. This U(8) +symmetry must be broken when other interaction terms, +e.g., J in Eq. (2), are taken into account, leading to a +multiplet splitting. +When the energy scale in the RG +is smaller than the multiplet splitting, the degeneracy +factor N should be reduced accordingly, and TK will be +further suppressed. (One can see section 17.2 of Ref. [99] +and Appendix B 3 for examples of how multiplet splitting +suppresses TK.) +The U(4) LM state at CNP is also supported by var- +ious experiments. In contrast to the Kondo resonance, +STM measurements have shown strong suppression of the +density of states at the zero energy at CNP [11, 12, 14– +17, 19, 21, 22]. Some transport experiments [4, 6, 7, 27] +also exhibits a gap behavior at CNP. Although there are +also transport experiments showing semimetal behavior, +the gaplessness can be explained if there are fluctuations +of the local moments from site to site, which is possi- +ble due to the Goldstone mode fluctuations [59, 60] and +possible inhomogeneity of the sample. +B. +Periodic Anderson model for ν > 0 states +We aim for an effective model describing the active ex- +citations upon the ground state |Ψ0⟩ (Eq. (6)) at CNP. +Let us first assume the valley-polarized state, where θµν’s +in Eq. (6) are all zero such that all the occupied f- +electrons are in the η = + valley. +As detailed in the +supplementary material of Ref. [61] and in Ref. [59], the +lowest particle and hole excitations are in the η = − +and η = + valleys, respectively. Thus, for ν > 0 states, +only particle excitations in the η = − valley will be in- +volved, and the electrons in the η = + valley can be +viewed as a static background. +The effective Hamil- +tonian can be obtained by replacing operators in the +η = + valley by their expectation values on |Ψ0⟩, which +are ⟨f † +Rα+sfR′α′+s′⟩ = δRR′δαα′δss′, ⟨c† +ka+sck′a′+s′⟩ ≈ +1 +2δkk′δaa′δss′, ⟨c† +ka+sfRα+s′⟩ = 0. Substituting these ex- +pectation values into ˆH0 + ˆHI, we obtain the effective +free Hamiltonian +ˆHeff +0 += +� +|k|<Λc +aa′s +(H(c) +aa′(k) − µδaa′)c† +kascka′s − µ +� +Rαs +nf +Rαs ++ +� +|k|<Λc +aα +� +e− 1 +2 λ2k2H(cf) +aα (k)c† +kasfkαs + h.c. +� +, +(9) +where nRαs = f † +RαsfRαs is the density operator of f- +electrons. Here we have dropped the valley index η as +they are limited to η = −. The H(c)(k) and H(cf)(k) +matrices are given by the H(c,−)(k) and H(cf,−)(k) ma- +trices defined after Eq. (1). We also obtain the effective +interaction Hamiltonian +ˆHeff +I += U1 +2 +� +R +nf +Rnf +R + U2 +2 +� +⟨RR′⟩ +nf +Rnf +R′ ++ +1 +2NM +� +qaa′ +V (q)δnc +−q,a′δnc +q,a + +1 +NM +� +Rqa +Wae−iq·Rnf +Rδnc +qa +− +J +NM +� +Rss′ +� +α +� +|k|,|k′|<Λc +e−i(k−k′)·R− λ2(k2+k′2) +2 +× (f † +Rαs′fRαs − 1 +2δss′)(c† +k,α+2,sck′,α+2,s′ − 1 +2δss′) , +(10) +where nf +R = � +αs nf +Rαs, δnc +qa = � +sk(c† +k+qasckas − 1 +2δq0) +with |k| and |k + q| being limited within the cutoff Λc. +The δnf +R operator in Eq. (2), which represents the density +deviation from the charge background at CNP, is now +replaced by nf +R, the total density in the η = − valley, +because the charge background is compensated by the +occupied η = + electrons. The J term in Eq. (2) is also +simplified: As active excitations are limited to η = −, +the factor ηη′ + (−1)α+α′ becomes 2δα,α′. +In the flat-band limit (M += 0), +ˆHeff +0 ++ ˆHeff +I +ap- +plies to arbitrary U(4) partners of the valley-polarized +state, including the so-called Krammers intervalley co- +herent state. To be specific, for a generic |Ψ0⟩ given in +Eq. (6), we can always define rotated operators fRαs = +UfRα−sU †, ckas = Ucka−sU †, where U = e−iθµν ˆΣµν is +the U(4) rotation defining |Ψ0⟩, such that the effective +Hamiltonian on the rotated basis is same as Eqs. (9) +and (10). +The effective Hamiltonian ˆHeff +0 ++ ˆHeff +I +respects all the +crystalline symmetries discussed in Sec. II. The effective +angular momenta of the active two f-orbitals and four +c-bands are L = 1, −1 and L = 1, −1, 0, 0, respectively. +And, the six-by-six representations of C3z, C2x, C2zT +on these orbitals are e−i 2π +3 σz ⊕ e−i 2π +3 σz ⊕ σ0, 13×3 ⊗ σx, +and 13×3 ⊗ σxK, respectively, with K being the complex +conjugation. +In the flat-band limit (M = 0), as |Ψ0⟩ +respects a U(2)×U(2) subgroup of the U(4) group, e.g., +independent spin-charge rotations in the two valleys for +the valley polarized |Ψ0⟩, one may expect a U(2)×U(2) +symmetry of ˆHeff +0 ++ ˆHeff +I . +However, since the effective +Hamiltonian only involves half of the d.o.f., e.g., the ac- +tive η = − valley for the valley polarized |Ψ0⟩, only a +single U(2) factor is meaningful for ˆHeff +0 ++ ˆHeff +I . There- +fore, hereafter we will say that ˆHeff +0 + ˆHeff +I +respects a U(2) +symmetry group. +When M ̸= 0, the U(4) symmetry is broken, and the +effective Hamiltonian will have an additional term. In +Appendix A we show that to the order of M 2, the cor- + +6 +ˆH0 + ˆHI +ˆHeff +0 ++ ˆHeff +I +ˆHSI +M = 0 +U(4) +U(2) +U(2)×U(2) +M ̸= 0 U(2)×U(2) +U(2) +U(2)×U(2) +TABLE I. Continuous symmetries of the Hamiltonians. ˆH0 + +ˆHI is the original THF model. For ν > 0 (ν < 0), ˆHeff +0 + ˆHeff +I +is the effective periodic Anderson model for the active particle +(hole) excitations upon the symmetry broken state at CNP. +ˆHSI is a single-impurity version of ˆHeff +0 ++ ˆHeff +I . +rection is simply an energy shift +M 2 +J +� +|k|<Λc +� +a=3,4 +� +s +c† +kasckas + O(M 4) . +(11) +Thus, the M-term breaks neither the crystalline symme- +try nor the U(2) symmetry, and it will play a minor role +in the effective theory. To avoid confusion, in Table I we +summarize the continuous symmetries of different Hamil- +tonians with M = 0 or M ̸= 0 discussed in this work. +The effective model for ν < 0 states, which only in- +volve hole excitations, can be obtained by applying the +particle-hole operation Pc [41, 61] to ˆHeff +0 ++ ˆHeff +I . +C. +Single impurity model for ν > 0 states +Hereafter we mainly focus on a single-impurity version +of ˆHeff +0 ++ ˆHeff +I , where only the correlation at the R = 0 +f-site is considered. The interactions involving other f- +sites will be treated at the mean-field level. The STM +spectra [11, 19, 21, 22] that show the zero-energy peaks +also clearly show a continuity between the gapped CNP +state and the gapless states at 1 ≲ |ν| < 2, implying +that they have the same symmetries. Therefore, in this +work, we assume no additional symmetry breaking. For +the completeness of the discussion, we also extend our +symmetric assumption to |ν| ≥ 2 states. One should be +aware that additional symmetry breaking may happen +at low temperatures in |ν| ≥ 2 states due to the effec- +tive RKKY interactions neglected in this work. Thus the +symmetric assumption is invalid for |ν| ≥ 2 states at low +temperatures. Nevertheless, the |ν| ≥ 2 states may re- +cover the symmetries at higher temperatures, where our +symmetric theory applies. +At a given filling ν, the symmetric mean field is char- +acterized by only a few parameters: νf = ⟨nf +R⟩, νc,a = +1 +NM ⟨δnc +q=0,a⟩, where νc,1 = νc,2, νc,3 = νc,4 due to the +crystalline symmetries. The actual values of νf and νc,a +can be determined self-consistently. The considered cor- +related site at R = 0 is described by the Hamiltonian +ˆHf = −µf +� +αs +nf +αs + U1 +� +(αs)<(α′s′) +nf +αsnf +α′s′ , +(12) +where the lattice index R is omitted for simplicity, µf = +−6νfU2−� +a νc,aWa− 1 +2U1+Jνc,3+µ is an effective chem- +ical potential for the f-site, and µ is the global chemical +potential determined by the total filling ν. The U2, Wa, J +terms in µf are contributed by the Hartree mean fields of +the U2, Wa, and J interactions in Eq. (10), respectively. +The U1 term in µf is from the diagonal U1 interactions +in Eq. (10), i.e., +1 +2U1 +� +αs nf +αsnf +αs. The U1 interaction +in ˆHf only contains the off-diagonal U1 interactions of +Eq. (10). +The effective Hamiltonian of c-electrons is given by +ˆHc = +� +|k|<Λc +� +aa′s +(H(c) +aa′(k) + ∆H(c) +aa′ − µcδaa′)c† +kascka′s , (13) +where H(c)(k) = −v⋆(σx ⊗ σ0kx + σy ⊗ σzky) is the free +Dirac Hamiltonian, ∆H(c) +aa′ = G·δaa′(δa3+δa4) is a mean- +field term that split the a = 1, 2 and a = 3, 4 c-electrons, +µc = −νfW1 −νc V +Ω0 +µ is an effective chemical potential +of c-electrons. The W1 and V terms in µc are contributed +by the W and V interactions in Eq. (10), respectively. +The mean field term G arises from two interaction terms: +(i) A mean field treatment of the J interaction in Eq. (10) +yields an energy shift J( 1 +2 − 1 +4νf) for a = 3, 4 c-electrons. +(ii) The Hartree mean fields of the Wa interactions in +Eq. (10) are νfW1 and νfW3 for a = 1, 2 and a = 3, 4 +c-electrons, respectively. As we have absorbed νfW1 to +µc, a = 3, 4 c-electrons have an extra energy shift (W3 − +W1)νf. Combining the two effects, the parameter G is +given by G = J +2 +(W3−W1− J +4 )νf. Since the interaction +V (q) of c-electrons is completely treated at the mean- +field level, ˆHc is an effective free-fermion system. +As +detailed in Appendix A, the band structure of Eq. (13) is +given by G/2± +� +G2/4 + v2⋆k2. In Fig. 1(c) we plot the c- +bands with µc = 0 and G = 10meV. It has a gap opened +by G, where, according to the symmetry representations +given in Sec. III B, the lowest conduction and highest +valence band states have angular momenta 0, 0 and 1, −1, +respectively. +The f-site is coupled to c-electrons via two terms. One +is the hybridization +ˆHhyb = +1 +√NM +� +|k|<Λc +� +aαs +� +e− λ2k2 +2 +H(cf) +aα (k)c† +kasfαs + h.c. +� +. +(14) +The other coupling term is the remaining ferromagnetic +exchange interaction +ˆHJ = − +J +NM +� +|k|,|k′|<Λc +� +αss′ +e− 1 +2 λ2(k2+k′2)(f † +αsfαs′ − 1 +4δss′νf) +× (c† +k′α+2s′ckα+2s − 1 +2δk,k′δss′νc,α+2) . +(15) +By “remaining” we mean that the mean field back- +grounds +1 +4νf and +1 +2νc,a are deducted in +ˆHJ. +As ex- +plained below, ˆHJ leads to an effective Hund’s coupling +that changes the symmetry of the single-impurity model. +There are also remaining density-density interactions be- +tween c- and f-electrons, i.e., Wa(nf − νf)(nc +q,a − νc,a). +However, these remaining density-density interactions do +not change the essence of the single-impurity problem as +ˆHJ does. + +7 +Thanks to the C3z symmetry, +ˆHhyb and ˆHJ couple +the f-electrons to two independent baths belonging to +different angular momenta. This allows us to treat the +two terms separately. In a polar coordinate, ˆHhyb only +couples f-electrons to +�ckαs = 1 +A +� +a +ˆ +dφ · H(cf) +αa (k)ckas +(α = 1, 2) , +(16) +where k = k(cos φ, sin φ), and A is a normalization fac- +tor. Explicitly, there are �ck1s ∼ +´ +dφ·(γck1s−v′ +⋆keiφck2s) +and �ck2s ∼ +´ +dφ · (γck2s − v′ +⋆ke−iφck1s). Under the C3z +operation (defined in Sec. III B), �ck1s and �ck2s have effec- +tive angular momenta 1, -1, respectively. On the other +hand, ˆHJ only couples f-electrons to +�ckas = 1 +A′ +ˆ +dφ · ckas +(a = 3, 4) . +(17) +Both ckas (a = 3, 4) have the effective angular momen- +tum 0 under C3z. +Because �ckas (a = 1, 2) and �ckas +(a = 3, 4) form different representations of C3z, they do +not couple to each other, hence the ˆHhyb-bath and the +ˆHJ-bath are indeed independent. +As a ferromagnetic coupling always flows to zero and +becomes irrelevant in low energy physics, we can inte- +grate out the ˆHJ-bath in a single attempt. This leads to +an effective Hund’s coupling (Appendix A) +ˆHH = JH +� +α +nα↑nα↓ , +(18) +where JH, estimated as 0.3meV, is the additional energy +that two electrons will acquire if they occupy the same +orbital. A nonzero M does not change the form of ˆHH. +Integrating out the ˆHhyb-bath leads to a self-energy +correction Σ(f) +αs,α′s′(ω) to the f-electrons, the imaginary +part of which defines the hybridization function ∆(ω), +i.e., Im[Σ(f) +αs,α′s′(ω)] = δαα′δss′sgn(ω)∆(ω). The identity +matrix structure of the self-energy is guaranteed by SU(2) +spin rotation symmetry and crystalline symmetries. In +Appendix A we derived an analytical expression of ∆(ω). +As shown in Fig. 1(d), where ∆(ω) for µc = 0 and G = +10meV is plotted, ∆(ω) has an abnormal ω-dependence +compared to those in usual metals. First, ∆(ω) = 0 for +ω in the gap of c-bands (Fig. 1(c)). Second, because f- +electrons (L = 1, −1) have different angular momenta +as the lowest conduction c-bands (L = 0), hybridization +between them vanishes as k → 0. +As a result, ∆(ω) +linearly approaches zero at the conduction band edge. +Third, as f-electrons have the same angular momenta as +the highest valence c-bands, ∆(ω) approaches a constant +at the valence band edge. In summary, around the gap +∆(ω) has the asymptotic behaviors +∆(ω) ∼ +� +� +� +� +� +|ω + µc − G|, +ω + µc → G + 0+ +0, +0 ≤ ω + µc ≤ G +const., +ω + µc → −0+ +. +(19) +A nonzero M does not change the asymptotic behav- +iors as these behaviors are guaranteed by the C3z sym- +metry that is also respected by M. Due to the damp- +ing factor e− 1 +2 λ2k2 in ˆHhyb, c-electrons with momenta +|k| ≫ 1/λ will not contribute to ∆(ω). Thus, ∆(ω) de- +cays exponentially when ω exceeds v⋆/λ ∼ 95meV. In +the rest of this work, we will restrict the hybridization to +|ω| < D = 100meV. +Baths giving rise to the same ∆(ω) are physically +equivalent. Therefore, we can choose a bath that is as +simple as possible. We introduce the following effective +single-impurity Hamiltonian +ˆHSI = ˆHf + ˆHH + +� +αs +ˆ D +−D +dϵ · ϵ · d† +αs(ϵ)dαs(ϵ) ++ +� +αs +ˆ D +−D +dϵ · +� +∆(ϵ) +π +(f † +αsdαs(ϵ) + h.c.) , +(20) +where ˆHf and ˆHH are given by Eqs. (12) and (18), re- +spectively, and dαs(ϵ) are the auxiliary bath fermions in- +troduced to reproduce the hybridization function. dαs(ϵ) +satisfy {dα′s′(ϵ′), d† +αs(ϵ)} = δα′αδs′sδ(ϵ′ − ϵ). ˆHSI is com- +pletely defined by the following parameters: µf the chem- +ical potential of f-electrons, U1 the Coulomb repulsion, +JH the Hund’s coupling, and ∆(ω) the hybridization +function, which further depends on µc the chemical po- +tential of c-electrons and G the gap of c-bands (Fig. 1(c)). +As explained at the beginning of this subsection, the ac- +tual values of µf, µc, and G depend on the occupations +νf, νc,a, which are further determined by self-consistent +calculations at given total fillings ν. We plot µf, µc, and +G as functions of ν in Fig. 2(a). For ν changing from 0 +to 4, µc changes from 0 to 64.70meV, µf changes from +-28.98meV to 124.13meV, and G changes from 8.19meV +to 14.21meV. We can now regard µc, µf, G as given pa- +rameters that define the single-impurity problem. +It is worth mentioning that Eq. (20) has an emergent +U(2)×U(2) symmetry - independent spin-charge rota- +tions in the α = 1, 2 orbitals - that is higher than the +U(2) symmetry of Eqs. (9) and (10). It is not surpris- +ing that a single-impurity model has a higher symmetry +than its lattice version. +For example, if J = 0, there +would be no Hund’s coupling JH, and ˆHSI would have +a U(4) symmetry, as expected in a four-flavor Anderson +impurity model without multiplet splitting. +IV. +PHASE DIAGRAM OF THE +SINGLE-IMPURITY MODEL +A. +Poor man’s scaling +Before going to numerical calculations, we first apply a +poor man’s scaling to the single impurity model Eq. (20). +The scaling theory helps us understand several features +of the data obtained by NRG, as will be discussed in +the following subsections. +And, it predicts the Kondo + +8 +temperatures in the same order as those predicted by +NRG. +We assume that the ground state of the isolated impu- +rity has nf (integer) occupied f-electrons. One should +not confuse nf with νf - the expectation value of f- +occupation after the impurity is coupled to the bath. The +chemical potential µf must be in the range (nf − 1)U1 < +µf < nfU1. +We apply a Schrieffer-Wolff transforma- +tion to Eq. (20) to obtain an effective Coqblin–Schrieff +model where the local Hilbert space of f-electrons is re- +stricted to nf particles. The transformation involves vir- +tual particle and hole excitations, the energies of which +are ∆E+ = nfU1 − µf and ∆E− = µf − (nf − 1)U1, re- +spectively. (We have ignored the JH term in ∆E± as it +is small compared to U1.) Adding the two contributions, +we have +ˆH = ˆHH + +� +αs +ˆ D +−D +dϵϵd† +αs(ϵ)dαs(ϵ) + 4g +πU1 +� +αα′ss′ +ˆ D +−D +dϵdϵ′ +� +× +� +∆(ϵ)∆(ϵ′)(f † +αsfα′s′ − xδαα′δss′)d† +α′s′(ϵ′)dαs(ϵ) +� +. +(21) +The parameters g, x are given by +g = U1 +4 +� +1 +∆E+ ++ +1 +∆E− +� +, +x = ∆E− +U1 +. +(22) +g is a dimensionless parameter characterizing the anti- +ferromagnetic coupling strength between the impurity +and the bath. x appears as a “charge background” of +the f-electrons. For µf = (νf − 1 +2)U1, there is g = 1, +x = 1 +2. For a generic (nf − 1)U1 < µf < nfU1, g ≥ 1 +and 0 < x < 1. Flow equations of g, x are derived in Ap- +pendix B 3, where the divergence of g indicates the strong +coupling fixed point that exhibits the Kondo screening. +We notice that x always flows to nf/4, i.e., the occupa- +tion fraction of f-electrons. +One should be careful about the cutoff D in Eq. (21) +First, it must be smaller than ∆E+ and ∆E− for the +Schrieffer-Wollf transformation to be valid. Second, for +analytical conveniences, we only keep the positive branch +of ∆(ω) (Eq. (19)) at ω > G − µc because when ν > +0 the negative branch is far away from the Fermi level +(Fig. 1(d)). Hence, we also require D < µc − G. We can +choose D = min(µc − G, ∆E+, ∆E−). +We first consider the case nf = 1. The flow equation +of g(t) as the cutoff is reduced to De−t is given by +dg +dt = 4∆(0) +πU +Ng2 + O(e−t) , +(23) +and the initial condition g(0) is given by Eq. (22). Here +N = 4 is the number of flavors. The local Hilbert space +for nf = 1 is four-fold. The Hund’s coupling JH does not +split the four-fold degeneracy and hence does not appear +in the flow equation. The O(e−t) terms are irrelevant +at small energy scales but they may affect the coupling +constant at an early stage of the RG process. Using a +linear approximation of ∆(ω), i.e., ∆(ω) ≈ ∆(0)(1 + +ω/(µc − G)), we obtain (Appendix B 3) +kBT (1) +K += Dey1 · e− +πU1 +4N ∆(0)g(0) +(24) +where y1 ≈ −0.75 +D +µc−G < 0 is factor contributed by the +irrelevant O(e−t) terms and will slightly suppress the +Kondo energy scale. The suppression factor ey1 appears +because, for the nf = 1 states, the virtual processes that +contribute to the RG equation involve more hole exci- +tations than particle excitations in the bath, such that +the smaller ∆(ω < 0) contributes more than the larger +∆(ω > 0). As a result, the resulting kBTK is smaller +than the standard case (y1 = 0) where the coupling is a +constant, i.e., ∆(ω) = ∆(0). +We then study the RG equations at nf = 2. Unlike +the nf = 1 case, Hund’s coupling JH splits the six- +dimensional local Hilbert space. According to Eq. (18), +the four states with (nf +1↑, nf +1↓; nf +2↑, nf +2↓) =(10;10), (10;01), +(01;10), (01;01) do not feel JH and have the energy +−2µf + U1, whereas the two states (11;00), (00;11) have +the energy −2µf + U1 + JH. We divide the RG process +into two stages: (i) a stage with an energy scale from D +to JH, (ii) a stage with an energy scale below JH. In the +first stage, JH plays a minor role; hence, a flow equation +similar to Eq. (23) applies. If g diverges before the energy +scale reaches JH, then the Kondo scale is given by +kBT (2)′ +K += D · e− +πU1 +4N ∆(0)g(0) +(25) +there is no y-factor as in Eq. (24) because, for the nf = 2 +states, the virtual processes that contribute to the RG +equations evolve equal holes and particles in the bath. +Otherwise, g will be renormalized to +g1 = +g(0) +1 − g(0) 4∆(0) +πU1 N ln D +JH +(26) +at the energy scale of JH. As detailed in Appendix B, RG +in the second stage is similar to Eq. (23) except that, due +to the multiplet splitting, the factor N = 4 is replaced +by 2. This leads to the Kondo energy scale +kBT (2)′′ +K += JH · e +− +πU1 +8∆(0)g1 = D +� D +JH +� N +2 −1 +e +− +πU1 +8∆(0)g(0) . (27) +Considering the g may diverge in either the first or the +second stage, the physical Kondo energy scale can be +written as +kBT (2) +K += +� +kBT (2)′ +K , +kBT (2)′ +K +> JH +kBT (2)′′ +K +, +otherwise +. +(28) +The theory at nf = 3 is almost the same as the theory +at nf = 1 except that the four single-particle states are +now replaced by the four single-hole states. Thus, the +local Hilbert space is also four-dimensional and will not +be split by Hund’s coupling JH. The Kondo energy scale +is given by +kBT (3) +K += Dey3 · e− +πU1 +4N ∆(0)g(0) +(29) +where y3 ≈ 0.75 +D +µc−G > 0 is a factor contributed by +the irrelevant O(e−t) terms and will slightly enhance the + +9 +ν +kBT (P ) +K +(meV) δK(meV) kBT (χ) +K +(meV) +0.75 +0.280 +0.165 +0.151 +1.00 +0.250 +0.158 +0.129 +1.25 +0.366 +0.305 +0.190 +1.50 +0.780 +0.558 +0.344 +1.75 +1.620 +1.674 +0.482 +2.00 +- +1.926 +0.675 +2.25 +2.73 +1.463 +0.670 +2.50 +3.22 +1.934 +0.736 +2.75 +4.18 +2.284 +0.872 +3.00 +4.12 +2.305 +1.024 +3.25 +4.05 +3.400 +1.271 +3.50 +2.86 +2.663 +1.391 +TABLE II. Kondo energy scales at various fillings ν. T (P ) +K , +δK, and T (χ) +K +are Kondo energy scales estimated by the poor +man’s scaling, the NRG spectral density, and the NRG spin +susceptibility, respectively. δK is defined as the half width at +half maximum of the spectral peak. T (χ) +K +is defined as the +turning temperature where χ(T) transitions from the Curie- +Weiss behavior to the Fermi liquid behavior. There is no data +of T (P ) +K +at ν = 2 because ν = 2 is close a mixed valence case +where µf ≈ U1 and the Schrieffer-Wolff transformation does +not apply. +Kondo temperature. The enhancement arises from the +fact that, in contrast to the case of nf = 1, for nf = 3 +states the virtual processes contributing to the RG equa- +tion evolve more particle excitations than hole excita- +tions in the bath, and particles have larger couplings than +holes. +In Table II we tabulate the Kondo energy scales ob- +tained by the poor man’s scaling at various fillings us- +ing the filling-dependent µc, µf, G parameters given in +Fig. 2(a), and compare them to the values obtained by +the NRG method. +B. +The NRG method +In the NRG method [81–83], the bath is alternatively +realized by a Wilson chain constructed in the way de- +scribed below. +First, the energy window [−D, D] is +discretized on a logarithmic scale, i.e., ωn = D/Λn−1 +(n = 1, 2 · · · ), where Λ > 1 is a scaling factor (cho- +sen as 3 in this work). +Then for each energy shell +ωn ≤ |ω| < ωn+1 two auxiliary bath electrons corre- +sponding to the positive and negative part of it are in- +troduced to reproduce the corresponding ∆(ω). These +auxiliary electrons are further recombined into a Wilson +chain, dnαs, such that (i) only the first site d1αs couples +to the impurity, (ii) the chain is a tight-binding model +with only on-site and nearest neighbor hopping terms. +The Hamiltonian Eq. (20) is now mapped to an impurity +plus a Wilson chain +ˆHN = ˆHf + ˆHH + +� +αs +t0(f † +αsd1αs + h.c.) ++ +N +� +n=1 +� +αs +ϵnd† +nαsdnαs + +N−1 +� +n=1 +� +αs +(tnd† +n+1αsdnαs + h.c.) , +(30) +where N is a large number. The parameters ϵn and tn +can be computed from ∆(ω) using a standard iterative +algorithm [83]. For n → ∞, ϵn ∼ Λ−n and tn ∼ Λ− 1 +2 n. +Thus, the right-most sites represent the lowest-lying bath +states. +One can define the Nth scaled Hamiltonian as �HN = +(Λ) +1 +2 N−1 ˆHN. They can be constructed iteratively +� +HN+1 =Λ +1 +2 � +HN + Λ +1 +2 (N−1) � +αs +� +ϵN+1d† +N+1,αsdN+1,αs ++ tNd† +N+1,αsdN,αs + tNd† +N,αsdN+1,αs +� +. +(31) +The Hilbert space dimension increases exponentially in +this iterative process. The NRG algorithm truncates the +Hilbert space by keeping a fixed number (chosen to be +∼1200 in this work) of the lowest-lying states at each +step. In order to keep the symmetry in the truncated +Hilbert space, in practice we keep all the states up to a +gap above the 1200th state. +C. +Phase diagram and fixed points +Two successive transformations (Eq. (31)) that take +�HN to �HN+2 can be thought as a renormalization group +operation [81, 82]. The system is said to achieve a fixed +point when �HN and �HN+2 have the same low-lying many- +body spectrum. We can obtain a zero temperature phase +diagram in the parameter space of µc, µf, G by analyz- +ing the fixed points. For the completeness of discussions, +here we let µf take value in [−0.5U1, 3.5U1] such that +the corresponding impurity occupation (in the decoupled +limit) nf = µf/U1 +1/2 takes value in [0, 4]. In Fig. 2(b) +and (c) we show the obtained phase diagrams in the pa- +rameter space of µc, µf for G = 8meV and G = 14meV, +respectively. 8meV and 14meV are chosen to be close to +the minimal (8.19meV) and maximal (14.21meV) values +of G (Fig. 2(a)), respectively. +Due to the U(2)×U(2) symmetry, all the many-body +levels can be classified into symmetry sectors labeled by +the good quantum numbers (Q1, Q2; S1, S2), where Qi +and Si are the charge and spin of the ith U(2) sym- +metry, respectively. +Here we take the convention that +Q1 + Q2 = 0 corresponds to a total occupation 2N + 2 +(2N) for odd (even) N. A fixed point is characterized by +low energy many-body levels and the associated quantum +numbers. In the whole phase diagram, we find two dis- +tinct types of stable fixed points: (i) the strong coupling +fixed point exhibiting a Fermi liquid behavior and (ii) the +LM fixed points exhibiting nonzero spin momenta. At a +strong coupling fixed point, as exampled in Fig. 2(d), +(e), for either even or odd N, the ground state is a sin- +glet and has (Q1, Q2; S1, S2) = (2k, 2k; 0, 0) for some or- +der one integer k, which in most cases equals to 0. The + +10 +(f) +(g) +(d) +(h) +EN (meV) +EN (meV) +(c) +(a) +(e) +200 +μf/U +μc (meV) +0 +1 +1 +2 +3 +60 +20 +0 +40 +LM1 +LM2 +LM3 +Kondo +FI +FI +δK (meV) +G=8meV +(b) +μf/U +60 +20 +0 +40 +LM1 +LM2 +LM3 +FI +Kondo +FI +μc (meV) +G=14meV +N +11 +21 +31 +41 +1 +(1, 1;1/2, 1/2) +(1, 1;1/2, 1/2) +(0, 0;0, 0) +(1, 0;1/2, 0) +(1, 0;1/2, 0) +(2, 0;0, 0) +-40 +40 +ω (meV) +N +11 +21 +31 +41 +1 +(2, 1;0, 1/2) +(2, 1;0, 1/2) +(1, 1;1/2, 1/2) +(1, 1;1/2, 1/2) +(2, 2;0, 0) +(1, 0;1/2, 0) +-40 +40 +ω (meV) +0 +100 +200 +150 +50 +11 +21 +31 +41 +N +(0, 0;0, 0) +(1, 0;1/2, 0) +(-1, 0;1/2, 0) +(-2, -1;0, 1/2) +(1, 0;1/2, 0) +(-1, 0;1/2, 0) +(0, 0;0, 0) +1 +-40 +40 +ω (meV) +E (meV) +0 +100 +150 +50 +11 +21 +31 +41 +1 +(-1, 0;1/2, 0) +(-1, 0;1/2, 0) +(0, 0;0, 0) +(0, 0;0, 0) +(1, 0;1/2, 0) +(1, 0;1/2, 0) +-40 +40 +ω (meV) +N +11 +21 +31 +41 +N +(0, 0;0, 0) +(1, 0;1/2, 0) +(1, 0;1/2, 0) +(-1, 0;1/2, 0) +1 +(1, 1;1/2, 1/2) +(1, 1;1/2, 1/2) +-40 +40 +ω (meV) +FIG. 2. Phase diagram and fixed points. (a) The self-consistent mean-field values of µc, µf, G as functions of the total filling ν +from ν = 0 to 4, where we have enforced the symmetries of the correlated insulator state at CNP. The left y-axis represents µc +and µf while the right y-axis represents G with a different range. (b) The phase diagram in the parameter space of µc, µf for +G = 8meV. The white lines are phase boundaries between the local moment (LM) phases and the strong coupling phase. The +dashed black lines are crossover boundaries between the frozen impurity (FI) and Kondo regimes of the strong coupling phase. +The color maps the half-width of the spectral density peak, reflecting the Kondo energy scale if in the Kondo regime. The +solid black line indicates the trajectory of µc and µf determined from a self-consistent calculation as ν changes from 0 to 4, +where the five arrows from left to right represent ν = 0, 1, 2, 3, 4, respectively. (c) is the same as (b) but a different parameter +G = 14meV is used. (d)(e) The RG flow of the many-body spectrum of the scaled Hamiltonian � +HN (N ∈ odd) in the Kondo +regime, where µc = 30.7meV, µf/U1 = 0.367, G = 9.49meV is the mean field value at ν = 1.25 for (d) and µc = 49.9meV, +µf/U1 = 1.286, G = 11.83meV is the mean field value at ν = 2.5 for (e). The spectral lines’ colors represent the symmetry +sectors labeled by good quantum numbers (Q1, Q2; S1, S2). Since the levels in sectors (Q1, Q2; S1, S2) and (Q2, Q1; S2, S1) are +identical, only |Q1| ≥ |Q2| sectors are shown for simplicity. The insets are the resulting single-particle spectral densities that +exhibit Kondo resonances. (f)(g)(h) The RG flow of many-body spectrum of the scaled Hamiltonian � +HN (N ∈ even) in the +LM1,2,3 phase, where µc = 5meV, µf/U1 = 0.5, 1.5, 2.5, G = 8meV respectively. The insets are the resulting single-particle +spectral densities that exhibit Hubbard bands. +low-lying many-body spectrum is identical to the one of +a free-fermion chain defined by ϵn and tn with an ad- +ditional chemical potential term. +In other words, the +impurity acts as if it was nonexistent. The underlying +mechanism is either the Kondo screening, where the im- +purity is an effective LM screened by the bath, or the +impurity freezing, where the impurity occupation νf is +effectively empty or full. We refer to the two cases as the +Kondo regime and the frozen impurity (FI) regime, re- +spectively, which are adiabatically connected. The fixed +points shown in Fig. 2(d), (e) are in the Kondo regime +because, if we continuously change µc to 0, they evolve +to the LM1 and LM2 states (discussed in the next para- +graph), respectively. There is a crossover between Kondo +and FI regimes as one changes µf, as indicated by the +dashed lines in Fig. 2(b), (c). Later we will determine +the crossover boundary using the spectral density. +At an LM fixed point, as exampled in Fig. 2(f), the low- +lying many-body spectrum is identical to a free-fermion +chain plus a detached LM. Depending on the represen- +tation of the ground state, the LM fixed points can be +further classified into LMn, where n = 1, 2, 3 is the +effective impurity occupation. +The flows of the spec- +tra towards these fixed points are shown in Fig. 2(f), +(g), (h), respectively. +LMn ground states have the +same SU(2)×SU(2) representations as ground states of +ˆHf + ˆHH (Eqs. (12) and (18)) with n impurity elec- +trons, where the Hubbard interaction freezes charge exci- +tations and the Hund’s coupling prefers states with elec- +trons lying in different orbitals. For n = 1, the ground +states are four-fold degenerate and belong to the sym- +metry sectors (Q1, Q2; S1, S2) = (2k + 1, 2k; 1 +2, 0) and +(2k, 2k + 1; 0, 1 +2), corresponding to the spin- 1 +2 states of +the two U(2)’s, respectively. The SU(2) representations +are the same as those of the four single-particle states of +ˆHf + ˆHH: (nf +1↑, nf +1↓; nf +2↑, nf +2↓) = (10;00), (01;00), (00;10), +(00;01). +For n = 2, the ground states are also four- +fold degenerate but belong to a different symmetry sec- +tor (2k + 1, 2k + 1; 1 +2, 1 +2). +One can understand them +as the product states of two spin- 1 +2 states of the two +U(2)’s. They have the same SU(2) representations as the +four two-particle states of ˆHf + ˆHH: (10;10), (10;01), +(01;10), (01;01). Without Hund’s coupling JH, the LM2 +ground states would be +�4 +2 +� += 6-fold degenerate. A fi- + +11 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-40 +0 +40 +(a) kBT=0 +-40 +0 +40 +0 +1 +2 +3 +4 +5 +(b) kBT=0 +(c) kBT=2meV +(d) kBT=5meV +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-40 +0 +40 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-40 +40 +-5 5 +A(𝜔,T) (meV-1) +𝜔 (meV) +𝜔 (meV) +𝜔 (meV) +𝜔 (meV) +𝜈=0 +𝜈=1 +𝜈=2 +𝜈=3 +𝜈=4 +FIG. 3. Spectral densities A(ω, T) at various fillings ν and +temperatures. (a) Spectral densities at the zero temperature +for ν = 0, 0.1, 0.2 · · · 4. The curves are offset by 0.1ν (meV−1) +for clarity. (b) is the same as (a) but is shown with a smaller +vertical scale for clarity of Hubbard bands, marked by in- +verted triangles. The curves are offset by 0.01ν (meV−1). (c) +and (d) are spectral densities at finite temperatures, where +the curves are offset by 0.01ν (meV−1). +nite JH raises the energy of the two many-body states +with both electrons occupying the same orbital, i.e., +(11;00), (00;11). For n = 3, the ground states are still +four-fold degenerate but belong to the symmetry sec- +tors (2k − 1, 2k; 1 +2, 0) and (2k, 2k − 1; 0, 1 +2). Their SU(2) +representations are same as the four single-hole states +ˆHf + ˆHH: (01;11), (10;11), (11;01), (11;10). +It is also helpful to look at the global U(2) symme- +try representations, where the two U(2) rotations are the +same. The total charge and spin of LM1,2,3 are 1, 2, 3 +(mod 4) and 1 +2, 1 +2 ± 1 +2, 1 +2, respectively. +Phase boundaries between different LM phases and the +strong coupling phase are described by unstable fixed +points where different ground states cross with each +other. +The phase boundaries are shown by the white +lines in Fig. 2(b), (c). Starting from an LMn phase, in- +creasing µc will eventually drive it into a strong coupling +phase due to the enhancement of hybridization. The crit- +ical µc, as expected, is close to G, the conduction band +edge (Fig. 1(c), (d)). +V. +SPECTRAL DENSITY +We calculate the spectral density of the f-electrons, +A(ω, T) = − 1 +π +� +αs Gαs(ω, T), with Gαs(ω, T) being the +retarded Green’s function of fαs at the temperature T. +We use the method described in Ref. [100] to collect +the many-body levels at different RG steps to compute +A(ω, T). +The fixed points in Fig. 2(d), (e) are in the +Kondo regime and hence have sharp zero-energy peaks +due to the Kondo resonance, as shown in the insets of +Fig. 2(d), (e). +The fixed points in Fig. 2(f), (g), (h) +are in the LM1,2,3 phases, respectively, thus, their spec- +tral density are dominated by the upper and lower Hub- +bard bands. We compute the spectral densities for all +the points in the phase diagrams in Fig. 2(b), (c). We +identify a central peak for every calculation and measure +its half-width δK at half maximum. (If there is no cen- +tral peak, e.g., Fig. 2(f), δK = 0.) δK is indicated by the +color in Fig. 2(b), (c). We can distinguish the Kondo and +FI regimes in the strong coupling phase through spectral +density. Intuitively, a state in the Kondo regime should +have a Kondo resonance. By contrast, a state in the FI +regime should have its main spectral weight away from +zero energy because the impurity occupation is either +empty or full. +Thus, we identify a phase point in the +Kondo regime if δK covers the zero energy and otherwise +in the FI regime. The crossover between the two regimes +is indicated by dashed lines in Fig. 2(b), (c). +Several features of δK in Fig. 2(b), (c) can be un- +derstood using the poor man’s scaling developed in +Sec. IV A. First, there are three domes around µf = +1 +2U1, 3 +2U1, 5 +2U1 where δK is relatively small. They cor- +respond to the nf = 1, 2, 3 cases discussed in Sec. IV A. +From the poor man’s scaling perspective, these three µf’s +correspond to the minimal initial value of the coupling +constant g (Eq. (22)), which then leads to smaller TK’s. +Second, when µc is small (≲ 30meV), δK in the middle +dome is significantly smaller than those of the other two +domes. The reason is that the Kondo energy scale TK +for nf = 2 will be strongly suppressed due to the multi- +plet splitting if TK is smaller than JH. One can see that +the N factor in the exponential function of Eq. (27) is +replaced by 2. Third, for the same µc, the first dome has +lower δK than the third dome. This difference is a result +of the suppression factor y1 for nf = 1 (Eq. (24)) and the +enhancement factor y3 for nf = 3 (Eq. (29)) due to the +particle-hole asymmetry of ∆(ω), as discussed in detail +in Sec. IV A. +In order to compare our results with STM measure- +ments, we need to adopt physical µc, µf, G parameters. +As discussed in Sec. III C, µc, µf, G can be determined as +functions of the filling ν via a symmetric self-consistent +calculation of Eqs. (10). µc, µf, G as functions of ν are +shown in Fig. 2(a). The obtained spectral densities at the +zero temperature are shown in Fig. 3(a), (b). For ν = 0, +the state is in the FI regime with an (almost) zero occu- +pation; hence the spectral weight is mainly distributed at +positive energy. As ν increases, the spectral peak moves +to the zero energy and is eventually pinned at the zero en- +ergy to form a Kondo resonance. This is precisely what is +seen in STM experiments at low temperatures (T < 1K) +[11, 19, 21, 22]. One can also observe the evolution of +Hubbard bands at T = 0 as ν changes (Fig. 3(b)), but +they are relatively weak compared to the Kondo reso- +nance peaks. +At finite temperatures (Fig. 3(c), (d)), +the Kondo resonance peaks are smeared by thermal fluc- +tuations and the evolution of Hubbard bands becomes +clearer. As ν increases from 0 to 4, the Hubbard bands +periodically pass through the zero energy, matching the +cascade of transitions seen in STM experiments at higher + +12 +(a) +kBT (meV) +4 +102 +101 +100 +10-1 +10-2 +10-3 +𝜈 +0 +1 +2 +3 +101 +100 +10-1 +10-2 +102 +101 +100 +10-1 +10-2 +10-3 +kBT (meV) +102 +101 +100 +10-1 +10-2 +10-3 +kBT (meV) +100 +10-1 +10-2 +10-3 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +𝜈 +(b) +(d) +(c) +102 +101 +100 +10-1 +10-2 +10-3 +kBT (meV) +𝜈 +0 +1 +2 +3 +4 +0 +1 +2 +3 +(e) +𝜔 (meV) +0 +20 +40 +-20 +-40 +0 +0.1 +0.2 +0.3 +A(ω,T) (meV-1) +𝜈=1 +0 +0.1 +0.5 +1 +2 +5 +kBT (meV) +kBT =0.82meV +(f) +𝜈 +0 +1 +2 +3 +4 +0 +0.4 +0.8 +1.2 +1.6 +Bloc(T) +0 +5 +10 +15 +20 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +𝜈 +0 +1 +2 +3 +4 +Simp +×kBln2 +FIG. 4. Spin susceptibility and entropy contributed by the im- +purity. (a) χloc(T)/χloc(0) as a function of filling ν and tem- +perature T. (b) The local spin susceptibilities χloc(T) at fill- +ings ν = 0, 0.5, 1 · · · 4. (c) The entropy contributed by the im- +purity as a function of ν and T. (d) The entropy contributed +by the impurity Simp(T)/(kB ln 2) at fillings ν = 0, 0.5, 1 · · · 4. +(e) The spectral densities at ν = 1 at various temperatures. +(f) The entropy contributed by the impurity as a function of +ν at B = 0, 5, 10, 15, 20 T and temperature kBT = 0.82 meV. +temperatures [17, 19]. +One can use δK to estimate the Kondo energy scale. +Using the ν-dependent µc, µf, G parameters given in +Fig. 2(a), we tabulate the δK’s at different fillings in Ta- +ble II. Comparing it to TK estimated by the poor man’s +scaling, denoted as T (P ) +K , we find T (P ) +K +is about δK ∼ 2δK. +VI. +LOCAL MOMENTS AND THE +POMERANCHUK EFFECT +At a temperature exceeding the Kondo energy scale, +the LM will become effectively decoupled from the bath +and visible in experimental measurements. This is the +mechanism of the Pomeranchuk effect [30, 31]. Refs. [30] +observed a higher entropy (∼ 1kB per moir´e cell with kB +being the Boltzmann’s constant) state at ν ≈ 1 at the +temperature T ≈ 10K. As this entropy can be quenched +by an in-plane magnetic field, it is ascribed to a free local +moment. Ref. [31] observed a similar effect at ν ≈ −1 +and showed that an additional resistivity peak that is +absent at T = 0 develops in the higher entropy state at +T ≈ 10K. These observations can be naturally explained +by the transition from the Fermi liquid phase to the LM +phase as the temperature increases. +To demonstrate the LM phase at higher temperatures, +we calculate the local spin susceptibilities χloc(T) us- +ing the filling-dependent µc, µf, G parameters given in +Fig. 2(a). +χloc(T) is defined as +dMloc +dBloc [101, 102], with +Mloc being the spin momenta contributed by the impu- +rity and Bloc a local magnetic field that only acts on the +impurity. As shown in Fig. 4(a), (b), for ν ≥ 0.5, χloc(T) +approaches a constant as T → 0, and obeys the Curie’s +law, i.e., χloc(T) ∼ T −1, when T is larger than the Kondo +energy scale. Obeying Curie’s law is a clear indication of +a free LM. One may notice that the T-dependences of +χloc(T)’s for ν < 0.5 are non-monotonous. The ν < 0.5 +states lie in the FI regime, thus the spin susceptibili- +ties are extremely small when T → 0, and will start to +increase when T is able to excite the LM1 states. For +ν > 0.5, we can define the Kondo temperature TK as +the turning point between Curie’s behavior and Fermi +liquid behavior. Specifically, it can be obtained as the +crossing of the extended T −1 line from the LM side and +the extended horizontal line from the Fermi liquid side +(Fig. 4(b)). We tabulate the resulting TK in Table II. As +shown in the table, such defined TK is about 1 +3δK ∼ δK +with δK being the half-width of the spectral density dis- +cussed in Sec. V. +We also calculate the impurity entropy Simp(T) for +comparison with experiments. Simp(T) is defined as the +difference of the entropy of �HN and that of a reference +free-fermion chain (without the impurity site) defined +by the same ϵn, tn parameters as in �HN. As shown in +Fig. 4(c) and (d), Simp(T) is zero in the Fermi liquid +phase at sufficiently low T and starts to increase when +T reaches the Kondo energy scale. For ν = 1, Simp(T) +climbs to about 2 ln 2 · kB at about kBT ≈ 1meV and +(approximately) stays at this value until kBT reaches +10meV. The entropy 2 ln 2 · kB ≈ 1.39kB is close to the +measured value (∼ 1kB) in Refs. [30, 31] and can be un- +derstood as contributed by the four degenerate states in +the LM1 phase. Higher excited states will be involved +when kBT is larger than 10meV, and the entropy contin- +ues to increase for larger kBT. We also show the spectral +density at ν = 1 in Fig. 4(e), one can see that the Kondo +resonance peak becomes weak for kBT > 1meV, which +is consistent with the entropy and spin susceptibility re- +sults. +An in-plane magnetic field will polarize the spin and +hence suppress the entropy. +However, as discussed in +Sec. IV C, the four-fold degenerate LM1 states consist of +two spin- 1 +2 states due to the orbital degeneracy, hence +a strong field will not completely quench the entropy. +Instead, due to the orbital degeneracy, the remaining en- +tropy will be ln 2 · kB ≈ 0.69kB. This is also consistent +with observations in Ref. [30]. In Fig. 4(f), we plot the + +13 +(a) 𝜈=1.2 +(b) 𝜈=1.8 +(c) 𝜈=2.4 +Band Energy (meV) +0 +20 +40 +60 +-20 +-40 +-60 +ΓM +KM +MM +ΓM +KM +MM +ΓM +KM +MM +0 +0.2 +0.4 +0.6 +0.8 +1 +FIG. 5. Heavy Fermi liquid bands at ν = 1.2 (a), 1.8 (b), 2.4 +(c). As explained in the text, the effective hybridization be- +tween c- and f-bands is suppressed by a factor z +1 +2 with z being +the quasi-particle weight of f-electrons, which are estimated +as 0.038, 0.27, 0.16 for (a), (b), and (c), respectively. The +color of the bands represents the total quasi-particle weight, +which is always larger than z. +impurity entropy as a function of the filling at various +magnetic fields Bloc. At ν = 1, the entropy saturates +to a constant ∼ ln 2 · kB under a strong field. One can +see that the entropy values, the shapes of the curves, and +their field-dependences in Fig. 4(f) are comparable to the +experimental results in Fig. 2(e) of Ref. [30]. +VII. +DISCUSSIONS +Based on the NRG calculations, the poor man’s scal- +ing, and various experimental observations, we have +shown that the gapless states at 1 ≲ |ν| < 2 are in the +Kondo regime. Considering the translation invariance of +the actual system, these states must be the heavy Fermi +liquid states. Here we estimate the heavy Fermi liquid +bands from the information provided by the NRG cal- +culation. At the zero temperature, a spectral density in +the Kondo regime possesses a Lorentz peak around the +zero energy, i.e., A(ω) ≈ 4z +π +δK +ω2+δ2 +K , where z is the quasi- +particle weight, δK is the half-width given in Fig. 2(b), +(c), and the factor 4 is from orbital and spin degenera- +cies. Then the quasi-particle weight can be estimated as +z = πA(0)δK/4. Substituting the quasi-particle part of +Gαs(iω), i.e., +z +iω, into Dyson’s equation of c-electrons +G(c)(iω, k) = G(c,0)(iω, k) ++ G(c,0)(iω, k)H(cf)(k) z +iω H(cf)†(k)G(c)(iω, k) , +(32) +one can see that the cf hybridization is effectively sup- +pressed by a factor of z +1 +2 . Here ω is the Matsubara fre- +quency, G(c,0)(iω, k) = (iω − H(c)(k))−1 is the free prop- +agator of c-electrons, and H(c)(k), H(cf)(k) are Hamilto- +nian matrices in Eq. (9). If z = 0 the effective hybridiza- +tion will be zero, corresponding to the LM phase where +the Fermi surface is solely contributed by the c-electrons. +Using this method we obtain the bands at ν = 1.2 and +1.8, as shown in Fig. 5(a), (b), respectively. One should +be aware that the Hubbard band information is com- +pletely neglected in this method. For future reference, +we also estimate the heavy Fermi liquid bands at ν = 2.4 +(Fig. 5(c)) by assuming the ν = 2.4 state in the Kondo +regime. +The heavy Fermi liquid states at 1 ≲ |ν| < 2 can be +further confirmed by future experimental research. For +example, the Fermi surface can dramatically change as +one tunes the filling and temperature or applies an ex- +ternal field. The Fermi surface change will be reflected +in spectral measurements such as quasi-particle interfer- +ence. It is also in principle possible to directly measure +the scattering phase shift [103]. +c-bands will induce RKKY interactions between LMs +at different f-sites, which can lead to further symme- +try breaking and should be crucial to stabilize the ob- +served correlated insulator states at |ν| = 2. We have +ignored these RKKY interactions and further symmetry +breaking in the current work. A full self-consistent treat- +ment including both RKKY and Kondo screening effects +may result in more complicated ν-dependencies of µc, µf +than those shown Fig. 2(a), (b), (c). +For example, at +ν = 2, µc given by a self-consistent symmetry breaking +Hartree-Fock mean field [61] is about 26meV, which is +significantly lower than the one (∼40meV) in Fig. 2(a). +Thus, a possible mechanism for the correlated insulators +to win the Kondo screening is that µc drops to a small +value such that the Kondo energy scale becomes irrele- +vant. (See Fig. 2(b), (c).) Observations in Refs. [18, 26] +also suggest that the ν-dependencies of µc, µf are com- +plicated. The competition between RKKY and Kondo +screening is also a potential mechanism for the observed +strange metal behaviors [27–29] and could play an impor- +tant role in the unconventional superconductivity [1, 4– +11]. We leave this for future studies. +Note added. +During the preparation of the current +work, a related work [104] appeared. This work studied +the symmetric Kondo state using a slave-fermion mean +field in a Kondo lattice model derived from the THF +model. Our theory is based on the symmetry-broken cor- +related state at CNP. We are also aware of related works +on the Kondo problem in MATBG by A. M. Tsvelik’s and +B. A. Bernevig’s group [105, 106] and P. Coleman’s group +[107] that will appear soon, and a generalization of the +THF model to the magic-angle twisted trilayer graphene +[108]. Ref. [105] also obtains a Kondo temperature about +1 ∼ 2K around |ν| ≈ 1. +ACKNOWLEDGMENTS +We are grateful to B. Andrei Bernevig, Ning-Hua +Tong, Xi Dai, Jia-Bin Yu, Xiao-Bo Lu, Yong-Long Xie, +Yi-Lin Wang, and Chang-Ming Yue for helpful discus- +sions. Z.-D. S. and G.-D. Z. were supported by National +Natural Science Foundation of China (General Program +No. 12274005), National Key Research and Development +Program of China (No. 2021YFA1401900). + +60 +0.9 +40 +0.8 +0.7 +20 +0.6 +0 +0.5 +0.4 +-20 +0.3 +0.2 +-40 +0.1 +-60 +0 +K +G +M60 +0.9 +40 +0.8 +0.7 +20 +0.6 +0 +0.5 +0.4 +-20 +0.3 +0.2 +-40 +0.1 +-60 +0 +K +G +M60 +0.9 +40 +0.8 +0.7 +20 +0.6 +0 +0.5 +0.4 +-20 +0.3 +0.2 +-40 +0.1 +-60 +0 +K +G +M60 +0.9 +40 +0.8 +0.7 +20 +0.6 +0 +0.5 +0.4 +-20 +0.3 +0.2 +-40 +0.1 +-60 +0 +K +G +M14 +Appendix A: More details about the effective +Hamiltonian +1. +Nonzero M term +A generic trial ground state at CNP is given by +(Eq. (6)) +|Ψ0⟩ = U +� +R +f † +R1+↑f † +R1+↓f † +R2+↑f † +R2+↓|FS⟩ , +(A1) +where U = exp(−iθµν ˆΣµν) is a U(4) rotation operator +and an implicit summation over repeated µ, ν indices is +assumed. We can always define the rotated fermion op- +erators �ckaηs = UckaηsU †, �fRaηs = UfRaηsU † such that +�fRα+s’s are occupied in |Ψ0⟩ and �fRα−s’s are empty in +|Ψ0⟩. According to the discussions in the supplementary +material section S4B of Ref. [61], in the flat-band limit +(M = 0), the lowest particle (hole) excitations only in- +volve �cka−s and �fRa−s (�cka+s and �fRa+s). +Thus, the +effective periodic Anderson model for ν > 0 derived +in Sec. III B is written in terms of ckas = �cka−s and +fRαs = �fRα−s. Here we give the explicit forms of the +rotated operators +�fRαηs = +� +α′η′s′ +� +eiθµνΣf +µν +� +αηs,α′η′s′ fRα′η′s′ , +(A2) +and +�ckaηs = +� +a′=1,2 +η′s′ +� +eiθµνΣc12 +µν +� +aηs,a′η′s′ cka′η′s′ +(a = 1, 2), (A3) +�ckaηs = +� +a′=3,4 +η′s′ +� +eiθµνΣc34 +µν +� +aηs,a′η′s′ cka′η′s′ +(a = 3, 4), (A4) +where the eight-by-eight matrices Σf +µν, Σc12 +µν , Σc34 +µν +are +defined in Eqs. (3) to (5). +The M-term in the original basis of the THF model +(Eq. (1)) is +M +� +aa′=3,4 +� +|k|<Λc +� +ηs +[σx]aa′c† +kaηscka′ηs . +(A5) +It favors the Kramers inter-valley coherent state dis- +cussed at the end of Sec. II, where θx0 and θy0 are nonzero +and satisfy θ2 +x0 + θ2 +y0 = (π/4)2. Without loss of general- +ity, we assume U = exp(−i π +4 ˆΣx0) for the Kramers inter- +valley coherent state. Writing this M-term in terms of +the rotated operators, we obtain +M +� +|k|<Λc +� +a,a′=3,4 +� +ηη′ss′ +�c† +kaηsOaηs,a′η′s′�cka′η′s′ , +(A6) +where O = ei π +4 Σc34 +x0 σxτ0ς0e−i π +4 Σc34 +x0 += −σzτxς0. The τx +matrix in O represents couplings between the empty +and occupied single-particle states. If we simply project +this M-term onto the empty states, it vanishes, i.e., +[Oa−s,a′−s′] = 0. +A better approximation is applying +a Schrieffer-Wolff transformation to decouple the η = ± +states, leading to a second-order correction to the effec- +tive Hamiltonian. As ⟨Ψ0| �f † +αηs �fαηs|Ψ0⟩ = (1 + η)/2, the +J term in Eq. (2) yields the following mean field term (see +also the supplementary material section S4B of Ref. [61]) +− J +2 +� +a=3,4 +� +ηs +η · �c† +aηs�caηs +(A7) +Then, regarding the J +2 term as the zeroth order Hamilto- +nian and M as a perturbation, a Schrieffer-Wolff trans- +formation leads to the correction +− M 2 +J +� +|k|<Λc +� +a=3,4 +� +ηs +η · �c† +kaηs�ckaηs + O(M 4) . +(A8) +The resulting energy levels ±(J/2 + M/J2) at k = 0 is +fully consistent with a Taylor expansion of the one-shot +energy levels ± +� +J2/4 + M 2 derived in Ref. [61]. Pro- +jecting the correction to the active d.o.f., i.e., ckas = +�cka−s, we obtain the correction to the effective Hamilto- +nian +M 2 +J +� +|k|<Λc +� +a=3,4 +� +s +c† +kasckas + O(M 4) . +(A9) +2. +Hund’s coupling +The four-by-four Hamiltonian matrix H(c)(k)+∆H(c) +in Eq. (13), i.e., −v⋆(σx⊗σ0kx+σy ⊗σzky)+02×2⊕Gσ0, +can be diagonalized analytically. As discussed at the end +of the last subsection, to O(M 2), the M term simply +shifts the energy of a = 3, 4 electrons by M 2/J. Thus, +all the analysis below applies to the M ̸= 0 after G is +replaced by G + M 2/J. We find the energy eigenvalues +and wave-functions of the H(c)(k) + ∆H(c) as +ϵ1(k) =ϵ+(k) = G +2 + +� +G2 +4 + v2⋆k2 +u1(k) = +� +sin θk +2 e−iφk 0 − cos θk +2 +0 +�T +, +(A10) +ϵ2(k) =ϵ+(k) = G +2 + +� +G2 +4 + v2⋆k2 +u2(k) = +� +0 sin θk +2 eiφk 0 − cos θk +2 +�T +, +(A11) +ϵ3(k) =ϵ−(k) = G +2 − +� +G2 +4 + v2⋆k2 +u3(k) = +� +cos θk +2 e−iφk 0 sin θk +2 +0 +�T +, +(A12) +ϵ4(k) =ϵ−(k) = G +2 − +� +G2 +4 + v2⋆k2 +u4(k) = +� +0 cos θk +2 eiφk 0 sin θk +2 +�T +. +(A13) + +15 +where +θk = arccos +G/2 +� +G2/4 + v2⋆k2 +(A14) +and φk = arg(kx + iky). +We now derive the effective Hund’s coupling ˆHH. Ap- +plying a second-order perturbation in terms of ˆHJ, we +obtained the correction to the Hamiltonian +∆ ˆH = − J2 +N 2 +M +� +I +� +α1α2 +s1s′ +1s2s′ +2 +� +k1,k′ +1 +k2,k′ +2 +(f † +α1s1fα1s′ +1 − νf +4 δs1s′ +1) +× (f † +α2s′ +2fα2s2 − νf +4 δs2s′ +2) · e− λ2 +2 (k2 +1+k′2 +1 +k2 +2+k′2 +2 ) +× +⟨Ψ0|c† +k′ +1α1+2s′ +1ck1α1+2s1|ΨI⟩⟨ΨI|c† +k2α2+2s2ck′ +2α2+2s′ +2|Ψ0⟩ +EI − E0 +, +(A15) +where |ΨI⟩ are excited states with a single particle-hole +pair and EI are the energies of the excited states. k1,2, +k′ +1,2 are limited within the cutoff Λc. Due to the mo- +mentum and spin conservation, for the matrix element +to be nonzero, there must be k1 = k2, s1 = s2, k′ +1 = k′ +2, +s′ +1 = s′ +2. For simplicity, we rewrite k1, k′ +1, s1, and s′ +1 +as k, k′, s, and s′, respectively. (k, s) and (k′, s′) label +the particle and the hole excitations, respectively. Then +the matrix element in the third line of Eq. (A15) can be +written as +nF (ϵ+(k′) − µc)(1 − nF (ϵ−(k) − µc)) +× ⟨Ψ0|c† +k′α1+2s′ckα1+2sc† +kα2+2sck′α2+2s′|Ψ0⟩ +(A16) +According +to +the +wave +functions +given +in +Eqs. +(A10)-(A13), +there are ⟨Ψ0|c† +k′α1+2s′ck′α2+2s′|Ψ0⟩ += +δα1α2 sin2 θk′ +2 , ⟨Ψ0|ckα1+2sc† +kα2+2s|Ψ0⟩ = δα1α2 cos2 θk +2 . +The excitation energy EI −E0 is given by ϵ+(k)−ϵ−(k′). +Thus, ∆ ˆH is simplified to +∆ ˆH = − J2 +N 2 +M +� +αss′ +kk′ +(f † +αsfαs′ − νf +4 δss′)(f † +αs′fαs − νf +4 δss′) +× nF (ϵ−(k′) − µc)(1 − nF (ϵ+(k) − µc)) sin2 θk′ +2 cos2 θk +2 +ϵ+(k) − ϵ−(k′) +× e−λ2(k2+k′2) +(A17) +The s = s′ contribution is an effective chemical potential +shift, estimated as 0.17meV at CNP, of the f-electrons. +As it is much smaller than U1, we will omit the s = s′ +contribution. The s ̸= s′ contribution can be written as +ˆHH = JH +� +α +nf +α↑nf +α↓ +(A18) +with JH given by +JH =2J2 +�Ω0 +2π +�2 ˆ Λc +0 +dk′ · k′ +ˆ Λc +k0 +dk · k · e−λ2(k2+k′2) +× sin2 θk′ +2 cos2 θk +2 +ϵ+(k) − ϵ−(k′) , +(A19) +where k0 is determined by ϵ+(k0) = µc. Here we have +made use of the fact that ϵ±(k) and θk only depends +on |k| but not φk. At CNP, µc = 0 and G = J/2 = +8.19meV, taking the limit Λc → ∞, we obtain JH ≈ +0.34meV. Using the self-consistent values of µc and G +shown in Fig. 2(a), we find JH at ν = 1, 2, 3, 4 are given +by 0.29meV, 0.26meV, 0.21meV, 0.19meV, respectively. +As JH is small and does not change significantly with ν, +in this work, we simply set JH = 0.34meV for simplicity. +3. +Hybridization function +By definition, the hybridization function ∆(ω) is given +by +∆(ω) = π +N +� +k +� +n +|Vnα(k)|2δ(ω − ϵn(k)) +(A20) +where Vnα(k) = � +a u∗ +an(k)H(cf) +aα (k)e− λ2k2 +2 +is the hy- +bridization between fαs and the n-th energy band of c- +electrons. +∆(ω) does not depend on α because of the +C2zT or C2x symmetry that flips the α index. Substitut- +ing ϵn(k) and uan(k) in Eqs. (A10)-(A13) into the above +equation, we obtain Vnα(k) for α = 1 as +V11(k) =γ sin θk +2 eiφk +V21(k) =v′ +⋆(−kx + iky) sin θk +2 e−iφk +V31(k) =γ cos θk +2 eiφk +V41(k) =v′ +⋆(−kx + iky) cos θk +2 e−iφk . +(A21) +Using the energy eigenvalues in Eqs. (A10)-(A13) and +the Vnα(k) matrix elements given above, it is direct to +obtain +∆(ω) = Ω0 +2v2⋆ +����ω + µc − G +2 +���� +� +γ2 + v′2 +⋆ k2 +F +� +e−k2 +F λ2 +� +θ(ω + µc − G) sin2 θkF +2 ++ θ(−ω − µc) cos2 θkF +2 +� +(A22) +where kF is given by +kF = 1 +v⋆ +� +(ω + µc − G/2)2 − (G/2)2 . +(A23) +We now verify the asymptotic behaviors of ∆(ω). +When ω + µc → G+, kF → 0 and only the first term +in the second line of Eq. (A22) contributes to ∆(ω). Ac- +cording to Eq. (A14), there is cos θkF = +G/2 +ω+µc−G/2 and +hence sin2 θkF +2 += 1 +2− 1 +2 cos θkF ≈ (ω+µc−G)/G. Then we +obtain the asymptotic behavior of ∆(ω) as ω + µc → G+ +∆(ω) = Ω0 +4v2⋆ +γ2 · (ω + µc − G) + O((ω + µc − G)2) . +(A24) +When ω + µc → −0+, kF → 0 and only the second +term in the second line of Eq. (A22) contributes to ∆(ω). + +16 +According to Eq. (A14), there is cos θkF = +G/2 +G/2−ω−µc and +hence cos2 θkF +2 += 1 +2 + 1 +2 cos θkF ≈ 1. Then we obtain the +asymptotic behavior of ∆(ω) as ω + µc → −0+ +∆(ω) = Ω0 +4v2⋆ +Gγ2 + O((ω + µc)) . +(A25) +Appendix B: Poor man’s scaling of Anderson models +with energy-dependent couplings +1. +Generic theory for U(N) models +We consider the Anderson impurity model with N +symmetric flavors +ˆH = − µf ˆ +Nf + U +2 +ˆ +Nf( ˆ +Nf − 1) + +N +� +µ=1 +ˆ D +−D +dϵ · ϵ · d† +µ(ϵ)dµ(ϵ) ++ +N +� +µ=1 +ˆ D +−D +dϵ +� +∆(ϵ) +π +(f † +µdµ(ϵ) + h.c.) , +(B1) +where µ is the flavor index and Nf = �N +µ=1 f † +µfµ. We +assume the ground state of the isolated impurity has nf +f-electrons, which can take the values 1, 2 · · · (N − 1). +(We do not consider the empty case (nf = 0), the full +case (nf = N), and the mixed valence case where ground +states with different nf are exactly degenerate.) We fur- +ther assume the charge gaps to nf − 1 and nf + 1 elec- +trons are ∆E− and ∆E+ = U − ∆E−, respectively. We +then apply a Schrieffer-Wolff transformation to obtain an +effective Coqblin–Schrieffer model for the Hilbert space +restricted to ˆNf = nf +ˆH = +N +� +µ=1 +ˆ D +−D +dϵ · ϵ · d† +µ(ϵ)dµ(ϵ) + 4g +πU +N +� +µ,µ′=1 +ˆ D +−D +dϵdϵ′ +� +� +∆(ϵ)∆(ϵ′)(f † +µfµ′ − xδµµ′)d† +µ′(ϵ′)dµ(ϵ) +� +. +(B2) +Terms that only involve ˆNf are omitted because they +only contribute to an energy shift. +The bandwidth D +should be smaller than min(∆E+, ∆E−), otherwise, the +Schrieffer-Wolff transformation is invalid. The parame- +ters g and x are given by +g = U +4 +� +1 +∆E+ ++ +1 +∆E− +� +, +x = ∆E− +U +, +(B3) +respectively. If µf = (nf − 1 +2)U, there is ∆E+ = ∆E− = +1 +2U and g = 1, x = 1 +2. +We now truncate the bandwidth at D−dD = D(1−dt) +(dt ≪ 1) and consider second order (in g) corrections +form the virtual particle (D − dD < ϵ < D) and hole +(−D < ϵ < −D + dD) excitations. The particle excita- +tion contributes to the correction +− (4g)2 +(πU)2 +1 +D +� +µ1µ2µ′ +1µ′ +2 +ˆ D−dD +−D+dD +dϵ1dϵ2d +ˆ D +D−dD +dϵ′ +1dϵ′ +2 +× +� +∆(ϵ1)∆(ϵ2)∆(ϵ′ +1)∆(ϵ′ +2)d† +µ1(ϵ1)⟨dµ′ +1(ϵ′ +1)d† +µ′ +2(ϵ′ +2)⟩dµ2(ϵ2) +× (f † +µ′ +1fµ1 − xδµ1µ′ +1)P(f † +µ2fµ′ +2 − xδµ2µ′ +2) . +(B4) +The denominator D in the factor is the excitation en- +ergy of a virtual particle. +P is a projector to the re- +stricted Hilbert space, where ˆNf = nf. +The expecta- +tion ⟨dµ′ +1(ϵ′ +1)d† +µ′ +2(ϵ′ +2)⟩ evaluated on the ground state is +δ(ϵ′ +1 − ϵ′ +2)δµ′ +1µ′ +2. Then we have +− (4g)2 +(πU)2 +dD +D ∆(D) +� +µ1µ2µ′ +ˆ D−dD +−D+dD +dϵ1dϵ2 +� +∆(ϵ1)∆(ϵ2) +× d† +µ1(ϵ1)dµ2(ϵ2)(f † +µ′fµ1 − xδµ1µ′)(f † +µ2fµ′ − xδµ2µ′) , (B5) +where P is omitted as it commutes with f † +µ2fµ′ and +f † +µ′fµ1. +After a few steps of algebra, the four-fermion +operator � +µ′ f † +µ′fµ1f † +µ2fµ′ can be simplified to +f † +µ2fµ1 + +� +µ′ +f † +µ′fµ′fµ1f † +µ2 = f † +µ2fµ1(1−nf)+nfδµ1µ2 , (B6) +where we have made use of the fact that the Hilbert space +is restricted to ˆNf = nf. Substituting this into Eq. (B5), +we obtain the corrections to parameters g and xg as +dg +dt +���� +p += 4∆(D(t)) +πU +((nf − 1) + 2x) g2 , +(B7) +d(xg) +dt +���� +p += 4∆(D(t)) +πU +� +x2 + nf +� +g2 . +(B8) +Here t is the RG parameter and D(t) = De−t is the +reduced bandwidth after successive t/dt RG steps. +We then calculate the contribution from virtual hole +excitation. +Following the same process as in the last +paragraph, we obtain +− (4g)2 +(πU)2 +dD +D ∆(−D) +� +µ1µ2µ′ +ˆ D−dD +−D+dD +dϵ1dϵ2 +� +∆(ϵ1)∆(ϵ2) +× dµ1(ϵ1)d† +µ2(ϵ2)(f † +µ1fµ′ − xδµ1µ′)P(f † +µ′fµ2 − xδµ2µ′) += (4g)2 +(πU)2 dt∆(−D) +� +µ1µ2µ′ +ˆ D−dD +−D+dD +dϵ1dϵ2 +� +∆(ϵ1)∆(ϵ2) +× d† +µ2(ϵ2)dµ1(ϵ1)(f † +µ1fµ′ − xδµ1µ′)(f † +µ′fµ2 − xδµ2µ′) . +(B9) +In the second equation, we have omitted an energy con- +stant term from the anti-commutator {d† +µ2(ϵ2), dµ1(ϵ1)}. +P is the projector to the restricted Hilbert space, where +ˆNf = nf. +It is omitted in the second equation be- +cause it commutes with f † +µ′fµ2 and f † +µ1fµ′. +The four- +fermion operator � +µ′ f † +µ1fµ′f † +µ′fµ2 can be simplified to +(N − nf + 1)f † +µ1fµ2 as the Hilbert space is restricted to +ˆNf = nf. Then the corrections to g, xg from Eq. (B9) +can be read out as +dg +dt +���� +h += 4∆(−D(t)) +πU +(N − nf + 1 − 2x) g2 , +(B10) +d(xg) +dt +���� +h += 4∆(−D(t)) +πU +� +−x2� +g2 . +(B11) + +17 +Adding up the particle and the hole contributions we +can obtain the RG equations for g and (xg). The Kondo +energy scale TK can be estimated as the reduced band- +width D(t) where g diverges. For a constant ∆(ω) = ∆0, +we obtain +dg +dt = 4∆0 +πU Ng2, +d(xg) +dt += 4∆0 +πU nfg2 +(B12) +and the solution +g(t) = +g(0) +1 − g(0) 4∆0 +πU N · t , +(B13) +x(t) = x(0)g(0) +g(t) + nf +N · g(t) − g(0) +g(t) +. +(B14) +where g(0) is the initial condition given in Eq. (B3). g(t) +diverges at tK = +πU +4N g(0)∆0 , corresponding the Kondo en- +ergy scale De−tK = De− +πU +4N g(0)∆0 . +As g(t) diverges as +t → tK, the second term in x(t) dominates and there +must be x → nf +N . In other words, x flows to the occupa- +tion fraction. +2. +Application to the symmetric state at CNP +We assume a symmetric state of the THF model at +CNP and examine its Kondo energy scale. Following the +calculations in Appendix A 3, we obtain the hybridiza- +tion function contributed by the fully symmetric c-bands +(Fig. 1(b)) +∆(ω) = Ω0 +4v2⋆ +� ����|ω| − M +2 +���� θ(|ω| − M) +� +γ2 + v′2 +⋆ k2 +F 1 +� +sin2 θkF 1 +2 +× e−k2 +F 1λ2 + +����|ω| + M +2 +���� +� +γ2 + v′2 +⋆ k2 +F 2 +� +cos2 θkF 2 +2 +e−k2 +F 2λ2� +, +(B15) +where +kF 1 = 1 +v⋆ +� +(|ω| − M/2)2 − (M/2)2, +(B16) +kF 2 = 1 +v⋆ +� +(|ω| + M/2)2 − (M/2)2, +(B17) +θk = arccos +M/2 +� +M 2/4 + v2⋆k2 . +(B18) +We should choose the initial cutoff D += +1 +2U1 be- +yond which the Schrieffer-Wolff transformation is invalid. +For these states kF 1,2 ≲ +U1 +2v⋆ and hence v′2 +⋆ k2 +F 1,2 ≲ +119.4meV2, which is significantly smaller than γ2 ≈ +612.6meV2. The damping factors e−λ2kF 1,22 ≳ 0.74 are +also large. Thus, in the following we approximate ∆(ω) +(|ω| < U1/2) as +∆(ω) ≈ Ω0 +4v2⋆ +� ����|ω| − M +2 +���� θ(|ω| − M)γ2 sin2 θkF 1 +2 ++ +����|ω| + M +2 +���� γ2 cos2 θkF 2 +2 +� +. +(B19) +We first consider the flat-band limit (M = 0), where +the parameter θk is always π +2 . Thus, we have +∆(ω) = b|ω|, +b = Ω0 +4v2⋆ +γ2 ≈ 0.1290 . +(B20) +We also assume that there is no multiplet splitting in the +symmetric state such that the effective Anderson model +should be a U(8) theory with nf = 4. Naively applying +the RG equations derived in the last subsection gives +d�g +dt = −�g + 4bD +πU1 +N �g2, +(B21) +where N = 8, D = U1/2, �g = ge−t. Due to the particle- +hole symmetry at CNP, the initial condition (Eq. (B3)) +is �g(0) = 1. It seems that there would be an unstable +fixed point �g∗ = +2π +4N b, the initial �g below (above) which +flows to zero (infinity). Using the actual parameters we +find g∗ ≈ 1.52, hence the system would not be in the +Kondo phase. This result differs from the standard case +with a constant ∆(ω), where a positive g always flows +to infinity. Furthermore, a more careful RG analysis [98] +shows that the fixed point �g∗ does not really exist. It is +a false result of the weak coupling expansion, which fails +for ∆(ω) ∼ |ω|r with r > 1 +2. Thus, a ∆(ω) ∼ |ω| bath +does not have a strong coupling phase. This conclusion +is also consistent with numerical studies [95–97]. +We then consider the case with M ̸= 0. We use the +value M = 3.697meV. The RG process can be divided +into two stages: (i) When D(t) = 1 +2U1e−t > M, there is +approximately ∆(D(t)) ≈ bD(t). (ii) When D(t) < M, +the first line of Eq. (B19) vanishes, and cos2 θkF 2 +2 +in the +second line equals to +M +4ω+2M + 1 +2. Then there is ∆(D(t)) ≈ +∆0(1 + D(t)/M), with ∆0 = Ω0Mγ2 +8v2⋆ +≈ 0.239meV. The +boundary between the two stages is t1 = ln U1 +2M . The RG +equation in the first stage reads +dg +dt = 2Nb +π +g2e−t ⇒ g(t) = +1 +1 − 2N b +π (1 − e−t) . +(B22) +Due to the particle-hole symmetry, the initial condition +given by Eq. (B3) is g(0) = 1. We have g1 = g(t1) ≈ 2.34. +The RG equation in the second stage is given by +dg +dt =4N∆0 +πU1 +g2 + 4N∆0 +πU1 +g2 · e−(t−t1) +⇒ g(t) = +1 +g−1 +1 +− 4N ∆0 +πU1 (t − t1) − 4N ∆0 +πU1 (1 − e−(t−t1)) . +(B23) + +18 +g(t) diverges at tK − t1 ≈ +πU1 +4g1N ∆0 − y with y = 1, corre- +sponding to the Kondo energy scale +kBTK = Mey · e− +πU1 +4g1N ∆0 ≈ 3.8 × 10−4meV. +(B24) +3. +Application to the effective model for ν > 0 +states +In the absence of the Hund’s coupling JH, we can re- +gard (α, s) as a composite index so that ˆHSI (Eq. (20)) +is a U(N) theory with N = 4. Then the flow equations +in Appendix B 1 apply. For simplicity, we omit the neg- +ative branch of ∆(ω) (Eq. (19)) at ω < −µc because +it is far away from the Fermi level for ν > 0 (Fig. 1(d)). +The positive branch of ∆(ω) can be well approximated by +∆(ω) = ∆(0)(1+ω/(µc−G)) for |ω| < µc−G (Fig. 1(d)). +We choose the initial cutoff D to be the minimum value +of µc − G and ∆E±. +Substituting this ∆(ω) into the +general RG equations in Appendix B 1, we obtain +dg +dt = 4∆(0) +πU1 Ng2 + +4∆(0)D +πU1(µc − G)(4x + 2nf − 2 − N)g2e−t +(B25) +and +d(xg) +dt += 4∆(0) +πU1 nfg2 + +4∆(0)D +πU1(µc − G)(2x2 + nf)g2e−t (B26) +The O(e−t) terms will eventually become irrelevant when +t is sufficiently large. +After the O(e−t) terms become +irrelevant, we have +d(xg) +dg += nf/N, implying x → +nf +N +at the divergence of g. We then approximate the flow +equation of g by setting x to its fixed point value +nf +N , +i.e., +dg +dt ≈ 4∆(0) +πU1 Ng2 + +4∆(0)D +πU1(µc − G)(3nf − 6)g2e−t +(B27) +The solution of g is +g(t) ≈ +1 +g−1(0) − 4∆(0) +πU1 N +� +t + ynf (1 − e−t) +� , +(B28) +where ynf = +D +µc−G(3nf − 6). The Kondo energy scale +is determined t = tK at which g diverges. +Assuming +tK ≫ 1, we have +tK ≈ +πU1 +4Ng(0)∆(0) − ynf +(B29) +and hence +kBTK ≈ D · eynf · e− +πU1 +4N g(0)∆(0) . +(B30) +a. +The nf = 1, 3 cases +In the presence of the Hund’s coupling, we have to +examine the derivations in Appendix B 1 carefully. The +most important effect of ˆHH is to change the local Hilbert +space at small energy scales. In general, JH leads to a +multiplet splitting. When the RG energy scale is smaller +than the splitting, the higher energy multiplet will be- +come inaccessible, and the local Hilbert space is effec- +tively reduced. A minor effect is that the charge gaps +∆E± will depend on JH and the resulted coupling be- +tween f-spin and d-spin in the Coqblin–Schrieffer model +will break the U(N) symmetry. +In the following, we study how ˆHH changes the RG +equations. We first consider the nf = 1 case. In the vir- +tual particle excitation process (Eq. (B5)), the interme- +diate f-multiplet is given by |F ′⟩ = (f † +µ2fµ′ − δµ2µ′)|F⟩, +where F is the initial f-multiplet. (µ should be regarded +as the composite index (α, s).) As |F ′⟩ has the same par- +ticle number as |F⟩, it must be one of the four states with +(n1↑, n1↓; n2↑, n2↓) = (10;00), (01;00), (00;10), (00;01). +All of the possible intermediate states do not feel the +Hund’s coupling (JH +� +α nα↑nα↓) and hence they have +the same energy as |F⟩. +Hence, the excitation energy +of the intermediate state is purely contributed by d- +electrons. Then all the following derivations apply. The +same argument applies to the virtual hole excitation +(Eq. (B9)). Therefore, the RG equations for nf = 1 will +not be affected by JH. For the same reason, RG equa- +tions for nf = 3 will also not be affected by JH, where +the initial and intermediate states are single-hole states +that do not feel JH. The TK for nf = 1, 3 is given by +Eq. (B30). +b. +The nf = 2 case +The Hilbert space with two particles has six states: +(n1↑, n1↓; n2↑, n2↓) = (10;10), (10;01), (01;10), (01;01), +(11;00), (00;11). The former four states have the energy +−2µf + U1, and the latter two states have the energy +−2µf + U1 + JH. Thus JH leads to a multiplet splitting. +We divide the RG into two stages. In the first stage D(t) +is larger than JH, then the splitting JH only plays a +minor role and can be neglected. Thus the RG equations +in the first stage are given by Eq. (B27). The first stage +ends at t1 = ln(D/JH). If g diverges before t reaches +t1, the Kondo energy scale should be given by Eq. (B30) +with y2 = 0, i.e., +kBT ′ +K = D · e− +πU1 +4N g(0)∆(0) . +(B31) +If g is still finite at t1 +g1 = +g(0) +1 − g(0) 4∆(0) +πU1 N ln D +JH +, +(B32) +then the RG goes into the second stage. +The effective cutoff and the initial g of the second stage +are JH and g1, respectively. We first examine the virtual +particle excitation process (Eq. (B5)), where the interme- +diate f-multiplet is given by |F ′⟩ = (f † +µ2fµ′ − δµ2µ′)|F⟩. + +19 +Here F is the initial f-multiplet. +µ′, µ2 should be re- +garded as the composite indices (α′, s′), (α2, s2), respec- +tively. Suppose |F⟩ is one of the four low energy states, +where each orbital (α = 1, 2) has one electron; then, for +|F ′⟩ to be a low energy state, the index µ′ must have the +same orbital index with µ2, i.e., α′ = α2, such that each +orbital (α = 1, 2) in |F ′⟩ still has one electron. +With +this restriction, the four-fermion operator in Eq. (B6) +becomes +f † +α2s2fα1s1 + +� +s′ +f † +α2s′fα2s′fα1s1f † +α2s2 +(B33) +� +s′ f † +α2s′fα2s′ acting on the bra state (final state) gives +nf +α2, which must equal to 1 given that the bra state is one +of the four low energy states. Thus the four-fermion op- +erator equals to δα2α1δs2s1. The resulting contributions +to the RG equation are +dg +dt +���� +p += 4∆(D(t)) +πU +(2x) g2 , +(B34) +d(xg) +dt +���� +p += 4∆(D(t)) +πU +� +x2 + 1 +� +g2 . +(B35) +We second examine the virtual hole excitation process +(Eq. (B9)), where the intermediate f-multiplet is given +by |F ′⟩ = (f † +µ′fµ2 − δµ′µ2)|F⟩. Suppose |F⟩ is one of the +four low energy states; then, for |F ′⟩ to be in the low +energy state, the index µ′ must have the same orbital +index with µ2, i.e., α′ = α2. With this restriction, the +four-fermion operator in Eq. (B9) can be written as +� +s′ +f † +α1s1fα2s′f † +α2s′fα2s2 . +(B36) +If |F⟩ is one of the four low energy states, it at most +occupies one electron in the α2 orbital. The α2 orbital +of fα2s2|F⟩ must be empty, implying � +s′ fα2s′f † +α2s′ = 2. +Thus the four-fermion operator equals 2f † +α1s1fα2s2. The +resulting contributions to the RG equation are +dg +dt +���� +h += 4∆(D(t)) +πU +(2 − 2x) g2 , +(B37) +d(xg) +dt +���� +h += 4∆(D(t)) +πU +� +−x2� +g2 . +(B38) +Eqs. (B34), (B35), (B37) and (B38) are identical to equa- +tions of the U(2) case where N = 2, nf = 1. Following +the steps of deriving Eq. (B30), we find x still flows to 1 +2, +and +kBT ′′ +K ≈ JH · e− +πU1 +8g1∆(0) . +(B39) +The final expression for the Kondo energy scale at nf = +2 is +kBTK = +� +kBT ′ +K, +kBTK > JH +kBT ′′ +K, +otherwise +. +(B40) + +20 +[1] Yuan Cao, Valla Fatemi, Shiang Fang, Kenji Watan- +abe, Takashi Taniguchi, Efthimios Kaxiras, and Pablo +Jarillo-Herrero, “Unconventional superconductivity in +magic-angle graphene superlattices,” Nature 556, 43– +50 (2018). +[2] Yuan Cao, Valla Fatemi, Ahmet Demir, Shiang Fang, +Spencer L. Tomarken, Jason Y. Luo, Javier D. Sanchez- +Yamagishi, +Kenji +Watanabe, +Takashi +Taniguchi, +Efthimios Kaxiras, Ray C. 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Crippa, T. Wehling, G. Sangio- +vanni, R. Valen´t, Alexei M. Tsvelik, +and B. Andrei +Bernevig, (2023), to appear. +[106] Haoyu Hu, B. Andrei Bernevig, and Alexei M. Tsvelik, +(2023), to appear. +[107] Piers Coleman, (2023), to appear. +[108] Jiabin Yu, +Ming Xie, +B. Andrei Bernevig, +and +Sankar Das Sarma, “Magic angle twisted symmetric tri- +layer graphene as a topological heavy fermion problem,” +(2023), to appear. + diff --git a/-tE3T4oBgHgl3EQfrgrp/content/tmp_files/load_file.txt b/-tE3T4oBgHgl3EQfrgrp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7ce2f8f3f10fcaf992630e7ded2091d6ddb8e6c6 --- /dev/null +++ b/-tE3T4oBgHgl3EQfrgrp/content/tmp_files/load_file.txt @@ -0,0 +1,1632 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf,len=1631 +page_content='Kondo Resonance, Pomeranchuk Effect, and Heavy Fermi Liquid in Twisted Bilayer Graphene - A Numerical Renormalization Group Study Geng-Dong Zhou1 and Zhi-Da Song1, ∗ 1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China (Dated: January 13, 2023) Low energy electron Hamiltonian in the magic-angle twisted bilayer graphene can be equivalently reformulated as a topological heavy fermion model [Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 129, 047601 (2022)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' It consists of effective localized f-electrons at AA-stacking regions and itinerant Dirac c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In this work, we applied systematic analytical and numerical renormalization group analyses to a single-impurity version of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We obtained a phase diagram consisting of a Fermi liquid phase in the Kondo regime, a Fermi liquid phase in the frozen impurity regime, and various local moment phases with different spin momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Remarkably,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' this single-impurity phase diagram explains a series of experimental discoveries reported recently: (i) the zero-energy peak at fillings 1 ≲ |ν| < 2 observed in STM at low temperatures (T < 1K) [Nature 588,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 610 (2020),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Nature Physics 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1375 (2021),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Nature 600,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 240 (2021),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Nature 589,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 536 (2021)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (ii) the cascade of transitions observed in STM at higher temperatures [Nature 582,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 198 (2020),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Nature Physics 17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1375 (2021)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (iii) the Pomeranchuk effect at ν ≈ ±1 observed in transport and compressibility measurements [Nature 592,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 214 (2021),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Nature 592,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 220 (2021)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' which show that the Fermi liquid ground state develops local moments upon heating,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' and (iv) various transport experiments showing resistance peaks but no gaps around ν ≈ ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For the first time, we point out that all these phenomena result from a simple unified mechanism - the Kondo effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The Fermi liquid state at ν ≈ ±1 exhibiting the zero-energy peak is stabilized by the Kondo screening with a Kondo temperature TK ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' A higher temperature will suppress the Kondo screening and favor a local moment phase that obeys Curie’s law and contributes to an entropy of the order of Boltzmann’s constant (per moir´e cell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We computed the spectral densities, entropies, and spin susceptibilities at various fillings and temperatures, and obtained results quantitatively comparable to experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We also predict the heavy Fermi liquid as the ground state in a wide range of fractional fillings and conjecture that it is the parent state for the observed unconventional superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' INTRODUCTION Since the first discovery of the superconductivity [1] and correlated insulators [2] in magic-angle twisted bi- layer graphene (MATBG) [3], MATBG has become a new platform to study novel correlation effects in flat-band systems and has attracted extensive attentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Remark- ably rich physics, including interplay between supercon- ductivity [4–11] and strong correlation [4, 6–8, 12–19], in- teraction driven Chern insulators [20–26], strange metal behaviors [27–29], and the Pomeranchuk effect [30, 31], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', have been observed in MATBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Several theoretical understandings of the correlated states have also been achieved recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The strong correlation arises from the two topological flat bands [32–37], each of which is four- fold degenerate due to the spin and valley d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' A large U(4) symmetry group [38–42] emerges in the flat-band limit, where the actual bandwidth is counted as negligi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then the observed correlated states at integer fillings ν = 0, ±1, ±2, ±3 can be understood as flavor polarized states [38–40, 42–58] that spontaneously break the U(4) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Here |ν| is the number of electrons (ν > 0) or holes (ν < 0) per moir´e cell counted from the charge neutrality point (CNP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The continuous U(4) degeneracy leads to Goldstone mode fluctuations [59, 60] that may ∗ songzd@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='cn destroy the long-range order due to the Mermin-Wagner theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Less theoretical understandings have been achieved for the gapless states, which are observed at both fractional and integer fillings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For example, at ν ≈ ±1, depend- ing on the experimental setup, both gapped correlated insulators [4, 6, 7] and gapless Fermi liquid states [6, 25– 27, 29–31] have been observed in transport experiments, suggesting that they are competing ground states with close energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Interestingly, the observed gapless Fermi liquid states around ν = ±1 usually exhibit resistivity peaks [25–27, 29, 31] above a few kelvins, and the peaks could become increasingly pronounced as temperature rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Scanning tunneling microscope (STM) measure- ments [11, 19, 21, 22] have constantly seen that, at low temperatures of about a few hundred millikelvins, the conduction (valence) band will be pinned at the Fermi level and form a sharp zero-energy peak for the fillings 1 ≲ ν < 2 (−2 < ν ≲ −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The sharp peak does not fit the intuition of Stoner instability given that the interac- tion is indeed strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' When the temperature increases to a few or ten kelvins, these peaks develop into a cascade of transitions like a quantum dot model [17, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In this work, based on the recently developed topolog- ical heavy fermion (THF) model [61, 62], we find that the zero-energy peak, as well as the gapless Fermi liq- uid states at ν ≈ ±1, are results of the Kondo resonance [63–80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' At a higher temperature exceeding the Kondo arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='04661v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='str-el] 11 Jan 2023 2 energy scale, the local moments (LMs) formed by local- ized electrons give rise to the transition cascades, and the resistance peaks around ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We have numerically reproduced the temperature-dependent features of the spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Our theory is also fully consistent with the Pomeranchuk effect observed around fillings ν = ±1 [30, 31], which show that local moments appear upon heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We have calculated LM entropies as functions of the temperature, filling, and an external field, and obtained curves comparable to the experimentally mea- sured data in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Our theory shows that the observed gapless states at the fillings 1 ≲ |ν| < 2 are the strongly correlated heavy Fermi liquid state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Since this filling range overlaps with the superconductivity [1, 4–11] and the strange metal [27–29] around ν = −2+δ (for small δ), the heavy Fermi liquid state could be the parent state for the unconven- tional superconductivity, as it is in the heavy fermion materials with 4f or 5f electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This opens a new perspective - with a solid theoretical and experimental basis at the same time - to study the superconductivity in MATBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This work is organized as the followings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For this work to be self-contained, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' II we will review the THF model and its symmetry shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' III, based on a poor man’s scaling analysis and experimental facts, we argue that the Kondo screening effect is irrelevant at CNP, and hence the ground state at CNP is the pre- viously identified symmetry-broken correlated insulator [38–40, 42–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then we derive a simpler effective peri- odic Anderson model describing active excitations upon the correlated ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We further simplify the model to a single-impurity version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' IV, by ap- plying poor man’s scaling and Wilson’s numerical renor- malization group (NRG) method [81–83] to the single- impurity problem, we obtain a phase diagram charac- terized by strong coupling fixed points and various LM fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The strong coupling phase is divided into a Kondo regime and a frozen impurity (FI) regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We also present a detailed analysis of the RG flows at these fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The gapless 1 ≲ |ν| < 2 states are found to be in the Kondo regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' V and VI we cal- culate the spectral densities, spin susceptibilities, and entropies as functions of the filling ν and the temper- ature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The spectral densities feature sharp Kondo res- onances at low temperatures smaller than the Kondo en- ergy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Whereas at higher temperatures, the Kondo resonances are suppressed and the Hubbard bands be- come clearer, which periodically cross the Fermi level as ν changes from 0 to 4 as that of a quantum dot model, matching the STM experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The spin susceptibili- ties obey Curie’s law at high temperatures, suggesting the existence of LM, and approach constants at lower temperatures, suggesting the Fermi liquid behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The Kondo temperature TK can be estimated as the turning temperature of the two behaviors of the spin susceptibil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For 1 ≲ |ν| < 2, we find kBTK ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='129meV to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='675meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (Here kB is Boltzmann’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=') At c c Energy (meV) 0 40 20 (b) ΓM KM MM 2|M| (a) 40 60 20 60 KM ΓM MM c 80 80 f f (c) G ΓM KM MM c L=0,0 L=1,-1 (d) G 15 10 0 ∆(ω) (meV) ω (meV) 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (a) Top: red spheres represent the effective f-electrons located at AA-stacking regions of MATBG, and blue spheres represent the itinerant c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Bottom: the moir´e Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (b) Black bands are given by the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The red and blue bands are the decou- pled f- and c-bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' M is a parameter that determines the bandwidth of the flat bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We focus on the M → 0 limit in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (c) Excitation spectrum of the ac- tive c-electrons upon the symmetry-breaking parent state for ν > 0, where M and µc are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (d) The hybridization function ∆(ω) contributed by the c-bands in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' A nonzero M will not change the asymptotic behavior of ∆(ω) around the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' |ν| = 1 and kBTK ≈ 1meV, the entropy contributed by the LM is around 2 ln 2 · kB ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='39kB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In the presence of a strong in-plane magnetic field, the entropy is quenched to about ln 2·kB ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='69kB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' These values will be appreci- ated if one notices that the measured entropies at ν ≈ 1 and T ≈ 10K in the absence and presence of a strong in- plane field are about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='2kB and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='6kB, respectively [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' VII, we discuss the heavy Fermi liquid states at 1 ≲ |ν| < 2 and propose experiments to confirm them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We estimate the heavy fermion bands and their quasi- particle weights using spectral information provided by the NRG calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The possible effects of the RKKY interactions are also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' THE THF MODEL One theoretical challenge in studying correlation physics in MATBG is the lack of a fully symmetric lat- tice model for low energy physics, which is forbidden by the band topology protected by a C2zT symmetry [32–34] and an emergent particle-hole symmetry P [37] even though extended Hubbard models [84–88] can be constructed at the sacrifice of either symmetry or local- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The band topology was thought as fragile [32–34] but was later shown to be a stable symmetry anomaly jointly protected by C2zT and P [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The THF model [61, 62] resolved this problem by ascribing the strong correlation to effective f-orbitals at the AA-stacking re- gions, which form a triangular lattice, and leaving the remaining low energy states to continuous c-bands de- scribed by a topological Dirac Hamiltonian (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The THF model faithfully reproduces the symmetry, topol- ogy, dispersion, and Coulomb interaction of the continu- ous Bistritzer-MacDonald model [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Its free part is given 3 by ˆH0 = −µ ˆ N + � ηs � aa′ � |k|<Λc H(c,η) aa′ (k)c† kaηsckaηs + � ηsαa � |k|<Λc � e− |k|2λ2 2 H(cf,η) aα (k)c† kaηsfkαηs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (1) Here µ is the chemical potential, ˆN is the particle-number operator, ckaηs is the fermion operator for the c-electron of the momentum k, orbital a (= 1, 2, 3, 4), valley η (= ±), and spin s (=↑, ↓), fkαηs is the corresponding fermion operator for the f-electron of the orbital α (=1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The summation over k for the c-bands is in principle limited within the cutoff Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' But the theory is well-defined and yields the same low energy physics if we take the Λc → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' H(c,η)(k) = v⋆(ησx⊗σ0kx−σy⊗σzky)+02×2⊕Mσx is the Dirac Hamiltonian of the c-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' When M ̸= 0, c-bands have a quadratic band touching at the zero en- ergy, whereas when M = 0, c-bands become linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The two-by-two block of H(cf,η) aα (k) for a = 1, 2 is given by γσ0 + v′ ⋆(ησxkx + σyky), and the two-by-two block of H(cf,η) aα (k) for a = 3, 4 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The parameter λ in the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (1) is the spread of the Wan- nier functions of f-electrons, and it truncates the hy- bridization at |k| ≫ λ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In this work we adopt the w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='8 parameters of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [61]: γ = −24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='75meV, v⋆ = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='303eV · ˚A, v′ ⋆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='623eV · ˚A, λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='4131/kθ, kθ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='703˚A−1 · 2 sin θm 2 with θm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='05◦ being the first magic angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (w0 and w1 are the interlayer couplings of MATBG at the AA-stacking and AB-stacking regions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Due to the corrugation effect [89–92], w0 is usually smaller than w1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [85] estimates w0/w1 as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=') The resulting band structure with a nonzero M (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='697meV) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' One can see that the topological flat bands result from the hybridization be- tween c- and f-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As explained in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [61] and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1(b), the parameter M determines the bandwidth of the flat-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In each valley, the Hamiltonian ˆH0 respects a magnetic space group P6′2′2 [32] (#177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='151 in the BNS setting [93]), generated by C3z, C2x, C2zT, and translation sym- metries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The two f-orbitals and the four c-bands have effective angular momenta L = −η, η and L = −η, η, 0, 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As shown in detail in [61], the six- by-six representations of C3z, C2x, C2zT symmetries on these orbitals are eiη 2π 3 σz ⊕ eiη 2π 3 σz ⊕ σ0, I3×3 ⊗ σx, and I3×3 ⊗ σxK, respectively, where σ0,x,y,z are Pauli matri- ces and K the complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' One can verify that the Hamiltonian matrices H(c,η) and H(cf,η) given in the last paragraph respect these crystalline symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The interaction Hamiltonian given in the following paragraph also respects these crystalline symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The interaction Hamiltonian is given by ˆHI =U1 2 � R δnf Rδnf R + U2 2 � ⟨RR′⟩ δnf Rδnf R′ + 1 2NM � qaa′ V (q)δnc −qa′δnc qa + 1 NM � Rqa Wae−iq·Rδnf Rδnc qa − J 2NM � ηη′αα′ ss′ � |k|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='|k′|<Λc R � (ηη′ + (−1)α+α′)e−i(k−k′)·R− λ2(k2+k′2) 2 (f † Rα′η′s′fRαηs − 1 2δηη′δαα′δss′)(c† k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='α+2ηsck′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='α′+2η′s′ − 1 2δkk′δηη′δαα′δss′) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (2) where NM is the number of moir´e cells,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' fRαηs is the real space fermion operator for the f-electrons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' R are the tri- angular lattice shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1(a), ⟨RR′⟩ represents near- est neighbor pairs (ordered), δnf R = � αηs(f † RαηsfRαηs − 1 2) is the density operator of f-electrons, δnc qa = � ηsk(c† k+qaηsckaηs − 1 2δq0) is the density operator for c- electrons of the orbital a with k and k + q being limited within the cutoff Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' U1,2, V (q), Wa are the density- density interaction between ff, cc, cf electrons, respec- tively, and J is an exchange interaction between cf elec- trons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We adopt the w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='8 parameters of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [61]: U1 = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='95meV, U2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='329meV, W1 = W2 = 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='03meV, W3 = W4 = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='20meV, J = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='38meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We choose V (q) as the double-gate-screened Coulomb interaction, πξ2Uξ Ω0 tanh(ξ|q|/2) ξ|q|/2 , with ξ = 10nm being distance between MATBG and the gates, Uξ = 24meV the Coulomb inter- action at the distance ξ, and Ω0 ≈ 156nm2 the area of the moir´e cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Hereafter, we mainly focus on the flat-band limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In this limit, an exact U(4) symmetry of ˆH0+ ˆHI between the spin, valley, and orbital flavors emerge, as previously recognized in the projected Coulomb Hamil- tonian of the continuous model [38–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We emphasize that this U(4) symmetry is not related to the so-called chiral limit [35, 94], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', w0 = 0, which leads a distinct U(4) symmetry [41, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The U(4) symmetry in the flat- band limit has been shown as a good approximation for realistic parameters such as w0/w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='8 [41, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The sixteen U(4) generators acting on fRαηs, ckaηs (a = 1, 2), and ckaηs (a = 3, 4) are Σf µν = {σ0τ0ςν, σyτxςν, σyτyςν, σ0τzςν} , (3) Σc12 µν = {σ0τ0ςν, σyτxςν, σyτyςν, σ0τzςν} , (4) and Σc34 µν = {σ0τ0ςν, −σyτxςν, −σyτyςν, σ0τzςν} , (5) 4 respectively, where ςν (ν = 0, x, y, z) are Pauli matrices acting in the spin subspace, τµ (µ = 0, x, y, z) are Pauli matrices acting in the valley subspace, and σ0,x,y,z are Pauli matrices acting in the orbital subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' With the help of U(4) symmetry, the J term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (2) can be writ- ten as a ferromagnetic coupling between the U(4) LM of f-electrons and the U(4) LM of c-electrons [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' When M ̸= 0, only the µ = 0, z U(4) generators commute with the Hamiltonian, leading to a lower U(2)×U(2) symme- try group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The rotation generated by µ = z, ν = 0 is referred to as the valley-U(1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Consistent with previous results [38–40, 42–44], a Hartree-Fock treatment of the THF model has predicted the ground state at CNP as a U(4) LM state that re- spects a U(2)×U(2) subgroup [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The LM forms a 20- fold multiplet belonging to the [2, 2]4 representation [43] of the U(4) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' These states can be approximately written as |Ψ0⟩ = e−iθµν ˆΣµν � R f † R1+↑f † R1+↓f † R2+↑f † R2+↓|FS⟩ , (6) where the |FS⟩ is the Fermi sea state with the half-filled c-bands, ˆΣµν are the U(4) generator operators defined by the matrices in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (3) to (5), and θµν are the ro- tation parameters, and an implicit summation over re- peated µ, ν indices is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' When θµν’s are zero, |Ψ0⟩ is the valley-polarized state because all the occupied f- electrons are in the η = + valley, and the U(2)×U(2) subgroup is generated by Σ0,ν and Σz,ν (ν = 0, x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For nonzero θµν’s, |Ψ0⟩ respects an equivalent U(2)×U(2) subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The Kramers inter-valley coherent states can be obtained by choosing nonzero θx0 and θy0 satisfying θ2 x0 + θ2 y0 = (π/4)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' When M ̸= 0, the Kramers inter- valley coherent states are the ground states, while the valley polarized states have higher energy (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1meV) [43, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' EFFECTIVE ANDERSON MODEL FOR ν > 0 STATES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Irrelevance of Kondo screening at CNP Here we argue that the Kondo screening effect is ir- relevant at CNP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' hence, the U(4) LM state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (6) is valid as an approximate ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We first exam- ine the energy scale of a fully symmetric Kondo state at CNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Since the f-sites are almost decoupled from each other, a reasonable approximation is to view each f-site as a single Anderson impurity coupled to a bath of c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' If we only consider the on-site U1 inter- action and the single-particle hybridization between f- and c-electrons (H(cf,η)(k) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (1)), then it is almost a standard Anderson model with eight flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The ef- fect of c-bath is described by the hybridization function ∆(ω), defined as the imaginary part of the self-energy of a free f-electron (in the absence of U1) coupled to the c- bath, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', Im[Σ(f) αηs,α′η′s′(ω)] = δα,α′δηη′δss′sgn(ω)∆(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The identity matrix form of Im[Σαηs,α′η′s′(ω)] is guar- anteed by the spin-SU(2) (δss′), the valley-U(1) and the time-reversal (δηη′), and crystalline (δαα′) symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Low energy c-bands (k → 0) are coupled to the impu- rity through the constant coupling γ in H(cf,η)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then ∆(ω) would be proportional to the density of states ρ(ω) of c-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In the flat-band limit (M = 0), c-bands have a linear dispersion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1(b)) and hence the density of states, as well as the hybridization function, linear in en- ergy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', ρ(ω) ∼ |ω|, ∆(ω) ∼ |ω|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As a consequence, low-lying states of the impurity will see vanishing bath electrons when the energy scale is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Both nu- merical [95–97] and analytical [98] RG studies have shown that Anderson impurity models with such a ∆(ω) ∼ |ω| hybridization function do not have the strong coupling fixed point that exhibits Kondo screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Instead, the only stable fixed point is the LM phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' With a finite M, the c-bands given by H(c,η)(k) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (1) have a quadratic touching at the zero energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', ±(−M/2 + � M 2/4 + v2⋆k2), leading to a finite den- sity of states at the zero energy and hence a finite ∆(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Nevertheless, as explained in the following, the Kondo energy scale resulting from realistic parameters is neg- ligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In Appendix B 2 we derived an analytical ex- pression of ∆(ω) for the symmetric state at CNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For |ω| > M, ∆(ω) is almost linear in |ω|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', ∆(ω) ≈ b·|ω|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For |ω| < M, ∆(ω) is a constant plus a linear term: ∆(ω) ≈ ∆(0)(1 + |ω|/M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Using the parameters given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' II and M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='697meV, we have b ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='129, ∆0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='239meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' A rough estimation of the Kondo en- ergy scale can be made by replacing the ω-dependent ∆(ω) with the constant ∆(0) at ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then naively applying the large-N formula at second order [99], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', kBTK ≈ De− πU1 4N ∆(0) with N = 8 being the number of flavors and D = U1/2 the energy scale up to which the perturbation theory applies, leads to an extremely small kBTK ≈ U1 · 2 × 10−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' A better estimation is given by a poor man’s scaling that considers the ω-dependence of ∆(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Readers may refer to Appendix B 2 for more de- tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Here we only present the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' There are two stages in the RG process: (i) a stage with energy scale from U1/2 - scale up to which the perturbation the- ory applies - to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (ii) a stage with an energy scale below M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' RG in the first stage effectively enhances ∆(0) to g1∆(0) with g1 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then, RG in the second stage gives kBTK ≈Meye− πU1 4N g1∆(0) ≈3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='8 × 10−4meV (2M = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='39meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (7) where y ≈ 1 is a factor contributed by the ω-dependence of ∆(ω) in the second stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This value is still much lower than the energy gain of the symmetry-broken correlated state [43, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The bandwidth of the Goldstone modes at CNP from ΓM to MM is about 8meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [59]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' If we understand this spectrum as a tight- binding band of the Holstein–Primakoff bosons on the f-sites, which form a triangular lattice, then the nearest 5 neighbor hopping is about 8meV/8=1meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This hopping indicates an RKKY interaction much larger than kBTK evaluated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The actual TK can even be much smaller than the value in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' First, as ∆(0) → 0 when M → 0, TK decays exponentially when M decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For example, a band- width 2M = 5meV corresponds to kBTK ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1 × 10−6meV (2M = 5meV) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (8) Second, because we only considered the U1 interaction and the cf hybridization that gives all flavors of f- electrons the same ∆(ω) (guaranteed by symmetries), the single-impurity model has a U(8) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This U(8) symmetry must be broken when other interaction terms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', J in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (2), are taken into account, leading to a multiplet splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' When the energy scale in the RG is smaller than the multiplet splitting, the degeneracy factor N should be reduced accordingly, and TK will be further suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (One can see section 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='2 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [99] and Appendix B 3 for examples of how multiplet splitting suppresses TK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=') The U(4) LM state at CNP is also supported by var- ious experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In contrast to the Kondo resonance, STM measurements have shown strong suppression of the density of states at the zero energy at CNP [11, 12, 14– 17, 19, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Some transport experiments [4, 6, 7, 27] also exhibits a gap behavior at CNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Although there are also transport experiments showing semimetal behavior, the gaplessness can be explained if there are fluctuations of the local moments from site to site, which is possi- ble due to the Goldstone mode fluctuations [59, 60] and possible inhomogeneity of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Periodic Anderson model for ν > 0 states We aim for an effective model describing the active ex- citations upon the ground state |Ψ0⟩ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (6)) at CNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Let us first assume the valley-polarized state, where θµν’s in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (6) are all zero such that all the occupied f- electrons are in the η = + valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As detailed in the supplementary material of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [61] and in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [59], the lowest particle and hole excitations are in the η = − and η = + valleys, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus, for ν > 0 states, only particle excitations in the η = − valley will be in- volved, and the electrons in the η = + valley can be viewed as a static background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The effective Hamil- tonian can be obtained by replacing operators in the η = + valley by their expectation values on |Ψ0⟩, which are ⟨f † Rα+sfR′α′+s′⟩ = δRR′δαα′δss′, ⟨c† ka+sck′a′+s′⟩ ≈ 1 2δkk′δaa′δss′, ⟨c† ka+sfRα+s′⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Substituting these ex- pectation values into ˆH0 + ˆHI, we obtain the effective free Hamiltonian ˆHeff 0 = � |k|<Λc aa′s (H(c) aa′(k) − µδaa′)c† kascka′s − µ � Rαs nf Rαs + � |k|<Λc aα � e− 1 2 λ2k2H(cf) aα (k)c† kasfkαs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' � , (9) where nRαs = f † RαsfRαs is the density operator of f- electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Here we have dropped the valley index η as they are limited to η = −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The H(c)(k) and H(cf)(k) matrices are given by the H(c,−)(k) and H(cf,−)(k) ma- trices defined after Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We also obtain the effective interaction Hamiltonian ˆHeff I = U1 2 � R nf Rnf R + U2 2 � ⟨RR′⟩ nf Rnf R′ + 1 2NM � qaa′ V (q)δnc −q,a′δnc q,a + 1 NM � Rqa Wae−iq·Rnf Rδnc qa − J NM � Rss′ � α � |k|,|k′|<Λc e−i(k−k′)·R− λ2(k2+k′2) 2 × (f † Rαs′fRαs − 1 2δss′)(c† k,α+2,sck′,α+2,s′ − 1 2δss′) , (10) where nf R = � αs nf Rαs, δnc qa = � sk(c† k+qasckas − 1 2δq0) with |k| and |k + q| being limited within the cutoff Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The δnf R operator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (2), which represents the density deviation from the charge background at CNP, is now replaced by nf R, the total density in the η = − valley, because the charge background is compensated by the occupied η = + electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The J term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (2) is also simplified: As active excitations are limited to η = −, the factor ηη′ + (−1)α+α′ becomes 2δα,α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In the flat-band limit (M = 0), ˆHeff 0 + ˆHeff I ap- plies to arbitrary U(4) partners of the valley-polarized state, including the so-called Krammers intervalley co- herent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' To be specific, for a generic |Ψ0⟩ given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (6), we can always define rotated operators fRαs = UfRα−sU †, ckas = Ucka−sU †, where U = e−iθµν ˆΣµν is the U(4) rotation defining |Ψ0⟩, such that the effective Hamiltonian on the rotated basis is same as Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (9) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The effective Hamiltonian ˆHeff 0 + ˆHeff I respects all the crystalline symmetries discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The effective angular momenta of the active two f-orbitals and four c-bands are L = 1, −1 and L = 1, −1, 0, 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' And, the six-by-six representations of C3z, C2x, C2zT on these orbitals are e−i 2π 3 σz ⊕ e−i 2π 3 σz ⊕ σ0, 13×3 ⊗ σx, and 13×3 ⊗ σxK, respectively, with K being the complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In the flat-band limit (M = 0), as |Ψ0⟩ respects a U(2)×U(2) subgroup of the U(4) group, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', independent spin-charge rotations in the two valleys for the valley polarized |Ψ0⟩, one may expect a U(2)×U(2) symmetry of ˆHeff 0 + ˆHeff I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' However, since the effective Hamiltonian only involves half of the d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', the ac- tive η = − valley for the valley polarized |Ψ0⟩, only a single U(2) factor is meaningful for ˆHeff 0 + ˆHeff I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' There- fore, hereafter we will say that ˆHeff 0 + ˆHeff I respects a U(2) symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' When M ̸= 0, the U(4) symmetry is broken, and the effective Hamiltonian will have an additional term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In Appendix A we show that to the order of M 2, the cor- 6 ˆH0 + ˆHI ˆHeff 0 + ˆHeff I ˆHSI M = 0 U(4) U(2) U(2)×U(2) M ̸= 0 U(2)×U(2) U(2) U(2)×U(2) TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Continuous symmetries of the Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' ˆH0 + ˆHI is the original THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For ν > 0 (ν < 0), ˆHeff 0 + ˆHeff I is the effective periodic Anderson model for the active particle (hole) excitations upon the symmetry broken state at CNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' ˆHSI is a single-impurity version of ˆHeff 0 + ˆHeff I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' rection is simply an energy shift M 2 J � |k|<Λc � a=3,4 � s c† kasckas + O(M 4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (11) Thus, the M-term breaks neither the crystalline symme- try nor the U(2) symmetry, and it will play a minor role in the effective theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' To avoid confusion, in Table I we summarize the continuous symmetries of different Hamil- tonians with M = 0 or M ̸= 0 discussed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The effective model for ν < 0 states, which only in- volve hole excitations, can be obtained by applying the particle-hole operation Pc [41, 61] to ˆHeff 0 + ˆHeff I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Single impurity model for ν > 0 states Hereafter we mainly focus on a single-impurity version of ˆHeff 0 + ˆHeff I , where only the correlation at the R = 0 f-site is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The interactions involving other f- sites will be treated at the mean-field level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The STM spectra [11, 19, 21, 22] that show the zero-energy peaks also clearly show a continuity between the gapped CNP state and the gapless states at 1 ≲ |ν| < 2, implying that they have the same symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Therefore, in this work, we assume no additional symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For the completeness of the discussion, we also extend our symmetric assumption to |ν| ≥ 2 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' One should be aware that additional symmetry breaking may happen at low temperatures in |ν| ≥ 2 states due to the effec- tive RKKY interactions neglected in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus the symmetric assumption is invalid for |ν| ≥ 2 states at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Nevertheless, the |ν| ≥ 2 states may re- cover the symmetries at higher temperatures, where our symmetric theory applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' At a given filling ν, the symmetric mean field is char- acterized by only a few parameters: νf = ⟨nf R⟩, νc,a = 1 NM ⟨δnc q=0,a⟩, where νc,1 = νc,2, νc,3 = νc,4 due to the crystalline symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The actual values of νf and νc,a can be determined self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The considered cor- related site at R = 0 is described by the Hamiltonian ˆHf = −µf � αs nf αs + U1 � (αs)<(α′s′) nf αsnf α′s′ , (12) where the lattice index R is omitted for simplicity, µf = −6νfU2−� a νc,aWa− 1 2U1+Jνc,3+µ is an effective chem- ical potential for the f-site, and µ is the global chemical potential determined by the total filling ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The U2, Wa, J terms in µf are contributed by the Hartree mean fields of the U2, Wa, and J interactions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The U1 term in µf is from the diagonal U1 interactions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (10), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', 1 2U1 � αs nf αsnf αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The U1 interaction in ˆHf only contains the off-diagonal U1 interactions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The effective Hamiltonian of c-electrons is given by ˆHc = � |k|<Λc � aa′s (H(c) aa′(k) + ∆H(c) aa′ − µcδaa′)c† kascka′s , (13) where H(c)(k) = −v⋆(σx ⊗ σ0kx + σy ⊗ σzky) is the free Dirac Hamiltonian, ∆H(c) aa′ = G·δaa′(δa3+δa4) is a mean- field term that split the a = 1, 2 and a = 3, 4 c-electrons, µc = −νfW1 −νc V Ω0 +µ is an effective chemical potential of c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The W1 and V terms in µc are contributed by the W and V interactions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (10), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The mean field term G arises from two interaction terms: (i) A mean field treatment of the J interaction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (10) yields an energy shift J( 1 2 − 1 4νf) for a = 3, 4 c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (ii) The Hartree mean fields of the Wa interactions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (10) are νfW1 and νfW3 for a = 1, 2 and a = 3, 4 c-electrons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As we have absorbed νfW1 to µc, a = 3, 4 c-electrons have an extra energy shift (W3 − W1)νf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Combining the two effects, the parameter G is given by G = J 2 +(W3−W1− J 4 )νf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Since the interaction V (q) of c-electrons is completely treated at the mean- field level, ˆHc is an effective free-fermion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As detailed in Appendix A, the band structure of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (13) is given by G/2± � G2/4 + v2⋆k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1(c) we plot the c- bands with µc = 0 and G = 10meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' It has a gap opened by G, where, according to the symmetry representations given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' III B, the lowest conduction and highest valence band states have angular momenta 0, 0 and 1, −1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The f-site is coupled to c-electrons via two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' One is the hybridization ˆHhyb = 1 √NM � |k|<Λc � aαs � e− λ2k2 2 H(cf) aα (k)c† kasfαs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (14) The other coupling term is the remaining ferromagnetic exchange interaction ˆHJ = − J NM � |k|,|k′|<Λc � αss′ e− 1 2 λ2(k2+k′2)(f † αsfαs′ − 1 4δss′νf) × (c† k′α+2s′ckα+2s − 1 2δk,k′δss′νc,α+2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (15) By “remaining” we mean that the mean field back- grounds 1 4νf and 1 2νc,a are deducted in ˆHJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As ex- plained below, ˆHJ leads to an effective Hund’s coupling that changes the symmetry of the single-impurity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' There are also remaining density-density interactions be- tween c- and f-electrons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', Wa(nf − νf)(nc q,a − νc,a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' However, these remaining density-density interactions do not change the essence of the single-impurity problem as ˆHJ does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 7 Thanks to the C3z symmetry, ˆHhyb and ˆHJ couple the f-electrons to two independent baths belonging to different angular momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This allows us to treat the two terms separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In a polar coordinate, ˆHhyb only couples f-electrons to �ckαs = 1 A � a ˆ dφ · H(cf) αa (k)ckas (α = 1, 2) , (16) where k = k(cos φ, sin φ), and A is a normalization fac- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Explicitly, there are �ck1s ∼ ´ dφ·(γck1s−v′ ⋆keiφck2s) and �ck2s ∼ ´ dφ · (γck2s − v′ ⋆ke−iφck1s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Under the C3z operation (defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' III B), �ck1s and �ck2s have effec- tive angular momenta 1, -1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' On the other hand, ˆHJ only couples f-electrons to �ckas = 1 A′ ˆ dφ · ckas (a = 3, 4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (17) Both ckas (a = 3, 4) have the effective angular momen- tum 0 under C3z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Because �ckas (a = 1, 2) and �ckas (a = 3, 4) form different representations of C3z, they do not couple to each other, hence the ˆHhyb-bath and the ˆHJ-bath are indeed independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As a ferromagnetic coupling always flows to zero and becomes irrelevant in low energy physics, we can inte- grate out the ˆHJ-bath in a single attempt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This leads to an effective Hund’s coupling (Appendix A) ˆHH = JH � α nα↑nα↓ , (18) where JH, estimated as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='3meV, is the additional energy that two electrons will acquire if they occupy the same orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' A nonzero M does not change the form of ˆHH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Integrating out the ˆHhyb-bath leads to a self-energy correction Σ(f) αs,α′s′(ω) to the f-electrons, the imaginary part of which defines the hybridization function ∆(ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', Im[Σ(f) αs,α′s′(ω)] = δαα′δss′sgn(ω)∆(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The identity matrix structure of the self-energy is guaranteed by SU(2) spin rotation symmetry and crystalline symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In Appendix A we derived an analytical expression of ∆(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1(d), where ∆(ω) for µc = 0 and G = 10meV is plotted, ∆(ω) has an abnormal ω-dependence compared to those in usual metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' First, ∆(ω) = 0 for ω in the gap of c-bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Second, because f- electrons (L = 1, −1) have different angular momenta as the lowest conduction c-bands (L = 0), hybridization between them vanishes as k → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As a result, ∆(ω) linearly approaches zero at the conduction band edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Third, as f-electrons have the same angular momenta as the highest valence c-bands, ∆(ω) approaches a constant at the valence band edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In summary, around the gap ∆(ω) has the asymptotic behaviors ∆(ω) ∼ � � � � � |ω + µc − G|, ω + µc → G + 0+ 0, 0 ≤ ω + µc ≤ G const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', ω + µc → −0+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (19) A nonzero M does not change the asymptotic behav- iors as these behaviors are guaranteed by the C3z sym- metry that is also respected by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Due to the damp- ing factor e− 1 2 λ2k2 in ˆHhyb, c-electrons with momenta |k| ≫ 1/λ will not contribute to ∆(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus, ∆(ω) de- cays exponentially when ω exceeds v⋆/λ ∼ 95meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In the rest of this work, we will restrict the hybridization to |ω| < D = 100meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Baths giving rise to the same ∆(ω) are physically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Therefore, we can choose a bath that is as simple as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We introduce the following effective single-impurity Hamiltonian ˆHSI = ˆHf + ˆHH + � αs ˆ D −D dϵ · ϵ · d† αs(ϵ)dαs(ϵ) + � αs ˆ D −D dϵ · � ∆(ϵ) π (f † αsdαs(ϵ) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=') , (20) where ˆHf and ˆHH are given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (12) and (18), re- spectively, and dαs(ϵ) are the auxiliary bath fermions in- troduced to reproduce the hybridization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' dαs(ϵ) satisfy {dα′s′(ϵ′), d† αs(ϵ)} = δα′αδs′sδ(ϵ′ − ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' ˆHSI is com- pletely defined by the following parameters: µf the chem- ical potential of f-electrons, U1 the Coulomb repulsion, JH the Hund’s coupling, and ∆(ω) the hybridization function, which further depends on µc the chemical po- tential of c-electrons and G the gap of c-bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As explained at the beginning of this subsection, the ac- tual values of µf, µc, and G depend on the occupations νf, νc,a, which are further determined by self-consistent calculations at given total fillings ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We plot µf, µc, and G as functions of ν in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For ν changing from 0 to 4, µc changes from 0 to 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='70meV, µf changes from 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='98meV to 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='13meV, and G changes from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='19meV to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='21meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We can now regard µc, µf, G as given pa- rameters that define the single-impurity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' It is worth mentioning that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (20) has an emergent U(2)×U(2) symmetry - independent spin-charge rota- tions in the α = 1, 2 orbitals - that is higher than the U(2) symmetry of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (9) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' It is not surpris- ing that a single-impurity model has a higher symmetry than its lattice version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For example, if J = 0, there would be no Hund’s coupling JH, and ˆHSI would have a U(4) symmetry, as expected in a four-flavor Anderson impurity model without multiplet splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' PHASE DIAGRAM OF THE SINGLE-IMPURITY MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Poor man’s scaling Before going to numerical calculations, we first apply a poor man’s scaling to the single impurity model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The scaling theory helps us understand several features of the data obtained by NRG, as will be discussed in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' And, it predicts the Kondo 8 temperatures in the same order as those predicted by NRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We assume that the ground state of the isolated impu- rity has nf (integer) occupied f-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' One should not confuse nf with νf - the expectation value of f- occupation after the impurity is coupled to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The chemical potential µf must be in the range (nf − 1)U1 < µf < nfU1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We apply a Schrieffer-Wolff transforma- tion to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (20) to obtain an effective Coqblin–Schrieff model where the local Hilbert space of f-electrons is re- stricted to nf particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The transformation involves vir- tual particle and hole excitations, the energies of which are ∆E+ = nfU1 − µf and ∆E− = µf − (nf − 1)U1, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (We have ignored the JH term in ∆E± as it is small compared to U1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=') Adding the two contributions, we have ˆH = ˆHH + � αs ˆ D −D dϵϵd† αs(ϵ)dαs(ϵ) + 4g πU1 � αα′ss′ ˆ D −D dϵdϵ′ � × � ∆(ϵ)∆(ϵ′)(f † αsfα′s′ − xδαα′δss′)d† α′s′(ϵ′)dαs(ϵ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (21) The parameters g, x are given by g = U1 4 � 1 ∆E+ + 1 ∆E− � , x = ∆E− U1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (22) g is a dimensionless parameter characterizing the anti- ferromagnetic coupling strength between the impurity and the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' x appears as a “charge background” of the f-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For µf = (νf − 1 2)U1, there is g = 1, x = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For a generic (nf − 1)U1 < µf < nfU1, g ≥ 1 and 0 < x < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Flow equations of g, x are derived in Ap- pendix B 3, where the divergence of g indicates the strong coupling fixed point that exhibits the Kondo screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We notice that x always flows to nf/4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', the occupa- tion fraction of f-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' One should be careful about the cutoff D in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (21) First, it must be smaller than ∆E+ and ∆E− for the Schrieffer-Wollf transformation to be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Second, for analytical conveniences, we only keep the positive branch of ∆(ω) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (19)) at ω > G − µc because when ν > 0 the negative branch is far away from the Fermi level (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Hence, we also require D < µc − G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We can choose D = min(µc − G, ∆E+, ∆E−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We first consider the case nf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The flow equation of g(t) as the cutoff is reduced to De−t is given by dg dt = 4∆(0) πU Ng2 + O(e−t) , (23) and the initial condition g(0) is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Here N = 4 is the number of flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The local Hilbert space for nf = 1 is four-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The Hund’s coupling JH does not split the four-fold degeneracy and hence does not appear in the flow equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The O(e−t) terms are irrelevant at small energy scales but they may affect the coupling constant at an early stage of the RG process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Using a linear approximation of ∆(ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', ∆(ω) ≈ ∆(0)(1 + ω/(µc − G)), we obtain (Appendix B 3) kBT (1) K = Dey1 · e− πU1 4N ∆(0)g(0) (24) where y1 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='75 D µc−G < 0 is factor contributed by the irrelevant O(e−t) terms and will slightly suppress the Kondo energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The suppression factor ey1 appears because, for the nf = 1 states, the virtual processes that contribute to the RG equation involve more hole exci- tations than particle excitations in the bath, such that the smaller ∆(ω < 0) contributes more than the larger ∆(ω > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As a result, the resulting kBTK is smaller than the standard case (y1 = 0) where the coupling is a constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', ∆(ω) = ∆(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We then study the RG equations at nf = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Unlike the nf = 1 case, Hund’s coupling JH splits the six- dimensional local Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (18), the four states with (nf 1↑, nf 1↓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' nf 2↑, nf 2↓) =(10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='10), (10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='01), (01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='10), (01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='01) do not feel JH and have the energy −2µf + U1, whereas the two states (11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='00), (00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='11) have the energy −2µf + U1 + JH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We divide the RG process into two stages: (i) a stage with an energy scale from D to JH, (ii) a stage with an energy scale below JH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In the first stage, JH plays a minor role;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' hence, a flow equation similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (23) applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' If g diverges before the energy scale reaches JH, then the Kondo scale is given by kBT (2)′ K = D · e− πU1 4N ∆(0)g(0) (25) there is no y-factor as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (24) because, for the nf = 2 states, the virtual processes that contribute to the RG equations evolve equal holes and particles in the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Otherwise, g will be renormalized to g1 = g(0) 1 − g(0) 4∆(0) πU1 N ln D JH (26) at the energy scale of JH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As detailed in Appendix B, RG in the second stage is similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (23) except that, due to the multiplet splitting, the factor N = 4 is replaced by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This leads to the Kondo energy scale kBT (2)′′ K = JH · e − πU1 8∆(0)g1 = D � D JH � N 2 −1 e − πU1 8∆(0)g(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (27) Considering the g may diverge in either the first or the second stage, the physical Kondo energy scale can be written as kBT (2) K = � kBT (2)′ K , kBT (2)′ K > JH kBT (2)′′ K , otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (28) The theory at nf = 3 is almost the same as the theory at nf = 1 except that the four single-particle states are now replaced by the four single-hole states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus, the local Hilbert space is also four-dimensional and will not be split by Hund’s coupling JH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The Kondo energy scale is given by kBT (3) K = Dey3 · e− πU1 4N ∆(0)g(0) (29) where y3 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='75 D µc−G > 0 is a factor contributed by the irrelevant O(e−t) terms and will slightly enhance the 9 ν kBT (P ) K (meV) δK(meV) kBT (χ) K (meV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='280 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='151 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='250 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='400 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='271 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='663 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='391 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Kondo energy scales at various fillings ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' T (P ) K , δK, and T (χ) K are Kondo energy scales estimated by the poor man’s scaling, the NRG spectral density, and the NRG spin susceptibility, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' δK is defined as the half width at half maximum of the spectral peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' T (χ) K is defined as the turning temperature where χ(T) transitions from the Curie- Weiss behavior to the Fermi liquid behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' There is no data of T (P ) K at ν = 2 because ν = 2 is close a mixed valence case where µf ≈ U1 and the Schrieffer-Wolff transformation does not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Kondo temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The enhancement arises from the fact that, in contrast to the case of nf = 1, for nf = 3 states the virtual processes contributing to the RG equa- tion evolve more particle excitations than hole excita- tions in the bath, and particles have larger couplings than holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In Table II we tabulate the Kondo energy scales ob- tained by the poor man’s scaling at various fillings us- ing the filling-dependent µc, µf, G parameters given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(a), and compare them to the values obtained by the NRG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The NRG method In the NRG method [81–83], the bath is alternatively realized by a Wilson chain constructed in the way de- scribed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' First, the energy window [−D, D] is discretized on a logarithmic scale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', ωn = D/Λn−1 (n = 1, 2 · · · ), where Λ > 1 is a scaling factor (cho- sen as 3 in this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then for each energy shell ωn ≤ |ω| < ωn+1 two auxiliary bath electrons corre- sponding to the positive and negative part of it are in- troduced to reproduce the corresponding ∆(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' These auxiliary electrons are further recombined into a Wilson chain, dnαs, such that (i) only the first site d1αs couples to the impurity, (ii) the chain is a tight-binding model with only on-site and nearest neighbor hopping terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The Hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (20) is now mapped to an impurity plus a Wilson chain ˆHN = ˆHf + ˆHH + � αs t0(f † αsd1αs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=') + N � n=1 � αs ϵnd† nαsdnαs + N−1 � n=1 � αs (tnd† n+1αsdnαs + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=') , (30) where N is a large number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The parameters ϵn and tn can be computed from ∆(ω) using a standard iterative algorithm [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For n → ∞, ϵn ∼ Λ−n and tn ∼ Λ− 1 2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus, the right-most sites represent the lowest-lying bath states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' One can define the Nth scaled Hamiltonian as �HN = (Λ) 1 2 N−1 ˆHN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' They can be constructed iteratively � HN+1 =Λ 1 2 � HN + Λ 1 2 (N−1) � αs � ϵN+1d† N+1,αsdN+1,αs + tNd† N+1,αsdN,αs + tNd† N,αsdN+1,αs � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (31) The Hilbert space dimension increases exponentially in this iterative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The NRG algorithm truncates the Hilbert space by keeping a fixed number (chosen to be ∼1200 in this work) of the lowest-lying states at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In order to keep the symmetry in the truncated Hilbert space, in practice we keep all the states up to a gap above the 1200th state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Phase diagram and fixed points Two successive transformations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (31)) that take �HN to �HN+2 can be thought as a renormalization group operation [81, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The system is said to achieve a fixed point when �HN and �HN+2 have the same low-lying many- body spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We can obtain a zero temperature phase diagram in the parameter space of µc, µf, G by analyz- ing the fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For the completeness of discussions, here we let µf take value in [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5U1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5U1] such that the corresponding impurity occupation (in the decoupled limit) nf = µf/U1 +1/2 takes value in [0, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(b) and (c) we show the obtained phase diagrams in the pa- rameter space of µc, µf for G = 8meV and G = 14meV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 8meV and 14meV are chosen to be close to the minimal (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='19meV) and maximal (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='21meV) values of G (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(a)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Due to the U(2)×U(2) symmetry, all the many-body levels can be classified into symmetry sectors labeled by the good quantum numbers (Q1, Q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' S1, S2), where Qi and Si are the charge and spin of the ith U(2) sym- metry, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Here we take the convention that Q1 + Q2 = 0 corresponds to a total occupation 2N + 2 (2N) for odd (even) N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' A fixed point is characterized by low energy many-body levels and the associated quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In the whole phase diagram, we find two dis- tinct types of stable fixed points: (i) the strong coupling fixed point exhibiting a Fermi liquid behavior and (ii) the LM fixed points exhibiting nonzero spin momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' At a strong coupling fixed point, as exampled in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(d), (e), for either even or odd N, the ground state is a sin- glet and has (Q1, Q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' S1, S2) = (2k, 2k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 0, 0) for some or- der one integer k, which in most cases equals to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The 10 (f) (g) (d) (h) EN (meV) EN (meV) (c) (a) (e) 200 μf/U μc (meV) 0 1 1 2 3 60 20 0 40 LM1 LM2 LM3 Kondo FI FI δK (meV) G=8meV (b) μf/U 60 20 0 40 LM1 LM2 LM3 FI Kondo FI μc (meV) G=14meV N 11 21 31 41 1 (1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 1/2) (1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 1/2) (0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0, 0) (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 0) (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 0) (2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0, 0) 40 40 ω (meV) N 11 21 31 41 1 (2, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0, 1/2) (2, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0, 1/2) (1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 1/2) (1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 1/2) (2, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0, 0) (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 0) 40 40 ω (meV) 0 100 200 150 50 11 21 31 41 N (0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0, 0) (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 0) (-1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 0) (-2, -1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0, 1/2) (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 0) (-1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 0) (0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0, 0) 1 40 40 ω (meV) E (meV) 0 100 150 50 11 21 31 41 1 (-1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 0) (-1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 0) (0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0, 0) (0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0, 0) (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 0) (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 0) 40 40 ω (meV) N 11 21 31 41 N (0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0, 0) (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 0) (1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 0) (-1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 0) 1 (1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 1/2) (1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1/2, 1/2) 40 40 ω (meV) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Phase diagram and fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (a) The self-consistent mean-field values of µc, µf, G as functions of the total filling ν from ν = 0 to 4, where we have enforced the symmetries of the correlated insulator state at CNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The left y-axis represents µc and µf while the right y-axis represents G with a different range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (b) The phase diagram in the parameter space of µc, µf for G = 8meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The white lines are phase boundaries between the local moment (LM) phases and the strong coupling phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The dashed black lines are crossover boundaries between the frozen impurity (FI) and Kondo regimes of the strong coupling phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The color maps the half-width of the spectral density peak, reflecting the Kondo energy scale if in the Kondo regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The solid black line indicates the trajectory of µc and µf determined from a self-consistent calculation as ν changes from 0 to 4, where the five arrows from left to right represent ν = 0, 1, 2, 3, 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (c) is the same as (b) but a different parameter G = 14meV is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (d)(e) The RG flow of the many-body spectrum of the scaled Hamiltonian � HN (N ∈ odd) in the Kondo regime, where µc = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='7meV, µf/U1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='367, G = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='49meV is the mean field value at ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='25 for (d) and µc = 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='9meV, µf/U1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='286, G = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='83meV is the mean field value at ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 for (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The spectral lines’ colors represent the symmetry sectors labeled by good quantum numbers (Q1, Q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' S1, S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Since the levels in sectors (Q1, Q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' S1, S2) and (Q2, Q1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' S2, S1) are identical, only |Q1| ≥ |Q2| sectors are shown for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The insets are the resulting single-particle spectral densities that exhibit Kondo resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (f)(g)(h) The RG flow of many-body spectrum of the scaled Hamiltonian � HN (N ∈ even) in the LM1,2,3 phase, where µc = 5meV, µf/U1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5, G = 8meV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The insets are the resulting single-particle spectral densities that exhibit Hubbard bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' low-lying many-body spectrum is identical to the one of a free-fermion chain defined by ϵn and tn with an ad- ditional chemical potential term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In other words, the impurity acts as if it was nonexistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The underlying mechanism is either the Kondo screening, where the im- purity is an effective LM screened by the bath, or the impurity freezing, where the impurity occupation νf is effectively empty or full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We refer to the two cases as the Kondo regime and the frozen impurity (FI) regime, re- spectively, which are adiabatically connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The fixed points shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(d), (e) are in the Kondo regime because, if we continuously change µc to 0, they evolve to the LM1 and LM2 states (discussed in the next para- graph), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' There is a crossover between Kondo and FI regimes as one changes µf, as indicated by the dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(b), (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Later we will determine the crossover boundary using the spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' At an LM fixed point, as exampled in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(f), the low- lying many-body spectrum is identical to a free-fermion chain plus a detached LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Depending on the represen- tation of the ground state, the LM fixed points can be further classified into LMn, where n = 1, 2, 3 is the effective impurity occupation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The flows of the spec- tra towards these fixed points are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(f), (g), (h), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' LMn ground states have the same SU(2)×SU(2) representations as ground states of ˆHf + ˆHH (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (12) and (18)) with n impurity elec- trons, where the Hubbard interaction freezes charge exci- tations and the Hund’s coupling prefers states with elec- trons lying in different orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For n = 1, the ground states are four-fold degenerate and belong to the sym- metry sectors (Q1, Q2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' S1, S2) = (2k + 1, 2k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1 2, 0) and (2k, 2k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 0, 1 2), corresponding to the spin- 1 2 states of the two U(2)’s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The SU(2) representations are the same as those of the four single-particle states of ˆHf + ˆHH: (nf 1↑, nf 1↓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' nf 2↑, nf 2↓) = (10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='00), (01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='00), (00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='10), (00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For n = 2, the ground states are also four- fold degenerate but belong to a different symmetry sec- tor (2k + 1, 2k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1 2, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' One can understand them as the product states of two spin- 1 2 states of the two U(2)’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' They have the same SU(2) representations as the four two-particle states of ˆHf + ˆHH: (10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='10), (10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='01), (01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='10), (01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Without Hund’s coupling JH, the LM2 ground states would be �4 2 � = 6-fold degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' A fi- 11 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 40 0 40 (a) kBT=0 40 0 40 0 1 2 3 4 5 (b) kBT=0 (c) kBT=2meV (d) kBT=5meV 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 40 0 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 40 40 5 5 A(𝜔,T) (meV-1) 𝜔 (meV) 𝜔 (meV) 𝜔 (meV) 𝜔 (meV) 𝜈=0 𝜈=1 𝜈=2 𝜈=3 𝜈=4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Spectral densities A(ω, T) at various fillings ν and temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (a) Spectral densities at the zero temperature for ν = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='2 · · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The curves are offset by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1ν (meV−1) for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (b) is the same as (a) but is shown with a smaller vertical scale for clarity of Hubbard bands, marked by in- verted triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The curves are offset by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='01ν (meV−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (c) and (d) are spectral densities at finite temperatures, where the curves are offset by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='01ν (meV−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' nite JH raises the energy of the two many-body states with both electrons occupying the same orbital, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', (11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='00), (00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For n = 3, the ground states are still four-fold degenerate but belong to the symmetry sec- tors (2k − 1, 2k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1 2, 0) and (2k, 2k − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 0, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Their SU(2) representations are same as the four single-hole states ˆHf + ˆHH: (01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='11), (10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='11), (11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='01), (11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' It is also helpful to look at the global U(2) symme- try representations, where the two U(2) rotations are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The total charge and spin of LM1,2,3 are 1, 2, 3 (mod 4) and 1 2, 1 2 ± 1 2, 1 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Phase boundaries between different LM phases and the strong coupling phase are described by unstable fixed points where different ground states cross with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The phase boundaries are shown by the white lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(b), (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Starting from an LMn phase, in- creasing µc will eventually drive it into a strong coupling phase due to the enhancement of hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The crit- ical µc, as expected, is close to G, the conduction band edge (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1(c), (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' SPECTRAL DENSITY We calculate the spectral density of the f-electrons, A(ω, T) = − 1 π � αs Gαs(ω, T), with Gαs(ω, T) being the retarded Green’s function of fαs at the temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We use the method described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [100] to collect the many-body levels at different RG steps to compute A(ω, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The fixed points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(d), (e) are in the Kondo regime and hence have sharp zero-energy peaks due to the Kondo resonance, as shown in the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(d), (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The fixed points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(f), (g), (h) are in the LM1,2,3 phases, respectively, thus, their spec- tral density are dominated by the upper and lower Hub- bard bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We compute the spectral densities for all the points in the phase diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(b), (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We identify a central peak for every calculation and measure its half-width δK at half maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (If there is no cen- tral peak, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(f), δK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=') δK is indicated by the color in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(b), (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We can distinguish the Kondo and FI regimes in the strong coupling phase through spectral density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Intuitively, a state in the Kondo regime should have a Kondo resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' By contrast, a state in the FI regime should have its main spectral weight away from zero energy because the impurity occupation is either empty or full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus, we identify a phase point in the Kondo regime if δK covers the zero energy and otherwise in the FI regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The crossover between the two regimes is indicated by dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(b), (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Several features of δK in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(b), (c) can be un- derstood using the poor man’s scaling developed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' First, there are three domes around µf = 1 2U1, 3 2U1, 5 2U1 where δK is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' They cor- respond to the nf = 1, 2, 3 cases discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' From the poor man’s scaling perspective, these three µf’s correspond to the minimal initial value of the coupling constant g (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (22)), which then leads to smaller TK’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Second, when µc is small (≲ 30meV), δK in the middle dome is significantly smaller than those of the other two domes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The reason is that the Kondo energy scale TK for nf = 2 will be strongly suppressed due to the multi- plet splitting if TK is smaller than JH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' One can see that the N factor in the exponential function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (27) is replaced by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Third, for the same µc, the first dome has lower δK than the third dome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This difference is a result of the suppression factor y1 for nf = 1 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (24)) and the enhancement factor y3 for nf = 3 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (29)) due to the particle-hole asymmetry of ∆(ω), as discussed in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In order to compare our results with STM measure- ments, we need to adopt physical µc, µf, G parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' III C, µc, µf, G can be determined as functions of the filling ν via a symmetric self-consistent calculation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' µc, µf, G as functions of ν are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The obtained spectral densities at the zero temperature are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 3(a), (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For ν = 0, the state is in the FI regime with an (almost) zero occu- pation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' hence the spectral weight is mainly distributed at positive energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As ν increases, the spectral peak moves to the zero energy and is eventually pinned at the zero en- ergy to form a Kondo resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This is precisely what is seen in STM experiments at low temperatures (T < 1K) [11, 19, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' One can also observe the evolution of Hubbard bands at T = 0 as ν changes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 3(b)), but they are relatively weak compared to the Kondo reso- nance peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' At finite temperatures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 3(c), (d)), the Kondo resonance peaks are smeared by thermal fluc- tuations and the evolution of Hubbard bands becomes clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As ν increases from 0 to 4, the Hubbard bands periodically pass through the zero energy, matching the cascade of transitions seen in STM experiments at higher 12 (a) kBT (meV) 4 102 101 100 10-1 10-2 10-3 𝜈 0 1 2 3 101 100 10-1 10-2 102 101 100 10-1 10-2 10-3 kBT (meV) 102 101 100 10-1 10-2 10-3 kBT (meV) 100 10-1 10-2 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0 𝜈 (b) (d) (c) 102 101 100 10-1 10-2 10-3 kBT (meV) 𝜈 0 1 2 3 4 0 1 2 3 (e) 𝜔 (meV) 0 20 40 20 40 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='3 A(ω,T) (meV-1) 𝜈=1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 1 2 5 kBT (meV) kBT =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='82meV (f) 𝜈 0 1 2 3 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='6 Bloc(T) 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='0 𝜈 0 1 2 3 4 Simp ×kBln2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Spin susceptibility and entropy contributed by the im- purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (a) χloc(T)/χloc(0) as a function of filling ν and tem- perature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (b) The local spin susceptibilities χloc(T) at fill- ings ν = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5, 1 · · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (c) The entropy contributed by the im- purity as a function of ν and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (d) The entropy contributed by the impurity Simp(T)/(kB ln 2) at fillings ν = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5, 1 · · · 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (e) The spectral densities at ν = 1 at various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (f) The entropy contributed by the impurity as a function of ν at B = 0, 5, 10, 15, 20 T and temperature kBT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='82 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' temperatures [17, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' One can use δK to estimate the Kondo energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Using the ν-dependent µc, µf, G parameters given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(a), we tabulate the δK’s at different fillings in Ta- ble II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Comparing it to TK estimated by the poor man’s scaling, denoted as T (P ) K , we find T (P ) K is about δK ∼ 2δK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' LOCAL MOMENTS AND THE POMERANCHUK EFFECT At a temperature exceeding the Kondo energy scale, the LM will become effectively decoupled from the bath and visible in experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This is the mechanism of the Pomeranchuk effect [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [30] observed a higher entropy (∼ 1kB per moir´e cell with kB being the Boltzmann’s constant) state at ν ≈ 1 at the temperature T ≈ 10K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As this entropy can be quenched by an in-plane magnetic field, it is ascribed to a free local moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [31] observed a similar effect at ν ≈ −1 and showed that an additional resistivity peak that is absent at T = 0 develops in the higher entropy state at T ≈ 10K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' These observations can be naturally explained by the transition from the Fermi liquid phase to the LM phase as the temperature increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' To demonstrate the LM phase at higher temperatures, we calculate the local spin susceptibilities χloc(T) us- ing the filling-dependent µc, µf, G parameters given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' χloc(T) is defined as dMloc dBloc [101, 102], with Mloc being the spin momenta contributed by the impu- rity and Bloc a local magnetic field that only acts on the impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 4(a), (b), for ν ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5, χloc(T) approaches a constant as T → 0, and obeys the Curie’s law, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', χloc(T) ∼ T −1, when T is larger than the Kondo energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Obeying Curie’s law is a clear indication of a free LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' One may notice that the T-dependences of χloc(T)’s for ν < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 are non-monotonous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The ν < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 states lie in the FI regime, thus the spin susceptibili- ties are extremely small when T → 0, and will start to increase when T is able to excite the LM1 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5, we can define the Kondo temperature TK as the turning point between Curie’s behavior and Fermi liquid behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Specifically, it can be obtained as the crossing of the extended T −1 line from the LM side and the extended horizontal line from the Fermi liquid side (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We tabulate the resulting TK in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As shown in the table, such defined TK is about 1 3δK ∼ δK with δK being the half-width of the spectral density dis- cussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We also calculate the impurity entropy Simp(T) for comparison with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Simp(T) is defined as the difference of the entropy of �HN and that of a reference free-fermion chain (without the impurity site) defined by the same ϵn, tn parameters as in �HN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 4(c) and (d), Simp(T) is zero in the Fermi liquid phase at sufficiently low T and starts to increase when T reaches the Kondo energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For ν = 1, Simp(T) climbs to about 2 ln 2 · kB at about kBT ≈ 1meV and (approximately) stays at this value until kBT reaches 10meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The entropy 2 ln 2 · kB ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='39kB is close to the measured value (∼ 1kB) in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [30, 31] and can be un- derstood as contributed by the four degenerate states in the LM1 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Higher excited states will be involved when kBT is larger than 10meV, and the entropy contin- ues to increase for larger kBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We also show the spectral density at ν = 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 4(e), one can see that the Kondo resonance peak becomes weak for kBT > 1meV, which is consistent with the entropy and spin susceptibility re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' An in-plane magnetic field will polarize the spin and hence suppress the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' However, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' IV C, the four-fold degenerate LM1 states consist of two spin- 1 2 states due to the orbital degeneracy, hence a strong field will not completely quench the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Instead, due to the orbital degeneracy, the remaining en- tropy will be ln 2 · kB ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='69kB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This is also consistent with observations in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 4(f), we plot the 13 (a) 𝜈=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='2 (b) 𝜈=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='8 (c) 𝜈=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='4 Band Energy (meV) 0 20 40 60 20 40 60 ΓM KM MM ΓM KM MM ΓM KM MM 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='8 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Heavy Fermi liquid bands at ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='2 (a), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='8 (b), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='4 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As explained in the text, the effective hybridization be- tween c- and f-bands is suppressed by a factor z 1 2 with z being the quasi-particle weight of f-electrons, which are estimated as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='038, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='27, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='16 for (a), (b), and (c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The color of the bands represents the total quasi-particle weight, which is always larger than z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' impurity entropy as a function of the filling at various magnetic fields Bloc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' At ν = 1, the entropy saturates to a constant ∼ ln 2 · kB under a strong field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' One can see that the entropy values, the shapes of the curves, and their field-dependences in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 4(f) are comparable to the experimental results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(e) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' DISCUSSIONS Based on the NRG calculations, the poor man’s scal- ing, and various experimental observations, we have shown that the gapless states at 1 ≲ |ν| < 2 are in the Kondo regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Considering the translation invariance of the actual system, these states must be the heavy Fermi liquid states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Here we estimate the heavy Fermi liquid bands from the information provided by the NRG cal- culation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' At the zero temperature, a spectral density in the Kondo regime possesses a Lorentz peak around the zero energy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', A(ω) ≈ 4z π δK ω2+δ2 K , where z is the quasi- particle weight, δK is the half-width given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(b), (c), and the factor 4 is from orbital and spin degenera- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then the quasi-particle weight can be estimated as z = πA(0)δK/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Substituting the quasi-particle part of Gαs(iω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', z iω, into Dyson’s equation of c-electrons G(c)(iω, k) = G(c,0)(iω, k) + G(c,0)(iω, k)H(cf)(k) z iω H(cf)†(k)G(c)(iω, k) , (32) one can see that the cf hybridization is effectively sup- pressed by a factor of z 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Here ω is the Matsubara fre- quency, G(c,0)(iω, k) = (iω − H(c)(k))−1 is the free prop- agator of c-electrons, and H(c)(k), H(cf)(k) are Hamilto- nian matrices in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' If z = 0 the effective hybridiza- tion will be zero, corresponding to the LM phase where the Fermi surface is solely contributed by the c-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Using this method we obtain the bands at ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='8, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 5(a), (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' One should be aware that the Hubbard band information is com- pletely neglected in this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For future reference, we also estimate the heavy Fermi liquid bands at ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='4 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 5(c)) by assuming the ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='4 state in the Kondo regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The heavy Fermi liquid states at 1 ≲ |ν| < 2 can be further confirmed by future experimental research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For example, the Fermi surface can dramatically change as one tunes the filling and temperature or applies an ex- ternal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The Fermi surface change will be reflected in spectral measurements such as quasi-particle interfer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' It is also in principle possible to directly measure the scattering phase shift [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' c-bands will induce RKKY interactions between LMs at different f-sites, which can lead to further symme- try breaking and should be crucial to stabilize the ob- served correlated insulator states at |ν| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We have ignored these RKKY interactions and further symmetry breaking in the current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' A full self-consistent treat- ment including both RKKY and Kondo screening effects may result in more complicated ν-dependencies of µc, µf than those shown Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(a), (b), (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For example, at ν = 2, µc given by a self-consistent symmetry breaking Hartree-Fock mean field [61] is about 26meV, which is significantly lower than the one (∼40meV) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus, a possible mechanism for the correlated insulators to win the Kondo screening is that µc drops to a small value such that the Kondo energy scale becomes irrele- vant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(b), (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=') Observations in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [18, 26] also suggest that the ν-dependencies of µc, µf are com- plicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The competition between RKKY and Kondo screening is also a potential mechanism for the observed strange metal behaviors [27–29] and could play an impor- tant role in the unconventional superconductivity [1, 4– 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We leave this for future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Note added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' During the preparation of the current work, a related work [104] appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This work studied the symmetric Kondo state using a slave-fermion mean field in a Kondo lattice model derived from the THF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Our theory is based on the symmetry-broken cor- related state at CNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We are also aware of related works on the Kondo problem in MATBG by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Tsvelik’s and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Bernevig’s group [105, 106] and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Coleman’s group [107] that will appear soon, and a generalization of the THF model to the magic-angle twisted trilayer graphene [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [105] also obtains a Kondo temperature about 1 ∼ 2K around |ν| ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' ACKNOWLEDGMENTS We are grateful to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Andrei Bernevig, Ning-Hua Tong, Xi Dai, Jia-Bin Yu, Xiao-Bo Lu, Yong-Long Xie, Yi-Lin Wang, and Chang-Ming Yue for helpful discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' were supported by National Natural Science Foundation of China (General Program No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 12274005), National Key Research and Development Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2021YFA1401900).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='9 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='7 20 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='2 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1 60 0 K G M60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='9 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='7 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='4 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='2 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1 60 0 K G M14 Appendix A: More details about the effective Hamiltonian 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Nonzero M term A generic trial ground state at CNP is given by (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (6)) |Ψ0⟩ = U � R f † R1+↑f † R1+↓f † R2+↑f † R2+↓|FS⟩ , (A1) where U = exp(−iθµν ˆΣµν) is a U(4) rotation operator and an implicit summation over repeated µ, ν indices is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We can always define the rotated fermion op- erators �ckaηs = UckaηsU †, �fRaηs = UfRaηsU † such that �fRα+s’s are occupied in |Ψ0⟩ and �fRα−s’s are empty in |Ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' According to the discussions in the supplementary material section S4B of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [61], in the flat-band limit (M = 0), the lowest particle (hole) excitations only in- volve �cka−s and �fRa−s (�cka+s and �fRa+s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus, the effective periodic Anderson model for ν > 0 derived in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' III B is written in terms of ckas = �cka−s and fRαs = �fRα−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Here we give the explicit forms of the rotated operators �fRαηs = � α′η′s′ � eiθµνΣf µν � αηs,α′η′s′ fRα′η′s′ , (A2) and �ckaηs = � a′=1,2 η′s′ � eiθµνΣc12 µν � aηs,a′η′s′ cka′η′s′ (a = 1, 2), (A3) �ckaηs = � a′=3,4 η′s′ � eiθµνΣc34 µν � aηs,a′η′s′ cka′η′s′ (a = 3, 4), (A4) where the eight-by-eight matrices Σf µν, Σc12 µν , Σc34 µν are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (3) to (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The M-term in the original basis of the THF model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (1)) is M � aa′=3,4 � |k|<Λc � ηs [σx]aa′c† kaηscka′ηs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A5) It favors the Kramers inter-valley coherent state dis- cussed at the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' II, where θx0 and θy0 are nonzero and satisfy θ2 x0 + θ2 y0 = (π/4)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Without loss of general- ity, we assume U = exp(−i π 4 ˆΣx0) for the Kramers inter- valley coherent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Writing this M-term in terms of the rotated operators, we obtain M � |k|<Λc � a,a′=3,4 � ηη′ss′ �c† kaηsOaηs,a′η′s′�cka′η′s′ , (A6) where O = ei π 4 Σc34 x0 σxτ0ς0e−i π 4 Σc34 x0 = −σzτxς0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The τx matrix in O represents couplings between the empty and occupied single-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' If we simply project this M-term onto the empty states, it vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', [Oa−s,a′−s′] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' A better approximation is applying a Schrieffer-Wolff transformation to decouple the η = ± states, leading to a second-order correction to the effec- tive Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As ⟨Ψ0| �f † αηs �fαηs|Ψ0⟩ = (1 + η)/2, the J term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (2) yields the following mean field term (see also the supplementary material section S4B of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [61]) − J 2 � a=3,4 � ηs η · �c† aηs�caηs (A7) Then, regarding the J 2 term as the zeroth order Hamilto- nian and M as a perturbation, a Schrieffer-Wolff trans- formation leads to the correction − M 2 J � |k|<Λc � a=3,4 � ηs η · �c† kaηs�ckaηs + O(M 4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A8) The resulting energy levels ±(J/2 + M/J2) at k = 0 is fully consistent with a Taylor expansion of the one-shot energy levels ± � J2/4 + M 2 derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Pro- jecting the correction to the active d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', ckas = �cka−s, we obtain the correction to the effective Hamilto- nian M 2 J � |k|<Λc � a=3,4 � s c† kasckas + O(M 4) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A9) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Hund’s coupling The four-by-four Hamiltonian matrix H(c)(k)+∆H(c) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (13), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', −v⋆(σx⊗σ0kx+σy ⊗σzky)+02×2⊕Gσ0, can be diagonalized analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As discussed at the end of the last subsection, to O(M 2), the M term simply shifts the energy of a = 3, 4 electrons by M 2/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus, all the analysis below applies to the M ̸= 0 after G is replaced by G + M 2/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We find the energy eigenvalues and wave-functions of the H(c)(k) + ∆H(c) as ϵ1(k) =ϵ+(k) = G 2 + � G2 4 + v2⋆k2 u1(k) = � sin θk 2 e−iφk 0 − cos θk 2 0 �T , (A10) ϵ2(k) =ϵ+(k) = G 2 + � G2 4 + v2⋆k2 u2(k) = � 0 sin θk 2 eiφk 0 − cos θk 2 �T , (A11) ϵ3(k) =ϵ−(k) = G 2 − � G2 4 + v2⋆k2 u3(k) = � cos θk 2 e−iφk 0 sin θk 2 0 �T , (A12) ϵ4(k) =ϵ−(k) = G 2 − � G2 4 + v2⋆k2 u4(k) = � 0 cos θk 2 eiφk 0 sin θk 2 �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A13) 15 where θk = arccos G/2 � G2/4 + v2⋆k2 (A14) and φk = arg(kx + iky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We now derive the effective Hund’s coupling ˆHH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Ap- plying a second-order perturbation in terms of ˆHJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' we obtained the correction to the Hamiltonian ∆ ˆH = − J2 N 2 M � I � α1α2 s1s′ 1s2s′ 2 � k1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='k′ 1 k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='k′ 2 (f † α1s1fα1s′ 1 − νf 4 δs1s′ 1) × (f † α2s′ 2fα2s2 − νf 4 δs2s′ 2) · e− λ2 2 (k2 1+k′2 1 +k2 2+k′2 2 ) × ⟨Ψ0|c† k′ 1α1+2s′ 1ck1α1+2s1|ΨI⟩⟨ΨI|c† k2α2+2s2ck′ 2α2+2s′ 2|Ψ0⟩ EI − E0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A15) where |ΨI⟩ are excited states with a single particle-hole pair and EI are the energies of the excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' k1,2, k′ 1,2 are limited within the cutoff Λc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Due to the mo- mentum and spin conservation, for the matrix element to be nonzero, there must be k1 = k2, s1 = s2, k′ 1 = k′ 2, s′ 1 = s′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For simplicity, we rewrite k1, k′ 1, s1, and s′ 1 as k, k′, s, and s′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (k, s) and (k′, s′) label the particle and the hole excitations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then the matrix element in the third line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A15) can be written as nF (ϵ+(k′) − µc)(1 − nF (ϵ−(k) − µc)) × ⟨Ψ0|c† k′α1+2s′ckα1+2sc† kα2+2sck′α2+2s′|Ψ0⟩ (A16) According to the wave functions given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A10)-(A13), there are ⟨Ψ0|c† k′α1+2s′ck′α2+2s′|Ψ0⟩ = δα1α2 sin2 θk′ 2 , ⟨Ψ0|ckα1+2sc† kα2+2s|Ψ0⟩ = δα1α2 cos2 θk 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The excitation energy EI −E0 is given by ϵ+(k)−ϵ−(k′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus, ∆ ˆH is simplified to ∆ ˆH = − J2 N 2 M � αss′ kk′ (f † αsfαs′ − νf 4 δss′)(f † αs′fαs − νf 4 δss′) × nF (ϵ−(k′) − µc)(1 − nF (ϵ+(k) − µc)) sin2 θk′ 2 cos2 θk 2 ϵ+(k) − ϵ−(k′) × e−λ2(k2+k′2) (A17) The s = s′ contribution is an effective chemical potential shift, estimated as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='17meV at CNP, of the f-electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As it is much smaller than U1, we will omit the s = s′ contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The s ̸= s′ contribution can be written as ˆHH = JH � α nf α↑nf α↓ (A18) with JH given by JH =2J2 �Ω0 2π �2 ˆ Λc 0 dk′ · k′ ˆ Λc k0 dk · k · e−λ2(k2+k′2) × sin2 θk′ 2 cos2 θk 2 ϵ+(k) − ϵ−(k′) , (A19) where k0 is determined by ϵ+(k0) = µc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Here we have made use of the fact that ϵ±(k) and θk only depends on |k| but not φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' At CNP, µc = 0 and G = J/2 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='19meV, taking the limit Λc → ∞, we obtain JH ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='34meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Using the self-consistent values of µc and G shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2(a), we find JH at ν = 1, 2, 3, 4 are given by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='29meV, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='26meV, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='21meV, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='19meV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As JH is small and does not change significantly with ν, in this work, we simply set JH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='34meV for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Hybridization function By definition, the hybridization function ∆(ω) is given by ∆(ω) = π N � k � n |Vnα(k)|2δ(ω − ϵn(k)) (A20) where Vnα(k) = � a u∗ an(k)H(cf) aα (k)e− λ2k2 2 is the hy- bridization between fαs and the n-th energy band of c- electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' ∆(ω) does not depend on α because of the C2zT or C2x symmetry that flips the α index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Substitut- ing ϵn(k) and uan(k) in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A10)-(A13) into the above equation, we obtain Vnα(k) for α = 1 as V11(k) =γ sin θk 2 eiφk V21(k) =v′ ⋆(−kx + iky) sin θk 2 e−iφk V31(k) =γ cos θk 2 eiφk V41(k) =v′ ⋆(−kx + iky) cos θk 2 e−iφk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A21) Using the energy eigenvalues in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A10)-(A13) and the Vnα(k) matrix elements given above, it is direct to obtain ∆(ω) = Ω0 2v2⋆ ����ω + µc − G 2 ���� � γ2 + v′2 ⋆ k2 F � e−k2 F λ2 � θ(ω + µc − G) sin2 θkF 2 + θ(−ω − µc) cos2 θkF 2 � (A22) where kF is given by kF = 1 v⋆ � (ω + µc − G/2)2 − (G/2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A23) We now verify the asymptotic behaviors of ∆(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' When ω + µc → G+, kF → 0 and only the first term in the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A22) contributes to ∆(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Ac- cording to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A14), there is cos θkF = G/2 ω+µc−G/2 and hence sin2 θkF 2 = 1 2− 1 2 cos θkF ≈ (ω+µc−G)/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then we obtain the asymptotic behavior of ∆(ω) as ω + µc → G+ ∆(ω) = Ω0 4v2⋆ γ2 · (ω + µc − G) + O((ω + µc − G)2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A24) When ω + µc → −0+, kF → 0 and only the second term in the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A22) contributes to ∆(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 16 According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A14), there is cos θkF = G/2 G/2−ω−µc and hence cos2 θkF 2 = 1 2 + 1 2 cos θkF ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then we obtain the asymptotic behavior of ∆(ω) as ω + µc → −0+ ∆(ω) = Ω0 4v2⋆ Gγ2 + O((ω + µc)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (A25) Appendix B: Poor man’s scaling of Anderson models with energy-dependent couplings 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Generic theory for U(N) models We consider the Anderson impurity model with N symmetric flavors ˆH = − µf ˆ Nf + U 2 ˆ Nf( ˆ Nf − 1) + N � µ=1 ˆ D −D dϵ · ϵ · d† µ(ϵ)dµ(ϵ) + N � µ=1 ˆ D −D dϵ � ∆(ϵ) π (f † µdµ(ϵ) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=') , (B1) where µ is the flavor index and Nf = �N µ=1 f † µfµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We assume the ground state of the isolated impurity has nf f-electrons, which can take the values 1, 2 · · · (N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (We do not consider the empty case (nf = 0), the full case (nf = N), and the mixed valence case where ground states with different nf are exactly degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=') We fur- ther assume the charge gaps to nf − 1 and nf + 1 elec- trons are ∆E− and ∆E+ = U − ∆E−, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We then apply a Schrieffer-Wolff transformation to obtain an effective Coqblin–Schrieffer model for the Hilbert space restricted to ˆNf = nf ˆH = N � µ=1 ˆ D −D dϵ · ϵ · d† µ(ϵ)dµ(ϵ) + 4g πU N � µ,µ′=1 ˆ D −D dϵdϵ′ � � ∆(ϵ)∆(ϵ′)(f † µfµ′ − xδµµ′)d† µ′(ϵ′)dµ(ϵ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B2) Terms that only involve ˆNf are omitted because they only contribute to an energy shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The bandwidth D should be smaller than min(∆E+, ∆E−), otherwise, the Schrieffer-Wolff transformation is invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The parame- ters g and x are given by g = U 4 � 1 ∆E+ + 1 ∆E− � , x = ∆E− U , (B3) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' If µf = (nf − 1 2)U, there is ∆E+ = ∆E− = 1 2U and g = 1, x = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We now truncate the bandwidth at D−dD = D(1−dt) (dt ≪ 1) and consider second order (in g) corrections form the virtual particle (D − dD < ϵ < D) and hole (−D < ϵ < −D + dD) excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The particle excita- tion contributes to the correction − (4g)2 (πU)2 1 D � µ1µ2µ′ 1µ′ 2 ˆ D−dD −D+dD dϵ1dϵ2d ˆ D D−dD dϵ′ 1dϵ′ 2 × � ∆(ϵ1)∆(ϵ2)∆(ϵ′ 1)∆(ϵ′ 2)d† µ1(ϵ1)⟨dµ′ 1(ϵ′ 1)d† µ′ 2(ϵ′ 2)⟩dµ2(ϵ2) × (f † µ′ 1fµ1 − xδµ1µ′ 1)P(f † µ2fµ′ 2 − xδµ2µ′ 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B4) The denominator D in the factor is the excitation en- ergy of a virtual particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' P is a projector to the re- stricted Hilbert space, where ˆNf = nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The expecta- tion ⟨dµ′ 1(ϵ′ 1)d† µ′ 2(ϵ′ 2)⟩ evaluated on the ground state is δ(ϵ′ 1 − ϵ′ 2)δµ′ 1µ′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then we have − (4g)2 (πU)2 dD D ∆(D) � µ1µ2µ′ ˆ D−dD −D+dD dϵ1dϵ2 � ∆(ϵ1)∆(ϵ2) × d† µ1(ϵ1)dµ2(ϵ2)(f † µ′fµ1 − xδµ1µ′)(f † µ2fµ′ − xδµ2µ′) , (B5) where P is omitted as it commutes with f † µ2fµ′ and f † µ′fµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' After a few steps of algebra, the four-fermion operator � µ′ f † µ′fµ1f † µ2fµ′ can be simplified to f † µ2fµ1 + � µ′ f † µ′fµ′fµ1f † µ2 = f † µ2fµ1(1−nf)+nfδµ1µ2 , (B6) where we have made use of the fact that the Hilbert space is restricted to ˆNf = nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Substituting this into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B5), we obtain the corrections to parameters g and xg as dg dt ���� p = 4∆(D(t)) πU ((nf − 1) + 2x) g2 , (B7) d(xg) dt ���� p = 4∆(D(t)) πU � x2 + nf � g2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B8) Here t is the RG parameter and D(t) = De−t is the reduced bandwidth after successive t/dt RG steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We then calculate the contribution from virtual hole excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Following the same process as in the last paragraph, we obtain − (4g)2 (πU)2 dD D ∆(−D) � µ1µ2µ′ ˆ D−dD −D+dD dϵ1dϵ2 � ∆(ϵ1)∆(ϵ2) × dµ1(ϵ1)d† µ2(ϵ2)(f † µ1fµ′ − xδµ1µ′)P(f † µ′fµ2 − xδµ2µ′) = (4g)2 (πU)2 dt∆(−D) � µ1µ2µ′ ˆ D−dD −D+dD dϵ1dϵ2 � ∆(ϵ1)∆(ϵ2) × d† µ2(ϵ2)dµ1(ϵ1)(f † µ1fµ′ − xδµ1µ′)(f † µ′fµ2 − xδµ2µ′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B9) In the second equation, we have omitted an energy con- stant term from the anti-commutator {d† µ2(ϵ2), dµ1(ϵ1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' P is the projector to the restricted Hilbert space, where ˆNf = nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' It is omitted in the second equation be- cause it commutes with f † µ′fµ2 and f † µ1fµ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The four- fermion operator � µ′ f † µ1fµ′f † µ′fµ2 can be simplified to (N − nf + 1)f † µ1fµ2 as the Hilbert space is restricted to ˆNf = nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then the corrections to g, xg from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B9) can be read out as dg dt ���� h = 4∆(−D(t)) πU (N − nf + 1 − 2x) g2 , (B10) d(xg) dt ���� h = 4∆(−D(t)) πU � −x2� g2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B11) 17 Adding up the particle and the hole contributions we can obtain the RG equations for g and (xg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The Kondo energy scale TK can be estimated as the reduced band- width D(t) where g diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For a constant ∆(ω) = ∆0, we obtain dg dt = 4∆0 πU Ng2, d(xg) dt = 4∆0 πU nfg2 (B12) and the solution g(t) = g(0) 1 − g(0) 4∆0 πU N · t , (B13) x(t) = x(0)g(0) g(t) + nf N · g(t) − g(0) g(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B14) where g(0) is the initial condition given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' g(t) diverges at tK = πU 4N g(0)∆0 , corresponding the Kondo en- ergy scale De−tK = De− πU 4N g(0)∆0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' As g(t) diverges as t → tK, the second term in x(t) dominates and there must be x → nf N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In other words, x flows to the occupa- tion fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Application to the symmetric state at CNP We assume a symmetric state of the THF model at CNP and examine its Kondo energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Following the calculations in Appendix A 3, we obtain the hybridiza- tion function contributed by the fully symmetric c-bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1(b)) ∆(ω) = Ω0 4v2⋆ � ����|ω| − M 2 ���� θ(|ω| − M) � γ2 + v′2 ⋆ k2 F 1 � sin2 θkF 1 2 × e−k2 F 1λ2 + ����|ω| + M 2 ���� � γ2 + v′2 ⋆ k2 F 2 � cos2 θkF 2 2 e−k2 F 2λ2� , (B15) where kF 1 = 1 v⋆ � (|ω| − M/2)2 − (M/2)2, (B16) kF 2 = 1 v⋆ � (|ω| + M/2)2 − (M/2)2, (B17) θk = arccos M/2 � M 2/4 + v2⋆k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B18) We should choose the initial cutoff D = 1 2U1 be- yond which the Schrieffer-Wolff transformation is invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For these states kF 1,2 ≲ U1 2v⋆ and hence v′2 ⋆ k2 F 1,2 ≲ 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='4meV2, which is significantly smaller than γ2 ≈ 612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='6meV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The damping factors e−λ2kF 1,22 ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='74 are also large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus, in the following we approximate ∆(ω) (|ω| < U1/2) as ∆(ω) ≈ Ω0 4v2⋆ � ����|ω| − M 2 ���� θ(|ω| − M)γ2 sin2 θkF 1 2 + ����|ω| + M 2 ���� γ2 cos2 θkF 2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B19) We first consider the flat-band limit (M = 0), where the parameter θk is always π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus, we have ∆(ω) = b|ω|, b = Ω0 4v2⋆ γ2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='1290 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B20) We also assume that there is no multiplet splitting in the symmetric state such that the effective Anderson model should be a U(8) theory with nf = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Naively applying the RG equations derived in the last subsection gives d�g dt = −�g + 4bD πU1 N �g2, (B21) where N = 8, D = U1/2, �g = ge−t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Due to the particle- hole symmetry at CNP, the initial condition (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B3)) is �g(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' It seems that there would be an unstable fixed point �g∗ = 2π 4N b, the initial �g below (above) which flows to zero (infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Using the actual parameters we find g∗ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='52, hence the system would not be in the Kondo phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This result differs from the standard case with a constant ∆(ω), where a positive g always flows to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Furthermore, a more careful RG analysis [98] shows that the fixed point �g∗ does not really exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' It is a false result of the weak coupling expansion, which fails for ∆(ω) ∼ |ω|r with r > 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus, a ∆(ω) ∼ |ω| bath does not have a strong coupling phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' This conclusion is also consistent with numerical studies [95–97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We then consider the case with M ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We use the value M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='697meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The RG process can be divided into two stages: (i) When D(t) = 1 2U1e−t > M, there is approximately ∆(D(t)) ≈ bD(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (ii) When D(t) < M, the first line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B19) vanishes, and cos2 θkF 2 2 in the second line equals to M 4ω+2M + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then there is ∆(D(t)) ≈ ∆0(1 + D(t)/M), with ∆0 = Ω0Mγ2 8v2⋆ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='239meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The boundary between the two stages is t1 = ln U1 2M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The RG equation in the first stage reads dg dt = 2Nb π g2e−t ⇒ g(t) = 1 1 − 2N b π (1 − e−t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B22) Due to the particle-hole symmetry, the initial condition given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B3) is g(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We have g1 = g(t1) ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The RG equation in the second stage is given by dg dt =4N∆0 πU1 g2 + 4N∆0 πU1 g2 · e−(t−t1) ⇒ g(t) = 1 g−1 1 − 4N ∆0 πU1 (t − t1) − 4N ∆0 πU1 (1 − e−(t−t1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B23) 18 g(t) diverges at tK − t1 ≈ πU1 4g1N ∆0 − y with y = 1, corre- sponding to the Kondo energy scale kBTK = Mey · e− πU1 4g1N ∆0 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='8 × 10−4meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B24) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Application to the effective model for ν > 0 states In the absence of the Hund’s coupling JH, we can re- gard (α, s) as a composite index so that ˆHSI (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (20)) is a U(N) theory with N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then the flow equations in Appendix B 1 apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For simplicity, we omit the neg- ative branch of ∆(ω) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (19)) at ω < −µc because it is far away from the Fermi level for ν > 0 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The positive branch of ∆(ω) can be well approximated by ∆(ω) = ∆(0)(1+ω/(µc−G)) for |ω| < µc−G (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 1(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We choose the initial cutoff D to be the minimum value of µc − G and ∆E±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Substituting this ∆(ω) into the general RG equations in Appendix B 1, we obtain dg dt = 4∆(0) πU1 Ng2 + 4∆(0)D πU1(µc − G)(4x + 2nf − 2 − N)g2e−t (B25) and d(xg) dt = 4∆(0) πU1 nfg2 + 4∆(0)D πU1(µc − G)(2x2 + nf)g2e−t (B26) The O(e−t) terms will eventually become irrelevant when t is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' After the O(e−t) terms become irrelevant, we have d(xg) dg = nf/N, implying x → nf N at the divergence of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We then approximate the flow equation of g by setting x to its fixed point value nf N , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', dg dt ≈ 4∆(0) πU1 Ng2 + 4∆(0)D πU1(µc − G)(3nf − 6)g2e−t (B27) The solution of g is g(t) ≈ 1 g−1(0) − 4∆(0) πU1 N � t + ynf (1 − e−t) � , (B28) where ynf = D µc−G(3nf − 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The Kondo energy scale is determined t = tK at which g diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Assuming tK ≫ 1, we have tK ≈ πU1 4Ng(0)∆(0) − ynf (B29) and hence kBTK ≈ D · eynf · e− πU1 4N g(0)∆(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B30) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The nf = 1, 3 cases In the presence of the Hund’s coupling, we have to examine the derivations in Appendix B 1 carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The most important effect of ˆHH is to change the local Hilbert space at small energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In general, JH leads to a multiplet splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' When the RG energy scale is smaller than the splitting, the higher energy multiplet will be- come inaccessible, and the local Hilbert space is effec- tively reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' A minor effect is that the charge gaps ∆E± will depend on JH and the resulted coupling be- tween f-spin and d-spin in the Coqblin–Schrieffer model will break the U(N) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In the following, we study how ˆHH changes the RG equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We first consider the nf = 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In the vir- tual particle excitation process (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B5)), the interme- diate f-multiplet is given by |F ′⟩ = (f † µ2fµ′ − δµ2µ′)|F⟩, where F is the initial f-multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (µ should be regarded as the composite index (α, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=') As |F ′⟩ has the same par- ticle number as |F⟩, it must be one of the four states with (n1↑, n1↓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' n2↑, n2↓) = (10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='00), (01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='00), (00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='10), (00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' All of the possible intermediate states do not feel the Hund’s coupling (JH � α nα↑nα↓) and hence they have the same energy as |F⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Hence, the excitation energy of the intermediate state is purely contributed by d- electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Then all the following derivations apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The same argument applies to the virtual hole excitation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Therefore, the RG equations for nf = 1 will not be affected by JH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' For the same reason, RG equa- tions for nf = 3 will also not be affected by JH, where the initial and intermediate states are single-hole states that do not feel JH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The TK for nf = 1, 3 is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The nf = 2 case The Hilbert space with two particles has six states: (n1↑, n1↓;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' n2↑, n2↓) = (10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='10), (10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='01), (01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='10), (01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='01), (11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='00), (00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The former four states have the energy −2µf + U1, and the latter two states have the energy −2µf + U1 + JH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus JH leads to a multiplet splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We divide the RG into two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' In the first stage D(t) is larger than JH, then the splitting JH only plays a minor role and can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus the RG equations in the first stage are given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The first stage ends at t1 = ln(D/JH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' If g diverges before t reaches t1, the Kondo energy scale should be given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B30) with y2 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', kBT ′ K = D · e− πU1 4N g(0)∆(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B31) If g is still finite at t1 g1 = g(0) 1 − g(0) 4∆(0) πU1 N ln D JH , (B32) then the RG goes into the second stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The effective cutoff and the initial g of the second stage are JH and g1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' We first examine the virtual particle excitation process (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B5)), where the interme- diate f-multiplet is given by |F ′⟩ = (f † µ2fµ′ − δµ2µ′)|F⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' 19 Here F is the initial f-multiplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' µ′, µ2 should be re- garded as the composite indices (α′, s′), (α2, s2), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Suppose |F⟩ is one of the four low energy states, where each orbital (α = 1, 2) has one electron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' then, for |F ′⟩ to be a low energy state, the index µ′ must have the same orbital index with µ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', α′ = α2, such that each orbital (α = 1, 2) in |F ′⟩ still has one electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' With this restriction, the four-fermion operator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B6) becomes f † α2s2fα1s1 + � s′ f † α2s′fα2s′fα1s1f † α2s2 (B33) � s′ f † α2s′fα2s′ acting on the bra state (final state) gives nf α2, which must equal to 1 given that the bra state is one of the four low energy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus the four-fermion op- erator equals to δα2α1δs2s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The resulting contributions to the RG equation are dg dt ���� p = 4∆(D(t)) πU (2x) g2 , (B34) d(xg) dt ���� p = 4∆(D(t)) πU � x2 + 1 � g2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B35) We second examine the virtual hole excitation process (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B9)), where the intermediate f-multiplet is given by |F ′⟩ = (f † µ′fµ2 − δµ′µ2)|F⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Suppose |F⟩ is one of the four low energy states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' then, for |F ′⟩ to be in the low energy state, the index µ′ must have the same orbital index with µ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=', α′ = α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' With this restriction, the four-fermion operator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B9) can be written as � s′ f † α1s1fα2s′f † α2s′fα2s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B36) If |F⟩ is one of the four low energy states, it at most occupies one electron in the α2 orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The α2 orbital of fα2s2|F⟩ must be empty, implying � s′ fα2s′f † α2s′ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Thus the four-fermion operator equals 2f † α1s1fα2s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' The resulting contributions to the RG equation are dg dt ���� h = 4∆(D(t)) πU (2 − 2x) g2 , (B37) d(xg) dt ���� h = 4∆(D(t)) πU � −x2� g2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B38) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B34), (B35), (B37) and (B38) are identical to equa- tions of the U(2) case where N = 2, nf = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' Following the steps of deriving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B30), we find x still flows to 1 2, and kBT ′′ K ≈ JH · e− πU1 8g1∆(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B39) The final expression for the Kondo energy scale at nf = 2 is kBTK = � kBT ′ K, kBTK > JH kBT ′′ K, otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE3T4oBgHgl3EQfrgrp/content/2301.04661v1.pdf'} +page_content=' (B40) 20 [1] Yuan Cao, Valla Fatemi, Shiang Fang, Kenji Watan- abe, Takashi Taniguchi, Efthimios Kaxiras, and Pablo Jarillo-Herrero, “Unconventional superconductivity in magic-angle 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Malyali1★, Z. Liu1, A. Rau1, I. Grotova1, A. Merloni1, A. J. Goodwin2, G. E. Anderson2, +J. C. A. Miller-Jones2, A. Kawka2, R. Arcodia1, J. Buchner1, K. Nandra1, D. Homan3, M. Krumpe3 +1Max-Planck-Institut für extraterrestrische Physik, Giessenbachstrasse 1, 85748 Garching, Germany +2International Centre for Radio Astronomy Research, Curtin University, GPO Box U1987, Perth, WA 6845, Australia +3Leibniz-Institut für Astrophysik Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +The ROSAT-selected tidal disruption event (TDE) candidate RX J133157.6-324319.7 (J1331), was detected in 1993 as a +bright (0.2–2 keV flux of (1.0 ± 0.1) × 10−12 erg s−1 cm−2), ultra-soft (𝑘𝑇 = 0.11 ± 0.03 keV) X-ray flare from a quiescent +galaxy (𝑧 = 0.05189). During its fifth All-Sky survey (eRASS5) in 2022, SRG/eROSITA detected the repeated flaring of +J1331, where it had rebrightened to an observed 0.2–2 keV flux of (6.0 ± 0.7) × 10−13 erg s−1 cm−2, with spectral properties +(𝑘𝑇 = 0.115 ± 0.007 keV) consistent with the ROSAT-observed flare ∼30 years earlier. In this work, we report on X-ray, UV, +optical, and radio observations of this system. During a pointed XMM observation ∼17 days after the eRASS5 detection, J1331 +was not detected in the 0.2–2 keV band, constraining the 0.2–2 keV flux to have decayed by a factor of ≳40 over this period. +Given the extremely low probability (∼ 5 × 10−6) of observing two independent full TDEs from the same galaxy over a 30 year +period, we consider the variability seen in J1331 to be likely caused by two partial TDEs involving a star on an elliptical orbit +around a black hole. J1331-like flares show faster rise and decay timescales (O(days)) compared to standard TDE candidates, +with neglible ongoing accretion at late times post-disruption between outbursts. +Key words: accretion, accretion discs – galaxies: nuclei – black hole physics – transients: tidal disruption events +1 INTRODUCTION +Benefitting from the latest generation of time-domain surveys, the +past decade has seen a vast growth in the diversity of observed tran- +sients originating from galactic nuclei. These events can be crudely +divided into, and described as, either ‘one-off’ or ‘repeating’ events, +depending on the observed evolution of their lightcurves. +‘One-off’ events, characterised by a single epoch of major tran- +sient behaviour over an observed monitoring campaign, comprise the +majority of newly reported nuclear transients. These include systems +where the variability is likely linked to changes in the accretion pro- +cess onto a supermassive black hole, such as has been reported in +previously known AGN (e.g. changing-state AGN; Frederick et al. +2019; Trakhtenbrot et al. 2019b; Ricci et al. 2020, 2021; Frederick +et al. 2021; short-rise, slowed-decay Bowen accretion flares, Trakht- +enbrot et al. 2019a), or due to stellar tidal disruption events (TDEs) in +quiescent galaxies1 (see Saxton et al. 2020; van Velzen et al. 2020, +2021a; Alexander et al. 2020 for recent reviews of X-ray, optical, +infrared and radio observations of TDEs, respectively). Other tran- +sients, which may occur so close to the centres of galaxies that they +are astrometically indistinguishable from SMBH accretion, have also +★ E-mail: amalyali@mpe.mpg.de +1 Strong TDE candidates have also been reported in galaxies showing signs +of previous AGN activity (e.g. Merloni et al. 2015; Blanchard et al. 2017; +Liu et al. 2020). +been reported (e.g. supernovae exploding in the narrow-line region of +AGN, Drake et al. 2011), or predicted to exist (e.g. stellar collisions +in nuclear star clusters; Dale et al. 2009). +Even more recently, the population of known ‘repeating’ events +has expanded. Several TDE candidates have now shown multiple +major outbursts, either through their strong, double-peaked optical +lightcurves (AT 2019avd, Malyali et al. 2021; Chen et al. 2022), +repeated X-ray outbursts (IC 3599, Grupe et al. 1995, 2001, 2015; +Campana et al. 2015; eRASSt J045650.3-203750, Liu et al. 2022; +AT 2018fyk, Wevers et al. 2022), or quasi-periodic optical outbursts +potentially associated with repeated partial TDEs (ASASSN-14ko, +Payne et al. 2021). Towards the more extreme end of known re- +peating transients lie the recently-discovered class of quasi-periodic +eruptions (QPEs; Miniutti et al. 2019; Giustini et al. 2020; Arcodia +et al. 2021, 2022), which show large amplitude, ultra-soft X-ray out- +bursts, with flare duration of the order of hours, and which recur over +timescales of hours to days. +In this work, we report on the SRG/eROSITA (Sunyaev et al. 2021; +Predehl et al. 2021) detection of the repeated flaring of a previously +reported, ROSAT-selected TDE candidate, RXJ133157.6-324319.7 +(Reiprich & Greiner 2001; Hampel et al. 2022), originating from +a quiescent galaxy at 𝑧 = 0.05189 (Moretti et al. 2017). In Sec- +tion 2, we report on the detection of this system with eROSITA and +follow-up observations performed with NICER (Section 2.2), XMM +(Section 2.3), and Swift XRT (Section 2.4), as well as archival X-ray +© 2015 The Authors +arXiv:2301.05501v1 [astro-ph.HE] 13 Jan 2023 + +2 +Adam Malyali et al. +observations (Section 2.5), UV, optical and mid-infrared photometry +(Section 2.6) and radio observations (Section 2.7). We discuss the +nature of the system in Section 3, before providing a summary in +Section 4. +All magnitudes are reported in the AB system and corrected for +Galactic extinction using 𝐴V = 0.142 mag, obtained from (Schlafly +& Finkbeiner 2011), 𝑅V = 3.1 and a Cardelli extinction law (Cardelli +et al. 1989), unless otherwise stated. The effective wavelength for +each filter was retrieved from the SVO Filter Profile Service2. All +dates/times will be reported in universal time (UT). +2 RE-DISCOVERY AND FOLLOW-UP +eRASSt J133157.9-324321 (herein J1331) was detected on 2022-01- +20 as a bright new X-ray point source in a systematic search for TDE +candidates during the fifth eROSITA All-Sky survey (eRASS5). The +eROSITA Science Analysis Software (eSASS; Brunner et al. 2022) +inferred source position was (RAJ2000, DecJ2000)=(13h31m57.9s, - +32◦43′21.2′′), with a 1𝜎 positional uncertainty of 1.6′′. No X-ray +point source was detected within 60" of this position in each of the +previous four eRASS. The eROSITA source position is consistent +with a quiescent host galaxy at 𝑧 = 0.05189, with total stellar mass, +log(𝑀★/𝑀⊙) = 10.15 ± 0.09, and an inferred black hole mass, +log(𝑀BH/𝑀⊙) = 6.5 ± 0.2 (appendix A). The quiescent nature of +the host is suggested by both the optical spectrum of its host galaxy +(appendix B; see also Hampel et al. 2022) and its AllWISE (Wright +et al. 2010; Mainzer et al. 2014) mid-infrared colour, W1-W2=0.05± +0.05 mag, far below the threshold of ≳0.7 for mid-infrared AGN +selection (Stern et al. 2012; Assef et al. 2018). After selecting J1331 +as a promising TDE candidate, it was also realised that the host galaxy +of J1331 was the same as that identified for the ROSAT-selected +TDE candidate, RXJ133157.6324319.7, first detected in outburst +in 1993, and recently presented in Hampel et al. (2022), with the +finder chart for these transients presented in Fig. A1. The eRASS5 +detection of J1331 thus suggested the remarkable rebrightening of +a previously known TDE candidate, ∼29 years after the outburst +detected by ROSAT. +2.1 eROSITA +Using the eSASS task SRCTOOL (eSASSusers_211214; Brunner et al. +2022), source (and background) spectra and lightcurves were ex- +tracted from a 60" radius source region centred on the eRASS5 +inferred position, with background counts extracted from a circular +annulus with inner and outer radii of 140" and 240", respectively. +eROSITA scanned the position of J1331 eight times during +eRASS5, with each scan separated by ∼4 hours, thus spanning a +∼28 hour window in total. During this time, J1331 was observed +to be persistently bright (Fig. D2), as opposed to showing a short- +lived flaring, and was clearly detected above background in each +observation. +The eRASS5 X-ray spectra were then fitted using the Bayesian X- +ray Analysis software (BXA; Buchner et al. 2014), which connects +the nested sampling algorithm UltraNest (Buchner 2021) with the +fitting environment XSPEC (Arnaud 1996). The source and back- +ground spectra were jointly fit with a source plus background model, +with the latter using the Principal Component Analysis (PCA) back- +ground modelling first described in Simmonds et al. (2018), and as +2 http://svo2.cab.inta-csic.es/theory/fps/ +also applied to AT 2019avd in Malyali et al. (2021). The eRASS5 +spectrum is well fitted by a tbabs*zbbody model (Fig. D1), with the +Galactic equivalent neutral hydrogen column density, 𝑁H, fixed to +3.84 × 1020 cm−2, the value along the line of sight to J1331 in HI4PI +Collaboration: et al. (2016), and 𝑘𝑇 = 0.115+0.007 +−0.007 keV. A fit with a +power-law (tbabs*zpowerlaw) leaves large residuals between the +observed data and model above 1 keV. When using the best fitting +tbabs*zbbody model described above, the eRASS5 observed (un- +absorbed) 0.2–2 keV flux for J1331 is (6.0±0.7)×10−13 erg s−1 cm−2 +((8 ± 1) × 10−13 erg s−1 cm−2), translating to an unabsorbed 0.2– +2 keV luminosity of (5.5 ± 0.7) × 1042 erg s−1. +J1331 was not detected in eRASS1–4, with 2𝜎 upper limits on +the 0.2–2 keV count rate of 0.016, 0.03, 0.07 and 0.03 cts s−1 in +each successive eRASS (see Table D1 for a full log of the X-ray +observations of J1331). These count rate upper limits were then +converted to 0.2–2 keV flux upper limits using the best fitting spectral +parameters to the eRASS5 spectrum described above. +2.2 NICER XTI +Follow-up observations of J1331 were obtained with the X-ray Tim- +ing Instrument (XTI) on board the Neutron Star Interior Composition +Explorer observatory (NICER; Gendreau et al. 2016) through pre- +approved ToOs (PI: Z. Liu). NICER observations commenced ∼4 +days after the last eRASS5 observation, and continued for the next +15 days on a near daily basis (Table D1). We first generated cleaned +and screened event files using the nicerl2 task (with default recom- +mended parameters), before using nibackgen3C50 (Remillard et al. +2022) to generate total and background spectra for each observation +ID (GTIs were filtered out using hbgcut=0.05 and s0cut=2, as +recommended in Remillard et al. 2022). ARF and RMF files were +subsequently generated using the tasks nicerarf and nicerrmf, +and the X-ray spectra were binned using the Kaastra & Bleeker +(2016) method to a minimum of 20 counts per bin. The total and +background count rates were then estimated in the 0.4–2 keV band3. +J1331 is not detected at 2 sigma above background in each OBSID +(Fig. D3), with 2𝜎 upper limits on the source count rates, inferred +using 𝐶𝑅tot +2𝜎, with 𝐶𝑅tot the total measured count rate, and 𝜎 the +estimated error on 𝐶𝑅tot. The 0.4–2 keV count rates were converted +to 0.2–2 keV fluxes (Table D1) assuming the eRASS5 spectral model +(section 2.1). NICER observations rule out a further brightening be- +yond eRASS5, or a persistently bright source that rapidly ‘cuts-off’ +in brightness by the time of the XMM observation (section 2.3). +2.3 XMM +J1331 was later observed by XMM (P.I. Z. Liu) on 2022-02-07 (de- +noted XMM1), ∼16 days after the last eRASS5 observation, and +also on 2022-08-06 (denoted XMM2). Observations were carried +out with the medium filter on PN, MOS1 and MOS2. The XMM data +were reduced using HEASOFT v6.29, SAS version 20211130_0941, +and the latest calibration data files (CALDB v20210915). Follow- +ing standard XMM data reduction procedures, calibrated event files +were first generated from the Observation Data Files (ODF) using +the SAS tasks emproc and epproc for the MOS and PN cameras +respectively. Then, periods of high background flaring were filtered +3 The 0.4 keV lower bound here was chosen to reduce contamination from +any incompletely modelled optical loading. +MNRAS 000, 1–9 (2015) + +Repeated partial tidal disruption flares from a quiescent galaxy +3 +10 +41 +10 +42 +10 +43 +LX [erg s +1] +48000 +50000 +52000 +54000 +56000 +58000 +60000 +MJD +10 +14 +10 +13 +10 +12 +FX [erg s +1 cm +2] +49005 +49010 +49015 +10 +13 +10 +12 +1992 +1996 +2000 +2004 +2008 +2012 +2016 +2020 +Year +ROSAT +XRT +eROSITA +XMM +Figure 1. Long-term 0.2–2 keV lightcurve of J1331, with circular and triangle markers representing observed fluxes and 2𝜎 upper limits, respectively. The +initial outburst was detected by ROSAT in 1993, before being observed by eROSITA in 2022 to have rebrightened to a similar 0.2–2 keV observed flux. The +X-ray spectra remained ultra-soft in each observation where the source was detected. For plotting clarity, we include the time-averaged flux measurement for +eRASS5, and omit the NICER upper limits. +out4. For XMM1 (XMM2), this resulted in only 4.1ks (25.7 ks), +12.8 ks (30.7 ks) and 11.8 ks (30.2 ks) of usable exposure time +for PN, MOS1 and MOS2, respectively. In the subsequent analysis, +only events with PATTERN<=4 and FLAG==0 were extracted for PN, +whilst PATTERN<=12 and FLAG==0 filtering was applied for MOS1 +and MOS2. +For XMM1, no source is detected within 30" of the host galaxy +position in PN and MOS1 with detection likelihood, DETML, above +3, when running the standard XMM source detection pipeline in the +0.2–2 keV band on the PN, MOS1, and MOS2 images. However, a +source was detected in MOS2 at (RAJ2000, DecJ2000)=(13h31m58s, +-32◦43′19′′), with a 1𝜎 positional uncertainty of 2′′, consistent with +the ROSAT and eROSITA positions (Fig. A1). The DETML for this +source is low (10.3), and the estimated observed 0.2–2 keV flux in +the emldetect output is (8±3) ×10−15 erg s−1 cm−2, ∼75× fainter +than the eRASS5 observed flux. +Given the uncertain detection of the system across all three EPIC +cameras, we computed a 2𝜎 upper limit on the 0.2–2 keV count +rate using the SAS task eupper. This was done using the 0.2–2 keV +band images, exposure and background maps for each camera, and +a 30" radius circular extraction region for the source counts (centred +on the Gaia position of the host galaxy). For XMM1, this yielded +upper limits of 0.006 ct s−1, 0.0014 ct s−1 and 0.002 ct s−1 for +PN, MOS1 and MOS2, respectively. We conservatively estimate the +upper limit for the XMM observation to that inferred from the MOS2 +data, which corresponds to a 0.2–2 keV observed (unabsorbed) flux +of 1 × 10−14 erg s−1 cm−2 (2 × 10−14 erg s−1 cm−2), assuming the +4 https://www.cosmos.esa.int/web/xmm-newton/ +sas-thread-epic-filterbackground +spectral model inferred from the eRASS5 observation. The same +procedure was repeated for XMM2, where we inferred upper limits +of 0.003 ct s−1, 0.0014 ct s−1 and 0.0010 ct s−1 for PN, MOS1 and +MOS2, respectively, translating to 2𝜎 upper limits on the observed +(unobserved) flux of 6×10−15 erg s−1 cm−2 (1×1014 erg s−1 cm−2). +2.4 Swift XRT +Additional Swift XRT (Burrows et al. 2005) observations of J1331 +were performed between 2022-02-27 and 2022-08-245. The XRT +observations were performed in photon counting mode, with the +data analysed using the UK Swift Science Data Centre’s (UKSSDC) +online XRT product building tool (Evans et al. 2007, 2009). No +source was detected in the 0.3–2 keV band at the position of J1331 in +any follow-up observation.The 0.3–2 keV count rates were converted +to 0.2–2 keV fluxes using webPIMMs6, assuming the same spectral +model as from the eROSITA eRASS5 detection, with the fluxes +presented in Table D1. +2.5 Archival X-ray observations +A +detailed +analysis +of +the +ultra-soft +outburst +from +RXJ133157.6324319.7, +detected +by +pointed +ROSAT +PSPC +observations in the early 1990s, was previously performed in +Hampel et al. (2022). In summary, the flare was characterised by an +5 The delay between the eRASS5 and Swift observations stemmed from the +January 2022 reaction wheel failure on-board the Swift observatory. +6 https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/ +w3pimms.pl +MNRAS 000, 1–9 (2015) + +4 +Adam Malyali et al. +8x increase in the 0.1–2.4 keV flux, relative to a 2𝜎 upper limit, over +an 8 day period (and a net increase in the same band by a factor of +at least 40 relative to the deepest upper limit available). The X-ray +spectrum at peak observed brightness was well fitted by a blackbody +with 𝑘𝑇 = 0.11 ± 0.03 keV. The system was then not detected in two +PSPC observations ∼165 days later, where it had faded by a factor +of at least 30 relative to the peak observed ROSAT flux. +To construct a long-term 0.2–2 keV lightcurve, the 0.1–2.4 keV +ROSAT PSPC lightcurve data in Table 1 of Hampel et al. (2022) was +converted into 0.2–2 keV band fluxes using webPIMMS, assuming +the best fitting spectral model to the ROSAT spectrum found in +Hampel et al. (2022). Then, the 2𝜎 upper limits from ROSAT Survey, +XMM Slew and Swift XRT observations were computed using the +High-Energy Lightcurve Generator server (HILIGT; Saxton et al. +2021; König et al. 2021); the archival fluxes are presented in Fig. 1 +and Table D1. +2.6 UV, optical and mid-infrared photometry +J1331 was observed both before (Section 2.5) and after (Section 2.4) +the eRASS5-detected outburst by Swift XRT and UVOT (UVM2 +filter; Roming et al. 2005). To search for transient UV emission, +aperture photometry was performed on the level 2 UVOT sky im- +ages (downloaded from the UKSSDC) using the uvotsource task +(HEASOFT v6.29, CALDB v20201215). Source counts were ex- +tracted from a circular aperture of 5′′ radius, centred on the Gaia +position of the host of J1331, and background counts were extracted +from a source-free region of radius 15′′. The measured UVM2 mag- +nitudes in the follow-up observations are consistent with the archival +measured UVM2 magnitudes on the 2018-04-18, 2018-04-22, 2018- +04-26 (Table E1). +No significant optical variability is seen in the ∼6 years before +the eRASS5 outburst (57500≲ MJD ≲59500) in the forced photom- +etry lightcurve provided by ATLAS (Tonry et al. 2018) (Fig. E1). +Lastly, we note that no major variability is detected above the host +galaxy emission within the NEOWISE mid-infrared lightcurve be- +tween MJD∼56680 and 59400 (Fig. E1), which was generated using +the procedure described in section 3.2 of Malyali et al. (2021). +2.7 Radio +We observed the coordinates of J1331 on 2022 Mar 02 with the +Australia Telescope Compact Array (ATCA) radio telescope in 6 km +configuration, using the 4cm dual receiver with central frequencies +5.5/9 GHz, each with a 2 GHz bandwidth split into 2049×1 MHz +spectral channels, and for a total of 150 min on source. Data were +reduced following standard procedures in the Common Astronomy +Software Applications (McMullin et al. 2007; CASA-TEAM et al. +2022). We used 1934-638 for flux and bandpass calibration and 1336- +260 for phase calibration. Images of the target field were created using +the CASA task tclean. No source was detected at the location of +J1331 at either frequency band with a 3𝜎 upper limit of 73.5𝜇Jy/bm +at 5.5 GHz and 54𝜇Jy/bm at 9 GHz. Additionally, no source was +detected in a stacked 5.5 and 9 GHz image, with a 3𝜎 upper limit of +57.9𝜇Jy/bm at a central frequency of 7.3 GHz. +3 DISCUSSION +Comparing the X-ray lightcurve of J1331 with other ultra-soft nu- +clear transients (Fig. D4) from galaxies that were recently quiescent, +or hosted low luminosity AGN, then J1331 decays faster than the +majority of other X-ray bright TDEs7, but decays over much longer +timescales than the bursts typically seen in QPEs (burst durations +≲30 ks, or ≲0.3 days; Miniutti et al. 2019; Giustini et al. 2020; +Arcodia et al. 2021, 2022). +Given the quiescent nature of the host galaxy, and the ultra-soft +X-ray spectrum, an AGN origin for J1331 is disfavoured. We also +rule out a mechanism similar to that producing the X-ray flares ob- +served in Sgr A* (e.g. Neilsen et al. 2013; Ponti et al. 2015; Yuan & +Wang 2016; Ponti et al. 2017; Mossoux et al. 2020), as the latter are +clearly observationally distinct to J1331, with respect to the flaring +timescales (Sgr A* flare durations ≲ 104 s; Mossoux et al. 2020), +spectral properties (flaring X-ray emission in Sgr A* is hard and +likely synchrotron, e.g. Ponti et al. 2017), and peak observed lumi- +nosity (bolometric luminosity of Sgr A* is ∼ 1036 erg s−1; Genzel +et al. 2010). Arguments against a Galactic origin for this system have +previously been presented in Hampel et al. (2022). +Ultra-soft X-ray flares from quiescent galaxies have previously +been considered as a reliable signature of a TDE (e.g. Zabludoff et al. +2021). However, the current theoretically predicted TDE rates are +≳ 10−4 yr−1 galaxy−1 (Stone et al. 2020), so it would be exceptionally +unlikely to have observed two independent tidal disruption flares +occuring within the same galaxy over a ∼30 year timescale (Poisson +probability ∼ 5 × 10−6; Fig. C2); a more exotic class of TDE would +need to be invoked to explain J1331. +One such possibility, discussed in Hampel et al. (2022), is that +J1331 was produced by a TDE involving a supermassive black hole +binary (SMBHB). This scenario was partly proposed in an attempt +to explain the fast X-ray brightening observed by ROSAT, since +such TDEs may have highly non-monotonic decays of their X-ray +lightcurves. This stems from the gravitational interaction between +the companion BH and the debris streams, which may cause large +perturbations to the orbits of the less bound debris and cause their +chaotic evolution, as well as a complex evolution of the accretion +rate over time. Liu et al. (2014); Ricarte et al. (2016); Coughlin et al. +(2017) predict these systems to show sharp dips and rises in the X- +ray lightcurve rate (of ∼1–2 orders of magnitude), on timescales of +the order of the binary orbital period (Liu et al. 2014; Ricarte et al. +2016), although Coughlin et al. (2017) find highly variable accretion +rates between different simulation runs and over timescales shorter +than the SMBHB orbital periods (i.e. there still seems to be quite +large uncertainties in the theoretically predicted lightcurves of TDEs +involving SMBHBs). +Under the SMBHB scenario, both the eROSITA and ROSAT obser- +vations would have had to have sampled a ‘dipping’, or ‘brightening +from a dip’, phase of the X-ray lightcurve, respectively. For binary +orbital periods of the order of ∼months, assuming ∼mpc binary sep- +aration as in Liu et al. (2014), then it would be quite fortuitous for us +to have observed such behaviour. Furthermore, there is importantly +no evidence for late time X-ray rebrightening episodes in the months +after each outburst, as seen by XMM and Swift (Fig. 1), which one +might expect to have observed given that the accretion rate is pre- +dicted to eventually revert back to the 𝑡−5/3 decay following ‘dips’ +(e.g. Fig. 12 in Coughlin et al. 2017). We would therefore disfavour +J1331 being caused by a full TDE around a SMBHB, given the fine +tuning needed in order to match observations. +A more feasible scenario is that both outbursts were driven by a +partial tidal disruption event (pTDE), potentially of the same object. +Unless the pTDE rate is orders of magnitude larger than currently +7 Ignoring short timescale flaring behaviour seen in some TDE candidates, +such as AT 2019ehz (van Velzen et al. 2021b). +MNRAS 000, 1–9 (2015) + +Repeated partial tidal disruption flares from a quiescent galaxy +5 +10 +40 +10 +41 +10 +42 +10 +43 +LX [erg s +1] +10 +1 +10 +2 +MJD - 59581 +10 +15 +10 +14 +10 +13 +10 +12 +FX [erg s +1 cm +2] +t +5/3 +t +9/4 +t +4 +Figure 2. Zoom-in on the first eROSITA-detected outburst in 2022, along +with multiple power-law decay slopes plotted in grey dashed lines. The decay +slope appears to be much steeper than the canonical 𝑡−5/3 decay predicted +for TDEs with a uniform distribution of specific energies, and appears more +consistent with a 𝑡−4 decay, as predicted in Ryu et al. (2020). We assume a +peak MJD of 59593 for the X-ray outburst, and roughly estimate the MJD of +disruption to be 59581 (section C). The markers follow the same legend as +for Fig. 1. +estimated in the literature (Stone & Metzger 2016; Chen & Shen +2021; Zhong et al. 2022), then both outbursts would likely be re- +lated to the same star being disrupted by the same black hole (i.e. +the star should have survived the initial encounter). Considering that +the recurrence timescale of J1331 is ≲ 30 years, then it is also diffi- +cult to reconcile this with theoretical predictions for the recurrence +timescales of flares in pTDEs where the star was initially scattered +onto a parabolic orbit around the black hole (≳ 400 years, e.g. Ryu +et al. 2020). Instead, the flaring may have been driven by the repeated +stripping of a star on an elliptical orbit by the disrupting SMBH (see +Hayasaki et al. 2013 for a discussion on potential origins for such +stars). This scenario would be further supported by both the relatively +small amount of inferred energy emitted in the eROSITA-detected +outburst8 of (5+6 +−3) × 1049 erg, corresponding to an accreted mass of +(5+7 +−2) × 10−4(𝜖/0.05)−1 M⊙, where 𝜖 is the radiative efficiency of +accretion, and also by the extremely low 𝐿X at late-times (as sug- +gested by the non-detection and deep upper limits in XMM2), since +elliptical TDEs are predicted to produce short-lived, finite accretion +bursts (Hayasaki et al. 2013). Given this, and that the radio obser- +vations were taken ∼40 days after the eRASS5 flare (section 2.7), +then we note that we may have missed any associated jet or out- +flow launched in this event, as seen in other TDE candidates (e.g. +Goodwin et al. 2022). +The case for a repeated pTDE is further enhanced by the fast rise +and decay timescales seen with ROSAT and eROSITA. Compared +with full disruptions, pTDEs only strip the outermost layers of the +star, with the specific energy distribution of the debris, d𝑀/d𝐸, +differing from full TDEs (e.g. Coughlin & Nixon 2019; Miles et al. +2020; Ryu et al. 2020). Since the mass fallback rate, �𝑀fb(𝑡), scales +∝ d𝑀/d𝐸, then �𝑀fb(𝑡) is also predicted to differ between full and +pTDEs. Ryu et al. (2020) find that the narrower spreads in d𝑀/d𝐸 +for pTDEs can yield �𝑀fb(𝑡) ∝ 𝑡−𝑝, where 𝑝 ∼ 2−5, more consistent +with what is observed in J1331 (Fig. 2), and much steeper than a +canonical 𝑡−5/3 decline predicted for the mass fallback rate in full +TDEs (Rees 1988; Phinney 1989). +Lastly, although the mass fallback in weak pTDEs may evolve +over shorter timescales relative to full TDEs, the viscous timescale, +8 Assuming a similar temporal evolution for both the eROSITA-detected and +ROSAT-detected outbursts- see section C. +𝑡visc, still needs to be shorter than the minimum orbital period of the +stellar debris so that the X-ray luminosity traces the mass fallback +rate (assuming a constant radiative efficiency, negligible obscuration +of the soft X-rays, and negligible disc cooling). Considering 𝑡visc ∼ +𝛼−1(𝐻/𝑅)−2Ω−1(𝑟), where 𝛼 is the viscosity parameter (Shakura +& Sunyaev 1973), 𝐻 and 𝑅 the scale height and width of the disc, +and Ω−1(𝑟) the orbital period at distance 𝑟 from the black hole, +then 𝑡visc ∼ 0.4(𝛼/0.1)−1(𝐻/𝑅)−2 days at the circularisation radius +(∼ 2𝑅tidal/𝛽, where 𝑅tidal and 𝛽 are the tidal radius and impact +parameter for the disruption). A geometrically thick disc (𝐻/𝑅 ∼ 1), +as may be expected to form for super-Eddington mass fallback rates, +would be needed to reproduce accretion timescales of the order ∼days +as seen in J1331. However, it is currently unclear how the stellar +debris might circularise so efficiently in a weak pTDE (see Bonnerot +& Stone 2021 for a review on accretion flow formation in TDEs), +and we also highlight here that similar concerns have recently been +raised for explaining the short X-ray flare durations observed in QPEs +via an accretion origin (e.g. Krolik & Linial 2022; Lu & Quataert +2022). Although future simulations would likely be needed to explore +the debris circularisation in J1331-like events, alternative origins for +the X-ray emission may be from compression shocks of the debris +streams at pericentre (e.g. Steinberg & Stone 2022), or circularisation +shocks from debris stream collisions (Krolik & Linial 2022; Lu & +Quataert 2022). +4 SUMMARY +J1331 is a repeating X-ray transient associated to a quiescent galaxy +at 𝑧 = 0.05189, which we consider to be consistent with a scenario +involving two weak pTDEs. Whilst several previously reported pTDE +candidates have occurred in galaxies hosting an AGN, we highlight +that the host of J1331 is quiescent. The main properties of J1331 can +be summarised as follows: +(i) J1331 was first detected by ROSAT in 1993 (Hampel et al. +2022), where it had shown an ultra-soft (𝑘𝑇 = 0.11 ± 0.03 keV) +flaring by a factor of at least 40 relative to a previous 2𝜎 upper limit. +The outburst also showed a fast rise, where it had brightened by a +factor of eight over an 8 day period. The system was subsequently not +detected in a deep pointed ROSAT observation ∼165 days afterwards, +as well as in XMM Slew, and Swift XRT observations performed +between 2006 and 2018 (Table D1). +(ii) After not being detected by eROSITA in its first four eRASS, +J1331 was observed to have brightened in eRASS5 to a 0.2–2 keV +flux of (6.0 ± 0.7) × 10−13 erg s−1 cm−2. The eRASS5 spectrum +is ultra-soft (𝑘𝑇 = 0.115+0.007 +−0.007 keV), and is consistent with the 𝑘𝑇 +inferred from the ROSAT-observed flare in 1993. +(iii) J1331 was not detected during pointed XMM observations +and Swift XRT observations when followed up after the eRASS5 +detection; the first (second) XMM observation constrains the 0.2– +2 keV flux to decay by a factor of ≳40 (≳100) over a 17 (∼200) +day period after the eRASS5 observation. The faint 0.2–2 keV X-ray +luminosities (< 7×1040 erg s−1, unabsorbed) at ∼ 200 days post-peak +brightness, inferred via the second XMM observation (Table D1), +may be due to a late-time drop off in the mass fallback rate once the +disruption episode is over. +(iv) Combined with the fast rise timescale seen by ROSAT, then +J1331-like outbursts are short lived (rise and decay timescales of +6+1 +−1 days and 3.9+0.1 +−0.1 days, respectively; appendix C) and evolve over +shorter timescales relative to full TDEs. +(v) J1331 has only been observed to show transient emission in +MNRAS 000, 1–9 (2015) + +6 +Adam Malyali et al. +the 0.2–2 keV band, with no transient optical, UV, or radio emission +observed in follow-up observations. +We conclude by noting that J1331 appears to fill in the continuum +of observed soft X-ray outbursts from quiescent galaxies, lying in be- +tween QPEs and TDEs with respect to its rise and decay timescales +(Fig. D4), although the recurrence timescales are much longer than +in the current sample of QPEs. Additional follow-up observations +will be scheduled in order to more tightly constrain the recurrence +timescales of outbursts from J1331. Future planned X-ray missions +geared towards exploiting the X-ray transient sky, such as the Einstein +Probe (Yuan et al. 2018), will likely be sensitive towards detecting +similar partial disruptions; for these missions, the eROSITA All-Sky +survey data may play an important role by providing a long-term +baseline towards which new candidates can be identified. Given the +faster decay timescales of J1331-like systems, then we would advo- +cate promptly triggering high-cadence X-ray follow-up in order to +better constrain the evolution of the accretion rate in future candi- +dates. +ACKNOWLEDGEMENTS +AM thanks Taeho Ryu for very useful discussions whilst preparing +the manuscript. AM acknowledges support by DLR under the grant +50 QR 2110 (XMM_NuTra, PI: Z. Liu). This work was supported by +the Australian government through the Australian Research Council’s +Discovery Projects funding scheme (DP200102471). We would like +to thank the referee for a constructive report that improved the quality +of the paper. +This work is based on data from eROSITA, the soft X-ray instru- +ment aboard SRG, a joint Russian-German science mission supported +by the Russian Space Agency (Roskosmos), in the interests of the +Russian Academy of Sciences represented by its Space Research In- +stitute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt +(DLR). The SRG spacecraft was built by Lavochkin Association +(NPOL) and its subcontractors, and is operated by NPOL with sup- +port from the Max Planck Institute for Extraterrestrial Physics (MPE). +The development and construction of the eROSITA X-ray instru- +ment was led by MPE, with contributions from the Dr. Karl Re- +meis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), +the University of Hamburg Observatory, the Leibniz Institute for +Astrophysics Potsdam (AIP), and the Institute for Astronomy and +Astrophysics of the University of Tübingen, with the support of DLR +and the Max Planck Society. The Argelander Institute for Astronomy +of the University of Bonn and the Ludwig Maximilians Universität +Munich also participated in the science preparation for eROSITA. +The eROSITA data shown here were processed using the eSASS +software system developed by the German eROSITA consortium. +This work made use of data supplied by the UK Swift Science +Data Centre at the University of Leicester. +The Australia Telescope Compact Array is part of the Aus- +tralia Telescope National Facility (https://ror.org/05qajvd42) +which is funded by the Australian Government for operation as a +National Facility managed by CSIRO. We acknowledge the Gomeroi +people as the traditional owners of the Observatory site. +The Legacy Surveys consist of three individual and complemen- +tary projects: the Dark Energy Camera Legacy Survey (DECaLS; +Proposal ID 2014B-0404; PIs: David Schlegel and Arjun Dey), the +Beijing-Arizona Sky Survey (BASS; NOAO Prop. ID #2015A-0801; +PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Sur- +vey (MzLS; Prop. ID #2016A-0453; PI: Arjun Dey). DECaLS, BASS +and MzLS together include data obtained, respectively, at the Blanco +telescope, Cerro Tololo Inter-American Observatory, NSF’s NOIR- +Lab; the Bok telescope, Steward Observatory, University of Arizona; +and the Mayall telescope, Kitt Peak National Observatory, NOIR- +Lab. Pipeline processing and analyses of the data were supported by +NOIRLab and the Lawrence Berkeley National Laboratory (LBNL). +The Legacy Surveys project is honored to be permitted to conduct +astronomical research on Iolkam Du’ag (Kitt Peak), a mountain with +particular significance to the Tohono O’odham Nation. +NOIRLab is operated by the Association of Universities for Re- +search in Astronomy (AURA) under a cooperative agreement with +the National Science Foundation. LBNL is managed by the Regents +of the University of California under contract to the U.S. Department +of Energy. +This project used data obtained with the Dark Energy Camera +(DECam), which was constructed by the Dark Energy Survey (DES) +collaboration. Funding for the DES Projects has been provided by the +U.S. Department of Energy, the U.S. National Science Foundation, +the Ministry of Science and Education of Spain, the Science and +Technology Facilities Council of the United Kingdom, the Higher +Education Funding Council for England, the National Center for +Supercomputing Applications at the University of Illinois at Urbana- +Champaign, the Kavli Institute of Cosmological Physics at the Uni- +versity of Chicago, Center for Cosmology and Astro-Particle Physics +at the Ohio State University, the Mitchell Institute for Fundamental +Physics and Astronomy at Texas A&M University, Financiadora de +Estudos e Projetos, Fundacao Carlos Chagas Filho de Amparo, Fi- +nanciadora de Estudos e Projetos, Fundacao Carlos Chagas Filho +de Amparo a Pesquisa do Estado do Rio de Janeiro, Conselho Na- +cional de Desenvolvimento Cientifico e Tecnologico and the Minis- +terio da Ciencia, Tecnologia e Inovacao, the Deutsche Forschungs- +gemeinschaft and the Collaborating Institutions in the Dark Energy +Survey. The Collaborating Institutions are Argonne National Labo- +ratory, the University of California at Santa Cruz, the University of +Cambridge, Centro de Investigaciones Energeticas, Medioambien- +tales y Tecnologicas-Madrid, the University of Chicago, University +College London, the DES-Brazil Consortium, the University of Ed- +inburgh, the Eidgenossische Technische Hochschule (ETH) Zurich, +Fermi National Accelerator Laboratory, the University of Illinois at +Urbana-Champaign, the Institut de Ciencies de l’Espai (IEEC/CSIC), +the Institut de Fisica d’Altes Energies, Lawrence Berkeley National +Laboratory, the Ludwig Maximilians Universitat Munchen and the +associated Excellence Cluster Universe, the University of Michigan, +NSF’s NOIRLab, the University of Nottingham, the Ohio State Uni- +versity, the University of Pennsylvania, the University of Portsmouth, +SLAC National Accelerator Laboratory, Stanford University, the Uni- +versity of Sussex, and Texas A&M University. +BASS is a key project of the Telescope Access Program (TAP), +which has been funded by the National Astronomical Observatories +of China, the Chinese Academy of Sciences (the Strategic Prior- +ity Research Program “The Emergence of Cosmological Structures” +Grant # XDB09000000), and the Special Fund for Astronomy from +the Ministry of Finance. The BASS is also supported by the Exter- +nal Cooperation Program of Chinese Academy of Sciences (Grant +# 114A11KYSB20160057), and Chinese National Natural Science +Foundation (Grant # 12120101003, # 11433005). +The Legacy Survey team makes use of data products from the +Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), +which is a project of the Jet Propulsion Laboratory/California Insti- +tute of Technology. NEOWISE is funded by the National Aeronautics +and Space Administration. +The Legacy Surveys imaging of the DESI footprint is supported +by the Director, Office of Science, Office of High Energy Physics +MNRAS 000, 1–9 (2015) + +Repeated partial tidal disruption flares from a quiescent galaxy +7 +of the U.S. Department of Energy under Contract No. DE-AC02- +05CH1123, by the National Energy Research Scientific Comput- +ing Center, a DOE Office of Science User Facility under the same +contract; and by the U.S. National Science Foundation, Division of +Astronomical Sciences under Contract No. AST-0950945 to NOAO. +M.K. acknowledges support from DFG grant KR 3338/4-1. D.H. +is supported by DLR grant FKZ 50OR2003. +DATA AVAILABILITY +The eRASS1-4 data taken within the German half of the eROSITA +sky is currently planned to be made public by Q2 2024, whilst +the eRASS5 data is scheduled to become public by Q2 2026. The +Swift data is available to download through the UK Swift Data Sci- +ence website9, whilst the NICER data is accessible through NASA’s +HEASARC interface10. Publicly available ATLAS data can be ac- +cessed through the ATLAS forced photometry service11, and NEO- +WISE lightcurves can be accessed through the IRSA web portal12. +ATCA data are stored in the Australia Telescope Online Archive13, +and will become publicly accessible 18 months from the date of ob- +servation. The XMM data will become public after the propietory +period expires (2023-08-30). Follow-up optical spectra will likely +remain private at least until the release of the forthcoming eROSITA- +selected TDE population paper, but could be made available upon +reasonable request. +REFERENCES +Alexander K. D., van Velzen S., Horesh A., Zauderer B. A., 2020, Space +Science Reviews, 216, 81 +Antonini F., Lombardi J. 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W.-S., Onori F., Hung T., Arcavi I., 2020, Space +Science Reviews, 216, 124 +van Velzen S., Pasham D. R., Komossa S., Yan L., Kara E. A., 2021a, Space +Science Reviews, 217, 63 +van Velzen S., et al., 2021b, The Astrophysical Journal, 908, 4 +APPENDIX A: HOST GALAXY PROPERTIES +Using the correlation reported in Kettlety et al. (2018) between +galaxy total stellar mass, 𝑀★, and luminosity in the WISE 𝑊1-band, +13 +h32 +m00 +s +31 +m58 +s +57 +s +56 +s +-32°43'00" +15" +30" +45" +J2000 +J2000 +Figure A1. Legacy Survey DR10 (early) 𝑔-band cutout image of the sky +region surrounding eRASSt J133158-324321. The dark orange circle is the +error circle for RXJ133157.6324319.7 inferred from ROSAT pointed obser- +vations in Hampel et al. (2022), whilst the red and blue circles denote the 3𝜎 +error circles on the source position inferred from eROSITA and XMM MOS2 +observations (although the detection of J1331 in the first XMM observation is +uncertain and we quote upper limits on the count rates for this in section 2.3, +we include it in this finder chart for completeness). The cyan star marks the +Gaia EDR3 (Gaia Collaboration et al. 2021) position of the host galaxy. +𝐿W1, then we infer log(𝑀★/𝑀⊙) = 10.15 ± 0.09 for the host galaxy. +Combining this with 𝑀BH − 𝑀★ relation in Reines & Volonteri +(2015), suggests a black hole mass of log(𝑀BH/𝑀⊙) = 6.5 ± 0.2. +The finder chart for J1331 is presented in Fig A1. +APPENDIX B: OPTICAL SPECTROSCOPY +LCO spectrum (2022-02-12): J1331 was observed with the low +dispersion FLOYDS spectrograph on the LCOGT 2m telescope at +Siding Spring Observatory operated by the Las Cumbres Observa- +tory (LCO; Brown et al. 2013) on 2022 February 12 (proposal ID +CON2022A-001, PI: M. Salvato). We obtained an exposure of 1800 +seconds using the “red/blu” grism and the 2” slit oriented along the +parallactic angle. The spectrum has a wavelength range of 3200- +10000A with dispersions of 3.51A/pixel and 1.74 A/pixel in the blue +(3200-5700A) and red (5400-10000A) bands, respectively. The data +were reduced and calibrated using the automatic FLOYDS pipeline. +The HgAr and Zn lamps were used for wavelength calibration and +a Tungsten-Halogen + Xenon lamp for flat fielding. A sensitivity +function from the FLOYDS archive was used for flux calibration. +WiFeS spectrum (2022-05-09): We observed J1331 with the Wide +Field Spectrograph (WiFeS; Dopita et al. 2010) on the ANU 2.3m +telescope at Siding Spring Observatory on 2022 May 08 (proposal +ID 2220157, PI Miller-Jones). We obtained 2x2400 s exposures us- +ing the R3000 and B3000 gratings and a NeAr arc lamp exposure +immediately following the target exposures. The data were reduced +using standard procedures including the PyWiFeS reduction pipeline +(Childress et al. 2014). LTT4364 was used as the flux standard and +a quartz-iodine lamp was used for flat-fielding. We then chose the +slitlets with the most significant flux from the calibrated spectra +MNRAS 000, 1–9 (2015) + +Repeated partial tidal disruption flares from a quiescent galaxy +9 +5500 +6000 +6500 +7000 +7500 +8000 +Rest Wavelength [Å] +0.0 +0.5 +1.0 +1.5 +2.0 +F [10 +16 erg cm +2 s +1 Å +1] +2022-02-12: LCO +2022-05-09: WiFeS +Figure B1. Optical spectra of J1331, with the first follow-up spectrum being +obtained on 2022-02-12, ∼23 days after the last eRASS5 detection. +obtained from the pipeline and performed background subtraction, +resulting in a spectrum with spectral range 3500 to 9000 Å. +Each follow-up optical spectrum appears to be consistent with a +quiescent host galaxy (Fig. B1), with no TDE-like optical emission +features detected, nor any transient features relative to the NOT spec- +trum taken on 1999-01-26 and presented in Hampel et al. (2022). +APPENDIX C: INFERRING THE OUTBURST +PROPERTIES +To obtain a coarse reconstruction of the 2022 outburst, we perform a +joint fit of the rising lightcurve from 1993, observed by ROSAT, and +the decay lightcurve from 2022, observed by eROSITA and XMM, +using: +𝐹X(𝑡) = 𝐹X,max × +� +exp +� +−(𝑡 − 𝑡peak,1)2/2𝜎2� +if 𝑡 < 𝑡peak,1 +exp +� +−(𝑡 − 𝑡peak,2)/𝜏 +� +if 𝑡 > 𝑡peak,2 +(C1) +where the free parameters of this model are 𝜎 (the rise timescale), +𝑡peak,1 and 𝑡peak,2 (the peak time of the ROSAT and eROSITA out- +bursts, respectively), 𝜏 (the decay timescale), and 𝐹X,max (the peak +flux of both outbursts), with the priors on these parameters listed in +Table C1. We assume that the upper bound on the peak luminosity +must be less than the Eddington luminosity for the SMBH, and that +both outbursts have the same peak luminosity. We then assume that +the rise for 2022 outburst was similar to the 1993 outburst (see below), +and use its modelled rise to approximate that of the unobserved rise of +the 2022 outburst. From this fittedlightcurve model (Fig. C1), we then +computed the integrated 0.2–2 keV luminosity, and corrected this to +a bolometric luminosity using the best fitting X-ray spectral model. +The inferred energy emitted in each outburst is (5+6 +−3) × 1049 erg, +corresponding to an accreted mass of (5+7 +−2) × 10−4(𝜖/0.05)−1 M⊙, +where 𝜖 is the radiative efficiency of accretion, whilst the inferred +peak MJD for each outburst are 49024+6 +−6 and 59593+3 +−2. The inferred +𝜎 and 𝜏 are 6+1 +−1 days and 3.9+0.1 +−0.1 days, respectively, and we roughly +estimate the MJD of disruption to be 59593 − 2 ∗ 𝜎 ∼ 59581. +It is of course extremely important to consider that these estimates +are subject to a number of caveats, mainly related to our observations +not covering the rise of the 2022 outburst, such that the estimated +values here should be treated with caution. For example, it is assumed +that the outburst can be well modelled by equation C1, and that +both the 1993 and 2022 outbursts are similar, whereas the actual +Table C1. Priors adopted in the fitting of the 1993 and 2022 outbursts. The +rise and decay timescales are in units of days. 𝑡peak,1 and 𝑡peak,2 are in MJD, +whilst 𝐹max is the maximum 0.2–2 keV flux of each outburst (with upper +bound set by the Eddington luminosity of the system). +Parameter +Prior +log[𝜎] +∼ U(0, log[50]) +𝑡peak,1 +∼ U(49006, 49178) +𝑡peak,2 +∼ U(58450, 58650) +log[𝜏] +∼ U(0, log[50]) +log[𝐹X,max] +∼ U(log[5 × 10−13], log[4 × 10−11]) +10 +40 +10 +42 +10 +44 +LX [erg s +1] +59560 +59580 +59600 +59620 +59640 +MJD +10 +16 +10 +14 +10 +12 +FX [erg s +1 cm +2] +Figure C1. Inferred full outburst (red) for the flaring observed by eROSITA +in 2022, assuming the model described in equation C1. The markers follow +the same legend as for Fig. 1. The darker and lighter shaded red bands enclose +the inner 68% and 98% of the posterior. +lightcurve may have had an extended plateau phase prior to the +eROSITA detection (so our estimated fluence and accreted mass +would be underestimated). +However, if the 2022 outburst does evolve relatively closely to the +functional form in equation C1, then it may be reasonable to consider +that the rise timescale for the flare in 1993 is similar to that observed +in 2022 (under a tidal disruption scenario), due to the approximately +constant eccentricity of the stellar remnant after repeated partial +disruptions (Antonini et al. 2011), and the weak dependence of the +period of the most bound debris on the stellar mass (Hayasaki et al. +2013). +APPENDIX D: ADDITIONAL X-RAY INFORMATION +The BXA fitted model to the eRASS5 spectrum is shown in Fig. D1, +and the eRASS5 lightcurve is shown in Fig. D2. The NICER count +rate lightcurve is plotted in Fig. D3, whilst the full X-ray lightcurve of +J1331 is presented in Table D1. A comparison of the X-ray lightcurve +of J1331 with other nuclear transients is presented in Fig D4. +APPENDIX E: ADDITIONAL PHOTOMETRIC +INFORMATION +Table E1 contains the Swift UVOT aperture photometry of the host +galaxy of J1331, whilst Fig. E1 shows the long term ATLAS and +NEOWISE lightcurves of J1331. +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–9 (2015) + +10 +Adam Malyali et al. +Table D1. X-ray lightcurve table for J1331. The fluxes from the ROSAT +pointed observations were derived from Hampel et al. (2022). The first four +eROSITA observations listed, between MJD 58868 and 59419, are upper +limits estimated from eRASS1, 2, 3 and 4, respectively; eROSITA fluxes +outside of this window have been computed from the individual visits within +eRASS5. +MJD +Observation +𝐹0.2−2keV,obs +𝐹0.2−2keV,unabs +[10−13 erg cm−2 s−1] +[10−13 erg cm−2 s−1] +48260.000 +ROSAT/ RASS +< 2.9 +< 4.5 +48844.598 +ROSAT/ Pointed +< 0.2 +< 0.4 +49006.094 +ROSAT/ Pointed +< 1.2 +< 1.9 +49012.146 +ROSAT/ Pointed +6.1 ± 0.7 +9.4 ± 1.0 +49012.180 +ROSAT/ Pointed +8.9 ± 1.9 +13.8 ± 2.9 +49013.591 +ROSAT/ Pointed +10.0 ± 1.1 +15.5 ± 1.7 +49178.555 +ROSAT/ Pointed +< 0.7 +< 1.0 +49178.766 +ROSAT/ Pointed +< 0.3 +< 0.5 +53745.291 +XMM/ Slew +< 3.8 +< 5.9 +57056.039 +XMM/ Slew +< 5.4 +< 8.3 +57241.869 +XMM/ Slew +< 8.3 +< 12.8 +58226.719 +Swift/ XRT +< 0.9 +< 1.4 +58230.707 +Swift/ XRT +< 0.5 +< 0.8 +58234.028 +Swift/ XRT +< 0.8 +< 1.2 +58868.114 +SRG/ eROSITA +< 0.3 +< 0.4 +59051.625 +SRG/ eROSITA +< 0.5 +< 0.7 +59229.875 +SRG/ eROSITA +< 1.3 +< 1.7 +59418.532 +SRG/ eROSITA +< 0.5 +< 0.7 +59599.448 +SRG/ eROSITA +10.8 ± 8.0 +14.4 ± 10.6 +59599.614 +SRG/ eROSITA +3.4 ± 1.7 +4.6 ± 2.2 +59599.781 +SRG/ eROSITA +5.7 ± 1.5 +7.7 ± 2.0 +59599.948 +SRG/ eROSITA +4.9 ± 1.2 +6.5 ± 1.6 +59600.114 +SRG/ eROSITA +5.1 ± 1.3 +6.9 ± 1.7 +59600.281 +SRG/ eROSITA +2.6 ± 1.1 +3.5 ± 1.5 +59600.448 +SRG/ eROSITA +9.0 ± 2.4 +12.0 ± 3.2 +59600.614 +SRG/ eROSITA +6.4 ± 3.5 +8.5 ± 4.6 +59604.892 +NICER/ XTI +<8.6 +<13.8 +59605.566 +NICER/ XTI +<10.3 +<16.6 +59606.082 +NICER/ XTI +<9.5 +<15.3 +59607.533 +NICER/ XTI +<7.7 +<12.3 +59608.280 +NICER/ XTI +<6.6 +<10.6 +59609.473 +NICER/ XTI +<6.3 +<10.1 +59610.119 +NICER/ XTI +<7.3 +<11.7 +59611.432 +NICER/ XTI +<6.5 +<10.4 +59612.210 +NICER/ XTI +<8.1 +<13.0 +59613.305 +NICER/ XTI +<7.6 +<12.3 +59614.210 +NICER/ XTI +<8.1 +<13.1 +59615.500 +NICER/ XTI +<6.9 +<11.1 +59616.889 +NICER/ XTI +<6.2 +<10.0 +59617.287 +XMM/ Pointed +< 0.1 +< 0.2 +59617.598 +NICER/ XTI +<5.5 +<8.8 +59618.630 +NICER/ XTI +<5.5 +<8.8 +59619.666 +NICER/ XTI +<5.6 +<9.0 +59620.463 +NICER/ XTI +<5.8 +<9.4 +59621.229 +NICER/ XTI +<9.5 +<15.3 +59622.488 +NICER/ XTI +<9.8 +<15.8 +59623.102 +NICER/ XTI +<12.2 +<19.6 +59624.362 +NICER/ XTI +<6.5 +<10.5 +59638.031 +Swift/ XRT +< 0.8 +< 1.4 +59766.375 +Swift/ XRT +< 0.7 +< 1.2 +59773.061 +Swift/ XRT +< 24.6 +< 43.7 +59774.292 +Swift/ XRT +< 2.2 +< 3.9 +59778.974 +Swift/ XRT +< 0.8 +< 1.4 +59780.760 +Swift/ XRT +< 0.8 +< 1.5 +59787.468 +Swift/ XRT +< 0.8 +< 1.4 +59794.352 +Swift/ XRT +< 0.8 +< 1.4 +59797.916 +XMM/ Pointed +< 0.06 +< 0.10 +59801.282 +Swift/ XRT +< 0.5 +< 1.0 +59808.180 +Swift/ XRT +< 0.9 +< 1.6 +59815.534 +Swift/ XRT +< 0.8 +< 1.4 +10 +3 +10 +2 +10 +1 +10 +0 +10 +1 +TDE rate, [30 yr +1 gal +1] +10 +6 +10 +4 +10 +2 +p(N +2)| ) += 0.01 += 0.05 += 0.15 += 0.003 +Figure C2. Poisson probability of 𝑁 ≥ 2 TDEs occurring within a 30 year +period for a given galaxy. The red dotted lines mark the estimated probability +for current theoretical estimates for TDE rates (10−4 yr−1 gal−1; Stone et al. +2020). The grey dashed lines mark out the TDE rates of 0.15, 0.05 and 0.01 +per +30 yr−1 gal−1, required to produce probabilities of 0.01, 0.001, and +0.0001, respectively. +0.3 +1.0 +2.0 +5.0 +Energy [keV] +10 +4 +10 +2 +100 +Counts s +1 keV +1 +Figure D1. BXA fit of a tbabs*zbbody model to the eRASS5 spectrum. +The solid red line represents the median model fit, whilst the shaded red +region encloses the inner 98% of the credible region. The X-ray spectrum is +ultra-soft with 𝑘𝑇 = 0.115+0.007 +−0.007 keV. +Table E1. Swift UVM2 photometry of the host galaxy of J1331. +MJD +Magnitude +58226.727 +23.1 ±1.0 +58230.747 +22.9 ±0.6 +58234.068 +22.3 ±0.4 +59638.032 +22.2 ±0.3 +59766.376 +23.0 ±0.7 +59774.294 +22.8 ±1.0 +59778.975 +22.5 ±0.6 +59780.762 +22.9 ±0.8 +59794.353 +22.7 ±0.6 +59801.283 +22.5 ±0.4 +59808.181 +22.8 ±0.6 +59815.535 +22.5 ±0.5 +MNRAS 000, 1–9 (2015) + +Repeated partial tidal disruption flares from a quiescent galaxy +11 +0 +5 +10 +15 +20 +25 +30 +t +teRASS5, 0 [hr] +10 +3 +10 +2 +10 +1 +10 +0 +Rate [cts/s] +Figure D2. 0.2–2 keV band eRASS5 lightcurve of J1331. The blue and grey +markers denote the inferred source and background count rates in the source +aperture, respectively. Times are measured relative to the start of the earliest +observation of J1331 in eRASS5, 𝑡eRASS5,0. J1331 is clearly detected above +background in each visit. +59605 +59610 +59615 +59620 +59625 +MJD - 0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Rate 0.4-2.0 keV [cts s +1] +3C50 background +Total +Figure D3. NICER count rate lightcurve in the 0.4-2 keV band, with blue +markers denoting the total observed count rate (source and background), and +grey markers representing the estimated background rate inferred using the +3C50 background model (Remillard et al. 2022). The system is not detected +at 2𝜎 above background in each NICER OBSID. +10 +2 +10 +1 +10 +0 +10 +1 +10 +2 +10 +3 +t +tpeak [days] +10 +40 +10 +41 +10 +42 +10 +43 +10 +44 +LX [erg s +1] +Figure D4. Comparison of the 0.2–2 keV X-ray lightcurve evolution of J1331 +(red markers) with other soft nuclear transients from quiescent galaxies (or +those recently hosting low luminosity AGN). J1331 decays in 𝐿X over longer +timescales than QPEs (orange for eROQPE1; Arcodia et al. 2021), but still +over much shorter timescales than previously reported TDEs in the literature, +such as ASAS-SN 14li (grey, Bright et al. 2018), AT 2019azh decay phase +(blue, Hinkle et al. 2020), AT 2019dsg (pink, Cannizzaro et al. 2021). The +𝑡peak for J1331 was set to MJD=59592.9, following the assumptions described +in Section C. +MNRAS 000, 1–9 (2015) + +12 +Adam Malyali et al. +57500 +58000 +58500 +59000 +59500 +MJD +50 +0 +50 +100 +150 +200 +F [Jy] +ATLAS o +ATLAS c +57000 +57500 +58000 +58500 +59000 +59500 +MJD +13.8 +14.0 +14.2 +14.4 +14.6 +Vega Magnitude +W1 +W2 +Figure E1. No major variability is seen within the ATLAS forced photometry +generated on the difference imaging (top), nor within the NEOWISE lightcurve +(bottom). +MNRAS 000, 1–9 (2015) + diff --git a/5NE5T4oBgHgl3EQfPA41/content/tmp_files/load_file.txt b/5NE5T4oBgHgl3EQfPA41/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9025eeabeaa645b618087a9de685ddce646b01d9 --- /dev/null +++ b/5NE5T4oBgHgl3EQfPA41/content/tmp_files/load_file.txt @@ -0,0 +1,1444 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf,len=1443 +page_content='MNRAS 000, 1–9 (2015) Preprint 16 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 The rebrightening of a ROSAT-selected tidal disruption event: repeated weak partial disruption flares from a quiescent galaxy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Malyali1★, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Liu1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Rau1, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Grotova1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Merloni1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Goodwin2, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Anderson2, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Miller-Jones2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Kawka2, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Arcodia1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Buchner1, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Nandra1, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Homan3, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Krumpe3 1Max-Planck-Institut für extraterrestrische Physik, Giessenbachstrasse 1, 85748 Garching, Germany 2International Centre for Radio Astronomy Research, Curtin University, GPO Box U1987, Perth, WA 6845, Australia 3Leibniz-Institut für Astrophysik Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' in original form ZZZ ABSTRACT The ROSAT-selected tidal disruption event (TDE) candidate RX J133157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6-324319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7 (J1331), was detected in 1993 as a bright (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV flux of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1) × 10−12 erg s−1 cm−2), ultra-soft (𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='03 keV) X-ray flare from a quiescent galaxy (𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='05189).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' During its fifth All-Sky survey (eRASS5) in 2022, SRG/eROSITA detected the repeated flaring of J1331, where it had rebrightened to an observed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV flux of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7) × 10−13 erg s−1 cm−2, with spectral properties (𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='115 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='007 keV) consistent with the ROSAT-observed flare ∼30 years earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' In this work, we report on X-ray, UV, optical, and radio observations of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' During a pointed XMM observation ∼17 days after the eRASS5 detection, J1331 was not detected in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV band, constraining the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV flux to have decayed by a factor of ≳40 over this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Given the extremely low probability (∼ 5 × 10−6) of observing two independent full TDEs from the same galaxy over a 30 year period, we consider the variability seen in J1331 to be likely caused by two partial TDEs involving a star on an elliptical orbit around a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' J1331-like flares show faster rise and decay timescales (O(days)) compared to standard TDE candidates, with neglible ongoing accretion at late times post-disruption between outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Key words: accretion, accretion discs – galaxies: nuclei – black hole physics – transients: tidal disruption events 1 INTRODUCTION Benefitting from the latest generation of time-domain surveys, the past decade has seen a vast growth in the diversity of observed tran- sients originating from galactic nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' These events can be crudely divided into, and described as, either ‘one-off’ or ‘repeating’ events, depending on the observed evolution of their lightcurves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' ‘One-off’ events, characterised by a single epoch of major tran- sient behaviour over an observed monitoring campaign, comprise the majority of newly reported nuclear transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' These include systems where the variability is likely linked to changes in the accretion pro- cess onto a supermassive black hole, such as has been reported in previously known AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' changing-state AGN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Frederick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Trakhtenbrot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Ricci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Frederick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' short-rise, slowed-decay Bowen accretion flares, Trakht- enbrot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2019a), or due to stellar tidal disruption events (TDEs) in quiescent galaxies1 (see Saxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' van Velzen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020, 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Alexander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020 for recent reviews of X-ray, optical, infrared and radio observations of TDEs, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Other tran- sients, which may occur so close to the centres of galaxies that they are astrometically indistinguishable from SMBH accretion, have also ★ E-mail: amalyali@mpe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='de 1 Strong TDE candidates have also been reported in galaxies showing signs of previous AGN activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Merloni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Blanchard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' been reported (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' supernovae exploding in the narrow-line region of AGN, Drake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2011), or predicted to exist (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' stellar collisions in nuclear star clusters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Dale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Even more recently, the population of known ‘repeating’ events has expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Several TDE candidates have now shown multiple major outbursts, either through their strong, double-peaked optical lightcurves (AT 2019avd, Malyali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2022), repeated X-ray outbursts (IC 3599, Grupe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 1995, 2001, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Campana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' eRASSt J045650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='3-203750, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' AT 2018fyk, Wevers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2022), or quasi-periodic optical outbursts potentially associated with repeated partial TDEs (ASASSN-14ko, Payne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Towards the more extreme end of known re- peating transients lie the recently-discovered class of quasi-periodic eruptions (QPEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Giustini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Arcodia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2021, 2022), which show large amplitude, ultra-soft X-ray out- bursts, with flare duration of the order of hours, and which recur over timescales of hours to days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' In this work, we report on the SRG/eROSITA (Sunyaev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Predehl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2021) detection of the repeated flaring of a previously reported, ROSAT-selected TDE candidate, RXJ133157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6-324319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7 (Reiprich & Greiner 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Hampel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2022), originating from a quiescent galaxy at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='05189 (Moretti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' In Sec- tion 2, we report on the detection of this system with eROSITA and follow-up observations performed with NICER (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2), XMM (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='3), and Swift XRT (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4), as well as archival X-ray © 2015 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='05501v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='HE] 13 Jan 2023 2 Adam Malyali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' observations (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5), UV, optical and mid-infrared photometry (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6) and radio observations (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We discuss the nature of the system in Section 3, before providing a summary in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' All magnitudes are reported in the AB system and corrected for Galactic extinction using 𝐴V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='142 mag, obtained from (Schlafly & Finkbeiner 2011), 𝑅V = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1 and a Cardelli extinction law (Cardelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 1989), unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The effective wavelength for each filter was retrieved from the SVO Filter Profile Service2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' All dates/times will be reported in universal time (UT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2 RE-DISCOVERY AND FOLLOW-UP eRASSt J133157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='9-324321 (herein J1331) was detected on 2022-01- 20 as a bright new X-ray point source in a systematic search for TDE candidates during the fifth eROSITA All-Sky survey (eRASS5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The eROSITA Science Analysis Software (eSASS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Brunner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2022) inferred source position was (RAJ2000, DecJ2000)=(13h31m57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='9s, - 32◦43′21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2′′), with a 1𝜎 positional uncertainty of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' No X-ray point source was detected within 60" of this position in each of the previous four eRASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The eROSITA source position is consistent with a quiescent host galaxy at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='05189, with total stellar mass, log(𝑀★/𝑀⊙) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='09, and an inferred black hole mass, log(𝑀BH/𝑀⊙) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2 (appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The quiescent nature of the host is suggested by both the optical spectrum of its host galaxy (appendix B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' see also Hampel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2022) and its AllWISE (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Mainzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2014) mid-infrared colour, W1-W2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='05± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='05 mag, far below the threshold of ≳0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7 for mid-infrared AGN selection (Stern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Assef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' After selecting J1331 as a promising TDE candidate, it was also realised that the host galaxy of J1331 was the same as that identified for the ROSAT-selected TDE candidate, RXJ133157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6324319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7, first detected in outburst in 1993, and recently presented in Hampel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2022), with the finder chart for these transients presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The eRASS5 detection of J1331 thus suggested the remarkable rebrightening of a previously known TDE candidate, ∼29 years after the outburst detected by ROSAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1 eROSITA Using the eSASS task SRCTOOL (eSASSusers_211214;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Brunner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2022), source (and background) spectra and lightcurves were ex- tracted from a 60" radius source region centred on the eRASS5 inferred position, with background counts extracted from a circular annulus with inner and outer radii of 140" and 240", respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' eROSITA scanned the position of J1331 eight times during eRASS5, with each scan separated by ∼4 hours, thus spanning a ∼28 hour window in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' During this time, J1331 was observed to be persistently bright (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' D2), as opposed to showing a short- lived flaring, and was clearly detected above background in each observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The eRASS5 X-ray spectra were then fitted using the Bayesian X- ray Analysis software (BXA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Buchner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2014), which connects the nested sampling algorithm UltraNest (Buchner 2021) with the fitting environment XSPEC (Arnaud 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The source and back- ground spectra were jointly fit with a source plus background model, with the latter using the Principal Component Analysis (PCA) back- ground modelling first described in Simmonds et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2018), and as 2 http://svo2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='inta-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='es/theory/fps/ also applied to AT 2019avd in Malyali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The eRASS5 spectrum is well fitted by a tbabs*zbbody model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' D1), with the Galactic equivalent neutral hydrogen column density, 𝑁H, fixed to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='84 × 1020 cm−2, the value along the line of sight to J1331 in HI4PI Collaboration: et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2016), and 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='115+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='007 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' A fit with a power-law (tbabs*zpowerlaw) leaves large residuals between the observed data and model above 1 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' When using the best fitting tbabs*zbbody model described above, the eRASS5 observed (un- absorbed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV flux for J1331 is (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7)×10−13 erg s−1 cm−2 ((8 ± 1) × 10−13 erg s−1 cm−2), translating to an unabsorbed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2– 2 keV luminosity of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7) × 1042 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' J1331 was not detected in eRASS1–4, with 2𝜎 upper limits on the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV count rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='016, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='07 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='03 cts s−1 in each successive eRASS (see Table D1 for a full log of the X-ray observations of J1331).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' These count rate upper limits were then converted to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV flux upper limits using the best fitting spectral parameters to the eRASS5 spectrum described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2 NICER XTI Follow-up observations of J1331 were obtained with the X-ray Tim- ing Instrument (XTI) on board the Neutron Star Interior Composition Explorer observatory (NICER;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Gendreau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2016) through pre- approved ToOs (PI: Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Liu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' NICER observations commenced ∼4 days after the last eRASS5 observation, and continued for the next 15 days on a near daily basis (Table D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We first generated cleaned and screened event files using the nicerl2 task (with default recom- mended parameters), before using nibackgen3C50 (Remillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2022) to generate total and background spectra for each observation ID (GTIs were filtered out using hbgcut=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='05 and s0cut=2, as recommended in Remillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' ARF and RMF files were subsequently generated using the tasks nicerarf and nicerrmf, and the X-ray spectra were binned using the Kaastra & Bleeker (2016) method to a minimum of 20 counts per bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The total and background count rates were then estimated in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4–2 keV band3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' J1331 is not detected at 2 sigma above background in each OBSID (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' D3), with 2𝜎 upper limits on the source count rates, inferred using 𝐶𝑅tot +2𝜎, with 𝐶𝑅tot the total measured count rate, and 𝜎 the estimated error on 𝐶𝑅tot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4–2 keV count rates were converted to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV fluxes (Table D1) assuming the eRASS5 spectral model (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' NICER observations rule out a further brightening be- yond eRASS5, or a persistently bright source that rapidly ‘cuts-off’ in brightness by the time of the XMM observation (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='3 XMM J1331 was later observed by XMM (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Liu) on 2022-02-07 (de- noted XMM1), ∼16 days after the last eRASS5 observation, and also on 2022-08-06 (denoted XMM2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Observations were carried out with the medium filter on PN, MOS1 and MOS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The XMM data were reduced using HEASOFT v6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='29, SAS version 20211130_0941, and the latest calibration data files (CALDB v20210915).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Follow- ing standard XMM data reduction procedures, calibrated event files were first generated from the Observation Data Files (ODF) using the SAS tasks emproc and epproc for the MOS and PN cameras respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Then, periods of high background flaring were filtered 3 The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4 keV lower bound here was chosen to reduce contamination from any incompletely modelled optical loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' MNRAS 000, 1–9 (2015) Repeated partial tidal disruption flares from a quiescent galaxy 3 10 41 10 42 10 43 LX [erg s 1] 48000 50000 52000 54000 56000 58000 60000 MJD 10 14 10 13 10 12 FX [erg s 1 cm 2] 49005 49010 49015 10 13 10 12 1992 1996 2000 2004 2008 2012 2016 2020 Year ROSAT XRT eROSITA XMM Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Long-term 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV lightcurve of J1331, with circular and triangle markers representing observed fluxes and 2𝜎 upper limits, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The initial outburst was detected by ROSAT in 1993, before being observed by eROSITA in 2022 to have rebrightened to a similar 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV observed flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The X-ray spectra remained ultra-soft in each observation where the source was detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' For plotting clarity, we include the time-averaged flux measurement for eRASS5, and omit the NICER upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' out4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' For XMM1 (XMM2), this resulted in only 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1ks (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7 ks), 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 ks (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7 ks) and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 ks (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2 ks) of usable exposure time for PN, MOS1 and MOS2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' In the subsequent analysis, only events with PATTERN<=4 and FLAG==0 were extracted for PN, whilst PATTERN<=12 and FLAG==0 filtering was applied for MOS1 and MOS2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' For XMM1, no source is detected within 30" of the host galaxy position in PN and MOS1 with detection likelihood, DETML, above 3, when running the standard XMM source detection pipeline in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV band on the PN, MOS1, and MOS2 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' However, a source was detected in MOS2 at (RAJ2000, DecJ2000)=(13h31m58s, 32◦43′19′′), with a 1𝜎 positional uncertainty of 2′′, consistent with the ROSAT and eROSITA positions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The DETML for this source is low (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='3), and the estimated observed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV flux in the emldetect output is (8±3) ×10−15 erg s−1 cm−2, ∼75× fainter than the eRASS5 observed flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Given the uncertain detection of the system across all three EPIC cameras, we computed a 2𝜎 upper limit on the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV count rate using the SAS task eupper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' This was done using the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV band images, exposure and background maps for each camera, and a 30" radius circular extraction region for the source counts (centred on the Gaia position of the host galaxy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' For XMM1, this yielded upper limits of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='006 ct s−1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0014 ct s−1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='002 ct s−1 for PN, MOS1 and MOS2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We conservatively estimate the upper limit for the XMM observation to that inferred from the MOS2 data, which corresponds to a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV observed (unabsorbed) flux of 1 × 10−14 erg s−1 cm−2 (2 × 10−14 erg s−1 cm−2), assuming the 4 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='int/web/xmm-newton/ sas-thread-epic-filterbackground spectral model inferred from the eRASS5 observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The same procedure was repeated for XMM2, where we inferred upper limits of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='003 ct s−1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0014 ct s−1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0010 ct s−1 for PN, MOS1 and MOS2, respectively, translating to 2𝜎 upper limits on the observed (unobserved) flux of 6×10−15 erg s−1 cm−2 (1×1014 erg s−1 cm−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4 Swift XRT Additional Swift XRT (Burrows et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2005) observations of J1331 were performed between 2022-02-27 and 2022-08-245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The XRT observations were performed in photon counting mode, with the data analysed using the UK Swift Science Data Centre’s (UKSSDC) online XRT product building tool (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2007, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' No source was detected in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='3–2 keV band at the position of J1331 in any follow-up observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='3–2 keV count rates were converted to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV fluxes using webPIMMs6, assuming the same spectral model as from the eROSITA eRASS5 detection, with the fluxes presented in Table D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 Archival X-ray observations A detailed analysis of the ultra-soft outburst from RXJ133157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6324319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7, detected by pointed ROSAT PSPC observations in the early 1990s, was previously performed in Hampel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' In summary, the flare was characterised by an 5 The delay between the eRASS5 and Swift observations stemmed from the January 2022 reaction wheel failure on-board the Swift observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 6 https://heasarc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='gov/cgi-bin/Tools/w3pimms/ w3pimms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='pl MNRAS 000, 1–9 (2015) 4 Adam Malyali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 8x increase in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4 keV flux, relative to a 2𝜎 upper limit, over an 8 day period (and a net increase in the same band by a factor of at least 40 relative to the deepest upper limit available).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The X-ray spectrum at peak observed brightness was well fitted by a blackbody with 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='03 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The system was then not detected in two PSPC observations ∼165 days later, where it had faded by a factor of at least 30 relative to the peak observed ROSAT flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' To construct a long-term 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV lightcurve, the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4 keV ROSAT PSPC lightcurve data in Table 1 of Hampel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2022) was converted into 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV band fluxes using webPIMMS, assuming the best fitting spectral model to the ROSAT spectrum found in Hampel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Then, the 2𝜎 upper limits from ROSAT Survey, XMM Slew and Swift XRT observations were computed using the High-Energy Lightcurve Generator server (HILIGT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Saxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' König et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the archival fluxes are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 1 and Table D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6 UV, optical and mid-infrared photometry J1331 was observed both before (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5) and after (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4) the eRASS5-detected outburst by Swift XRT and UVOT (UVM2 filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Roming et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' To search for transient UV emission, aperture photometry was performed on the level 2 UVOT sky im- ages (downloaded from the UKSSDC) using the uvotsource task (HEASOFT v6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='29, CALDB v20201215).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Source counts were ex- tracted from a circular aperture of 5′′ radius, centred on the Gaia position of the host of J1331, and background counts were extracted from a source-free region of radius 15′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The measured UVM2 mag- nitudes in the follow-up observations are consistent with the archival measured UVM2 magnitudes on the 2018-04-18, 2018-04-22, 2018- 04-26 (Table E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' No significant optical variability is seen in the ∼6 years before the eRASS5 outburst (57500≲ MJD ≲59500) in the forced photom- etry lightcurve provided by ATLAS (Tonry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2018) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Lastly, we note that no major variability is detected above the host galaxy emission within the NEOWISE mid-infrared lightcurve be- tween MJD∼56680 and 59400 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' E1), which was generated using the procedure described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2 of Malyali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7 Radio We observed the coordinates of J1331 on 2022 Mar 02 with the Australia Telescope Compact Array (ATCA) radio telescope in 6 km configuration, using the 4cm dual receiver with central frequencies 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5/9 GHz, each with a 2 GHz bandwidth split into 2049×1 MHz spectral channels, and for a total of 150 min on source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Data were reduced following standard procedures in the Common Astronomy Software Applications (McMullin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' CASA-TEAM et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We used 1934-638 for flux and bandpass calibration and 1336- 260 for phase calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Images of the target field were created using the CASA task tclean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' No source was detected at the location of J1331 at either frequency band with a 3𝜎 upper limit of 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5𝜇Jy/bm at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 GHz and 54𝜇Jy/bm at 9 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Additionally, no source was detected in a stacked 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 and 9 GHz image, with a 3𝜎 upper limit of 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='9𝜇Jy/bm at a central frequency of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='3 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 3 DISCUSSION Comparing the X-ray lightcurve of J1331 with other ultra-soft nu- clear transients (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' D4) from galaxies that were recently quiescent, or hosted low luminosity AGN, then J1331 decays faster than the majority of other X-ray bright TDEs7, but decays over much longer timescales than the bursts typically seen in QPEs (burst durations ≲30 ks, or ≲0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='3 days;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Giustini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Arcodia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Given the quiescent nature of the host galaxy, and the ultra-soft X-ray spectrum, an AGN origin for J1331 is disfavoured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We also rule out a mechanism similar to that producing the X-ray flares ob- served in Sgr A* (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Neilsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Ponti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Yuan & Wang 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Ponti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Mossoux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020), as the latter are clearly observationally distinct to J1331, with respect to the flaring timescales (Sgr A* flare durations ≲ 104 s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Mossoux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020), spectral properties (flaring X-ray emission in Sgr A* is hard and likely synchrotron, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Ponti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2017), and peak observed lumi- nosity (bolometric luminosity of Sgr A* is ∼ 1036 erg s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Arguments against a Galactic origin for this system have previously been presented in Hampel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Ultra-soft X-ray flares from quiescent galaxies have previously been considered as a reliable signature of a TDE (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Zabludoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' However, the current theoretically predicted TDE rates are ≳ 10−4 yr−1 galaxy−1 (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020), so it would be exceptionally unlikely to have observed two independent tidal disruption flares occuring within the same galaxy over a ∼30 year timescale (Poisson probability ∼ 5 × 10−6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' C2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' a more exotic class of TDE would need to be invoked to explain J1331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' One such possibility, discussed in Hampel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2022), is that J1331 was produced by a TDE involving a supermassive black hole binary (SMBHB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' This scenario was partly proposed in an attempt to explain the fast X-ray brightening observed by ROSAT, since such TDEs may have highly non-monotonic decays of their X-ray lightcurves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' This stems from the gravitational interaction between the companion BH and the debris streams, which may cause large perturbations to the orbits of the less bound debris and cause their chaotic evolution, as well as a complex evolution of the accretion rate over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Ricarte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Coughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2017) predict these systems to show sharp dips and rises in the X- ray lightcurve rate (of ∼1–2 orders of magnitude), on timescales of the order of the binary orbital period (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Ricarte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2016), although Coughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2017) find highly variable accretion rates between different simulation runs and over timescales shorter than the SMBHB orbital periods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' there still seems to be quite large uncertainties in the theoretically predicted lightcurves of TDEs involving SMBHBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Under the SMBHB scenario, both the eROSITA and ROSAT obser- vations would have had to have sampled a ‘dipping’, or ‘brightening from a dip’, phase of the X-ray lightcurve, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' For binary orbital periods of the order of ∼months, assuming ∼mpc binary sep- aration as in Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2014), then it would be quite fortuitous for us to have observed such behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Furthermore, there is importantly no evidence for late time X-ray rebrightening episodes in the months after each outburst, as seen by XMM and Swift (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 1), which one might expect to have observed given that the accretion rate is pre- dicted to eventually revert back to the 𝑡−5/3 decay following ‘dips’ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 12 in Coughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We would therefore disfavour J1331 being caused by a full TDE around a SMBHB, given the fine tuning needed in order to match observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' A more feasible scenario is that both outbursts were driven by a partial tidal disruption event (pTDE), potentially of the same object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Unless the pTDE rate is orders of magnitude larger than currently 7 Ignoring short timescale flaring behaviour seen in some TDE candidates, such as AT 2019ehz (van Velzen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' MNRAS 000, 1–9 (2015) Repeated partial tidal disruption flares from a quiescent galaxy 5 10 40 10 41 10 42 10 43 LX [erg s 1] 10 1 10 2 MJD - 59581 10 15 10 14 10 13 10 12 FX [erg s 1 cm 2] t 5/3 t 9/4 t 4 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Zoom-in on the first eROSITA-detected outburst in 2022, along with multiple power-law decay slopes plotted in grey dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The decay slope appears to be much steeper than the canonical 𝑡−5/3 decay predicted for TDEs with a uniform distribution of specific energies, and appears more consistent with a 𝑡−4 decay, as predicted in Ryu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We assume a peak MJD of 59593 for the X-ray outburst, and roughly estimate the MJD of disruption to be 59581 (section C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The markers follow the same legend as for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' estimated in the literature (Stone & Metzger 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Chen & Shen 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2022), then both outbursts would likely be re- lated to the same star being disrupted by the same black hole (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the star should have survived the initial encounter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Considering that the recurrence timescale of J1331 is ≲ 30 years, then it is also diffi- cult to reconcile this with theoretical predictions for the recurrence timescales of flares in pTDEs where the star was initially scattered onto a parabolic orbit around the black hole (≳ 400 years, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Ryu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Instead, the flaring may have been driven by the repeated stripping of a star on an elliptical orbit by the disrupting SMBH (see Hayasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2013 for a discussion on potential origins for such stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' This scenario would be further supported by both the relatively small amount of inferred energy emitted in the eROSITA-detected outburst8 of (5+6 −3) × 1049 erg, corresponding to an accreted mass of (5+7 −2) × 10−4(𝜖/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='05)−1 M⊙, where 𝜖 is the radiative efficiency of accretion, and also by the extremely low 𝐿X at late-times (as sug- gested by the non-detection and deep upper limits in XMM2), since elliptical TDEs are predicted to produce short-lived, finite accretion bursts (Hayasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Given this, and that the radio obser- vations were taken ∼40 days after the eRASS5 flare (section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7), then we note that we may have missed any associated jet or out- flow launched in this event, as seen in other TDE candidates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Goodwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The case for a repeated pTDE is further enhanced by the fast rise and decay timescales seen with ROSAT and eROSITA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Compared with full disruptions, pTDEs only strip the outermost layers of the star, with the specific energy distribution of the debris, d𝑀/d𝐸, differing from full TDEs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Coughlin & Nixon 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Miles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Ryu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Since the mass fallback rate, �𝑀fb(𝑡), scales ∝ d𝑀/d𝐸, then �𝑀fb(𝑡) is also predicted to differ between full and pTDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Ryu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2020) find that the narrower spreads in d𝑀/d𝐸 for pTDEs can yield �𝑀fb(𝑡) ∝ 𝑡−𝑝, where 𝑝 ∼ 2−5, more consistent with what is observed in J1331 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2), and much steeper than a canonical 𝑡−5/3 decline predicted for the mass fallback rate in full TDEs (Rees 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Phinney 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Lastly, although the mass fallback in weak pTDEs may evolve over shorter timescales relative to full TDEs, the viscous timescale, 8 Assuming a similar temporal evolution for both the eROSITA-detected and ROSAT-detected outbursts- see section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 𝑡visc, still needs to be shorter than the minimum orbital period of the stellar debris so that the X-ray luminosity traces the mass fallback rate (assuming a constant radiative efficiency, negligible obscuration of the soft X-rays, and negligible disc cooling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Considering 𝑡visc ∼ 𝛼−1(𝐻/𝑅)−2Ω−1(𝑟), where 𝛼 is the viscosity parameter (Shakura & Sunyaev 1973), 𝐻 and 𝑅 the scale height and width of the disc, and Ω−1(𝑟) the orbital period at distance 𝑟 from the black hole, then 𝑡visc ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4(𝛼/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1)−1(𝐻/𝑅)−2 days at the circularisation radius (∼ 2𝑅tidal/𝛽, where 𝑅tidal and 𝛽 are the tidal radius and impact parameter for the disruption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' A geometrically thick disc (𝐻/𝑅 ∼ 1), as may be expected to form for super-Eddington mass fallback rates, would be needed to reproduce accretion timescales of the order ∼days as seen in J1331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' However, it is currently unclear how the stellar debris might circularise so efficiently in a weak pTDE (see Bonnerot & Stone 2021 for a review on accretion flow formation in TDEs), and we also highlight here that similar concerns have recently been raised for explaining the short X-ray flare durations observed in QPEs via an accretion origin (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Krolik & Linial 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Lu & Quataert 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Although future simulations would likely be needed to explore the debris circularisation in J1331-like events, alternative origins for the X-ray emission may be from compression shocks of the debris streams at pericentre (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Steinberg & Stone 2022), or circularisation shocks from debris stream collisions (Krolik & Linial 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Lu & Quataert 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 4 SUMMARY J1331 is a repeating X-ray transient associated to a quiescent galaxy at 𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='05189, which we consider to be consistent with a scenario involving two weak pTDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Whilst several previously reported pTDE candidates have occurred in galaxies hosting an AGN, we highlight that the host of J1331 is quiescent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The main properties of J1331 can be summarised as follows: (i) J1331 was first detected by ROSAT in 1993 (Hampel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2022), where it had shown an ultra-soft (𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='03 keV) flaring by a factor of at least 40 relative to a previous 2𝜎 upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The outburst also showed a fast rise, where it had brightened by a factor of eight over an 8 day period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The system was subsequently not detected in a deep pointed ROSAT observation ∼165 days afterwards, as well as in XMM Slew, and Swift XRT observations performed between 2006 and 2018 (Table D1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (ii) After not being detected by eROSITA in its first four eRASS, J1331 was observed to have brightened in eRASS5 to a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV flux of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7) × 10−13 erg s−1 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The eRASS5 spectrum is ultra-soft (𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='115+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='007 keV), and is consistent with the 𝑘𝑇 inferred from the ROSAT-observed flare in 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (iii) J1331 was not detected during pointed XMM observations and Swift XRT observations when followed up after the eRASS5 detection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the first (second) XMM observation constrains the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2– 2 keV flux to decay by a factor of ≳40 (≳100) over a 17 (∼200) day period after the eRASS5 observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The faint 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV X-ray luminosities (< 7×1040 erg s−1, unabsorbed) at ∼ 200 days post-peak brightness, inferred via the second XMM observation (Table D1), may be due to a late-time drop off in the mass fallback rate once the disruption episode is over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (iv) Combined with the fast rise timescale seen by ROSAT, then J1331-like outbursts are short lived (rise and decay timescales of 6+1 −1 days and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1 days, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' appendix C) and evolve over shorter timescales relative to full TDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (v) J1331 has only been observed to show transient emission in MNRAS 000, 1–9 (2015) 6 Adam Malyali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV band, with no transient optical, UV, or radio emission observed in follow-up observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We conclude by noting that J1331 appears to fill in the continuum of observed soft X-ray outbursts from quiescent galaxies, lying in be- tween QPEs and TDEs with respect to its rise and decay timescales (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' D4), although the recurrence timescales are much longer than in the current sample of QPEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Additional follow-up observations will be scheduled in order to more tightly constrain the recurrence timescales of outbursts from J1331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Future planned X-ray missions geared towards exploiting the X-ray transient sky, such as the Einstein Probe (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2018), will likely be sensitive towards detecting similar partial disruptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' for these missions, the eROSITA All-Sky survey data may play an important role by providing a long-term baseline towards which new candidates can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Given the faster decay timescales of J1331-like systems, then we would advo- cate promptly triggering high-cadence X-ray follow-up in order to better constrain the evolution of the accretion rate in future candi- dates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' ACKNOWLEDGEMENTS AM thanks Taeho Ryu for very useful discussions whilst preparing the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' AM acknowledges support by DLR under the grant 50 QR 2110 (XMM_NuTra, PI: Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Liu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' This work was supported by the Australian government through the Australian Research Council’s Discovery Projects funding scheme (DP200102471).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We would like to thank the referee for a constructive report that improved the quality of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' This work is based on data from eROSITA, the soft X-ray instru- ment aboard SRG, a joint Russian-German science mission supported by the Russian Space Agency (Roskosmos), in the interests of the Russian Academy of Sciences represented by its Space Research In- stitute (IKI), and the Deutsches Zentrum für Luft- und Raumfahrt (DLR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The SRG spacecraft was built by Lavochkin Association (NPOL) and its subcontractors, and is operated by NPOL with sup- port from the Max Planck Institute for Extraterrestrial Physics (MPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The development and construction of the eROSITA X-ray instru- ment was led by MPE, with contributions from the Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Karl Re- meis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The eROSITA data shown here were processed using the eSASS software system developed by the German eROSITA consortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The Australia Telescope Compact Array is part of the Aus- tralia Telescope National Facility (https://ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='org/05qajvd42) which is funded by the Australian Government for operation as a National Facility managed by CSIRO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We acknowledge the Gomeroi people as the traditional owners of the Observatory site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The Legacy Surveys consist of three individual and complemen- tary projects: the Dark Energy Camera Legacy Survey (DECaLS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Proposal ID 2014B-0404;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' PIs: David Schlegel and Arjun Dey), the Beijing-Arizona Sky Survey (BASS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' NOAO Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' ID #2015A-0801;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' PIs: Zhou Xu and Xiaohui Fan), and the Mayall z-band Legacy Sur- vey (MzLS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' ID #2016A-0453;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' PI: Arjun Dey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' DECaLS, BASS and MzLS together include data obtained, respectively, at the Blanco telescope, Cerro Tololo Inter-American Observatory, NSF’s NOIR- Lab;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the Bok telescope, Steward Observatory, University of Arizona;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' and the Mayall telescope, Kitt Peak National Observatory, NOIR- Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Pipeline processing and analyses of the data were supported by NOIRLab and the Lawrence Berkeley National Laboratory (LBNL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The Legacy Surveys project is honored to be permitted to conduct astronomical research on Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the Tohono O’odham Nation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' NOIRLab is operated by the Association of Universities for Re- search in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' LBNL is managed by the Regents of the University of California under contract to the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Department of Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Funding for the DES Projects has been provided by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Department of Energy, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' National Science Foundation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the Ministry of Science and Education of Spain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the Science and Technology Facilities Council of the United Kingdom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the Higher Education Funding Council for England,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the National Center for Supercomputing Applications at the University of Illinois at Urbana- Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the Kavli Institute of Cosmological Physics at the Uni- versity of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Center for Cosmology and Astro-Particle Physics at the Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Financiadora de Estudos e Projetos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Fundacao Carlos Chagas Filho de Amparo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Fi- nanciadora de Estudos e Projetos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Fundacao Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Conselho Na- cional de Desenvolvimento Cientifico e Tecnologico and the Minis- terio da Ciencia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Tecnologia e Inovacao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the Deutsche Forschungs- gemeinschaft and the Collaborating Institutions in the Dark Energy Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The Collaborating Institutions are Argonne National Labo- ratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the University of California at Santa Cruz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Centro de Investigaciones Energeticas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Medioambien- tales y Tecnologicas-Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' University College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the DES-Brazil Consortium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the University of Ed- inburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the Eidgenossische Technische Hochschule (ETH) Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Fermi National Accelerator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the University of Illinois at Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the Institut de Ciencies de l’Espai (IEEC/CSIC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the Institut de Fisica d’Altes Energies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the Ludwig Maximilians Universitat Munchen and the associated Excellence Cluster Universe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' NSF’s NOIRLab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the University of Nottingham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the Ohio State Uni- versity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the University of Pennsylvania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' SLAC National Accelerator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Stanford University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' the Uni- versity of Sussex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' and Texas A&M University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' BASS is a key project of the Telescope Access Program (TAP), which has been funded by the National Astronomical Observatories of China, the Chinese Academy of Sciences (the Strategic Prior- ity Research Program “The Emergence of Cosmological Structures” Grant # XDB09000000), and the Special Fund for Astronomy from the Ministry of Finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The BASS is also supported by the Exter- nal Cooperation Program of Chinese Academy of Sciences (Grant # 114A11KYSB20160057), and Chinese National Natural Science Foundation (Grant # 12120101003, # 11433005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The Legacy Survey team makes use of data products from the Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), which is a project of the Jet Propulsion Laboratory/California Insti- tute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' NEOWISE is funded by the National Aeronautics and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The Legacy Surveys imaging of the DESI footprint is supported by the Director, Office of Science, Office of High Energy Physics MNRAS 000, 1–9 (2015) Repeated partial tidal disruption flares from a quiescent galaxy 7 of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Department of Energy under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' DE-AC02- 05CH1123, by the National Energy Research Scientific Comput- ing Center, a DOE Office of Science User Facility under the same contract;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' and by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' National Science Foundation, Division of Astronomical Sciences under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' AST-0950945 to NOAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' acknowledges support from DFG grant KR 3338/4-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' is supported by DLR grant FKZ 50OR2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' DATA AVAILABILITY The eRASS1-4 data taken within the German half of the eROSITA sky is currently planned to be made public by Q2 2024, whilst the eRASS5 data is scheduled to become public by Q2 2026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The Swift data is available to download through the UK Swift Data Sci- ence website9, whilst the NICER data is accessible through NASA’s HEASARC interface10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Publicly available ATLAS data can be ac- cessed through the ATLAS forced photometry service11, and NEO- WISE lightcurves can be accessed through the IRSA web portal12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' ATCA data are stored in the Australia Telescope Online Archive13, and will become publicly accessible 18 months from the date of ob- servation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The XMM data will become public after the propietory period expires (2023-08-30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Follow-up optical spectra will likely remain private at least until the release of the forthcoming eROSITA- selected TDE population paper, but could be made available upon reasonable request.' metadata={'source': 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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=', Komossa S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=', Yan L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=', Kara E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=', 2021a, Space Science Reviews, 217, 63 van Velzen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=', 2021b, The Astrophysical Journal, 908, 4 APPENDIX A: HOST GALAXY PROPERTIES Using the correlation reported in Kettlety et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2018) between galaxy total stellar mass, 𝑀★, and luminosity in the WISE 𝑊1-band, 13 h32 m00 s 31 m58 s 57 s 56 s 32°43\'00" 15" 30" 45" J2000 J2000 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Legacy Survey DR10 (early) 𝑔-band cutout image of the sky region surrounding eRASSt J133158-324321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The dark orange circle is the error circle for RXJ133157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6324319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7 inferred from ROSAT pointed obser- vations in Hampel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2022), whilst the red and blue circles denote the 3𝜎 error circles on the source position inferred from eROSITA and XMM MOS2 observations (although the detection of J1331 in the first XMM observation is uncertain and we quote upper limits on the count rates for this in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='3, we include it in this finder chart for completeness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The cyan star marks the Gaia EDR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2021) position of the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 𝐿W1, then we infer log(𝑀★/𝑀⊙) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='09 for the host galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Combining this with 𝑀BH − 𝑀★ relation in Reines & Volonteri (2015), suggests a black hole mass of log(𝑀BH/𝑀⊙) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The finder chart for J1331 is presented in Fig A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' APPENDIX B: OPTICAL SPECTROSCOPY LCO spectrum (2022-02-12): J1331 was observed with the low dispersion FLOYDS spectrograph on the LCOGT 2m telescope at Siding Spring Observatory operated by the Las Cumbres Observa- tory (LCO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2013) on 2022 February 12 (proposal ID CON2022A-001, PI: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Salvato).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We obtained an exposure of 1800 seconds using the “red/blu” grism and the 2” slit oriented along the parallactic angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The spectrum has a wavelength range of 3200- 10000A with dispersions of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='51A/pixel and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='74 A/pixel in the blue (3200-5700A) and red (5400-10000A) bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The data were reduced and calibrated using the automatic FLOYDS pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The HgAr and Zn lamps were used for wavelength calibration and a Tungsten-Halogen + Xenon lamp for flat fielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' A sensitivity function from the FLOYDS archive was used for flux calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' WiFeS spectrum (2022-05-09): We observed J1331 with the Wide Field Spectrograph (WiFeS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Dopita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2010) on the ANU 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='3m telescope at Siding Spring Observatory on 2022 May 08 (proposal ID 2220157, PI Miller-Jones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We obtained 2x2400 s exposures us- ing the R3000 and B3000 gratings and a NeAr arc lamp exposure immediately following the target exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The data were reduced using standard procedures including the PyWiFeS reduction pipeline (Childress et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' LTT4364 was used as the flux standard and a quartz-iodine lamp was used for flat-fielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We then chose the slitlets with the most significant flux from the calibrated spectra MNRAS 000, 1–9 (2015) Repeated partial tidal disruption flares from a quiescent galaxy 9 5500 6000 6500 7000 7500 8000 Rest Wavelength [Å] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 F [10 16 erg cm 2 s 1 Å 1] 2022-02-12: LCO 2022-05-09: WiFeS Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Optical spectra of J1331, with the first follow-up spectrum being obtained on 2022-02-12, ∼23 days after the last eRASS5 detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' obtained from the pipeline and performed background subtraction, resulting in a spectrum with spectral range 3500 to 9000 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Each follow-up optical spectrum appears to be consistent with a quiescent host galaxy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' B1), with no TDE-like optical emission features detected, nor any transient features relative to the NOT spec- trum taken on 1999-01-26 and presented in Hampel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' APPENDIX C: INFERRING THE OUTBURST PROPERTIES To obtain a coarse reconstruction of the 2022 outburst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' we perform a joint fit of the rising lightcurve from 1993,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' observed by ROSAT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' and the decay lightcurve from 2022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' observed by eROSITA and XMM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' using: 𝐹X(𝑡) = 𝐹X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='max × � exp � −(𝑡 − 𝑡peak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1)2/2𝜎2� if 𝑡 < 𝑡peak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1 exp � −(𝑡 − 𝑡peak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2)/𝜏 � if 𝑡 > 𝑡peak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2 (C1) where the free parameters of this model are 𝜎 (the rise timescale),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 𝑡peak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1 and 𝑡peak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2 (the peak time of the ROSAT and eROSITA out- bursts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' respectively),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 𝜏 (the decay timescale),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' and 𝐹X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='max (the peak flux of both outbursts),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' with the priors on these parameters listed in Table C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We assume that the upper bound on the peak luminosity must be less than the Eddington luminosity for the SMBH, and that both outbursts have the same peak luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' We then assume that the rise for 2022 outburst was similar to the 1993 outburst (see below), and use its modelled rise to approximate that of the unobserved rise of the 2022 outburst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' From this fittedlightcurve model (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' C1), we then computed the integrated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV luminosity, and corrected this to a bolometric luminosity using the best fitting X-ray spectral model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The inferred energy emitted in each outburst is (5+6 −3) × 1049 erg, corresponding to an accreted mass of (5+7 −2) × 10−4(𝜖/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='05)−1 M⊙, where 𝜖 is the radiative efficiency of accretion, whilst the inferred peak MJD for each outburst are 49024+6 −6 and 59593+3 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The inferred 𝜎 and 𝜏 are 6+1 −1 days and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1 days, respectively, and we roughly estimate the MJD of disruption to be 59593 − 2 ∗ 𝜎 ∼ 59581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' It is of course extremely important to consider that these estimates are subject to a number of caveats, mainly related to our observations not covering the rise of the 2022 outburst, such that the estimated values here should be treated with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' For example, it is assumed that the outburst can be well modelled by equation C1, and that both the 1993 and 2022 outbursts are similar, whereas the actual Table C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Priors adopted in the fitting of the 1993 and 2022 outbursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The rise and decay timescales are in units of days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 𝑡peak,1 and 𝑡peak,2 are in MJD, whilst 𝐹max is the maximum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV flux of each outburst (with upper bound set by the Eddington luminosity of the system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Parameter Prior log[𝜎] ∼ U(0, log[50]) 𝑡peak,1 ∼ U(49006, 49178) 𝑡peak,2 ∼ U(58450, 58650) log[𝜏] ∼ U(0, log[50]) log[𝐹X,max] ∼ U(log[5 × 10−13], log[4 × 10−11]) 10 40 10 42 10 44 LX [erg s 1] 59560 59580 59600 59620 59640 MJD 10 16 10 14 10 12 FX [erg s 1 cm 2] Figure C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Inferred full outburst (red) for the flaring observed by eROSITA in 2022, assuming the model described in equation C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The markers follow the same legend as for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The darker and lighter shaded red bands enclose the inner 68% and 98% of the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' lightcurve may have had an extended plateau phase prior to the eROSITA detection (so our estimated fluence and accreted mass would be underestimated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' However, if the 2022 outburst does evolve relatively closely to the functional form in equation C1, then it may be reasonable to consider that the rise timescale for the flare in 1993 is similar to that observed in 2022 (under a tidal disruption scenario), due to the approximately constant eccentricity of the stellar remnant after repeated partial disruptions (Antonini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2011), and the weak dependence of the period of the most bound debris on the stellar mass (Hayasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' APPENDIX D: ADDITIONAL X-RAY INFORMATION The BXA fitted model to the eRASS5 spectrum is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' D1, and the eRASS5 lightcurve is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The NICER count rate lightcurve is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' D3, whilst the full X-ray lightcurve of J1331 is presented in Table D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' A comparison of the X-ray lightcurve of J1331 with other nuclear transients is presented in Fig D4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' APPENDIX E: ADDITIONAL PHOTOMETRIC INFORMATION Table E1 contains the Swift UVOT aperture photometry of the host galaxy of J1331, whilst Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' E1 shows the long term ATLAS and NEOWISE lightcurves of J1331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' MNRAS 000, 1–9 (2015) 10 Adam Malyali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Table D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' X-ray lightcurve table for J1331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The fluxes from the ROSAT pointed observations were derived from Hampel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The first four eROSITA observations listed, between MJD 58868 and 59419, are upper limits estimated from eRASS1, 2, 3 and 4, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' eROSITA fluxes outside of this window have been computed from the individual visits within eRASS5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' MJD Observation 𝐹0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2−2keV,obs 𝐹0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2−2keV,unabs [10−13 erg cm−2 s−1] [10−13 erg cm−2 s−1] 48260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='000 ROSAT/ RASS < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='9 < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 48844.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='598 ROSAT/ Pointed < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4 49006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='094 ROSAT/ Pointed < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='9 49012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='146 ROSAT/ Pointed 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 49012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='180 ROSAT/ Pointed 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='9 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='9 49013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='591 ROSAT/ Pointed 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7 49178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='555 ROSAT/ Pointed < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 49178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='766 ROSAT/ Pointed < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 53745.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='532 SRG/ eROSITA < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7 59599.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='448 SRG/ eROSITA 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7 59600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='281 SRG/ eROSITA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 59600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='448 SRG/ eROSITA 9.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6 59604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='892 NICER/ XTI <8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6 <13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 59605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='566 NICER/ XTI <10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 <15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 59623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='102 NICER/ XTI <12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2 <19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6 59624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='362 NICER/ XTI <6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 <10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 59638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='031 Swift/ XRT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4 59766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='375 Swift/ XRT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2 59773.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='061 Swift/ XRT < 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6 < 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='7 59774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='292 Swift/ XRT < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='9 59778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='974 Swift/ XRT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4 59780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='760 Swift/ XRT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 59787.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='468 Swift/ XRT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4 59794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='352 Swift/ XRT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4 59797.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='916 XMM/ Pointed < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='06 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='10 59801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='282 Swift/ XRT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='5 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 59808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='180 Swift/ XRT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='9 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6 59815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='534 Swift/ XRT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4 10 3 10 2 10 1 10 0 10 1 TDE rate, [30 yr 1 gal 1] 10 6 10 4 10 2 p(N 2)| ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='05 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='15 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='003 Figure C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Poisson probability of 𝑁 ≥ 2 TDEs occurring within a 30 year period for a given galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The red dotted lines mark the estimated probability for current theoretical estimates for TDE rates (10−4 yr−1 gal−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The grey dashed lines mark out the TDE rates of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='01 per 30 yr−1 gal−1, required to produce probabilities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='001, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0001, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 Energy [keV] 10 4 10 2 100 Counts s 1 keV 1 Figure D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' BXA fit of a tbabs*zbbody model to the eRASS5 spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The solid red line represents the median model fit, whilst the shaded red region encloses the inner 98% of the credible region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The X-ray spectrum is ultra-soft with 𝑘𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='115+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='007 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='007 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Table E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Swift UVM2 photometry of the host galaxy of J1331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' MJD Magnitude 58226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='727 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='1 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 58230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='747 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='9 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6 58234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='068 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='3 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4 59638.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV band eRASS5 lightcurve of J1331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The blue and grey markers denote the inferred source and background count rates in the source aperture, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Times are measured relative to the start of the earliest observation of J1331 in eRASS5, 𝑡eRASS5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' J1331 is clearly detected above background in each visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 59605 59610 59615 59620 59625 MJD - 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 keV [cts s 1] 3C50 background Total Figure D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' NICER count rate lightcurve in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4-2 keV band, with blue markers denoting the total observed count rate (source and background), and grey markers representing the estimated background rate inferred using the 3C50 background model (Remillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The system is not detected at 2𝜎 above background in each NICER OBSID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 10 2 10 1 10 0 10 1 10 2 10 3 t tpeak [days] 10 40 10 41 10 42 10 43 10 44 LX [erg s 1] Figure D4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Comparison of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2–2 keV X-ray lightcurve evolution of J1331 (red markers) with other soft nuclear transients from quiescent galaxies (or those recently hosting low luminosity AGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' J1331 decays in 𝐿X over longer timescales than QPEs (orange for eROQPE1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' Arcodia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2021), but still over much shorter timescales than previously reported TDEs in the literature, such as ASAS-SN 14li (grey, Bright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2018), AT 2019azh decay phase (blue, Hinkle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2020), AT 2019dsg (pink, Cannizzaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' The 𝑡peak for J1331 was set to MJD=59592.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='9, following the assumptions described in Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' MNRAS 000, 1–9 (2015) 12 Adam Malyali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' 57500 58000 58500 59000 59500 MJD 50 0 50 100 150 200 F [Jy] ATLAS o ATLAS c 57000 57500 58000 58500 59000 59500 MJD 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='4 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content='6 Vega Magnitude W1 W2 Figure E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' No major variability is seen within the ATLAS forced photometry generated on the difference imaging (top), nor within the NEOWISE lightcurve (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} +page_content=' MNRAS 000, 1–9 (2015)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE5T4oBgHgl3EQfPA41/content/2301.05501v1.pdf'} diff --git a/6dE1T4oBgHgl3EQf7AUK/content/tmp_files/2301.03528v1.pdf.txt b/6dE1T4oBgHgl3EQf7AUK/content/tmp_files/2301.03528v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ef9809a32ffb4dfe0e8f16bf4f93422edd17c572 --- /dev/null +++ b/6dE1T4oBgHgl3EQf7AUK/content/tmp_files/2301.03528v1.pdf.txt @@ -0,0 +1,717 @@ +Multi-point Padè for the study of phase transitions: from +the Ising model to lattice QCD +Francesco Di Renzo∗ and Simran Singh +Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma +and INFN, Gruppo Collegato di Parma, I-43100, Parma, Italy +E-mail: francesco.direnzo@unipr.it, simran.singh@unipr.it +The Bielefeld Parma collaboration has recently put forward a method to investigate the QCD phase +diagram based on the computation of Taylor series coefficients at both zero and imaginary values +of the baryonic chemical potential. The method is based on the computation of multi-point Padé +approximants. We review the methodological aspects of the computation and, in order to gain +confidence in the approach, we report on the application of the method to the two-dimensional +Ising model (probably the most popular arena for testing tools in the study of phase transitions). +Besides showing the effectiveness of the multi-point Padé approach, we discuss what these results +can suggest in view of further progress in the study of the QCD phase diagram. We finally report +on very preliminary results in which we look for Padé approximants at different temperatures and +fixed values of the (imaginary) baryonic chemical potential. +The 39th International Symposium on Lattice Field Theory (Lattice2022), +8-13 August, 2022 +Bonn, Germany +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.03528v1 [hep-lat] 9 Jan 2023 + +Multi-point Padè for the study of phase transitions +Francesco Di Renzo +1. +How it all began: from Taylor expansions on thimbles to imaginary 𝜇𝐵 LQCD +The QCD phase diagram is still to a large extent elusive: in particular, due to the so-called sign +problem, the lattice (the non-perturbative tool which would be supposed to provide valuable insight) +cannot probe the relevant regions in the 𝑇 − 𝜇𝐵 (Temperature-baryonic chemical potential) plane. +In the last couple of years, the Bielefeld-Parma collaboration put forward a method to compute +finite-density QCD thermodynamic observables in the region to which access would be precluded +by the sign problem; this approach is also able to probe the singualrity structure of the theory in the +complex 𝜇𝐵 plane [1–4]. The method is based on the computation of Taylor series coefficients at +both zero and imaginary values of the baryonic chemical potential, which enables the computation +of multi-point Padé approximants. This work aims to assess the effectiveness of the method by +making use of it in the context of a very standard playground for the physics of phase transitions (e.g. +the 2d Ising model). At the same time, we present (very) preliminary results on new applications +in the context of finite-density QCD. +Before entering the main subject, it is useful to recall when the idea of applying multi-point +Padé rational approximants first came to our mind; that was in the context of thimble regularisation. +The latter [5, 6] was introduced to solve (or at least tame) the sign problem by re-expressing the +path integral as a sum of integrals computed on manifolds different from the original one. After +complexifying the degrees of freedom, one considers the so-called Lefschetz thimbles, i.e. the +manifolds that are the union of the steepest ascent paths stemming from the various stationary +points of the action. On such manifolds the imaginary part of the action stays constant, so that +the sign problem reduces to the so-called residual phase which is there due to the Jacobian of +the change of variables. There is a thimble attached to each stationary point and in principle all +can give a contribution to the path integral. This is referred to as the thimble decomposition. To +make a long story short, we recall that (a) not all the thimbles give a non-null contribution, (b) +this picture changes in different regions of the parameters space of the theory (i.e. a given thimble +can contribute to the path integral in a region and not in another one) and (c) there are cases in +which a single thimble (usually the so called dominant one, attached to the stationary point with +the lowest action) is enough to compute the answer one is interested in. The latter observation +gave raise to the single thimble dominance hypothesis, which was shown to hold in a few cases, +but failed in others. The first example of a failure was provided by the 1-D Thirring model [7, 8], +where it was clearly shown that a single thimble is not enough to account for the known analytic +result. It is nevertheless important to remark that there are regions in which one single thimble is +enough, and this was the logical starting point for the success of a computation based on multi-point +Padé rational approximants. The success of such approach [9] can be recognised in Fig. 1. On +the left, we display the known analytic result for the chiral condensate ¯𝜒𝜒 of the 1-D Thirring +model (𝐿 = 8, 𝑚 = 1, 𝛽 = 1) at various values of the chemical potential by mass ratio 𝜇 +𝑚. This +is plotted together with the numerical results which we got: triangles are results computed on one +single thimble at points where we are able to show that this is enough; dots are results taken from +the multi-point Padé method that we will better describe in the next section. Here it is enough to +say that a few Taylor expansion coefficients were computed at the points marked by triangles and +from those the multi-point Padé approximant was computed. The right panel of the figure shows +2 + +Multi-point Padè for the study of phase transitions +Francesco Di Renzo +how the singularity pattern of the solution was reconstructed: the rational approximant displayed a +singularity which falls on top of the analytic one. Convergence radii of the Taylor expansions we +computed can be spotted, showing that there is an intersection of convergence disks, validating the +procedure of bridging the two regions where we were able to compute single thimble results: all in +all, while the thimble decomposition is discontinuous, the physical observable is not. The figure +refers to a given choice of lattice size, mass and 𝛽-value; we were able to show [10] that the method +can successfully account for the extraction of the continuum limit. +μ. We can obtain a dimensionless quantity by taking the +ratio μ +m ¼ ˆμ +ˆm. Since the analytic result is known, the single +thimble approximation was shown not to account for the +correct result on the entire μ +m axis. In our new approach the +problem is solved and in Fig. 2 we display the essential +features of our results: as an example, we show results for +the chiral condensate h¯χχi (parameters are L ¼ 8, β ¼ 1, +m ¼ 2). We can argue that all the requirements of the +program that we sketched above can be met. There is a +preliminary point we have to make. For real β a Stokes +phenomenon is potentially present up to a given value of μ +m: +this involves the dominant thimble pσ0 and another critical +point. We denote the latter pσ¯0, following the notation of +[19]. The problem can be easily solved by adding a small +imaginary part to β: in this way a Stokes phenomenon does +not take place, a thimble decomposition is in place and +while pσ¯0 could in principle give a contribution to the +result, this is de facto negligible due to the huge difference +SRðpσ¯0Þ ≫ SRðpσ0Þ. This solves the problem and any +further reference to this point will be omitted in the +following. +(1) A first value of +μ +m for which only the dominant +thimble pσ0 accounts for the correct result can be +found in a very fundamental, yet simple way. The +range of values SI can take on the real axis depends +on the values of ˆμ and ˆm and, below a given value of +μ +m, this range is limited. By explicit computation of +the SðσÞ +I ðμ +mÞ we can show that no unstable thimble +associated to a critical point pσ other that the +dominant one can intersect the original domain of +integration below a given value μ0 +m.7 Thus for μ +m < μ0 +m +we can easily select a first point at which the +dominant thimble provides the only contribution +to the result. We picked μ +m ¼ 0.4 and computed the +Taylor expansion up to the second derivative. +We now need to find a second value of μ +m at which +the dominant thimble accounts for the complete +result and compute the Taylor expansion on it. In +principle we could study the crossing mechanism +between the different curves SðσÞ +I ðμ +mÞ (see subsec- +tion II B). In practice there is a much simpler way to +proceed. First of all, we point out that the asymptotic +value of h¯χχi is known: for large enough values of μ +the chiral condensate is zero. We notice that for μ +m ¼ +1.4 the value of h¯χχi computed on the dominant +thimble is very close to zero. By inspecting the +values of SRðpσÞ for thimbles other than the funda- +mental one, we find that, for μ +m ¼ 1.4, SRðpσÞ ≫ +SRðpσ0Þ for all the critical points but three, that we +denote σ1, σ¯1, σ¯2.8 Two of them (σ¯1 and σ¯2) have +values of the real action which are lower than Smin, +which is the minimum value SR takes on the original +domain of integration: because of this, the unstable +thimbles associated to them can’t intersect the +original domain of integration. As for σ1, in this +simple model it does not take that much to show that +the unstable thimble attached to it does not intersect +the original domain of integration (see the left panel +of Fig. 2). We conclude that the dominant thimble σ0 +can account for the complete result at this value of μ +m. +We have thus selected the second point we were +looking for; at this point the series has been +computed up to the fifth derivative. One might +object that we made use of the explicit query for +intersections between the original domain of inte- +gration and a given unstable thimble, which thing is +FIG. 2. +(Left panel) The flow lines highlighting the thimbles structure of the 1-dim Thirring model at μ +m ¼ 1.4: stable thimbles are +depicted in blue, unstable thimbles in magenta. The dominant thimble is associated to the critical point sitting at ℜðzÞ ¼ 0. The critical +point σ1 is the closest to the latter to the right (there is a mirror image to the left as well): notice that the unstable thimble associated to it +does not intersect the original domain of integration (which is on the real axis). (Center panel) The chiral condensate as obtained from +the analytic solution (continuous black line) and from our Pad´e approximant (we plot points instead of a continuum line so that the size +of errors are easier to spot.). The points providing input to the evaluation of Pad´e are marked as triangles. (Right panel) Singularity of the +solution in the complex plane: red point computed from the analytic solution, green point is the only pole of our Pad´e approximant. We +plot the radii of convergence which are relevant for the expansions at hand: our analytic continuation indeed stands on firm ground. +7The value of ˆm is held fixed. +8We once again adhere to the notation of [19]. +F. DI RENZO, S. SINGH, and K. ZAMBELLO +PHYS. REV. D 103, 034513 (2021) +034513-6 +Figure 1: Left panel: (continuum line) analytic solution for the condensate ¯𝜒𝜒 of the 1-D Thirring model +(𝐿 = 8, 𝑚 = 1, 𝛽 = 1) at various values of the chemical potential by mass ratio 𝜇 +𝑚; (triangles) numerical +results obtained on one single thimble; (dots) numerical results taken from the rational approximant. Right +panel: we plot in the complex 𝜇 +𝑚 plane the singularity we got from the rational approximant; it is depicted +on top of the known analytic one. +2. +Multi-point Padè method for finite density Lattice QCD +2.1 Basics of the multi-point Padè method +Suppose we know a few Taylor expansion coefficients of a given function 𝑓 (𝑧) at different +points {𝑧𝑘 | 𝑘 = 1 . . . 𝑁}. The basic idea of our multi-point Padé approach is to approximate 𝑓 (𝑧) +by a rational function 𝑅𝑚 +𝑛 (𝑧), which we call a [𝑚/𝑛] Padé approximant +𝑅𝑚 +𝑛 (𝑧) = 𝑃𝑚(𝑧) +˜𝑄𝑛(𝑧) += +𝑃𝑚(𝑧) +1 + 𝑄𝑛(𝑧) = +𝑚� +𝑖=0 +𝑎𝑖 𝑧𝑖 +1 + +𝑛� +𝑗=1 +𝑏 𝑗 𝑧 𝑗 +. +(1) +𝑅𝑚 +𝑛 (𝑧) (i.e. the 𝑎𝑖, 𝑏 𝑗 coefficients defining it) can be fixed by requiring that it reproduces the values +of 𝑓 and a few of its derivatives at the given points {𝑧𝑘}. Provided that 𝑛 + 𝑚 + 1 = 𝑁𝑠 ( 𝑓 (𝑠−1) +being the highest order derivative we computed at each point), this is possible by requiring that +. . . +𝑃𝑚(𝑧𝑘) − 𝑓 (𝑧𝑘)𝑄𝑛(𝑧𝑘) = 𝑓 (𝑧𝑘) +𝑃′ +𝑚(𝑧𝑘) − 𝑓 ′(𝑧𝑘)𝑄𝑛(𝑧𝑘) − 𝑓 (𝑧𝑘)𝑄′ +𝑛(𝑧𝑘) = 𝑓 ′(𝑧𝑘) +. . . +(2) +3 + +0.6 +1.0 +3.0/ +0.5 +2.0 +0.4 +0.5 +0.3 +21.0 +m +0.2 +0.0 +0.0 +0.1 +0.0 +-0.5 +-1.0 +-0.1 +2.0 +-0.2 +-1.0 +3.0 +-2.0 +-1.0 +0.0 +1.0 +2.0 +3.0 +0.0 +0.5 +1.0 +1.5 +2.0 +1.0 +-0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +Re z +μ/m +Re (μ / m)Multi-point Padè for the study of phase transitions +Francesco Di Renzo +In Eq. (2) we only wrote 2 out of 𝑠 equations for 1 out of 𝑁 points. It should be clear what the +overall problem amounts to: we have to solve a linear system, the unknowns being the {𝑎𝑖, 𝑏 𝑗 | 𝑖 = +1 . . . 𝑚, 𝑗 = 1 . . . 𝑛}. This is not the only possible way to solve for 𝑅𝑚 +𝑛 (𝑧), but for the purpose of +understanding our approach it suffices (the interested reader can refer to [4] for other alternatives1). +It should be clear that +• Not only 𝑅𝑚 +𝑛 (𝑧) can reproduce our input pieces of information; by a natural analytic continu- +ation it can predict values of 𝑓 in an extended region (to the extent we do not exit the region +in which the approximation holds, which thing of course deserves care of its own): left panel +of Fig. 1 is an example. +• When a zero in the denominator of 𝑅𝑚 +𝑛 (𝑧) is not canceled by a corresponding zero of the +numerator, we face a singularity of the rational approximation, which is supposed to teach us +something on the singularity structure of 𝑓 ; quite obviously, singularities live in the complex +𝑧 plane: right panel of Fig. 1 is an example. +2.2 First application of the multi-point Padè method to finite density LQCD +In [4] the Bielefeld Parma collaboration applied the multi-point Padè method to finite density +LQCD. In the example of section 1 we did not have a way to safely compute the 1D Thirring +condensate in regions where more than one thimble give a contribution; on the other hand, we +could safely compute (on a single thimble) at given values of 𝜇 +𝑚. This is the same as in LQCD: +the sign problem does not allow us to compute observables at real values of the baryonic chemical +potential 𝜇𝐵, but computations are safe at 𝜇𝐵 = 0 and at imaginary values of 𝜇𝐵 (in particular, we +can compute a few orders of the Taylor expansion of an observable). For (2+1)-flavor of highly +improved staggered quarks (HISQ) [11] with imaginary chemical potential, we computed cumulants +of the net baryon number density, given as +𝜒𝑛𝐵(𝑇,𝑉, 𝜇𝐵) = +� 𝜕 +𝜕 ˆ𝜇𝐵 +�𝑛 ln 𝑍(𝑇,𝑉, 𝜇𝑙, 𝜇𝑠) +𝑉𝑇3 +, +(3) +with ˆ𝜇𝐵 = 𝜇𝐵/𝑇 and 𝑙, 𝑠 referring to light and strange flavors. Dependence on masses is not made +explicit: the light to strange ratio is the physical one. By computing at different imaginary values of +ˆ𝜇𝐵 (including ˆ𝜇𝐵 = 0) we could implement the program of subsection 2.1. Fig. 2 is the counterpart +of Fig. 1. We point out that +• In the left panel we can see how well the rational approximants for the number density 𝜒1𝐵 +describe data at different temperatures. Actually we show two different rational approximants +(enforcing parity or not): they are both fine. The big spike is expected to be there: it is related +to the Roberge Weiss transition, and it occurs at the temperature which is supposed to be the +relevant one (𝑇𝑅𝑊 ). Minor spikes can be also spotted: they are harmless, and they can be +understood in terms of what we will explain in the next section (partial cancellation of zeros +between numerator and denominator). +1Notice that this is the simplest setting also with respect to another point: there is no reason for strictly asking +knowledge of the same number of derivatives at each point. +4 + +Multi-point Padè for the study of phase transitions +Francesco Di Renzo +0 +1 +2 +3 +4 +5 +Re[µB/T] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +Im[µB/T] +ˆµLY +RW scaling +chiral scaling +CEP scaling +Figure 2: (Left panel) The number density 𝜒1𝐵 at various values of ˆ𝜇𝐵 and different temperatures 𝑇. Data +are shown together with two different rational approximants (enforcing parity or not): both describe data very +well. The big spike is expected: it is the hint for the Roberge Weiss transition. (Right panel) The singularity +pattern in the complex ˆ𝜇𝐵, highlighting their expected overall compliance with Roberge Weiss, chiral and +Critical End Point scaling. +• In the right panel we display the singularities we found at different temperatures, relating them +to the expected singularity scaling pattern. These are the expected Lee-Yang singularities: +one expects a given scaling for the singularities connected to the Roberge Weiss transition, +to the chiral transition and to the QCD Critical End Point. While the last two are still under +investigation2, one can clearly see a consistent picture for the Roberge Weiss scaling: indeed +in [4] we were able to show that it is the expected one. +All in all, results are intriguing. That’s why we now want to show that the machinery is under +control for the the most popular arena for testing tools in the study of phase transitions, i.e. the +two-dimensional Ising model. +3. +Testing the method on the 2d Ising model +Lee-Yang theory is one of the possible approach to the study of phase transitions. For an +example of its application, we refer the interested reader to [12], where the authors study the 2d +Ising model. We will basically follow their program, but will not rely on the study of many different +cumulants (as they do). We will instead make use of our multi-point Padè method and study only +two different cumulants at different values of temperature and magnetic field. The hamiltonian is +the well-known one, based on interactions between nearest neighbours and with external magnetic +field ℎ +𝐻 = −𝐽 +∑︁ +<𝑖, 𝑗> +𝜎𝑖𝜎𝑗 − ℎ +∑︁ +𝑖 +𝜎𝑖 +(4) +2Indeed we now have an estimate for the CEP Temperature. +5 + +RataprxST=167MeV +0.8 +RataprxNST=167MeV +RataprxST=186MeV +0.6 +RataprxNST=186MeV +RataprxSTRW +0.4 +RataprxNSTRW +Nt4T=167MeVdata +Nt4T=186MeVdata +0.2 +Nt4TRWdata +0 +-0.2 +-0.4 +-0.6 +-0.8 +1 +0 +1 +2 +3 +4 +5 +6 +Im[μg/T]Multi-point Padè for the study of phase transitions +Francesco Di Renzo +with the only possible values 𝜎𝑖 = ±1. In the following 𝐽 will be set to 𝐽 = 1. The partition function +can be written in terms of its zeros {𝛽𝑘} +𝑍(𝛽, ℎ) = 𝑍(0, ℎ) 𝑒 𝛽𝑐 � +𝑘 +(1 − 𝛽 +𝛽𝑘 +) +(5) +𝑐 being a constant. If we define thermal cumulants by +⟨⟨𝑈𝑛⟩⟩ = +𝜕𝑛 +𝜕(−𝛽)𝑛 ln 𝑍(𝛽, ℎ) +it is easy to show that they can be expressed as +⟨⟨𝑈𝑛⟩⟩ = (−1)(𝑛−1) ∑︁ +𝑘 +(𝑛 − 1)! +(𝛽𝑘 − 𝛽)𝑛 +(𝑛 > 1) +(6) +Furthermore, scaling relations describe the approach of leading zeros to critical inverse temperature +|𝛽0 − 𝛽𝑐| ∼ 𝐿−1/𝜈 +Im(𝛽0) ∼ 𝐿−1/𝜈. +(7) +In Eq. (7) 𝛽0 is the Fisher zero, that is the closest zero of the partition function to the real axis, +resulting in the closest singularity of cumulants to the real axis3, 𝛽𝑐 is the critical inverse temperature +and 𝜈 is the relevant critical exponent. +Our program now entails four steps: (1) we compute the 𝑛 = 2 thermal cumulant (i.e. the specific +heat) at various inverse temperatures 𝛽 and lattice sizes 𝐿; (2) for each 𝐿 we compute the rational +approximant 𝑅𝑚 +𝑛 (𝛽) by our multi-point Padè method; (3) at each 𝐿 we find the Fisher zero 𝛽0, which +is obtained as the the closest singularity of the cumulant to the real axis; (4) we study the finite size +scaling of the values of 𝛽0. The result of the procedure can be inspected in Fig. 3. +Figure 3: (Left panel) The scaling in 1/𝐿 of Im(𝛽0), i.e. the imaginary part of the Fisher zero, detected as +that the closest singularity of the cumulant to the real axis. The correct critical exponent 𝜈 = 1 is got with +fairly good accuracy. (Right panel) Once 𝜈 has been recognised to be the right one, one can fit the value of +the critical inverse temperature 𝛽𝑐, which is reconstructed to per mille accuracy. +3𝛽0 shows up together with its complex conjugate 𝛽∗ +0. +6 + +V = 1.03(3) +0.08 +0.07 +0.06 +0.05 +Im(βo) +0.04 +0.03 +0.02 +0.01 +0 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +1/Lβ。= 0.4405(5) +0.45 +0.4 +0.35 +0.3 +0.25 +0.2 +0.15 +0.1 +0.05 +0 +-0.05 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +1/ LMulti-point Padè for the study of phase transitions +Francesco Di Renzo +• In the left panel we display the scaling in 1/𝐿 of Im(𝛽0). Errors are computed by varying +results with respect to statistical errors for the cumulant and functional form for the rational +approximant. As one can see, the value of the relevant critical exponent 𝜈 = 1 is got with +fairly good accuracy (1.03(3)). +• Once 𝜈 = 1 has been recognised, we can fit the scaling of the real part Re(𝛽0) (right panel), +thus finding the value of the critical inverse temperature. We get the very accurate result +𝛽𝑐 = 0.4405(5). +Once the critical inverse temperature is known, one can sit on top of it and study the scaling in 𝐿 +of Im(ℎ0), ℎ0 being the Lee Yang zero, that is the closest singularity of a magnetic cumulant to +the real axis. Explicitly, our program again entails four steps: (1) we compute the 𝑛 = 1 magnetic +cumulant (i.e. the magnetisation) at 𝛽 = 𝛽𝑐 and various values of external magnetic field ℎ and +lattice size 𝐿; (2) for each 𝐿 we compute the rational approximant 𝑅𝑚 +𝑛 (ℎ) for the magnetisation by +our multi-point Padè method; (3) at each 𝐿 we find the Lee Yang zero ℎ0, which is the singularity +of the rational approximant for the magnetisation which is the closest to the real axis; (4) we study +the finite size scaling of the values of Im(ℎ0) (as we will see, ℎ0 always sits at Re(ℎ0) = 0). +Before we inspect this scaling behaviour, it is useful to have a closer look at the singularity pattern +in the complex ℎ plane at given values of 𝐿. In Fig 4 we depict the zeros of the numerator (blue +crosses) and of the denominator (red circles) of our 𝑅𝑚 +𝑛 (ℎ) at different values of the lattice size 𝐿, +i.e. 𝐿 = 15 (left panel) and 𝐿 = 30 (right panel). We can easily make a couple of key observations. +• A few zeros of the denominator are canceled by corresponding zeros of the numerator. These +are not genuine pieces of information: actually their location vary when varying e.g. the order +of the Padé approximant [𝑚, 𝑛]. On the other hand, genuine pieces of information (i.e. actual +zeros and poles) stay constant to a very good precision. Notice that this is the explanation for +the small spikes in Fig. 2: they are simply the shadow of cancellations which are indeed very +good, but not good enough to be invisible when plotting the rational approximant. +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +Figure 4: (Left panel) Zeros of the numerator (blue crosses) and of the denominator (red circles) of the +rational approximant 𝑅𝑚 +𝑛 (ℎ) for the magnetisation on 𝐿 = 15 (left panel) and 𝐿 = 30 (right panel). We +highlight the closest singularity to the real axis, which is getting closer to the real axis itself as 𝐿 gets larger, +with real parts being Re(ℎ0) = 0. Plots are in the complex ℎ plane. +7 + +Multi-point Padè for the study of phase transitions +Francesco Di Renzo +• We can clearly see that, as the lattice size 𝐿 gets larger, the closest singularity (Lee Yang +zero, highlighted in the plot) gets closer to the real axis, with real parts being Re(ℎ0) = 0. +Finally, in Fig. 5 we plot the finite size scaling of Im(ℎ0). As one can see, the critical exponent in +is got with very good accuracy (this time, less than percent: −1.880(16) vs −1.875). The steps we +could take in the (much simpler) case of the Ising model would be the preferred conceptual path to +follow also for LQCD. Needless to say, it will take time before we can be in a position to do that. +4. +Back to LQCD: a T-Padé application +We finally go back to LQCD for a (very) preliminary account of a new application. Till now +we have seen multi-point Padè approximants from data taken at a given temperature 𝑇 and different +values of ˆ𝜇𝐵: with this we mean that we obtained different 𝑅𝑚 +𝑛 ( ˆ𝜇𝐵) at different 𝑇 values. With +the very same data, we can think of going the other way around, that is we can obtain 𝑅𝑚 +𝑛 (𝑇) at +different ˆ𝜇𝐵 values. Fig. 6 is an example of what we can get following this path. Of course, this +time singularities emerge in the complex 𝑇 plane. +5. +Conclusions +The multi-point Padè method for the study of phase transitions has already proved to be quite +effective in the case of LQCD. Here we showed how the approach can provide very accurate results +when collecting a rich statistics is not such a hard numerical task (as it was the case for the 2d Ising +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +L1/8-2 +10-3 +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +Im(h0) +1.880(16) +Figure 5: Finite size scaling of Im(ℎ0). To guide the eye, we plot data versus 𝐿1/8−2, where the correct +critical exponent is taken. As the figure title we report the absolute value of the one we got, which turns out +to be a very accurate estimate, to less than percent. +8 + +Multi-point Padè for the study of phase transitions +Francesco Di Renzo +Figure 6: (Top-left panel) An example of 𝑅𝑚 +𝑛 (𝑇) for 𝜒1𝐵 at a given value of ˆ𝜇𝐵 on top of data taken at +different temperatures 𝑇 at the same given value of ˆ𝜇𝐵. (Top-right) Actual measurements of 𝜒1𝐵( ˆ𝜇𝐵) at a +given temperature 𝑇 plotted together with interpolating data obtained from 𝑅𝑚 +𝑛 (𝑇). Everything looks pretty +smooth; we plot in a different colour the only data point possibly not falling smoothly on top of actual data. +(Bottom-left) Zeros of denominator (red) and zeros of numerator (blue) of 𝑅𝑚 +𝑛 (𝑇) in the complex 𝑇 plane at +a low value of ˆ𝜇𝐵. (Bottom-right) The same plot at a value of ˆ𝜇𝐵 close to ˆ𝜇𝐵 = 𝑖𝜋 (𝑇 is expressed in GeV) +model). This is at same time a proof of concept of the reliability of the method and a stimulus to +do better in the case of finite density LQCD. +Acknowledgements +This work has received funding from the European Union’s Horizon 2020 research and inno- +vation programme under the Marie Skłodowska-Curie grant agreement No. 813942 (EuroPLEx). +We also acknowledge support from I.N.F.N. under the research project i.s. QCDLAT. This work +benefits from the HPC facility of the University of Parma, Italy. +References +[1] C. Schmidt, J. Goswami, G. Nicotra, F. Ziesché, P. Dimopoulos, F. Di Renzo et al., +Net-baryon Number Fluctuations, Acta Physica Polonica B Proceedings Supplement 14 +(2021) 241. +9 + +0.4 +0.35 +0.3 +0.25 +B +Im(X1 +0.2 +0.15 +0.1 +0.05 +0.12 +0.13 +0.14 +0.15 +0.16 +0.17 +0.18 +0.19 +T0.2 +0.15 +0.1 +Im(X1B) +0.05 +0$ +-0.05 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +μ/T0.06 +0.04 +0.02 +(L)w) +-0.02 +-0.04 +-0.06 +0.115 +0.12 +0.125 +0.13 +0.135 +0.14 +0.145 +0.15 +0.155 +Re(T)0.06 +0.04 +0.02 +(D)w) +-0.02 +-0.04 +-0.06 +0.11 +0.12 +0.13 +0.14 +0.15 +0.16 +0.17 +0.18 +0.19 +0.2 +0.21 +Re(T)Multi-point Padè for the study of phase transitions +Francesco Di Renzo +[2] S. Singh, P. Dimopoulos, L. Dini, F. Di Renzo, J. Goswami, G. Nicotra et al., Lee-Yang edge +singularities in lattice QCD : A systematic study of singularities in the complex 𝜇𝐵 plane +using rational approximations, Proceedings of The 38th International Symposium on Lattice +Field Theory — PoS(LATTICE2021) (2022) 544. +[3] G. Nicotra, P. Dimopoulos, L. Dini, F. Di Renzo, J. Goswami, C. Schmidt et al., Lee-Yang +edge singularities in 2 + 1 flavor QCD with imaginary chemical potential, Proceedings of The +38th International Symposium on Lattice Field Theory — PoS(LATTICE2021) (2022) 260. +[4] P. Dimopoulos, L. Dini, F. Di Renzo, J. Goswami, G. Nicotra, C. Schmidt et al., Contribution +to understanding the phase structure of strong interaction matter: Lee-Yang edge +singularities from lattice QCD, Phys. Rev. D 105 (2022) 034513 [2110.15933]. +[5] AuroraScience collaboration, New approach to the sign problem in quantum field theories: +High density QCD on a Lefschetz thimble, Phys. Rev. D 86 (2012) 074506 [1205.3996]. +[6] H. Fujii, D. Honda, M. Kato, Y. Kikukawa, S. Komatsu and T. Sano, Hybrid Monte Carlo on +Lefschetz thimbles - A study of the residual sign problem, JHEP 10 (2013) 147 [1309.4371]. +[7] H. Fujii, S. Kamata and Y. Kikukawa, Monte Carlo study of Lefschetz thimble structure in +one-dimensional Thirring model at finite density, JHEP 12 (2015) 125 [1509.09141]. +[8] A. Alexandru, G. Basar, P.F. Bedaque, G.W. Ridgway and N.C. Warrington, Sign problem +and Monte Carlo calculations beyond Lefschetz thimbles, JHEP 05 (2016) 053 +[1512.08764]. +[9] F. Di Renzo, S. Singh and K. Zambello, Taylor expansions on Lefschetz thimbles, Phys. Rev. +D 103 (2021) 034513 [2008.01622]. +[10] F. Di Renzo and K. Zambello, Solution of the Thirring model in thimble regularization, Phys. +Rev. D 105 (2022) 054501 [2109.02511]. +[11] HPQCD, UKQCD collaboration, Highly improved staggered quarks on the lattice, with +applications to charm physics, Phys. Rev. D 75 (2007) 054502 [hep-lat/0610092]. +[12] A. Deger and C. Flindt, Determination of universal critical exponents using Lee-Yang theory, +Phys. Rev. Research. 1 (2019) 023004 [1905.02379]. +10 + diff --git a/6dE1T4oBgHgl3EQf7AUK/content/tmp_files/load_file.txt b/6dE1T4oBgHgl3EQf7AUK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2fdc51465a1b5b3a3aec006928b6f73895abb274 --- /dev/null +++ b/6dE1T4oBgHgl3EQf7AUK/content/tmp_files/load_file.txt @@ -0,0 +1,526 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf,len=525 +page_content='Multi-point Padè for the study of phase transitions: from the Ising model to lattice QCD Francesco Di Renzo∗ and Simran Singh Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma and INFN, Gruppo Collegato di Parma, I-43100, Parma, Italy E-mail: francesco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='direnzo@unipr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='it, simran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='singh@unipr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='it The Bielefeld Parma collaboration has recently put forward a method to investigate the QCD phase diagram based on the computation of Taylor series coefficients at both zero and imaginary values of the baryonic chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The method is based on the computation of multi-point Padé approximants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We review the methodological aspects of the computation and, in order to gain confidence in the approach, we report on the application of the method to the two-dimensional Ising model (probably the most popular arena for testing tools in the study of phase transitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Besides showing the effectiveness of the multi-point Padé approach, we discuss what these results can suggest in view of further progress in the study of the QCD phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We finally report on very preliminary results in which we look for Padé approximants at different temperatures and fixed values of the (imaginary) baryonic chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The 39th International Symposium on Lattice Field Theory (Lattice2022), 8-13 August, 2022 Bonn, Germany ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='03528v1 [hep-lat] 9 Jan 2023 Multi-point Padè for the study of phase transitions Francesco Di Renzo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' How it all began: from Taylor expansions on thimbles to imaginary 𝜇𝐵 LQCD The QCD phase diagram is still to a large extent elusive: in particular, due to the so-called sign problem, the lattice (the non-perturbative tool which would be supposed to provide valuable insight) cannot probe the relevant regions in the 𝑇 − 𝜇𝐵 (Temperature-baryonic chemical potential) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' In the last couple of years, the Bielefeld-Parma collaboration put forward a method to compute finite-density QCD thermodynamic observables in the region to which access would be precluded by the sign problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' this approach is also able to probe the singualrity structure of the theory in the complex 𝜇𝐵 plane [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The method is based on the computation of Taylor series coefficients at both zero and imaginary values of the baryonic chemical potential, which enables the computation of multi-point Padé approximants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' This work aims to assess the effectiveness of the method by making use of it in the context of a very standard playground for the physics of phase transitions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' the 2d Ising model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' At the same time, we present (very) preliminary results on new applications in the context of finite-density QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Before entering the main subject, it is useful to recall when the idea of applying multi-point Padé rational approximants first came to our mind;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' that was in the context of thimble regularisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The latter [5, 6] was introduced to solve (or at least tame) the sign problem by re-expressing the path integral as a sum of integrals computed on manifolds different from the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' After complexifying the degrees of freedom, one considers the so-called Lefschetz thimbles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' the manifolds that are the union of the steepest ascent paths stemming from the various stationary points of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' On such manifolds the imaginary part of the action stays constant, so that the sign problem reduces to the so-called residual phase which is there due to the Jacobian of the change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' There is a thimble attached to each stationary point and in principle all can give a contribution to the path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' This is referred to as the thimble decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' To make a long story short, we recall that (a) not all the thimbles give a non-null contribution, (b) this picture changes in different regions of the parameters space of the theory (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' a given thimble can contribute to the path integral in a region and not in another one) and (c) there are cases in which a single thimble (usually the so called dominant one, attached to the stationary point with the lowest action) is enough to compute the answer one is interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The latter observation gave raise to the single thimble dominance hypothesis, which was shown to hold in a few cases, but failed in others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The first example of a failure was provided by the 1-D Thirring model [7, 8], where it was clearly shown that a single thimble is not enough to account for the known analytic result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' It is nevertheless important to remark that there are regions in which one single thimble is enough, and this was the logical starting point for the success of a computation based on multi-point Padé rational approximants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The success of such approach [9] can be recognised in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' On the left, we display the known analytic result for the chiral condensate ¯𝜒𝜒 of the 1-D Thirring model (𝐿 = 8, 𝑚 = 1, 𝛽 = 1) at various values of the chemical potential by mass ratio 𝜇 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' This is plotted together with the numerical results which we got: triangles are results computed on one single thimble at points where we are able to show that this is enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' dots are results taken from the multi-point Padé method that we will better describe in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Here it is enough to say that a few Taylor expansion coefficients were computed at the points marked by triangles and from those the multi-point Padé approximant was computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The right panel of the figure shows 2 Multi-point Padè for the study of phase transitions Francesco Di Renzo how the singularity pattern of the solution was reconstructed: the rational approximant displayed a singularity which falls on top of the analytic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Convergence radii of the Taylor expansions we computed can be spotted, showing that there is an intersection of convergence disks, validating the procedure of bridging the two regions where we were able to compute single thimble results: all in all, while the thimble decomposition is discontinuous, the physical observable is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The figure refers to a given choice of lattice size, mass and 𝛽-value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' we were able to show [10] that the method can successfully account for the extraction of the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' μ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We can obtain a dimensionless quantity by taking the ratio μ m ¼ ˆμ ˆm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Since the analytic result is known, the single thimble approximation was shown not to account for the correct result on the entire μ m axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' In our new approach the problem is solved and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 2 we display the essential features of our results: as an example, we show results for the chiral condensate h¯χχi (parameters are L ¼ 8, β ¼ 1, m ¼ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We can argue that all the requirements of the program that we sketched above can be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' There is a preliminary point we have to make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' For real β a Stokes phenomenon is potentially present up to a given value of μ m: this involves the dominant thimble pσ0 and another critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We denote the latter pσ¯0, following the notation of [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The problem can be easily solved by adding a small imaginary part to β: in this way a Stokes phenomenon does not take place, a thimble decomposition is in place and while pσ¯0 could in principle give a contribution to the result, this is de facto negligible due to the huge difference SRðpσ¯0Þ ≫ SRðpσ0Þ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' This solves the problem and any further reference to this point will be omitted in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (1) A first value of μ m for which only the dominant thimble pσ0 accounts for the correct result can be found in a very fundamental, yet simple way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The range of values SI can take on the real axis depends on the values of ˆμ and ˆm and, below a given value of μ m, this range is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' By explicit computation of the SðσÞ I ðμ mÞ we can show that no unstable thimble associated to a critical point pσ other that the dominant one can intersect the original domain of integration below a given value μ0 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='7 Thus for μ m < μ0 m we can easily select a first point at which the dominant thimble provides the only contribution to the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We picked μ m ¼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='4 and computed the Taylor expansion up to the second derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We now need to find a second value of μ m at which the dominant thimble accounts for the complete result and compute the Taylor expansion on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' In principle we could study the crossing mechanism between the different curves SðσÞ I ðμ mÞ (see subsec- tion II B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' In practice there is a much simpler way to proceed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' First of all, we point out that the asymptotic value of h¯χχi is known: for large enough values of μ the chiral condensate is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We notice that for μ m ¼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='4 the value of h¯χχi computed on the dominant thimble is very close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' By inspecting the values of SRðpσÞ for thimbles other than the funda- mental one, we find that, for μ m ¼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='4, SRðpσÞ ≫ SRðpσ0Þ for all the critical points but three, that we denote σ1, σ¯1, σ¯2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='8 Two of them (σ¯1 and σ¯2) have values of the real action which are lower than Smin, which is the minimum value SR takes on the original domain of integration: because of this, the unstable thimbles associated to them can’t intersect the original domain of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' As for σ1, in this simple model it does not take that much to show that the unstable thimble attached to it does not intersect the original domain of integration (see the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We conclude that the dominant thimble σ0 can account for the complete result at this value of μ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We have thus selected the second point we were looking for;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' at this point the series has been computed up to the fifth derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' One might object that we made use of the explicit query for intersections between the original domain of inte- gration and a given unstable thimble, which thing is FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (Left panel) The flow lines highlighting the thimbles structure of the 1-dim Thirring model at μ m ¼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='4: stable thimbles are depicted in blue, unstable thimbles in magenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The dominant thimble is associated to the critical point sitting at ℜðzÞ ¼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The critical point σ1 is the closest to the latter to the right (there is a mirror image to the left as well): notice that the unstable thimble associated to it does not intersect the original domain of integration (which is on the real axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (Center panel) The chiral condensate as obtained from the analytic solution (continuous black line) and from our Pad´e approximant (we plot points instead of a continuum line so that the size of errors are easier to spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The points providing input to the evaluation of Pad´e are marked as triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (Right panel) Singularity of the solution in the complex plane: red point computed from the analytic solution, green point is the only pole of our Pad´e approximant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We plot the radii of convergence which are relevant for the expansions at hand: our analytic continuation indeed stands on firm ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 7The value of ˆm is held fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 8We once again adhere to the notation of [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' DI RENZO, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' SINGH, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' ZAMBELLO PHYS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' REV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' D 103, 034513 (2021) 034513-6 Figure 1: Left panel: (continuum line) analytic solution for the condensate ¯𝜒𝜒 of the 1-D Thirring model (𝐿 = 8, 𝑚 = 1, 𝛽 = 1) at various values of the chemical potential by mass ratio 𝜇 𝑚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (triangles) numerical results obtained on one single thimble;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (dots) numerical results taken from the rational approximant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Right panel: we plot in the complex 𝜇 𝑚 plane the singularity we got from the rational approximant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' it is depicted on top of the known analytic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Multi-point Padè method for finite density Lattice QCD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='1 Basics of the multi-point Padè method Suppose we know a few Taylor expansion coefficients of a given function 𝑓 (𝑧) at different points {𝑧𝑘 | 𝑘 = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 𝑁}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The basic idea of our multi-point Padé approach is to approximate 𝑓 (𝑧) by a rational function 𝑅𝑚 𝑛 (𝑧), which we call a [𝑚/𝑛] Padé approximant 𝑅𝑚 𝑛 (𝑧) = 𝑃𝑚(𝑧) ˜𝑄𝑛(𝑧) = 𝑃𝑚(𝑧) 1 + 𝑄𝑛(𝑧) = 𝑚� 𝑖=0 𝑎𝑖 𝑧𝑖 1 + 𝑛� 𝑗=1 𝑏 𝑗 𝑧 𝑗 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (1) 𝑅𝑚 𝑛 (𝑧) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' the 𝑎𝑖, 𝑏 𝑗 coefficients defining it) can be fixed by requiring that it reproduces the values of 𝑓 and a few of its derivatives at the given points {𝑧𝑘}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Provided that 𝑛 + 𝑚 + 1 = 𝑁𝑠 ( 𝑓 (𝑠−1) being the highest order derivative we computed at each point), this is possible by requiring that .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 𝑃𝑚(𝑧𝑘) − 𝑓 (𝑧𝑘)𝑄𝑛(𝑧𝑘) = 𝑓 (𝑧𝑘) 𝑃′ 𝑚(𝑧𝑘) − 𝑓 ′(𝑧𝑘)𝑄𝑛(𝑧𝑘) − 𝑓 (𝑧𝑘)𝑄′ 𝑛(𝑧𝑘) = 𝑓 ′(𝑧𝑘) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (2) 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='0 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='0 Re z μ/m Re (μ / m)Multi-point Padè for the study of phase transitions Francesco Di Renzo In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (2) we only wrote 2 out of 𝑠 equations for 1 out of 𝑁 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' It should be clear what the overall problem amounts to: we have to solve a linear system, the unknowns being the {𝑎𝑖, 𝑏 𝑗 | 𝑖 = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 𝑚, 𝑗 = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 𝑛}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' This is not the only possible way to solve for 𝑅𝑚 𝑛 (𝑧), but for the purpose of understanding our approach it suffices (the interested reader can refer to [4] for other alternatives1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' It should be clear that Not only 𝑅𝑚 𝑛 (𝑧) can reproduce our input pieces of information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' by a natural analytic continu- ation it can predict values of 𝑓 in an extended region (to the extent we do not exit the region in which the approximation holds, which thing of course deserves care of its own): left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 1 is an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' When a zero in the denominator of 𝑅𝑚 𝑛 (𝑧) is not canceled by a corresponding zero of the numerator, we face a singularity of the rational approximation, which is supposed to teach us something on the singularity structure of 𝑓 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' quite obviously, singularities live in the complex 𝑧 plane: right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 1 is an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='2 First application of the multi-point Padè method to finite density LQCD In [4] the Bielefeld Parma collaboration applied the multi-point Padè method to finite density LQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' In the example of section 1 we did not have a way to safely compute the 1D Thirring condensate in regions where more than one thimble give a contribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' on the other hand, we could safely compute (on a single thimble) at given values of 𝜇 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' This is the same as in LQCD: the sign problem does not allow us to compute observables at real values of the baryonic chemical potential 𝜇𝐵, but computations are safe at 𝜇𝐵 = 0 and at imaginary values of 𝜇𝐵 (in particular, we can compute a few orders of the Taylor expansion of an observable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' For (2+1)-flavor of highly improved staggered quarks (HISQ) [11] with imaginary chemical potential, we computed cumulants of the net baryon number density, given as 𝜒𝑛𝐵(𝑇,𝑉, 𝜇𝐵) = � 𝜕 𝜕 ˆ𝜇𝐵 �𝑛 ln 𝑍(𝑇,𝑉, 𝜇𝑙, 𝜇𝑠) 𝑉𝑇3 , (3) with ˆ𝜇𝐵 = 𝜇𝐵/𝑇 and 𝑙, 𝑠 referring to light and strange flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Dependence on masses is not made explicit: the light to strange ratio is the physical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' By computing at different imaginary values of ˆ𝜇𝐵 (including ˆ𝜇𝐵 = 0) we could implement the program of subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 2 is the counterpart of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We point out that In the left panel we can see how well the rational approximants for the number density 𝜒1𝐵 describe data at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Actually we show two different rational approximants (enforcing parity or not): they are both fine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The big spike is expected to be there: it is related to the Roberge Weiss transition, and it occurs at the temperature which is supposed to be the relevant one (𝑇𝑅𝑊 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Minor spikes can be also spotted: they are harmless, and they can be understood in terms of what we will explain in the next section (partial cancellation of zeros between numerator and denominator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 1Notice that this is the simplest setting also with respect to another point: there is no reason for strictly asking knowledge of the same number of derivatives at each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 4 Multi-point Padè for the study of phase transitions Francesco Di Renzo 0 1 2 3 4 5 Re[µB/T] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='5 Im[µB/T] ˆµLY RW scaling chiral scaling CEP scaling Figure 2: (Left panel) The number density 𝜒1𝐵 at various values of ˆ𝜇𝐵 and different temperatures 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Data are shown together with two different rational approximants (enforcing parity or not): both describe data very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The big spike is expected: it is the hint for the Roberge Weiss transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (Right panel) The singularity pattern in the complex ˆ𝜇𝐵, highlighting their expected overall compliance with Roberge Weiss, chiral and Critical End Point scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' In the right panel we display the singularities we found at different temperatures, relating them to the expected singularity scaling pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' These are the expected Lee-Yang singularities: one expects a given scaling for the singularities connected to the Roberge Weiss transition, to the chiral transition and to the QCD Critical End Point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' While the last two are still under investigation2, one can clearly see a consistent picture for the Roberge Weiss scaling: indeed in [4] we were able to show that it is the expected one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' All in all, results are intriguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' That’s why we now want to show that the machinery is under control for the the most popular arena for testing tools in the study of phase transitions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' the two-dimensional Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Testing the method on the 2d Ising model Lee-Yang theory is one of the possible approach to the study of phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' For an example of its application, we refer the interested reader to [12], where the authors study the 2d Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We will basically follow their program, but will not rely on the study of many different cumulants (as they do).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We will instead make use of our multi-point Padè method and study only two different cumulants at different values of temperature and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The hamiltonian is the well-known one, based on interactions between nearest neighbours and with external magnetic field ℎ 𝐻 = −𝐽 ∑︁ <𝑖, 𝑗> 𝜎𝑖𝜎𝑗 − ℎ ∑︁ 𝑖 𝜎𝑖 (4) 2Indeed we now have an estimate for the CEP Temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 5 RataprxST=167MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='8 RataprxNST=167MeV RataprxST=186MeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='6 RataprxNST=186MeV RataprxSTRW 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='4 RataprxNSTRW Nt4T=167MeVdata Nt4T=186MeVdata 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='2 Nt4TRWdata 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='8 1 0 1 2 3 4 5 6 Im[μg/T]Multi-point Padè for the study of phase transitions Francesco Di Renzo with the only possible values 𝜎𝑖 = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' In the following 𝐽 will be set to 𝐽 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The partition function can be written in terms of its zeros {𝛽𝑘} 𝑍(𝛽, ℎ) = 𝑍(0, ℎ) 𝑒 𝛽𝑐 � 𝑘 (1 − 𝛽 𝛽𝑘 ) (5) 𝑐 being a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' If we define thermal cumulants by ⟨⟨𝑈𝑛⟩⟩ = 𝜕𝑛 𝜕(−𝛽)𝑛 ln 𝑍(𝛽, ℎ) it is easy to show that they can be expressed as ⟨⟨𝑈𝑛⟩⟩ = (−1)(𝑛−1) ∑︁ 𝑘 (𝑛 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (𝛽𝑘 − 𝛽)𝑛 (𝑛 > 1) (6) Furthermore, scaling relations describe the approach of leading zeros to critical inverse temperature |𝛽0 − 𝛽𝑐| ∼ 𝐿−1/𝜈 Im(𝛽0) ∼ 𝐿−1/𝜈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (7) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (7) 𝛽0 is the Fisher zero, that is the closest zero of the partition function to the real axis, resulting in the closest singularity of cumulants to the real axis3, 𝛽𝑐 is the critical inverse temperature and 𝜈 is the relevant critical exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Our program now entails four steps: (1) we compute the 𝑛 = 2 thermal cumulant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' the specific heat) at various inverse temperatures 𝛽 and lattice sizes 𝐿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (2) for each 𝐿 we compute the rational approximant 𝑅𝑚 𝑛 (𝛽) by our multi-point Padè method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (3) at each 𝐿 we find the Fisher zero 𝛽0, which is obtained as the the closest singularity of the cumulant to the real axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (4) we study the finite size scaling of the values of 𝛽0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The result of the procedure can be inspected in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Figure 3: (Left panel) The scaling in 1/𝐿 of Im(𝛽0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' the imaginary part of the Fisher zero, detected as that the closest singularity of the cumulant to the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The correct critical exponent 𝜈 = 1 is got with fairly good accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (Right panel) Once 𝜈 has been recognised to be the right one, one can fit the value of the critical inverse temperature 𝛽𝑐, which is reconstructed to per mille accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 3𝛽0 shows up together with its complex conjugate 𝛽∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 6 V = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='03(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='05 Im(βo) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='01 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='12 1/Lβ。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='4405(5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='12 1/ LMulti-point Padè for the study of phase transitions Francesco Di Renzo In the left panel we display the scaling in 1/𝐿 of Im(𝛽0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Errors are computed by varying results with respect to statistical errors for the cumulant and functional form for the rational approximant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' As one can see, the value of the relevant critical exponent 𝜈 = 1 is got with fairly good accuracy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='03(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Once 𝜈 = 1 has been recognised, we can fit the scaling of the real part Re(𝛽0) (right panel), thus finding the value of the critical inverse temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We get the very accurate result 𝛽𝑐 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='4405(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Once the critical inverse temperature is known, one can sit on top of it and study the scaling in 𝐿 of Im(ℎ0), ℎ0 being the Lee Yang zero, that is the closest singularity of a magnetic cumulant to the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Explicitly, our program again entails four steps: (1) we compute the 𝑛 = 1 magnetic cumulant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' the magnetisation) at 𝛽 = 𝛽𝑐 and various values of external magnetic field ℎ and lattice size 𝐿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (2) for each 𝐿 we compute the rational approximant 𝑅𝑚 𝑛 (ℎ) for the magnetisation by our multi-point Padè method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (3) at each 𝐿 we find the Lee Yang zero ℎ0, which is the singularity of the rational approximant for the magnetisation which is the closest to the real axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (4) we study the finite size scaling of the values of Im(ℎ0) (as we will see, ℎ0 always sits at Re(ℎ0) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Before we inspect this scaling behaviour, it is useful to have a closer look at the singularity pattern in the complex ℎ plane at given values of 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' In Fig 4 we depict the zeros of the numerator (blue crosses) and of the denominator (red circles) of our 𝑅𝑚 𝑛 (ℎ) at different values of the lattice size 𝐿, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 𝐿 = 15 (left panel) and 𝐿 = 30 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We can easily make a couple of key observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' A few zeros of the denominator are canceled by corresponding zeros of the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' These are not genuine pieces of information: actually their location vary when varying e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' the order of the Padé approximant [𝑚, 𝑛].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' On the other hand, genuine pieces of information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' actual zeros and poles) stay constant to a very good precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Notice that this is the explanation for the small spikes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 2: they are simply the shadow of cancellations which are indeed very good, but not good enough to be invisible when plotting the rational approximant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='1 0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='2 Figure 4: (Left panel) Zeros of the numerator (blue crosses) and of the denominator (red circles) of the rational approximant 𝑅𝑚 𝑛 (ℎ) for the magnetisation on 𝐿 = 15 (left panel) and 𝐿 = 30 (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We highlight the closest singularity to the real axis, which is getting closer to the real axis itself as 𝐿 gets larger, with real parts being Re(ℎ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Plots are in the complex ℎ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 7 Multi-point Padè for the study of phase transitions Francesco Di Renzo We can clearly see that, as the lattice size 𝐿 gets larger, the closest singularity (Lee Yang zero, highlighted in the plot) gets closer to the real axis, with real parts being Re(ℎ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 5 we plot the finite size scaling of Im(ℎ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' As one can see, the critical exponent in is got with very good accuracy (this time, less than percent: −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='880(16) vs −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='875).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' The steps we could take in the (much simpler) case of the Ising model would be the preferred conceptual path to follow also for LQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Needless to say, it will take time before we can be in a position to do that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Back to LQCD: a T-Padé application We finally go back to LQCD for a (very) preliminary account of a new application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Till now we have seen multi-point Padè approximants from data taken at a given temperature 𝑇 and different values of ˆ𝜇𝐵: with this we mean that we obtained different 𝑅𝑚 𝑛 ( ˆ𝜇𝐵) at different 𝑇 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' With the very same data, we can think of going the other way around, that is we can obtain 𝑅𝑚 𝑛 (𝑇) at different ˆ𝜇𝐵 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 6 is an example of what we can get following this path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Of course, this time singularities emerge in the complex 𝑇 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Conclusions The multi-point Padè method for the study of phase transitions has already proved to be quite effective in the case of LQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Here we showed how the approach can provide very accurate results when collecting a rich statistics is not such a hard numerical task (as it was the case for the 2d Ising 0 2 4 6 8 10 12 14 16 18 L1/8-2 10-3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='06 Im(h0) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='880(16) Figure 5: Finite size scaling of Im(ℎ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' To guide the eye, we plot data versus 𝐿1/8−2, where the correct critical exponent is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' As the figure title we report the absolute value of the one we got, which turns out to be a very accurate estimate, to less than percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 8 Multi-point Padè for the study of phase transitions Francesco Di Renzo Figure 6: (Top-left panel) An example of 𝑅𝑚 𝑛 (𝑇) for 𝜒1𝐵 at a given value of ˆ𝜇𝐵 on top of data taken at different temperatures 𝑇 at the same given value of ˆ𝜇𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (Top-right) Actual measurements of 𝜒1𝐵( ˆ𝜇𝐵) at a given temperature 𝑇 plotted together with interpolating data obtained from 𝑅𝑚 𝑛 (𝑇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Everything looks pretty smooth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' we plot in a different colour the only data point possibly not falling smoothly on top of actual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (Bottom-left) Zeros of denominator (red) and zeros of numerator (blue) of 𝑅𝑚 𝑛 (𝑇) in the complex 𝑇 plane at a low value of ˆ𝜇𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' (Bottom-right) The same plot at a value of ˆ𝜇𝐵 close to ˆ𝜇𝐵 = 𝑖𝜋 (𝑇 is expressed in GeV) model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' This is at same time a proof of concept of the reliability of the method and a stimulus to do better in the case of finite density LQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Acknowledgements This work has received funding from the European Union’s Horizon 2020 research and inno- vation programme under the Marie Skłodowska-Curie grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' 813942 (EuroPLEx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' We also acknowledge support from I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' under the research project i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' QCDLAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' This work benefits from the HPC facility of the University of Parma, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Schmidt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Goswami, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Nicotra, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Ziesché, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Dimopoulos, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Di 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Dini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Di Renzo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Goswami, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=' Nicotra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE1T4oBgHgl3EQf7AUK/content/2301.03528v1.pdf'} +page_content=', Lee-Yang edge singularities in lattice QCD : A systematic study of singularities in the complex 𝜇𝐵 plane using rational approximations, Proceedings of The 38th International Symposium on Lattice Field Theory — PoS(LATTICE2021) (2022) 544.' metadata={'source': 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--git a/ANAzT4oBgHgl3EQfTPxG/content/tmp_files/2301.01245v1.pdf.txt b/ANAzT4oBgHgl3EQfTPxG/content/tmp_files/2301.01245v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c20edb58a56d34d1ba02ea0b646b7f873c5cca01 --- /dev/null +++ b/ANAzT4oBgHgl3EQfTPxG/content/tmp_files/2301.01245v1.pdf.txt @@ -0,0 +1,934 @@ +RegTraffic: A Regression Based Traffic Simulator +for Spatiotemporal Traffic Modeling, Simulation +and Visualization +Sifatul Mostafi, Taghreed Alghamdi, Khalid Elgazzar +IoT Research Lab, ECSE, Ontario Tech University, Oshawa, ON, Canada +{sifatul.mostafi, Taghreed Alghamdi, khalid.elgazzar}@ontariotechu.ca +Abstract—Traffic simulation is a great tool to demonstrate +complex traffic structures which can be extremely useful for +the planning, development, and management of road traffic +networks. Current traffic simulators offer limited features when +it comes to interactive and adaptive traffic modeling. This paper +presents RegTraffic, a novel interactive traffic simulator that +integrates dynamic regression-based spatiotemporal traffic anal- +ysis to predict congestion of intercorrelated road segments. The +simulator models traffic congestion of road segments depending +on neighboring road links and temporal features of the dynamic +traffic flow. The simulator provides a user-friendly web interface +to select road segments of interest, receive user-defined traffic +parameters, and visualize the traffic for the flow of correlated +road links based on the user inputs and the underlying correlation +of these road links. Performance evaluation shows that RegTraffic +can effectively predict traffic congestion with a Mean Squared +Error of 1.3 Km/h and a Root Mean Squared Error of 1.71 +Km/h. RegTraffic can effectively simulate the results and provide +visualization on interactive geographical maps. +Index Terms—Road traffic, simulator, regression, visualization, +software +I. INTRODUCTION +With the advancement of computer technologies and soft- +ware engineering, computer-based traffic simulation has be- +come a popular approach for traffic analysis in support of the +evaluation and design of Intelligent Transport Systems (ITS) +[2]. Traffic simulation software supported by the ability to +emulate the variability of spatial and temporal components in +traffic flows is a practical tool for capturing and explaining +complex traffic systems [6]. +The purpose of developing Traffic simulation tools is to +experiment with varieties of strategies in traffic modeling [3]. +Traffic simulation software tools and models built on real- +life traffic data are widely applied to support real-time traffic +decisions and management solutions. +Regression analysis in the traffic domain is a well- +established approach that facilitates traffic modeling and pre- +diction [1]. Regression-based traffic modeling helps in ana- +lyzing complex traffic structures which is a useful method for +the development and planning of traffic systems and networks. +Hence, traffic congestion estimation and computerized simula- +tion are suitable options for policymakers to analyze different +complex traffic scenarios and take actions accordingly [5]. +A lot of microscopic and macroscopic traffic simulators +have been developed including SUMO [10], Aimsun [13], +Traffsim [14], SUMMIT [15], SifTraffic [16] and VISSIM [7]. +These simulators have practical use cases in traffic analysis +including traffic flow measurement, multi-agent simulation, +particle-based simulation, and so on. Although these state-of- +the-art simulators have many practical traffic use cases, they +face challenges in the application of simulating road traffic +congestion in heterogeneous road transportation networks with +a small amount of real-time data [3]. Also, these simulators +lack the feature to adopt regression-based traffic modeling and +simulate traffic congestion of a road link depending on traffic +congestion of neighboring road links. Some of the simulators +do not provide visualization for the simulation results in +interactive geographical maps. +We aim to develop a regression-based traffic simulator for +spatiotemporal traffic modeling to predict traffic congestion +of a road link depending on neighboring road segments and +provide features to simulate and visualize the results using +interactive geographical maps. +The remainder of this paper is organized as follows. Section +II briefly reviews the state-of-the-art traffic simulators and +their scope in traffic modeling and simulation. The traffic +modeling approach of RegTraffic is described in Section III. +Section IV outlines the processing pipeline of the different +components of RegTraffic. Section V provides a step-by-step +simulation scenario of a traffic use case. Section VI shows +the performance analysis of RegTraffic. Lastly, Section VII +concludes this work and provides future research directions. +II. BACKGROUND AND RELATED WORK +Traffic simulator software is commonly divided into two +categories: microscopic traffic simulators [7], [10] and macro- +scopic traffic simulators [8]. +FreeSim [9] is a traffic simulator designed to conduct real- +time freeway traffic simulation. SUMO (Simulation of Urban +Mobility) [10] is a microscopic traffic simulator that is devel- +oped to process complex and large road networks. SUMO is +widely used in many applications including traffic flow model- +ing [11] and color mapping Google Maps routes [12]. Aimsun +[13] is a traffic simulator for modeling smart mobility. Traffsim +[14] simulator is widely used for modeling isolated traffic +control strategies. SUMMIT [15] provides functionalities to +simulate urban driving in large traffic scenarios with massive +and mixed traffic. SifTraffic [16] is a practical software tool to +arXiv:2301.01245v1 [cs.NI] 23 Nov 2022 + +TABLE I +COMPARISON OF TRAFFIC SIMULATORS +Comparison Category Simulator +RegTraffic +FreeSim [9] +SUMO [10] +Aimsun [13] +TraffSim [14] +SUMMIT [15] +SimTraffic [16] +VISSIM [7] +Spatiotemporal Traffic Modeling +Yes +Yes +Yes +Yes +Yes +Yes +Yes +Yes +Regression Modeling +Yes +No +No +No +No +No +No +No +Interactive Geographical Maps +Yes +No +Yes +Yes +No +Yes +Yes +Yes +Web Interface +Yes +No +No +No +No +No +No +No +conduct simulations of practical traffic applications. VISSIM +[7] is a microscopic traffic simulator for behavior-based multi- +purpose traffic flow simulation. +Wang et al. [17] explored different methods of correct- +ing the traffic simulation models based on linear regression. +Golovnin et al. [20] took a web-oriented approach to simulate +road traffic, especially in urban settings. Mizuta et al. [21] +evaluated the traffic flow near intersections of a metropolitan +city to understand how agent-based traffic simulators work to +approximate vehicle behaviors. +A comparison among the existing traffic simulators along +with RegTraffic is listed in Table I in terms of some key +characteristics and features. +III. MATHEMATICAL MODELING +A. Spatial Feature +Figure 1 shows a traffic road intersection. In this intersec- +tion, we consider a road link as the spatial road feature that is +dependent on one or several connected spatial road features. +For example, for the intersection shown in Figure 1, the road +link ˆy is a spatial feature and is modeled as a dependent +variable in our regression modeling. The road links xs +1, xs +2 up +to the road link xs +n are the independent spatial features. It’s +worth noting that the dependent road link ˆy is an outbound +while all the independent road links xs +1, xs +2, ..., xs +n inbound +to the intersection. Our proposed traffic modeling approach +described in [18] indicates that the dependent spatial feature +must be an outbound road link and the independent spatial +features must be inbound road links. The model incorporates +a set of temporal features that can be extracted from both in- +dependent and dependent spatial features through exploratory +data analysis. The specific number of temporal features and +independent spatial features are arbitrary and dependent on the +specific road intersection and their orientation. +Here, XS is defined as a set of independent spatial features +� +xs +1, xs +2, .., xs +ns +� +as shown in Eq. (1). +XS = +� +xs +1, xs +2, .., xs +ns +� +(1) +The cardinality of set XS is defined as ns as shown in Eq. +(2). +ns = |XS| +(2) +Fig. 1. Traffic Road intersection +B. Temporal Feature Extraction +In our modeling, we convert temporal features into categori- +cal features using one hot encoding. To simplify our modeling, +temporal features are encoded using only two values. Here, +XT is a set of temporal features +� +xt +1, xt +2, .., xt +nt +� +as shown in +Eq. (3). +XT = +� +xt +1, xt +2, .., xt +nt +� +(3) +The cardinality of set XT is defined as nt as shown in Eq. +(4). +nt = |XT | +(4) +The set of temporal features is extracted from spatial fea- +tures using exploratory data analysis. Here, XT is the output +of function f which takes in the set of spatial features XS as +input. The function f is a many to many function that takes in +a set of spatial features and conducts exploratory data analysis +to extract a set of temporal features as shown in Eq. (5) +XT = fns→nt(XS) +(5) +We define the set X as a union of the temporal features XT +and spatial features XS as shown in Eq. (6). + +X = XT ∪ XS +(6) +C. Regression Modeling +1) Regression Formation: RegTraffic forms a regression +model through a linear combination of both temporal and +spatial explanatory features to explain the dependent spatial +feature ˆy as shown in Eq. (7). In this equation, all the +independent features are associated with their corresponding +regression coefficient. α indicates the bias and ϵ refers to the +error term. +ˆy = +nt +� +i=1 +βt +ixt +i + +ns +� +i=1 +βs +i xs +i + α + ϵ +(7) +In the regression Eq. (7), every explanatory temporal feature +from setting XT is associated with a regression coefficient +from set βT as shown in Eq. (8). +βT = +� +βt +1, βt +2, .., βt +nt +� +(8) +Similarly, in the regression Eq. (7), every explanatory spatial +feature from set XS is associated with a regression coefficient +from set βS as shown in Eq. (9). +βS = +� +βs +1, βs +2, .., βs +ns +� +(9) +Here, β is defined as the union of set βT and βS +β = βT ∪ βS +(10) +2) Posterior Probability Distribution: +We use a novel +Bayesian linear regression approach for spatiotemporal traffic +modeling of a road link proposed in [18]. Bayesian linear +regression formulates a posterior probability distribution of +the model parameters rather than just finding a single point +estimate. The response variable is drawn from a probability +distribution instead of a single value estimation. A Bayesian +linear regression model samples the response variable from a +normal distribution as shown in Eq. (11). +y ∼ N(βT X, σ2I) +(11) +In Eq. (11), the response variable y is generated from a +Gaussian normal distribution, which is characterized by a +mean and variance. Eq. (12) refers to the Bayes Theorem +which is the fundamental building block of Bayesian linear +regression. Here, P(β | ˆy, X) is the posterior probability +distribution of the model parameters, P(ˆy | β, X) is the +likelihood of the data, P(β | X) is the prior probability of the +parameters and P(ˆy | X) is the normalization constant. The +posterior distribution of the model parameters is proportional +to the multiplication of the likelihood of the data and the prior +probability of the parameters. A detailed description of the +model is described in [18]. +P(β | ˆy, X) = P(ˆy | β, X) ∗ P(β | X) +P(ˆy | X) +(12) +Once the regression model is built, the user can provide +new observations for independent spatial features XS and +independent temporal features XT into the model. Based on +the new observation, the model incorporates the regression +coefficients associated with the explanatory variables and +predicts the output for the dependent variable ˆy. An event +can be associated with a specific value as an input for any +independent spatial feature. +D. Event Integration +Here, XE is defined as a set of events +� +XE +1 , XE +2 , .., XE +nE +� +as shown in Eq. (13). +XE = +� +XE +1 , XE +2 , .., XE +nE +� +(13) +The cardinality of set XE is defined as nE as shown in Eq. +(14). +nE = |XE| +(14) +After event integration, the independent spatial features +associated with an event are integrated into Eq. (7). If any +independent spatial feature is associated with an event, we +need to replace the value for the independent spatial feature +xS with the events xE. Spatial features which are not affected +by any specific event are represented by xS′ along with their +model parameter βS′ as shown in Eq. (16). The amount of +spatial features unaffected by any specific event is denoted as +shown in Eq. (15). +nS′ = nS − nE +(15) +ˆy = +nt +� +i=1 +βt +ixs +i + +nS′ +� +i=1 +βS′ +i xS′ +i + +nE +� +i=1 +βE +i xE +i + α + ϵ +(16) +However, we add a time constraint in association with the +temporal components for adding a specific event into the +regression equation. For any specific event XE occurred at +time TE, the value of spatial feature XS will be replaced by +the value of XE if TE ⊂ XT . +IV. PROCESSING PIPELINE +The processing pipeline of RegTraffic is shown in Figure +- 2. The spatial feature consists of a unique name of the +feature, the corresponding time series dataset of that spatial +feature, and a set of latitude and longitude as the waypoints +of the route of that spatial feature. The traffic data extraction +is described in [19]. In this process, a user selects the starting +and ending points of the route of interest and specifies the +time range. The traffic data extraction tool gathers time-series +information of the “congestion index” of that road link every +15 minutes throughout the time range from Google Maps. +The congestion index is defined by the average speed of that +road link in terms of kilometers per hour. At the end of the +process, the tool generates a time series dataset that has a + +Fig. 2. Processing Pipeline. +unique name as provided by the user when adding a spatial +feature in RegTraffic. +RegTraffic also constructs a temporal feature component +with three core input values. These are the unique name of +the temporal feature, the corresponding time series dataset, +and the time range of that temporal feature. RegTraffic takes +a set of input preferences from the user as part of the model +selection. It also allows users to choose the dependent feature +for the regression model. Once the regression model is built, +RegTraffic passes the regression coefficients to a visualization +interface where a user can input new observations for the +independent features that can be both spatial and temporal. +V. SIMULATION +A. Spatial Feature +We conduct our experiment on four connected road links in +Oshawa, Ontario, Canada as shown in Figure 3(a). The ending +point of road links 2, 3 and 4 are connected with the origin of +the road link 1. A connected road network is formed by these +road links. We represent the traffic congestion level of these 4 +road links as Road1, Road2, Road3, and Road4, respectively. +Road1 is the dependent link where Road2, Road3 and Road4 +are the independent links that collectively affect Road1 during +a specific time of the day. +We collect the average traffic speed of each road link every +15 minutes for an entire week from 12:00 am March 01, 2020, +to 11:45 pm March 07, 2020. As a result, there are a total +of 672 observations over 7 days of time-series data for each +road link. Figure 3(b) shows the time series of the average +traffic speed of all four road links for the first two days. The +y axis represents the average traffic speed in km/h, which is +considered the traffic congestion index in our analysis. We can +see that the time series has a cycle as the average traffic speed +shows regular and predictable changes that recur every day +within a certain time interval. The higher the average speed, +the low the traffic congestion, and vice versa. +B. Temporal Feature Extraction +Figure 3(c) shows the hourly mean of the average speed for +each road link. The mean values show very little variance +(a) Intersection of Simcoe and Conlin +Road in Oshawa +(b) Time series data of 4 road links +(c) Hourly average speed throughout a +day +(d) Identifying threshold for Peakhour +Fig. 3. Average speed throughout a day +compared to each other as they seem to move together +throughout the day. The average of the different means of all +road links is plotted in Figure 3(d). The horizontal line at a +speed of 11.75 km/h divides the plot evenly and intersects with +the total average speed at two points, one at daytime 8:00 and +the other one at 23:00. From this exploratory data analysis, +a new categorical feature called Peakhour is extracted that +indicates a certain time interval during a day where the average +traffic speed remains below 11.75 km/h. From 9:00 am to +12:00 pm, the value of Peakhour would be 1, otherwise 0. +Another temporal component is considered in the analysis as +a categorical variable which is AM. The value of AM would +be 1 when the meridiem is AM and 0 when it is PM. +C. Regression Modeling +The outcome of our Bayesian linear regression is the distri- +bution of the model parameters. The model does not provide +an exact estimate for a feature, but the mean value of the +distribution can be considered as an estimate for the feature. +The benefit of having a posterior probability distribution is +that the model also provides an entire range of values that +shows the uncertainty of the true values. The mean of a +posterior probability distribution is taken as the best estimate +of that model parameter. These mean estimates of these model +parameters are put together to derive a new Eq. (17). +Road1 = 7.4163 ∗ Intercept + 1.7561 ∗ AM +−2.7517 ∗ Peakhour − 0.0477 ∗ Road2 +−0.0479 ∗ Road3 + 0.7139 ∗ Road4 + 1.7003 ∗ SD +(17) + +Name +Name +Model Selection +Time Series +Spatial +Temporal +Time Series +Data +Feature +Feature +Data +Regression +Waypoints +Time Range +Modelling +User Input +User Input +Visualization +for +for +Scheme +Spatial Features +Temporal FeaturesN +W +Road 1 +S +2 +Road 2 +Con +CopperBrang +Vegan · ss +Road 4 +Road 3 +Subway +Sandwich shop +S +C +S +SmileRoadl +20 +Road2 +Road3 +18 +Road4 +16 +Speed (Km/h) +14 +12 +10 +8 +6 +01-Mar +06:00 +12:00 +18:00 +02-Mar +06:00 +12:00 +18:00 +00:00 +00:00 +2020 +Datetime20 +18 +16 +Speed (Km/h) +14 +12 +10 +Roadl +Road2 +8 +Road3 +Road4 +6 +5 +10 +15 +20 +0 +Hours (0-23)18 +Hourly Mean +- Threshold Speed +16 +14 +Speed (Km/h) +Peakhour +Peakhour +starts +ends +12 +S +10 +8 +0 +5 +10 +15 +20 +Hours (0-23)Fig. 4. Regression Analysis in RegTraffic Simulator +D. Visualization +Figure 4 describes a sample simulation procedure of a road +intersection where Road1 is considered as a dependent road +link and Road2, Road3 and Road4 are independent road +links. Based on the spatial features, two new temporal features +are extracted which are Peakhour and AM. RegTraffic shows +the location of the road links on an interactive geographical +map where the user can provide new observations for indepen- +dent road links and temporal features to predict the outcome of +the dependent road link. As shown in the figure, the user sets +the congestion index of Road2, Road3, and Road4 to 18.05, +4.4, and 10.45 kilometers per hour, respectively. The user also +needs to provide the specific time as an input for the temporal +features Peakhour and AM. RegTraffic calculates the value +for the temporal features from the time input provided by the +user and incorporates these values along with the input values +for independent spatial features to predict the congestion index +of dependent road link Road1. Based on the input values +provided by the user, RegTraffic predicts the congestion index +of the road link Road1, which is 13.3 kilometers per hour in +this case. +VI. PERFORMANCE EVALUATION +A. Test Observations +To evaluate the performance, the model is tested on a +testing dataset of traffic observations. Figure 5 shows four +random test observations from the testing dataset along with +the probability density function of Road1. The true value of +Road1 is represented by the dotted line and the mean of the +probability distribution is represented by the straight line. The +mean of the probability distribution is considered as the best +estimate for the distributions. The estimated value provided by +the model is very close to the true value in Figures 5(a), 5(b), +5(c) and 5(d). +B. New Observations +To see how the model performs for new and modified obser- +vations, we test the model with a set of new observations with +random values for both the spatial and temporal components +as shown in Figure 6. For every new observation, the model +TABLE II +MODEL COMPARISON BASED ON DIFFERENT FEATURES +Mean Absolute +Error +Root Mean Squared +Error +Multiple Linear Regression +1.31269 +1.71981 +Elastic Net Regression +1.33501 +1.91345 +Bayesian Linear Regression +1.3123 +1.71962 +Baseline +3.75357 +5.09258 +(a) +(b) +(c) +(d) +Fig. 5. Test observations +provides a new posterior distribution with the mean estimate. +The vertical straight line represents the mean estimate of +the posterior probability distribution for a new observation. +We can see the highest probability density near the mean +estimation of all posterior probability distributions as shown +in Figures 6(a), 6(b), and 6(c) and 6(d). +C. Comparison With Other Approaches +The performance of the Bayesian linear regression model +is compared in terms of Mean Absolute Error (MAE) and +Root Mean Squared Error (RMSE) with two state-of-the-art +frequentist models: Multiple Linear Regression and Elastic Net +Regression as shown in Table II. We also develop a comparison +baseline which is the mean of all possible observations of +the traffic congestion. Here, Bayesian linear regression out- +performs the state-of the-art-approaches in terms of accuracy +as it has the lowest MAE and RMSE values. + +Estimated Dist. +0.25 +True Value +Mean Estimate +0.20 - +0.1 +Density +0.10 +0.05 +0.00 +4 +6 +8 +10 +12 +14 +Speed (Km/h)Estimated Dist. +0.25 +True Value +Mean Estimate +0.20 - +0.15 +Density +0.10 +0.05 +0.00 +4 +6 +8 +10 +12 +14 +16 +18 +Speed (Km/h)RegTraffic +@loT Research Lab,Ontario Tech University +Provide Time Input: +Admin Panel +3 +14:54 +Show Regressions +【2] +3 +16] +Submit +ShowCorrelations +Winchester Rd E +33 +Feature +Value +Winchester Rd W +Winchester Golf Club +Peakhour +1 +Windfiel +Farms +Shopping +entre +Kedron Dells +Golf Course +AM +KingMeadow +0 +adoGolf Club +Road1:Dependent +13.3Km/h +33 +Associate Event +The Fields +ofConlin +Windfields +Road2:Independent +18.05Km/h +33 +Road4:Independent +10.45Km/h +[16] +CampSamac +Fresh Food +Road3:Independent +Conlin Rd +4.4Km/h +Gar +Sobeys +P +F +35 +[26] +The Waltzing Weasel +WI +Mili Express +MetroEstimated Dist. +0.25 +True Value +Mean Estimate +0.20 - +0.1 +Density +0.10 +0.05 +0.00 +12 +14 +16 +18 +20 +22 +24 +Speed (Km/h)Estimated Dist. +0.25 +True Value +Mean Estimate +0.20 - +0.15 +Density +0.10 - +0.05 +0.00 +2 +4 +6 +8 +10 +12 +14 +16 +Speed (Km/h)(a) +(b) +(c) +(d) +Fig. 6. New observations +VII. CONCLUSION +This paper presents RegTraffic, a new dynamic traffic sim- +ulator for spatiotemporal traffic modeling for intercorrelated +road links. RegTraffic builds a regression-based spatiotemporal +traffic model to predict traffic congestion of a road link +depending on neighboring road links and temporal compo- +nents extracted through exploratory data analysis. RegTraffic +provides a dynamic interface for a user to provide new obser- +vations for independent features of the regression model and +provides visualization on interactive geographical maps. The +Mean Absolute Error and Root Mean Squared Error metrics +are used to evaluate the performance of the regression-based +predictive model integrated into RegTraffic. Performance eval- +uation shows that RegTraffic can effectively predict traffic +congestion of intercorrelated road links. In the current version +of RegTraffic, we apply a Bayesian linear regression model for +better interpretation and uncertainty evaluation. In the future, +we plan to enhance RegTraffic by supporting other regression- +based spatiotemporal traffic modeling approaches. +REFERENCES +[1] X. Yan and X. Su, Linear regression analysis: theory and computing. +Singapore, Hackensack, NJ: World Scientific, 2009. +[2] J. Barcelo, Ed., Fundamentals of Traffic Simulation. New York: Springer, +2010 +[3] A. Pell, A. Meingast, and O. Schauer, “Trends in real time Traffic Sim- +ulation,” in Transportation Research Procedia, vol. 25, pp. 1477–1484, +2017. +[4] G. Kotusevski and K. A. Hawick, ”A review of traffic simula- +tion software”. 2009, Accessed: Jul. 30, 2021. [Online]. Available: +https://mro.massey.ac.nz/handle/10179/4506 +[5] M. M. Mubasher and J. 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Elgazzar, ”A Bayesian Linear Regression +Approach to Predict Traffic Congestion,” in 2020 IEEE 6th World Forum +on Internet of Things (WF-IoT), 2020, pp. 1-6. +[19] S. Mostafi and K. Elgazzar, ”An Open Source Tool to Extract Traffic +Data from Google Maps: Limitations and Challenges,” in International +Symposium on Networks, Computers and Communications (ISNCC), +2021. +[20] O. K. Golovnin, K. V. Pupynin and A. S. Privalov, ”A Web-Oriented +Approach for Urban Road Traffic Simulation,” in International Multi- +Conference on Industrial Engineering and Modern Technologies (Far- +EastCon), 2019, pp. 1-4. +[21] H. Mizuta, ”Evaluation of metropolitan traffic flow with agent based +traffic simulator and approximated vehicle behavior model near intersec- +tions,” in Winter Simulation Conference (WSC), 2015, pp. 3925-3936. + +0.25 +Estimated Dist. +Mean Estimate +0.20 - +0.15 +Density +0.10 : +0.05 +0.00 +4 +6 +8 +10 +12 +14 +16 +18 +Speed (Km/h)0.25 +Estimated Dist. +Mean Estimate +0.20 - +0.15 +Density +0.10 +0.05 +0.00 +4 +6 +8 +10 +12 +14 +16 +Speed (Km/h)0.25 +Estimated Dist +Mean Estimate +0.20 - +0.15 +Density +0.10 +0.05 +0.00 +10 +12 +14 +16 +18 +20 +22 +24 +Speed (Km/h)Estimated Dist. +0.25 - +Mean Estimate +0.20 +0.15 +Density +0.10 +0.05 +0.00 +8 +10 +12 +14 +16 +18 +20 +Speed (Km/h) \ No newline at end of file diff --git a/ANAzT4oBgHgl3EQfTPxG/content/tmp_files/load_file.txt b/ANAzT4oBgHgl3EQfTPxG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..884481bed8d0d723486fb59b3a9ae94ba5c10809 --- /dev/null +++ b/ANAzT4oBgHgl3EQfTPxG/content/tmp_files/load_file.txt @@ -0,0 +1,529 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf,len=528 +page_content='RegTraffic: A Regression Based Traffic Simulator for Spatiotemporal Traffic Modeling, Simulation and Visualization Sifatul Mostafi, Taghreed Alghamdi, Khalid Elgazzar IoT Research Lab, ECSE, Ontario Tech University, Oshawa, ON, Canada {sifatul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='mostafi, Taghreed Alghamdi, khalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='elgazzar}@ontariotechu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='ca Abstract—Traffic simulation is a great tool to demonstrate complex traffic structures which can be extremely useful for the planning, development, and management of road traffic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Current traffic simulators offer limited features when it comes to interactive and adaptive traffic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' This paper presents RegTraffic, a novel interactive traffic simulator that integrates dynamic regression-based spatiotemporal traffic anal- ysis to predict congestion of intercorrelated road segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The simulator models traffic congestion of road segments depending on neighboring road links and temporal features of the dynamic traffic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The simulator provides a user-friendly web interface to select road segments of interest, receive user-defined traffic parameters, and visualize the traffic for the flow of correlated road links based on the user inputs and the underlying correlation of these road links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Performance evaluation shows that RegTraffic can effectively predict traffic congestion with a Mean Squared Error of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='3 Km/h and a Root Mean Squared Error of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='71 Km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' RegTraffic can effectively simulate the results and provide visualization on interactive geographical maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Index Terms—Road traffic, simulator, regression, visualization, software I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' INTRODUCTION With the advancement of computer technologies and soft- ware engineering, computer-based traffic simulation has be- come a popular approach for traffic analysis in support of the evaluation and design of Intelligent Transport Systems (ITS) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Traffic simulation software supported by the ability to emulate the variability of spatial and temporal components in traffic flows is a practical tool for capturing and explaining complex traffic systems [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The purpose of developing Traffic simulation tools is to experiment with varieties of strategies in traffic modeling [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Traffic simulation software tools and models built on real- life traffic data are widely applied to support real-time traffic decisions and management solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Regression analysis in the traffic domain is a well- established approach that facilitates traffic modeling and pre- diction [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Regression-based traffic modeling helps in ana- lyzing complex traffic structures which is a useful method for the development and planning of traffic systems and networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Hence, traffic congestion estimation and computerized simula- tion are suitable options for policymakers to analyze different complex traffic scenarios and take actions accordingly [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' A lot of microscopic and macroscopic traffic simulators have been developed including SUMO [10], Aimsun [13], Traffsim [14], SUMMIT [15], SifTraffic [16] and VISSIM [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' These simulators have practical use cases in traffic analysis including traffic flow measurement, multi-agent simulation, particle-based simulation, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Although these state-of- the-art simulators have many practical traffic use cases, they face challenges in the application of simulating road traffic congestion in heterogeneous road transportation networks with a small amount of real-time data [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Also, these simulators lack the feature to adopt regression-based traffic modeling and simulate traffic congestion of a road link depending on traffic congestion of neighboring road links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Some of the simulators do not provide visualization for the simulation results in interactive geographical maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' We aim to develop a regression-based traffic simulator for spatiotemporal traffic modeling to predict traffic congestion of a road link depending on neighboring road segments and provide features to simulate and visualize the results using interactive geographical maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Section II briefly reviews the state-of-the-art traffic simulators and their scope in traffic modeling and simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The traffic modeling approach of RegTraffic is described in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Section IV outlines the processing pipeline of the different components of RegTraffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Section V provides a step-by-step simulation scenario of a traffic use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Section VI shows the performance analysis of RegTraffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Lastly, Section VII concludes this work and provides future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' BACKGROUND AND RELATED WORK Traffic simulator software is commonly divided into two categories: microscopic traffic simulators [7], [10] and macro- scopic traffic simulators [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' FreeSim [9] is a traffic simulator designed to conduct real- time freeway traffic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' SUMO (Simulation of Urban Mobility) [10] is a microscopic traffic simulator that is devel- oped to process complex and large road networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' SUMO is widely used in many applications including traffic flow model- ing [11] and color mapping Google Maps routes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Aimsun [13] is a traffic simulator for modeling smart mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Traffsim [14] simulator is widely used for modeling isolated traffic control strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' SUMMIT [15] provides functionalities to simulate urban driving in large traffic scenarios with massive and mixed traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' SifTraffic [16] is a practical software tool to arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='01245v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='NI] 23 Nov 2022 TABLE I COMPARISON OF TRAFFIC SIMULATORS Comparison Category Simulator RegTraffic FreeSim [9] SUMO [10] Aimsun [13] TraffSim [14] SUMMIT [15] SimTraffic [16] VISSIM [7] Spatiotemporal Traffic Modeling Yes Yes Yes Yes Yes Yes Yes Yes Regression Modeling Yes No No No No No No No Interactive Geographical Maps Yes No Yes Yes No Yes Yes Yes Web Interface Yes No No No No No No No conduct simulations of practical traffic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' VISSIM [7] is a microscopic traffic simulator for behavior-based multi- purpose traffic flow simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' [17] explored different methods of correct- ing the traffic simulation models based on linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Golovnin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' [20] took a web-oriented approach to simulate road traffic, especially in urban settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Mizuta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' [21] evaluated the traffic flow near intersections of a metropolitan city to understand how agent-based traffic simulators work to approximate vehicle behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' A comparison among the existing traffic simulators along with RegTraffic is listed in Table I in terms of some key characteristics and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' MATHEMATICAL MODELING A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Spatial Feature Figure 1 shows a traffic road intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' In this intersec- tion, we consider a road link as the spatial road feature that is dependent on one or several connected spatial road features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' For example, for the intersection shown in Figure 1, the road link ˆy is a spatial feature and is modeled as a dependent variable in our regression modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The road links xs 1, xs 2 up to the road link xs n are the independent spatial features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' It’s worth noting that the dependent road link ˆy is an outbound while all the independent road links xs 1, xs 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=', xs n inbound to the intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Our proposed traffic modeling approach described in [18] indicates that the dependent spatial feature must be an outbound road link and the independent spatial features must be inbound road links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The model incorporates a set of temporal features that can be extracted from both in- dependent and dependent spatial features through exploratory data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The specific number of temporal features and independent spatial features are arbitrary and dependent on the specific road intersection and their orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Here, XS is defined as a set of independent spatial features � xs 1, xs 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='., xs ns � as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' XS = � xs 1, xs 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='., xs ns � (1) The cardinality of set XS is defined as ns as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' ns = |XS| (2) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Traffic Road intersection B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Temporal Feature Extraction In our modeling, we convert temporal features into categori- cal features using one hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' To simplify our modeling, temporal features are encoded using only two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Here, XT is a set of temporal features � xt 1, xt 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='., xt nt � as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' XT = � xt 1, xt 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='., xt nt � (3) The cardinality of set XT is defined as nt as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' nt = |XT | (4) The set of temporal features is extracted from spatial fea- tures using exploratory data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Here, XT is the output of function f which takes in the set of spatial features XS as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The function f is a many to many function that takes in a set of spatial features and conducts exploratory data analysis to extract a set of temporal features as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (5) XT = fns→nt(XS) (5) We define the set X as a union of the temporal features XT and spatial features XS as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' X = XT ∪ XS (6) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Regression Modeling 1) Regression Formation: RegTraffic forms a regression model through a linear combination of both temporal and spatial explanatory features to explain the dependent spatial feature ˆy as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' In this equation, all the independent features are associated with their corresponding regression coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' α indicates the bias and ϵ refers to the error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' ˆy = nt � i=1 βt ixt i + ns � i=1 βs i xs i + α + ϵ (7) In the regression Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (7), every explanatory temporal feature from setting XT is associated with a regression coefficient from set βT as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' βT = � βt 1, βt 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='., βt nt � (8) Similarly, in the regression Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (7), every explanatory spatial feature from set XS is associated with a regression coefficient from set βS as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' βS = � βs 1, βs 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='., βs ns � (9) Here, β is defined as the union of set βT and βS β = βT ∪ βS (10) 2) Posterior Probability Distribution: We use a novel Bayesian linear regression approach for spatiotemporal traffic modeling of a road link proposed in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Bayesian linear regression formulates a posterior probability distribution of the model parameters rather than just finding a single point estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The response variable is drawn from a probability distribution instead of a single value estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' A Bayesian linear regression model samples the response variable from a normal distribution as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' y ∼ N(βT X, σ2I) (11) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (11), the response variable y is generated from a Gaussian normal distribution, which is characterized by a mean and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (12) refers to the Bayes Theorem which is the fundamental building block of Bayesian linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Here, P(β | ˆy, X) is the posterior probability distribution of the model parameters, P(ˆy | β, X) is the likelihood of the data, P(β | X) is the prior probability of the parameters and P(ˆy | X) is the normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The posterior distribution of the model parameters is proportional to the multiplication of the likelihood of the data and the prior probability of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' A detailed description of the model is described in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' P(β | ˆy, X) = P(ˆy | β, X) ∗ P(β | X) P(ˆy | X) (12) Once the regression model is built, the user can provide new observations for independent spatial features XS and independent temporal features XT into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Based on the new observation, the model incorporates the regression coefficients associated with the explanatory variables and predicts the output for the dependent variable ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' An event can be associated with a specific value as an input for any independent spatial feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Event Integration Here, XE is defined as a set of events � XE 1 , XE 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='., XE nE � as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' XE = � XE 1 , XE 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='., XE nE � (13) The cardinality of set XE is defined as nE as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' nE = |XE| (14) After event integration, the independent spatial features associated with an event are integrated into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' If any independent spatial feature is associated with an event, we need to replace the value for the independent spatial feature xS with the events xE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Spatial features which are not affected by any specific event are represented by xS′ along with their model parameter βS′ as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The amount of spatial features unaffected by any specific event is denoted as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' nS′ = nS − nE (15) ˆy = nt � i=1 βt ixs i + nS′ � i=1 βS′ i xS′ i + nE � i=1 βE i xE i + α + ϵ (16) However, we add a time constraint in association with the temporal components for adding a specific event into the regression equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' For any specific event XE occurred at time TE, the value of spatial feature XS will be replaced by the value of XE if TE ⊂ XT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' PROCESSING PIPELINE The processing pipeline of RegTraffic is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The spatial feature consists of a unique name of the feature, the corresponding time series dataset of that spatial feature, and a set of latitude and longitude as the waypoints of the route of that spatial feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The traffic data extraction is described in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' In this process, a user selects the starting and ending points of the route of interest and specifies the time range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The traffic data extraction tool gathers time-series information of the “congestion index” of that road link every 15 minutes throughout the time range from Google Maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The congestion index is defined by the average speed of that road link in terms of kilometers per hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' At the end of the process, the tool generates a time series dataset that has a Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Processing Pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' unique name as provided by the user when adding a spatial feature in RegTraffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' RegTraffic also constructs a temporal feature component with three core input values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' These are the unique name of the temporal feature, the corresponding time series dataset, and the time range of that temporal feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' RegTraffic takes a set of input preferences from the user as part of the model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' It also allows users to choose the dependent feature for the regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Once the regression model is built, RegTraffic passes the regression coefficients to a visualization interface where a user can input new observations for the independent features that can be both spatial and temporal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' SIMULATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Spatial Feature We conduct our experiment on four connected road links in Oshawa, Ontario, Canada as shown in Figure 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The ending point of road links 2, 3 and 4 are connected with the origin of the road link 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' A connected road network is formed by these road links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' We represent the traffic congestion level of these 4 road links as Road1, Road2, Road3, and Road4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Road1 is the dependent link where Road2, Road3 and Road4 are the independent links that collectively affect Road1 during a specific time of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' We collect the average traffic speed of each road link every 15 minutes for an entire week from 12:00 am March 01, 2020, to 11:45 pm March 07, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' As a result, there are a total of 672 observations over 7 days of time-series data for each road link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Figure 3(b) shows the time series of the average traffic speed of all four road links for the first two days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The y axis represents the average traffic speed in km/h, which is considered the traffic congestion index in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' We can see that the time series has a cycle as the average traffic speed shows regular and predictable changes that recur every day within a certain time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The higher the average speed, the low the traffic congestion, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Temporal Feature Extraction Figure 3(c) shows the hourly mean of the average speed for each road link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The mean values show very little variance (a) Intersection of Simcoe and Conlin Road in Oshawa (b) Time series data of 4 road links (c) Hourly average speed throughout a day (d) Identifying threshold for Peakhour Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Average speed throughout a day compared to each other as they seem to move together throughout the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The average of the different means of all road links is plotted in Figure 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The horizontal line at a speed of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='75 km/h divides the plot evenly and intersects with the total average speed at two points, one at daytime 8:00 and the other one at 23:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' From this exploratory data analysis, a new categorical feature called Peakhour is extracted that indicates a certain time interval during a day where the average traffic speed remains below 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='75 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' From 9:00 am to 12:00 pm, the value of Peakhour would be 1, otherwise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Another temporal component is considered in the analysis as a categorical variable which is AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The value of AM would be 1 when the meridiem is AM and 0 when it is PM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Regression Modeling The outcome of our Bayesian linear regression is the distri- bution of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The model does not provide an exact estimate for a feature, but the mean value of the distribution can be considered as an estimate for the feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The benefit of having a posterior probability distribution is that the model also provides an entire range of values that shows the uncertainty of the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The mean of a posterior probability distribution is taken as the best estimate of that model parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' These mean estimates of these model parameters are put together to derive a new Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Road1 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='4163 ∗ Intercept + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='7561 ∗ AM −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='7517 ∗ Peakhour − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='0477 ∗ Road2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='0479 ∗ Road3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='7139 ∗ Road4 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='7003 ∗ SD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='(17) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Model Selection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Time Series ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Spatial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Temporal ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Time Series ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Feature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Regression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Waypoints ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Time Range ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Modelling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='User Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='User Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Visualization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Scheme ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Spatial Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Temporal FeaturesN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Road 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Road 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Con ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='CopperBrang ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Vegan · ss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Road 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Road 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Subway ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Sandwich shop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='SmileRoadl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Road2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Road3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Road4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Speed (Km/h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='01-Mar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='06:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='12:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='18:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='02-Mar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='06:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='12:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='18:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='00:00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='2020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Datetime20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Speed (Km/h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Roadl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Road2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Road3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Road4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Hours (0-23)18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Hourly Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Threshold Speed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Speed (Km/h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Peakhour ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Peakhour ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='starts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='ends ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='Hours (0-23)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Regression Analysis in RegTraffic Simulator D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Visualization Figure 4 describes a sample simulation procedure of a road intersection where Road1 is considered as a dependent road link and Road2, Road3 and Road4 are independent road links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Based on the spatial features, two new temporal features are extracted which are Peakhour and AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' RegTraffic shows the location of the road links on an interactive geographical map where the user can provide new observations for indepen- dent road links and temporal features to predict the outcome of the dependent road link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' As shown in the figure, the user sets the congestion index of Road2, Road3, and Road4 to 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='05, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='4, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='45 kilometers per hour, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The user also needs to provide the specific time as an input for the temporal features Peakhour and AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' RegTraffic calculates the value for the temporal features from the time input provided by the user and incorporates these values along with the input values for independent spatial features to predict the congestion index of dependent road link Road1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Based on the input values provided by the user, RegTraffic predicts the congestion index of the road link Road1, which is 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='3 kilometers per hour in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' PERFORMANCE EVALUATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Test Observations To evaluate the performance, the model is tested on a testing dataset of traffic observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Figure 5 shows four random test observations from the testing dataset along with the probability density function of Road1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The true value of Road1 is represented by the dotted line and the mean of the probability distribution is represented by the straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The mean of the probability distribution is considered as the best estimate for the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The estimated value provided by the model is very close to the true value in Figures 5(a), 5(b), 5(c) and 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' New Observations To see how the model performs for new and modified obser- vations, we test the model with a set of new observations with random values for both the spatial and temporal components as shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' For every new observation, the model TABLE II MODEL COMPARISON BASED ON DIFFERENT FEATURES Mean Absolute Error Root Mean Squared Error Multiple Linear Regression 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='31269 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='71981 Elastic Net Regression 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='33501 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='91345 Bayesian Linear Regression 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='3123 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='71962 Baseline 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='75357 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='09258 (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Test observations provides a new posterior distribution with the mean estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The vertical straight line represents the mean estimate of the posterior probability distribution for a new observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' We can see the highest probability density near the mean estimation of all posterior probability distributions as shown in Figures 6(a), 6(b), and 6(c) and 6(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Comparison With Other Approaches The performance of the Bayesian linear regression model is compared in terms of Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE) with two state-of-the-art frequentist models: Multiple Linear Regression and Elastic Net Regression as shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' We also develop a comparison baseline which is the mean of all possible observations of the traffic congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Here, Bayesian linear regression out- performs the state-of the-art-approaches in terms of accuracy as it has the lowest MAE and RMSE values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Estimated Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='25 True Value Mean Estimate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='1 Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='00 4 6 8 10 12 14 Speed (Km/h)Estimated Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='25 True Value Mean Estimate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='15 Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='00 4 6 8 10 12 14 16 18 Speed (Km/h)RegTraffic @loT Research Lab,Ontario Tech University Provide Time Input: Admin Panel 3 14:54 Show Regressions 【2] 3 16] Submit ShowCorrelations Winchester Rd E 33 Feature Value Winchester Rd W Winchester Golf Club Peakhour 1 Windfiel Farms Shopping entre Kedron Dells Golf Course AM KingMeadow 0 adoGolf Club Road1:Dependent 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='3Km/h 33 Associate Event The Fields ofConlin Windfields Road2:Independent 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='05Km/h 33 Road4:Independent 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='45Km/h [16] CampSamac Fresh Food Road3:Independent Conlin Rd 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='4Km/h Gar Sobeys P F 35 [26] The Waltzing Weasel WI Mili Express MetroEstimated Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='25 True Value Mean Estimate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='1 Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='00 12 14 16 18 20 22 24 Speed (Km/h)Estimated Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='25 True Value Mean Estimate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='15 Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='10 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content='00 2 4 6 8 10 12 14 16 Speed (Km/h)(a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' New observations VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' CONCLUSION This paper presents RegTraffic, a new dynamic traffic sim- ulator for spatiotemporal traffic modeling for intercorrelated road links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' RegTraffic builds a regression-based spatiotemporal traffic model to predict traffic congestion of a road link depending on neighboring road links and temporal compo- nents extracted through exploratory data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' RegTraffic provides a dynamic interface for a user to provide new obser- vations for independent features of the regression model and provides visualization on interactive geographical maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' The Mean Absolute Error and Root Mean Squared Error metrics are used to evaluate the performance of the regression-based predictive model integrated into RegTraffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' Performance eval- uation shows that RegTraffic can effectively predict traffic congestion of intercorrelated road links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' In the current version of RegTraffic, we apply a Bayesian linear regression model for better interpretation and uncertainty evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' In the future, we plan to enhance RegTraffic by supporting other regression- based spatiotemporal traffic modeling approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfTPxG/content/2301.01245v1.pdf'} +page_content=' REFERENCES [1] X.' metadata={'source': 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a/BtE4T4oBgHgl3EQfFQxm/content/tmp_files/2301.04884v1.pdf.txt b/BtE4T4oBgHgl3EQfFQxm/content/tmp_files/2301.04884v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b31c1fd172c1c5aae549b6e126d560f09aecc596 --- /dev/null +++ b/BtE4T4oBgHgl3EQfFQxm/content/tmp_files/2301.04884v1.pdf.txt @@ -0,0 +1,1067 @@ +Performance of an ultra-pure NaI(Tl) detector +produced by an indigenously-developed +purification method and crystal growth for the +COSINE-200 experiment +H. Lee 1,2,B.J. Park 1,2,J.J. Choi 2,3, O. Gileva 2, C. Ha 4, A. Iltis 5, E.J. Jeon 2,1, +D.Y. Kim 2, K.W. Kim 2, S.H. Kim 2, S.K. Kim 3, Y.D. Kim 2,1, Y.J. Ko 2, C.H. Lee 2, +H.S. Lee 2,1, I.S. Lee 2,∗, M.H. Lee 2,1, S.J. Ra 2, J.K. Son 2, K.A. Shin 2 +1IBS School, University of Science and Technology (UST), Daejeon 34113, Republic +of Korea +2 Center for Underground Physics, Institute for Basic Science (IBS), Daejeon 34126, +Republic of Korea +3 Department of Physics and Astronomy, Seoul National University, Seoul 08826, +Republic of Korea +4 Department of Physics, Chung-Ang University, Seoul 06973, Republic of Korea +5 Damavan Imaging, Troyes 10430, France +Correspondence*: +I.S. Lee +islee@ibs.re.kr +ABSTRACT +The COSINE-100 experiment has been +operating with 106 kg of low-background +NaI(Tl) detectors to test the results from the +DAMA/LIBRA experiment, which claims to +have observed dark matter. However, since the +background of the NaI(Tl) crystals used in the +COSINE-100 experiment is 2–3 times higher +than that in the DAMA detectors, no conclusion +regarding the claimed observation from the +DAMA/LIBRA experiment could be reached. +Therefore, we plan to upgrade the current +COSINE-100 experiment to the next phase, +COSINE-200, by using ultra-low background +NaI(Tl) detectors. The basic principle was +already proved with the commercially available +Astro-grade NaI powder from Sigma-Aldrich +company. However, we have developed a +mass production process of ultra-pure NaI +powder at the Center for Underground Physics +(CUP) of the Institute for Basic Science (IBS), +Korea, using the direct purification of the raw +NaI powder. We plan to produce more than +1,000 kg of ultra-pure powder for the COSINE- +200 experiment. +With our crystal grower +installed at CUP, we have successfully grown +a low-background crystal using our purification +technique for the NaI powder. +We have +assembled a low-background NaI(Tl) detector. +In this article, we report the performance of this +ultra-pure NaI(Tl) crystal detector produced at +IBS, Korea. +Keywords: +NaI(Tl) crystal; +Dark matter; +COSINE-200; +Low- +background detector; Purification +1 +INTRODUCTION +Numerous astronomical observations support the +theory that most of the matter in universe is the +invisible dark matter, although an understanding of +its nature and interactions remains elusive (1, 2, 3). +Even though tremendous efforts have been made +to search for dark matter, no definitive signals +have been observed (4, 5). The only exception is +the DAMA experiment, which has observed an +annual modulation of event rates using an array of +NaI(Tl) detectors (6, 7). This observation could be +interpreted as dark matter-nuclei interactions (8, 9). +However, this result has been the subject of a +continuing debate because no other experimental +searches have observed similar signals (5, 10). +1 +arXiv:2301.04884v1 [physics.ins-det] 12 Jan 2023 + +H. Lee et al. +Several experimental efforts using the same +NaI(Tl) target materials are currently underway (11, +12, 13, 14, 15, 16, 17). +The COSINE-100 +experiment is one such effort presently operating +at the Yangyang underground laboratory in Korea, +which has provided several exciting physics +results (9, 18, 19, 20, 21). However, due to +the approximately 2–3 times higher background +level, an unambiguous conclusion regarding the +observation in the DAMA experiment using the +same annual modulation signal has not been +observed yet (22, 23). +As an effort to upgrade the ongoing COSINE- +100 experiment for the next-phase COSINE- +200 experiment, we have conducted an R&D +program aimed at producing a low-background +NaI(Tl) detector to conclude on the observed +signals from DAMA/LIBRA unambiguously. It +includes the chemical purification of the raw NaI +powder (24, 25), its crystal growth (26), and +detector assembly (27). We have already proved +the principle of a low-background NaI(Tl) detector +using the commercially available low-background +Astro-grade NaI powder from Sigma-Aldrich (28). +As a next step, we have grown an NaI(Tl) crystal +using our own NaO power produced using the mass +purification process at IBS, Korea (29). This article +reports the characteristics and performance of this +indigenously-produced NaI(Tl) crystal. +2 +NAI PURIFICATION AND CRYSTAL +GROWTH +The COSINE-200 detector requires extremely +low levels of radioactive contamination in the +materials used in the detector production. The +major contributors to the background are the +decays of 40K and 210Pb in the bulk NaI(Tl) +crystal (30, 31). Because of the similarity in +its chemical properties to those of Na, which +is in the same periodic table group, K is the +primary impurity contaminant, and its selective +extraction from NaI powder is challenging. We +found that the fractional recrystallization method +effectively reduces the K and Pb impurities (24). In +addition, using this method, the Ba concentration +was significantly reduced, indicating a reduction +of Ra impurities (24). Thus,we constructed a +mass production facility at IBS, Daejeon, Korea, +for producing ultra-pure NaI powder using the +fractional recrystallization method on-site (25). +The facility has been operated with a maximum +production rate of 35 kg of ultra-pure powder +in a single processing cycle of two weeks (29). +Using our purification facility, we have performed +mass purification of the fractional recrystallization +process using raw NaI powder from Merck +(99.99(5)% purity). In this mass purification process +of the NaI power, we have achieved a concentration +of K of 6.4 ppb and that of Pb below 0.3 ppb (29), +which are consistent with contamination levels of +the Astro-grade powder. +The ultra-pure crystal was grown using a small- +volume Kyropouls grower (32), which is the same +grower used for growing the proof of principle low- +background NaI(Tl) crystals using the commercial +Astro-grade powder (28). In growing the crystal, +1.7 kg of the purified NaI powder was loaded in +a 12 cm diameter, 10 cm high quartz crucible. An +NaI(Tl) crystal ingot, as shown in Figure 1(a), of +∼70 mm diameter and ∼80 mm high and having +a 1.1 kg mass, was grown in ∼24 h . During the +crystal growth, N2 gas was continuously flushed +using a thallium trap with a flow rate of 10 L/m. +3 +EXPERIMENTAL SETUP +3.1 +NaI(Tl) crystal +The growth of the NaI(Tl) crystal (named NaI- +037) was completed on January 18, 2021, using +NaI powder purified IBS (24, 32). The top and +bottom sections of the crystal ingot were cut using +a diamond bandsaw, as shown in Figure 1(b). After +cutting the top and bottom, the NaI-037 crystal is +70 mm in diameter and 51 mm in height. The flat +top and bottom surfaces and a barrel-shaped side +surface were polished using aluminum oxide films +ranging from 400 to 8000 grit. After polishing, the +barrel was wrapped with a polytetrafluoroethylene +(PTFE) film in several layers as a diffusive reflector. +Frontiers +2 + +H. Lee et al. +Figure 1a. NaI(Tl) Crystal ingot +Figure 1b. Cut and polished NaI(Tl) +crystal +Figure 1. Bare NaI(Tl) (NaI-037) crystal +A 3 mm thick copper casing with quartz windows at +each end was encapsulated the crystal hermetically. +Hamamatsu 3 inch photomultiplier tubes (PMTs), +selected for high quantum efficiency (R12669SEL), +were coupled via an optical interface to each end +of the crystal. The entire assembly was performed +in a glovebox, where the humidity was maintained +to be less than 10 ppm (H2O) using Ar gas and +a molecular sieve trap. Before the assembly, all +parts were cleaned using diluted Citranox liquid +with sonication and baked in a vacuum oven for +more than 12 h. After assembly, the detector was +delivered to the Yangyang underground laboratory +(Y2L), which has ∼700 m of rock overburden (33). +From the crystal growth to Y2L delivery, it took less +than three weeks and minimized the cosmogenic +activation in the crystal. +3.2 +Shielding structure +The background contamination levels of the +NaI-037 crystal were evaluated using, the same +experimental apparatus as that used for the NaI(Tl) +crystal R&D at the Y2L (28, 34). It includes an +array of 12 CsI(Tl) crystals surrounded by 10 cm +copper, 5 cm polyethylene, 15 cm lead, and 30 cm +liquid scintillator-loaded mineral oil (35, 36) as a +radiation shield. The detector was installed inside +the CsI(Tl) array, as shown in Figure 2. +Figure 2. A schematic view of the Y2L setup. The +NaI-037 crystal (red circle) was installed inside the +CsI(Tl) crystal array (blue squares). +3.3 +Electronics +The PMTs attached to the NaI-037 crystal had +two readouts each, a high-gain signal from the +anode and a low-gain signal from the fifth-stage +dynode. The anode signal was amplified by a +factor of 30, whereas the dynode signal was +amplified by a factor of 100 using a custom- +made preamplifier. The amplified signals were +digitized by 500 MHz, 12-bit flash analog-to-digital +converters (FADCs). Triggers from the individual +Frontiers +3 + +YPARINEOY +NOUYPCCsl(T)Crystals +Nal(m)Crystal +Copper +Folvetnviene +Lead +MineralOilH. Lee et al. +channels were generated by the field-programmable +gate arrays (FPGAs) embedded in the FADC. The +final trigger was generated in the trigger and clock +board (TCB) when an anode signal corresponding +to one or more photoelectrons (PEs) occurred in +each PMT within a 200 ns time window. The anode +and dynode signals were recorded whenever the +anode signal produced a trigger. +Signals from the CsI(Tl) crystals were amplified +by a factor of 10 and digitized in a charge-sensitive +62.5 MHz FADC (SADC). The SADC provided the +integrated charge and the time of the signal. An +integration time of 2048 ns was used to record the +CsI(Tl) signals considering their decay time. The +SADC channels did not generate triggers. +If the trigger condition was satisfied, the TCB sent +trigger signals to the FADC and SADC to store the +signals from the NaI(Tl) and the CsI(Tl) crystals. +The FADC stored an 8 µs long waveform starting +approximately 2.4 µs before the time of the trigger +position. The SADC stored the maximum integrated +charge within an 8 µs search window. This system +is similar to the one used in the COSINE-100 data +acquisition (37). +Energy [keV] +0 +10 +20 +30 +40 +50 +60 +70 +80 +Entries +0 +50 +100 +150 +200 +Am +241 +Figure 3. Anode energy distribution obtained +using a 241Am source. +3.4 +Energy calibration and light yields +The energy calibration of the anode signal was +done using a 59.54 keV X-ray emitted from 241Am. +Figure 3 shows the anode energy spectrum. A clear +peak at 59.54 keV resulting from the 241Am source +is shown together with the 127I X-ray escape peak +around 30 keV. The dynode signal was calibrated +using the photopeaks corresponding to 214Bi(609 +keV) and 40K(1460 keV) contaminants in the +crystal. +The charge distribution of the single photoelectron +(SPE) was obtained by identifying the isolated +clusters at the decay tail of the 59.54 keV X-ray +signal from the 241Am source (3-5 µs after the +signal start). The light yield was determined from +the ratio of the total deposited charge and the mean +of the SPE charge for the 59.54 keV X-ray data. In +this crystal, a light yield of 17.8±0.6 number of +photoelectron (NPE)/keV was obtained. It is similar +to the result for the NaI-036 crystal, which has the +highest light yield among the previously developed +low-background NaI(Tl) crystals using the Astro- +grade powder (28). This light yield is also larger +than those of the detectors used in the COSINE-100 +and DAMA/LIBRA experiments, as summarized in +table 1. +4 +UNDERSTANDING THE +BACKGROUND IN THE SPECTRUM +4.1 +40K background +40K is one of the most problematic background +sources in the search for weakly interacting massive +particles (WIMP) using NaI(Tl) crystals. The X- +rays/Auger electrons from 40K decays produce +3.2 keV energy signals, similar to the energy signals +expected for a WIMP-nuclei interaction (30, 31, 38). +The 40K decays also emit 1460 keV γ rays, which +can escape from the NaI(Tl) crystal and hit the +surrounding CsI(Tl) crystals, leading to a double +coincidence with the 3.2 keV X-rays. +Figure 4 shows the tagged low-energy spectra +from the NaI(Tl) crystal by requiring the detection +of the 1460 keV γ ray in the CsI(Tl) crystals. The +Frontiers +4 + +H. Lee et al. +40K background level in the NaI(Tl) crystal was +determined by comparing the measured coincidence +rate from a GEANT4-based simulated data, as +described in Ref. (39). By accumulating more than +six months of data, the K level was measured to be +8.3±4.6 ppb, which was compared with the other +NaI(Tl) crystals listed in Table 1. It is well below +our goal of 20 ppb, consistent with the results from +the DAMA/LIBRA crystals (34, 40) and previously +developed NaI-035 and NaI-036 crystals with the +Astro-grade powder. +1 +2 +3 +4 +5 +6 +Energy (keV) +0 +1 +2 +3 +4 +5 +6 +7 +8 +Number of Events +Data +Fit(Gaussian+Constant) +Gaussian Component +Figure 4. Energy deposition of the 3.2 keV +40K coincidence events in the NaI-037 crystal. +The model of the energy spectrum assumes a +combination of a Gaussian 40K signal and a constant +background. +4.2 +α analysis +Alpha-induced events inside the NaI(Tl) crystals +can be identified using the fast decay times of +their corresponding signals. The charge-weighted +duration time, called the meantime, is defined as +⟨t⟩ = ΣiAiti +ΣiAi +, +(1) +where A and t are the charge and time of the i-th +digitized bin of a signal waveform, respectively. +The meantime is estimated within 1500 ns from the +pulse starting timing. Figure 5 shows a scatter plot +of the energy versus the meantime for the NaI-037 +0.2 +0.25 +0.3 +0.35 +s) +µ +Meantime ( +1 +2 +3 +4 +Energy (MeV) +Figure 5. Scatter plot of the meantime versus the +energy distribution events measured over 7.8 days +for the NaI-037 crystal. The α events (red dots) and +the γ/β events (black dots) are separated clearly. +crystal. In the figure, the populations of γ/β and +α events can be separated clearly due to the faster +decay times of the α-induced events. +4.3 +210Pb background +In the NaI(Tl) crystal experiments, the dominant +background source in the low-energy signal region +is from 210Pb (31, 41, 42). The 210Pb activity can be +estimated from the alpha-decay studies, because the +α decays of 210Po originate from the β decays of +210Pb. Due to the decay time of 200 days of 210Po, +the amount of 210Pb produced can be estimated +using a time-dependent fit of the alpha rate as +follows: +N = NPb210 +� +1 − e−(t−t0)/τP o210 +� ++ C, +(2) +where N is the total alpha rate, NPb210 is the +amount of 210Pb at the equilibrium state, t0 is the +time difference between 210Pb contamination and +the start time of data taking, τPo210 is the mean +lifetime of 210Po, approximately 200 days, and C +represents the long-lived components from 238U +and 232Th chains. Figure 6 shows the measured +total alpha rates in the NaI-037 crystal over the +detector running time. The 210Pb activity in the +crystal was estimated to be 0.38±0.10 mBq/kg, +Frontiers +5 + +H. Lee et al. +which is lower than the COSINE-100 crystals and +is consistent with the activity in the NaI-036 crystal +produced using the Astro-grade powder. However, +this activity is slightly higher than the DAMA +crystals and another crystal NaI-036 grown with +the same Astro-grade powder. The 0.4 mBq/kg +level contamination of 210Pb is enough to reach +1 count/kg/keV/day background level, similar to +the activity in the DAMA/LIBRA detectors, as +described in Ref. (28). +100 +150 +200 +Days from crystal growing +0 +0.1 +0.2 +0.3 +0.4 +0.5 +Activity (mBq/kg) +Data +Fit +Asymptotic Line +Figure 6. The total alpha rate in the NaI-037 +crystal as a function of time, modeled with 210Po +assuming contamination of 222Rn (and/or 210Pb). +The asymptotic line corresponds to the rate of total +alpha events in the equilibrium state. +4.4 +232Th background +Contaminants from the 228Th subchain in the +232Th family can be estimated by deploying the +time-delayed α–α coincident events of 220Rn and +216Po. The alpha decay of 216Po has a half-life of +0.145 s following its production via alpha decay of +220Rn. Owing to the short half-life of 216Po, it is +straightforward to select two successive α particles +with almost no random coincident events. +The presence of the coincident events is shown +in figure 7(a) as the distribution of the time gap +between those two α events. The exponential +component indicates the contamination from 232Th, +corresponding to below 0.39 ppt (90% confidence +level). The 232Th concentration in the NaI-037 +crystal is the lowest among the other NaI(Tl) +crystals, as summarized in table 1. +4.5 +238U background +238U is one of the common radioisotopes because +of its long half-life . The 238U content in the +background can be studied using the time-delayed +β–α coincident events, similar to the calculation of +the 232Th background. This method exploits the α +decay of 214Po with a half-life of 164.3 µs, while +214Bi, the parent particle of 214Po, undergoes β +decay. Due to the 50 µs dead time of the trigger +system, the coincident events with delay times +greater than 50 µs can be tagged. The results are +shown in figure 7. The 238U activity of NaI-037 was +1.02±0.58 ppt, similar to that observed for the other +NaI(Tl) crystals, as given in Table 1. +4.6 +External Background +Because of the small size of the NaI-037 crystal +and no liquid scintillator active veto, a significantly +higher background contribution is expected from +the external background compared to those found +in the COSINE-100 crystals (30, 41). The PMTs +attached to the NaI(Tl) and the CsI(Tl) crystals are +the primary sources of external background. In this +study, the external background contributions were +simulated using the GEANT4-based simulation +toolkit used for the COSINE-100 background +modeling (30, 41). +4.7 +Cosmogenic radionuclides +The +cosmogenic +production +of +radioactive +isotopes in the NaI(Tl) crystal is mainly due to long- +lived nuclides such as 3H and 22Na (30, 44). The +NaI-037 crystal was grown in Daejeon, Korea (70 +m in altitude) and delivered underground within a +month. Based on the previous study, one-month +exposure time near sea level can produce 0.004 +mBq/kg of 3H and 0.05 mBq/kg of 22Na (44), +respectively. +Frontiers +6 + +H. Lee et al. +0 +0.2 +0.4 +0.6 +0.8 +1 +Time (s) +0 +1 +2 +3 +4 +Number of Events +Data +Fit(Exponential+Constant) +Exponential Component +Constant +Figure 7a. Time differen ofbetween two α decays of +the 220Rn–216Po decay chain. +200 +400 +600 +800 +1000 +s) +µ +Time ( +0 +2 +4 +6 +8 +10 +Number of Events +Data +Fit(Exponential+Constant) +Exponential Component +Constant +Figure 7b. Time difference between the 214Po α +decay and 214Bi β decay. +Figure 7. Time difference distributions of data (black dots) and the exponential fits to them (red-solid +line). +Table 1. Measured radioactive contaminants in the NaI-037 crystal, C6 of COSINE-100 (30), DAMA +crystals (40, 43), and the previously grown NaI-035 and NaI-036 crystals using the Astro-grade +powder (28).The upper limits are given at a 90% confidence level. +Crystal +Mass (kg) +LY (NPE/keV) +40K (ppb) +210Pb (mBq/kg) +232Th (ppt) +238U (ppt) +NaI-037 +0.71 +17.8±0.6 +8.3±4.6 +0.44±0.09 +0.2±0.3 +1.0±0.6 +NaI-035 +0.61 +11.8±1.8 +<42 +0.01±0.02 +1.7±0.5 +0.9±0.3 +NaI-036 +0.78 +17.1±0.5 +<53 +0.42±0.27 +<4.9 +36.5±3.9 +COSINE-100 +12.5 +14.6±1.5 +16.8±2.5 +1.87±0.09 +0.7±0.2 +<0.02 +DAMA +9.7 +5–10 +<20 +0.01–0.03 +0.5–7.5 +0.7–10 +5 +BACKGROUND MODELING +For a quantitative understanding of the background +in the NaI-037 crystal, GEANT4-based simulation, +developed for the background modeling of the +COSINE-100 NaI(Tl) crystals (30, 41) and also +used in the previously grown crystals using the +Astro-grade powder (28), was performed. The input +values of the contamination levels are obtained +from Table 1. A simultaneous fit was done to +the single-hit low energy (3–60 keV), single- +hit high energy (60 keV–3 MeV), multiple-hit +low energy, and multiple-hit high energy events +using the log-likelihood method. A multiple-hit +event corresponds to one or more coincident +hits in any of the surrounding CsI(Tl) crystals. +The backgrounds from the PMTs attached to the +NaI(Tl) and CsI(Tl) crystals were measured using +a high-purity germanium detector (30, 31). These +values were constrained to be within 50% of +the measured result because the exact locations +of such radioisotopes are uncertain. The long- +lived cosmogenic radioisotopes were constrained +to be within 50% of their calculation production +values whereas the other short-lived cosmogenic +components were floated. Figure 9 and Table 2 +show the fitted results for the NaI-037 crystal +on all simulated background components and the +Frontiers +7 + +H. Lee et al. +Energy [keV] +10 +20 +30 +40 +50 +60 +Counts/da/kg/keV +2 +− +10 +1 +− +10 +1 +10 +2 +10 +Data +Internal +Cosmogenic +External +Figure 9a. Single-hit low-energy (2–60 keV) +Energy [keV] +500 +1000 +1500 +2000 +2500 +3000 +Counts/da/kg/keV +3 +− +10 +2 +− +10 +1 +− +10 +1 +10 +2 +10 +Data +Internal +Cosmogenic +External +Figure 9b. Single-hit high-energy (60–3000 keV) +Energy [keV] +10 +20 +30 +40 +50 +60 +Counts/da/kg/keV +2 +− +10 +1 +− +10 +1 +10 +2 +10 +Data +Internal +Cosmogenic +External +Figure 9c. Multiple-hit low-energy (2–60 keV) +Energy [keV] +500 +1000 +1500 +2000 +2500 +3000 +Counts/da/kg/keV +3 +− +10 +2 +− +10 +1 +− +10 +1 +10 +2 +10 +Data +Internal +Cosmogenic +External +Figure 9d. Multiple-hit high-energy (60–3000 keV) +Figure 9. Measured single-hit and multiple-hit background spectra of the NaI-037 (black point) crystal +fitted with the different simulated background components using a simultaneous fit of four channels using +the log-likelihood method. The external component (purple-hatched area) is the dominant contributor. +summary of the fitted radioactive contaminants, +respectively. The overall energy spectra match +the data for the single-hit and multiple-hit events +satisfactorily. +The level of the fitted internal +components is similar to the previously grown +NaI-036 crystal (28). +The expected background level in the COSINE- +200 crystal can be studied from the simulated +background by assuming a 12.5 kg detector in the +COSINE-100 shielding, as described in Ref. (28). +If the measured backgrounds, given in Table 2 for +the simulated study, are considered, a background +level of approximately 0.5 counts/kg/keV/day in the +1–6 keV energy region is obtained, which is similar +to the result for the NaI-036 crystal in the previous +study (28). This is a slightly higher background +level than observed from the NaI-035 crystal owing +to the higher 210Pb contamination. However, it +is still less than 1 count/kg/keV/day, the target +background level for the COSINE-200 experiment. +6 +CONCLUSION +In this article, we presented the performance of the +first ultra-low background NaI(Tl) crystal produced +Frontiers +8 + +H. Lee et al. +Table +2. Summary of the fitted radioactive +contaminants in the modeling of the NaI-037 +crystal. +Background source +Isotope +Activity (mBq/kg) +Internal +238U +0.025 ± 0.35 +228Th +0.0065 ± 0.00025 +40K +0.17 ± 0.047 +210Pb +0.36 ± 0.11 +Cosmogenic +125I +0.40 ± 0.0015 +121Te +0.80 ± 0.0029 +121mTe +0.063 ± 0.0096 +123mTe +0.045 ± 0.099 +125mTe +0.14 ± 0.011 +127mTe +0.16 ± 0.10 +109Cd +0.0071 ± 0.0010 +113Sn +0.020 ± 0.00094 +22Na +0.050 ± 0.010 +3H +0.0037 ± 0.0097 +NaI PMTs +238U +48.83 ± 5.90 +228Th +23.80 ± 5.70 +40K +58.07 ± 17.82 +CsI PMTs +238U +27.64 ± 6.15 +228Th +24.18 ± 6.10 +40K +378.28 ± 17.74 +using the direct purification of the NaI powder in +our facility as a part of a program for the next- +generation COSINE-200 experiment. The results +of this study show a similar quantity of internal +background contamination in the crystals grown +using commercial Astro-grade powder. It indicates +that the direct powder purification and crystal +growth procedures employed at our facility can +provide suitable NaI(Tl) crystals for the COSINE- +200 experiment. Based on the experience of +developing ultra-pure NaI(Tl) crystals, we are +moving to full-size crystal growth with our purified +powder. +ACKNOWLEDGMENTS +We thank Korea Hydro and Nuclear Power Co., Ltd. +(KHNP) for providing the underground laboratory +space at Yangyang. This work is supported by the +Institute for Basic Science (IBS) under the project +code IBS-R016-A1. +Frontiers +9 + +H. Lee et al. +REFERENCES +1 .Clowe D, et al. +A direct empirical proof of +the existence of dark matter. Astrophys. J. 648 +(2006) L109. doi:10.1086/508162. +2 .Aghanim N, et al. +Planck 2018 results. VI. +Cosmological parameters. Astron. Astrophys. +641 (2020) A6. +doi:10.1051/0004-6361/ +201833910. [Erratum: Astron.Astrophys. 652, +C4 (2021)]. +3 .Bertone G, Hooper D. History of dark matter. +Rev. Mod. 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Phys. 115 (2020) 102390. +doi:10.1016/j.astropartphys.2019.102390. +Frontiers +11 + diff --git a/BtE4T4oBgHgl3EQfFQxm/content/tmp_files/load_file.txt b/BtE4T4oBgHgl3EQfFQxm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0cfe769f750c04cd1c2062bfe2b4c03b74d53d52 --- /dev/null +++ b/BtE4T4oBgHgl3EQfFQxm/content/tmp_files/load_file.txt @@ -0,0 +1,774 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf,len=773 +page_content='Performance of an ultra-pure NaI(Tl) detector produced by an indigenously-developed purification method and crystal growth for the COSINE-200 experiment H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee 1,2,B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Jeon 2,1, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Kim 2, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Kim 2, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Kim 2, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Kim 3, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Kim 2,1, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Ko 2, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee 2, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee 2,1, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee 2,∗, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee 2,1, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Ra 2, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Son 2, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Shin 2 1IBS School, University of Science and Technology (UST), Daejeon 34113, Republic of Korea 2 Center for Underground Physics, Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea 3 Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea 4 Department of Physics, Chung-Ang University, Seoul 06973, Republic of Korea 5 Damavan Imaging, Troyes 10430, France Correspondence*: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee islee@ibs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='kr ABSTRACT The COSINE-100 experiment has been operating with 106 kg of low-background NaI(Tl) detectors to test the results from the DAMA/LIBRA experiment, which claims to have observed dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' However, since the background of the NaI(Tl) crystals used in the COSINE-100 experiment is 2–3 times higher than that in the DAMA detectors, no conclusion regarding the claimed observation from the DAMA/LIBRA experiment could be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Therefore, we plan to upgrade the current COSINE-100 experiment to the next phase, COSINE-200, by using ultra-low background NaI(Tl) detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The basic principle was already proved with the commercially available Astro-grade NaI powder from Sigma-Aldrich company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' However, we have developed a mass production process of ultra-pure NaI powder at the Center for Underground Physics (CUP) of the Institute for Basic Science (IBS), Korea, using the direct purification of the raw NaI powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' We plan to produce more than 1,000 kg of ultra-pure powder for the COSINE- 200 experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' With our crystal grower installed at CUP, we have successfully grown a low-background crystal using our purification technique for the NaI powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' We have assembled a low-background NaI(Tl) detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' In this article, we report the performance of this ultra-pure NaI(Tl) crystal detector produced at IBS, Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Keywords: NaI(Tl) crystal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Dark matter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' COSINE-200;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Low- background detector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Purification 1 INTRODUCTION Numerous astronomical observations support the theory that most of the matter in universe is the invisible dark matter, although an understanding of its nature and interactions remains elusive (1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Even though tremendous efforts have been made to search for dark matter, no definitive signals have been observed (4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The only exception is the DAMA experiment, which has observed an annual modulation of event rates using an array of NaI(Tl) detectors (6, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' This observation could be interpreted as dark matter-nuclei interactions (8, 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' However, this result has been the subject of a continuing debate because no other experimental searches have observed similar signals (5, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='04884v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='ins-det] 12 Jan 2023 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Several experimental efforts using the same NaI(Tl) target materials are currently underway (11, 12, 13, 14, 15, 16, 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The COSINE-100 experiment is one such effort presently operating at the Yangyang underground laboratory in Korea, which has provided several exciting physics results (9, 18, 19, 20, 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' However, due to the approximately 2–3 times higher background level, an unambiguous conclusion regarding the observation in the DAMA experiment using the same annual modulation signal has not been observed yet (22, 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' As an effort to upgrade the ongoing COSINE- 100 experiment for the next-phase COSINE- 200 experiment, we have conducted an R&D program aimed at producing a low-background NaI(Tl) detector to conclude on the observed signals from DAMA/LIBRA unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' It includes the chemical purification of the raw NaI powder (24, 25), its crystal growth (26), and detector assembly (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' We have already proved the principle of a low-background NaI(Tl) detector using the commercially available low-background Astro-grade NaI powder from Sigma-Aldrich (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' As a next step, we have grown an NaI(Tl) crystal using our own NaO power produced using the mass purification process at IBS, Korea (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' This article reports the characteristics and performance of this indigenously-produced NaI(Tl) crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 2 NAI PURIFICATION AND CRYSTAL GROWTH The COSINE-200 detector requires extremely low levels of radioactive contamination in the materials used in the detector production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The major contributors to the background are the decays of 40K and 210Pb in the bulk NaI(Tl) crystal (30, 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Because of the similarity in its chemical properties to those of Na, which is in the same periodic table group, K is the primary impurity contaminant, and its selective extraction from NaI powder is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' We found that the fractional recrystallization method effectively reduces the K and Pb impurities (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' In addition, using this method, the Ba concentration was significantly reduced, indicating a reduction of Ra impurities (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Thus,we constructed a mass production facility at IBS, Daejeon, Korea, for producing ultra-pure NaI powder using the fractional recrystallization method on-site (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The facility has been operated with a maximum production rate of 35 kg of ultra-pure powder in a single processing cycle of two weeks (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Using our purification facility, we have performed mass purification of the fractional recrystallization process using raw NaI powder from Merck (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='99(5)% purity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' In this mass purification process of the NaI power, we have achieved a concentration of K of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='4 ppb and that of Pb below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='3 ppb (29), which are consistent with contamination levels of the Astro-grade powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The ultra-pure crystal was grown using a small- volume Kyropouls grower (32), which is the same grower used for growing the proof of principle low- background NaI(Tl) crystals using the commercial Astro-grade powder (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' In growing the crystal, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='7 kg of the purified NaI powder was loaded in a 12 cm diameter, 10 cm high quartz crucible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' An NaI(Tl) crystal ingot, as shown in Figure 1(a), of ∼70 mm diameter and ∼80 mm high and having a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='1 kg mass, was grown in ∼24 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' During the crystal growth, N2 gas was continuously flushed using a thallium trap with a flow rate of 10 L/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 3 EXPERIMENTAL SETUP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='1 NaI(Tl) crystal The growth of the NaI(Tl) crystal (named NaI- 037) was completed on January 18, 2021, using NaI powder purified IBS (24, 32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The top and bottom sections of the crystal ingot were cut using a diamond bandsaw, as shown in Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' After cutting the top and bottom, the NaI-037 crystal is 70 mm in diameter and 51 mm in height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The flat top and bottom surfaces and a barrel-shaped side surface were polished using aluminum oxide films ranging from 400 to 8000 grit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' After polishing, the barrel was wrapped with a polytetrafluoroethylene (PTFE) film in several layers as a diffusive reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Frontiers 2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' NaI(Tl) Crystal ingot Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Cut and polished NaI(Tl) crystal Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Bare NaI(Tl) (NaI-037) crystal A 3 mm thick copper casing with quartz windows at each end was encapsulated the crystal hermetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Hamamatsu 3 inch photomultiplier tubes (PMTs), selected for high quantum efficiency (R12669SEL), were coupled via an optical interface to each end of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The entire assembly was performed in a glovebox, where the humidity was maintained to be less than 10 ppm (H2O) using Ar gas and a molecular sieve trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Before the assembly, all parts were cleaned using diluted Citranox liquid with sonication and baked in a vacuum oven for more than 12 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' After assembly, the detector was delivered to the Yangyang underground laboratory (Y2L), which has ∼700 m of rock overburden (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' From the crystal growth to Y2L delivery, it took less than three weeks and minimized the cosmogenic activation in the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='2 Shielding structure The background contamination levels of the NaI-037 crystal were evaluated using, the same experimental apparatus as that used for the NaI(Tl) crystal R&D at the Y2L (28, 34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' It includes an array of 12 CsI(Tl) crystals surrounded by 10 cm copper, 5 cm polyethylene, 15 cm lead, and 30 cm liquid scintillator-loaded mineral oil (35, 36) as a radiation shield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The detector was installed inside the CsI(Tl) array, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' A schematic view of the Y2L setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The NaI-037 crystal (red circle) was installed inside the CsI(Tl) crystal array (blue squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='3 Electronics The PMTs attached to the NaI-037 crystal had two readouts each, a high-gain signal from the anode and a low-gain signal from the fifth-stage dynode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The anode signal was amplified by a factor of 30, whereas the dynode signal was amplified by a factor of 100 using a custom- made preamplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The amplified signals were digitized by 500 MHz, 12-bit flash analog-to-digital converters (FADCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Triggers from the individual Frontiers 3 YPARINEOY NOUYPCCsl(T)Crystals Nal(m)Crystal Copper Folvetnviene Lead MineralOilH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' channels were generated by the field-programmable gate arrays (FPGAs) embedded in the FADC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The final trigger was generated in the trigger and clock board (TCB) when an anode signal corresponding to one or more photoelectrons (PEs) occurred in each PMT within a 200 ns time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The anode and dynode signals were recorded whenever the anode signal produced a trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Signals from the CsI(Tl) crystals were amplified by a factor of 10 and digitized in a charge-sensitive 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='5 MHz FADC (SADC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The SADC provided the integrated charge and the time of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' An integration time of 2048 ns was used to record the CsI(Tl) signals considering their decay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The SADC channels did not generate triggers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' If the trigger condition was satisfied, the TCB sent trigger signals to the FADC and SADC to store the signals from the NaI(Tl) and the CsI(Tl) crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The FADC stored an 8 µs long waveform starting approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='4 µs before the time of the trigger position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The SADC stored the maximum integrated charge within an 8 µs search window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' This system is similar to the one used in the COSINE-100 data acquisition (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Energy [keV] 0 10 20 30 40 50 60 70 80 Entries 0 50 100 150 200 Am 241 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Anode energy distribution obtained using a 241Am source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='4 Energy calibration and light yields The energy calibration of the anode signal was done using a 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='54 keV X-ray emitted from 241Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Figure 3 shows the anode energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' A clear peak at 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='54 keV resulting from the 241Am source is shown together with the 127I X-ray escape peak around 30 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The dynode signal was calibrated using the photopeaks corresponding to 214Bi(609 keV) and 40K(1460 keV) contaminants in the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The charge distribution of the single photoelectron (SPE) was obtained by identifying the isolated clusters at the decay tail of the 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='54 keV X-ray signal from the 241Am source (3-5 µs after the signal start).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The light yield was determined from the ratio of the total deposited charge and the mean of the SPE charge for the 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='54 keV X-ray data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' In this crystal, a light yield of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='6 number of photoelectron (NPE)/keV was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' It is similar to the result for the NaI-036 crystal, which has the highest light yield among the previously developed low-background NaI(Tl) crystals using the Astro- grade powder (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' This light yield is also larger than those of the detectors used in the COSINE-100 and DAMA/LIBRA experiments, as summarized in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 4 UNDERSTANDING THE BACKGROUND IN THE SPECTRUM 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='1 40K background 40K is one of the most problematic background sources in the search for weakly interacting massive particles (WIMP) using NaI(Tl) crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The X- rays/Auger electrons from 40K decays produce 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='2 keV energy signals, similar to the energy signals expected for a WIMP-nuclei interaction (30, 31, 38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The 40K decays also emit 1460 keV γ rays, which can escape from the NaI(Tl) crystal and hit the surrounding CsI(Tl) crystals, leading to a double coincidence with the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='2 keV X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Figure 4 shows the tagged low-energy spectra from the NaI(Tl) crystal by requiring the detection of the 1460 keV γ ray in the CsI(Tl) crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The Frontiers 4 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 40K background level in the NaI(Tl) crystal was determined by comparing the measured coincidence rate from a GEANT4-based simulated data, as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' By accumulating more than six months of data, the K level was measured to be 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='3±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='6 ppb, which was compared with the other NaI(Tl) crystals listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' It is well below our goal of 20 ppb, consistent with the results from the DAMA/LIBRA crystals (34, 40) and previously developed NaI-035 and NaI-036 crystals with the Astro-grade powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 1 2 3 4 5 6 Energy (keV) 0 1 2 3 4 5 6 7 8 Number of Events Data Fit(Gaussian+Constant) Gaussian Component Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Energy deposition of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='2 keV 40K coincidence events in the NaI-037 crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The model of the energy spectrum assumes a combination of a Gaussian 40K signal and a constant background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='2 α analysis Alpha-induced events inside the NaI(Tl) crystals can be identified using the fast decay times of their corresponding signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The charge-weighted duration time, called the meantime, is defined as ⟨t⟩ = ΣiAiti ΣiAi , (1) where A and t are the charge and time of the i-th digitized bin of a signal waveform, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The meantime is estimated within 1500 ns from the pulse starting timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Figure 5 shows a scatter plot of the energy versus the meantime for the NaI-037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='35 s) µ Meantime ( 1 2 3 4 Energy (MeV) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Scatter plot of the meantime versus the energy distribution events measured over 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='8 days for the NaI-037 crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The α events (red dots) and the γ/β events (black dots) are separated clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' In the figure, the populations of γ/β and α events can be separated clearly due to the faster decay times of the α-induced events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='3 210Pb background In the NaI(Tl) crystal experiments, the dominant background source in the low-energy signal region is from 210Pb (31, 41, 42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The 210Pb activity can be estimated from the alpha-decay studies, because the α decays of 210Po originate from the β decays of 210Pb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Due to the decay time of 200 days of 210Po,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' the amount of 210Pb produced can be estimated using a time-dependent fit of the alpha rate as follows: N = NPb210 � 1 − e−(t−t0)/τP o210 � + C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' (2) where N is the total alpha rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' NPb210 is the amount of 210Pb at the equilibrium state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' t0 is the time difference between 210Pb contamination and the start time of data taking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' τPo210 is the mean lifetime of 210Po,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' approximately 200 days,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' and C represents the long-lived components from 238U and 232Th chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Figure 6 shows the measured total alpha rates in the NaI-037 crystal over the detector running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The 210Pb activity in the crystal was estimated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='10 mBq/kg, Frontiers 5 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' which is lower than the COSINE-100 crystals and is consistent with the activity in the NaI-036 crystal produced using the Astro-grade powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' However, this activity is slightly higher than the DAMA crystals and another crystal NaI-036 grown with the same Astro-grade powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='4 mBq/kg level contamination of 210Pb is enough to reach 1 count/kg/keV/day background level, similar to the activity in the DAMA/LIBRA detectors, as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 100 150 200 Days from crystal growing 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='5 Activity (mBq/kg) Data Fit Asymptotic Line Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The total alpha rate in the NaI-037 crystal as a function of time, modeled with 210Po assuming contamination of 222Rn (and/or 210Pb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The asymptotic line corresponds to the rate of total alpha events in the equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='4 232Th background Contaminants from the 228Th subchain in the 232Th family can be estimated by deploying the time-delayed α–α coincident events of 220Rn and 216Po.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The alpha decay of 216Po has a half-life of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='145 s following its production via alpha decay of 220Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Owing to the short half-life of 216Po, it is straightforward to select two successive α particles with almost no random coincident events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The presence of the coincident events is shown in figure 7(a) as the distribution of the time gap between those two α events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The exponential component indicates the contamination from 232Th, corresponding to below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='39 ppt (90% confidence level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The 232Th concentration in the NaI-037 crystal is the lowest among the other NaI(Tl) crystals, as summarized in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='5 238U background 238U is one of the common radioisotopes because of its long half-life .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The 238U content in the background can be studied using the time-delayed β–α coincident events, similar to the calculation of the 232Th background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' This method exploits the α decay of 214Po with a half-life of 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='3 µs, while 214Bi, the parent particle of 214Po, undergoes β decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Due to the 50 µs dead time of the trigger system, the coincident events with delay times greater than 50 µs can be tagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The results are shown in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The 238U activity of NaI-037 was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='58 ppt, similar to that observed for the other NaI(Tl) crystals, as given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='6 External Background Because of the small size of the NaI-037 crystal and no liquid scintillator active veto, a significantly higher background contribution is expected from the external background compared to those found in the COSINE-100 crystals (30, 41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The PMTs attached to the NaI(Tl) and the CsI(Tl) crystals are the primary sources of external background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' In this study, the external background contributions were simulated using the GEANT4-based simulation toolkit used for the COSINE-100 background modeling (30, 41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='7 Cosmogenic radionuclides The cosmogenic production of radioactive isotopes in the NaI(Tl) crystal is mainly due to long- lived nuclides such as 3H and 22Na (30, 44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The NaI-037 crystal was grown in Daejeon, Korea (70 m in altitude) and delivered underground within a month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Based on the previous study, one-month exposure time near sea level can produce 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='004 mBq/kg of 3H and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='05 mBq/kg of 22Na (44), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Frontiers 6 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='8 1 Time (s) 0 1 2 3 4 Number of Events Data Fit(Exponential+Constant) Exponential Component Constant Figure 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Time differen ofbetween two α decays of the 220Rn–216Po decay chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 200 400 600 800 1000 s) µ Time ( 0 2 4 6 8 10 Number of Events Data Fit(Exponential+Constant) Exponential Component Constant Figure 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Time difference between the 214Po α decay and 214Bi β decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Time difference distributions of data (black dots) and the exponential fits to them (red-solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Measured radioactive contaminants in the NaI-037 crystal, C6 of COSINE-100 (30), DAMA crystals (40, 43), and the previously grown NaI-035 and NaI-036 crystals using the Astro-grade powder (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='The upper limits are given at a 90% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Crystal Mass (kg) LY (NPE/keV) 40K (ppb) 210Pb (mBq/kg) 232Th (ppt) 238U (ppt) NaI-037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='71 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='3±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='6 NaI-035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='61 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='8±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='8 <42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='3 NaI-036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='78 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='5 <53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='27 <4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='9 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='5±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='9 COSINE-100 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='5 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='8±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='2 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='02 DAMA 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='7 5–10 <20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='01–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='5–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='7–10 5 BACKGROUND MODELING For a quantitative understanding of the background in the NaI-037 crystal, GEANT4-based simulation, developed for the background modeling of the COSINE-100 NaI(Tl) crystals (30, 41) and also used in the previously grown crystals using the Astro-grade powder (28), was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The input values of the contamination levels are obtained from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' A simultaneous fit was done to the single-hit low energy (3–60 keV), single- hit high energy (60 keV–3 MeV), multiple-hit low energy, and multiple-hit high energy events using the log-likelihood method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' A multiple-hit event corresponds to one or more coincident hits in any of the surrounding CsI(Tl) crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The backgrounds from the PMTs attached to the NaI(Tl) and CsI(Tl) crystals were measured using a high-purity germanium detector (30, 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' These values were constrained to be within 50% of the measured result because the exact locations of such radioisotopes are uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The long- lived cosmogenic radioisotopes were constrained to be within 50% of their calculation production values whereas the other short-lived cosmogenic components were floated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Figure 9 and Table 2 show the fitted results for the NaI-037 crystal on all simulated background components and the Frontiers 7 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Energy [keV] 10 20 30 40 50 60 Counts/da/kg/keV 2 − 10 1 − 10 1 10 2 10 Data Internal Cosmogenic External Figure 9a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Single-hit low-energy (2–60 keV) Energy [keV] 500 1000 1500 2000 2500 3000 Counts/da/kg/keV 3 − 10 2 − 10 1 − 10 1 10 2 10 Data Internal Cosmogenic External Figure 9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Single-hit high-energy (60–3000 keV) Energy [keV] 10 20 30 40 50 60 Counts/da/kg/keV 2 − 10 1 − 10 1 10 2 10 Data Internal Cosmogenic External Figure 9c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Multiple-hit low-energy (2–60 keV) Energy [keV] 500 1000 1500 2000 2500 3000 Counts/da/kg/keV 3 − 10 2 − 10 1 − 10 1 10 2 10 Data Internal Cosmogenic External Figure 9d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Multiple-hit high-energy (60–3000 keV) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Measured single-hit and multiple-hit background spectra of the NaI-037 (black point) crystal fitted with the different simulated background components using a simultaneous fit of four channels using the log-likelihood method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The external component (purple-hatched area) is the dominant contributor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' summary of the fitted radioactive contaminants, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The overall energy spectra match the data for the single-hit and multiple-hit events satisfactorily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The level of the fitted internal components is similar to the previously grown NaI-036 crystal (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The expected background level in the COSINE- 200 crystal can be studied from the simulated background by assuming a 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='5 kg detector in the COSINE-100 shielding, as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' If the measured backgrounds, given in Table 2 for the simulated study, are considered, a background level of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='5 counts/kg/keV/day in the 1–6 keV energy region is obtained, which is similar to the result for the NaI-036 crystal in the previous study (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' This is a slightly higher background level than observed from the NaI-035 crystal owing to the higher 210Pb contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' However, it is still less than 1 count/kg/keV/day, the target background level for the COSINE-200 experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 6 CONCLUSION In this article, we presented the performance of the first ultra-low background NaI(Tl) crystal produced Frontiers 8 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Summary of the fitted radioactive contaminants in the modeling of the NaI-037 crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Background source Isotope Activity (mBq/kg) Internal 238U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='025 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='35 228Th 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='0065 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='00025 40K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='047 210Pb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='11 Cosmogenic 125I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='0015 121Te 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='0029 121mTe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='063 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='0096 123mTe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='045 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='099 125mTe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='011 127mTe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='10 109Cd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='0071 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='0010 113Sn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='020 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='00094 22Na 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='050 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='010 3H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='0037 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='0097 NaI PMTs 238U 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='83 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='90 228Th 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='80 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='70 40K 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='07 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='82 CsI PMTs 238U 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='64 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='15 228Th 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='18 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='10 40K 378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='28 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='74 using the direct purification of the NaI powder in our facility as a part of a program for the next- generation COSINE-200 experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' The results of this study show a similar quantity of internal background contamination in the crystals grown using commercial Astro-grade powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' It indicates that the direct powder purification and crystal growth procedures employed at our facility can provide suitable NaI(Tl) crystals for the COSINE- 200 experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Based on the experience of developing ultra-pure NaI(Tl) crystals, we are moving to full-size crystal growth with our purified powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Korea Hydro and Nuclear Power Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=', Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' (KHNP) for providing the underground laboratory space at Yangyang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' This work is supported by the Institute for Basic Science (IBS) under the project code IBS-R016-A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Frontiers 9 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' REFERENCES 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='Clowe D, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' A direct empirical proof of the existence of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Astrophys.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='045002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content='Undagoitia TM, Rauch L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Dark matter direct- detection experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE4T4oBgHgl3EQfFQxm/content/2301.04884v1.pdf'} 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a/F9E1T4oBgHgl3EQf-wbm/content/tmp_files/2301.03574v1.pdf.txt b/F9E1T4oBgHgl3EQf-wbm/content/tmp_files/2301.03574v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e824bf250a8f7dcf795f54c76631aa141428b62b --- /dev/null +++ b/F9E1T4oBgHgl3EQf-wbm/content/tmp_files/2301.03574v1.pdf.txt @@ -0,0 +1,1963 @@ +arXiv:2301.03574v1 [math.NA] 9 Jan 2023 +SHARP PREASYMPTOTIC ERROR BOUNDS FOR THE +HELMHOLTZ h-FEM +J. GALKOWSKI∗ AND E. A. SPENCE† +Abstract. +In the analysis of the h-version of the finite-element method (FEM), with fixed +polynomial degree p, applied to the Helmholtz equation with wavenumber k ≫ 1, the asymptotic +regime is when (hk)pCsol is sufficiently small and the sequence of Galerkin solutions are quasioptimal; +here Csol is the norm of the Helmholtz solution operator, normalised so that Csol ∼ k for nontrapping +problems. The preasymptotic regime is when (hk)2pCsol is sufficiently small, and (for physical data) +one expects the relative error of the Galerkin solution to be controllably small. +In this paper, we prove the natural error bounds in the preasymptotic regime for the variable- +coefficient Helmholtz equation in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or +combinations of these) and with the radiation condition approximated either by a radial perfectly- +matched layer (PML) or an impedance boundary condition. Previously, such bounds for p > 1 were +only available for Dirichlet obstacles with the radiation condition approximated by an impedance +boundary condition. +Our result is obtained via a novel generalisation of the “elliptic-projection” +argument (the argument used to obtain the result for p = 1) which can be applied to a wide variety +of abstract Helmholtz-type problems. +AMS subject classifications. 35J05, 65N15, 65N30, 78A45 +Key words. Helmholtz, FEM, high order, pollution effect, preasymptotic, perfectly-matched +layer, elliptic projection. +1. Introduction. +1.1. Informal statement of the main result. We consider the h-version of +the finite-element method (h-FEM), where accuracy is increased by decreasing the +meshwidth h while keeping the polynomial degree p constant, applied to the Helmholtz +equation. +Theorem 1.1 (Informal statement of the main result). +Let u be the solution to +the variable-coefficient Helmholtz equation, with wavenumber k > 0, in the exterior +of a Dirichlet, or Neumann, or penetrable obstacle (or combinations of these) and +with the radiation condition approximated either by a perfectly-matched layer (PML) +or an impedance boundary condition. Let Csol be the norm of the solution operator, +normalised so that Csol ∼ k for nontrapping problems. +Under the natural regularity assumptions on the domain and coefficients, if +(1.1) +(hk)2pCsol is sufficiently small +then the Galerkin solution uh exists, is unique, and satisfies +∥u − uh∥H1 +k(Ω) ≤ C +� +1 + hk + (hk)pCsol +� +min +vh∈Hh ∥u − vh∥H1 +k(Ω) , +(1.2) +∥u − uh∥L2(Ω) ≤ C +� +hk + (hk)pCsol +� +min +vh∈Hh ∥u − vh∥H1 +k(Ω) . +(1.3) +Furthermore, if the data is k-oscillatory (in a sense made precise below), then +(1.4) +∥u − uh∥H1 +k(Ω) +∥u∥H1 +k(Ω) +≤ C +� +1 + hk + (hk)pCsol +� +(hk)p; +∗Department of Mathematics, University College London, 25 Gordon Street, London, WC1H +0AY, UK, J.Galkowski@ucl.ac.uk +†Department +of +Mathematical +Sciences, +University +of +Bath, +Bath, +BA2 +7AY, +UK, +E.A.Spence@bath.ac.uk +1 + +i.e., the relative H1 +k error can be made controllably small by making (hk)2pCsol suffi- +ciently small. +The norm ∥ · ∥H1 +k(Ω) in the bounds above is defined by +(1.5) +∥v∥2 +H1 +k(Ω) := k−2 ∥∇v∥2 +L2(Ω) + ∥v∥2 +L2(Ω) . +The fact that, for oscillatory data, the relative H1 +k error for the Helmholtz h-FEM +is controllably small if (hk)2pCsol is sufficiently small was famously identified for 1-d +nontrapping problems by the work of Ihlenburg and Babuˇska [25, 26]. The bounds +(1.2) and (1.3) have previously been obtained (i) for the Dirichlet obstacle problem +with impedance boundary conditions approximating the radiation condition [12, 40] +and (ii) for PML with constant-coefficients, no obstacle, and p = 1 [32]. +The present paper proves the bounds (1.2), (1.3), and (1.4) assuming only that +the sesquilinear form is continuous, satisfies a G˚arding inequality, and satisfies certain +standard elliptic-regularity assumptions, therefore covering a variety of scatterers and +methods for truncating the exterior domain (to approximate the radiation condition). +Regarding the latter: in this paper we consider truncating with a PML or an imped- +ance boundary condition, but truncating with the exact Dirichlet-to-Neumann map +is also, in principle, covered by the abstract framework; see Remark 5.4 below. +1.2. Statement of the main abstract result. Let H ⊂ H0 ⊂ H∗ be Hilbert +spaces with H0 identified with its dual and H ⊂ H0 compact. Let a : H × H → C be +a continuous sesquilinear form, i.e., +(1.6) |a(u, v)| ≤ Ccont ∥u∥H ∥v∥H +and +a(λu, µv) = λ¯µa(u, v) +for all u, v ∈ H, +satisfying the G˚arding inequality +(1.7) +ℜa(v, v) ≥ CG1 ∥v∥2 +H − CG2 ∥v∥2 +H0 +for all v ∈ H +for some CG1, CG2 > 0. We assume further that Ccont, c, C and all the other constants +in this section are independent of k. +Assumption 1.2 (“Elliptic regularity” assumptions on a). +Let Z0 = H0, Z1 = +H, and Zj ⊂ Zj−1 for j = 2, . . . , ℓ + 1 such that Zj is dense in Zj−1, and assume +that for all u ∈ H with +sup +v∈H, ∥v∥(Zj−2)∗=1 +|a(u, v)| < ∞, +u ∈ Zj and +(1.8) +∥u∥Zj ≤ C +� +∥u∥H0 + +sup +v∈H, ∥v∥(Zj−2)∗=1 +|a(u, v)| +� +, +j = 2, . . . , ℓ + 1. +Assume further that for any w ∈ H such that +sup +w∈H, ∥v∥(Zj−2)∗=1 +|(ℜa)(u, v)| < ∞, +w ∈ Zj with +(1.9) +∥w∥Zj ≤ C +� +∥u∥H0 + +sup +v∈H, ∥v∥(Zj−2)∗=1 +|(ℜa)(u, v)| +� +, +j = 2, . . . , ℓ + 1, +2 + +where the sesquilinear form ℜa is defined by +(1.10) +(ℜa)(u, v) := 1 +2 +� +a(u, v) + a(v, u) +� +. +Remark 1.3. Note that ℜa in (1.7) and (1.10) could be replaced by ℜ(eiωa), so +long as one uses the same value of ω in both conditions. Remark 4.4 below describes +a situation where this is useful. +Given g ∈ H∗, suppose that u ∈ H satisfies +(1.11) +a(u, v) = ⟨g, v⟩ +for all v ∈ H. +Given a sequence of finite dimensional subspace {Hh}h>0 with Hh ⊂ H, the +sequence of Galerkin approximations of u, {uh}h>0, are defined by +(1.12) +a(uh, vh) = ⟨g, vh⟩ for all vh ∈ Hh. +Example 1.4. For the Helmholtz equation outside a Dirichlet obstacle with PML +truncation and Ω the truncated exterior domain, H0 = L2(Ω), H = H1 +0(Ω), and +Zj = Hj(Ω) ∩ H1 +0(Ω). Assumption 1.2 is then elliptic regularity for the Helmholtz +PML operator and its real part, which both hold if the coefficients of the Helmholtz +equation are in Cℓ−1,1, the PML scaling function is Cℓ,1, and ∂Ω is Cℓ,1 (see Lemma +4.7 below). +Theorem 1.5 (Abstract generalisation of the elliptic-projection argument). +Let a : H × H → C satisfy (1.6), (1.7), and Assumption 1.2. +Suppose that +R∗ : H∗ → H defined by +(1.13) +a(w, R∗v) = ⟨w, v⟩ +for all w ∈ H, v ∈ H∗, +is well defined and let +(1.14) +η(Hh) := +sup +g∈H0,g̸=0 +∥(I − Π)R∗g∥H +∥g∥H0 +, +where Π : H → Hh is the orthogonal projection. Then the solution, u, to (1.11) exists +and is unique and there exist C1, C2, C3 > 0 such that if h satisfies +(1.15) +η(Hh)∥I − Π∥Zℓ+1→H ≤ C1, +then the solution uh to (1.12) exists, is unique, and satisfies +∥u − uh∥H ≤ C2 +� +1 + η(Hh) +� +min +wh∈Hh ∥u − vh∥H , +(1.16) +∥u − uh∥H0 ≤ C3 η(Hh) min +wh∈Hh ∥u − vh∥H . +(1.17) +If, in addition, +(1.18) +∥g∥Zℓ−1 ≤ C ∥g∥H∗ +for some C > 0, then there exists C4 > 0 such that if h satisfies (1.15) then +(1.19) +∥u − uh∥H +∥u∥H +≤ C4 +� +1 + η(Hh) +� +∥I − Π∥Zℓ+1→H ; +i.e., +the +relative +error +in +H +can +be +made +controllably +small +by +making +η(Hh) ∥I − Π∥Zℓ+1→H sufficiently small. +3 + +Theorem 1.5 includes the result that the sequence of Galerkin solutions are qua- +sioptimal with constant independent of k if η(Hh) is sufficiently small – with this the +so-called asymptotic regime (see the discussion in §1.3). +The bounds (1.16), (1.17), and (1.19) and the meshthreshold (1.15) in Theorem +1.5 all involve the quantity η(Hh), which measures how well solutions of the adjoint +problem are approximated in the space Hh. Bounds on η(Hh) are given in [37, 38, 36, +13, 6, 29, 19, 20, 3]; see the discussion in §1.3. The following bound on η(Hh) is proved +using the ideas in [6] (although the end result is phrased in a different way there); we +include it here both for completeness, and because it holds under the assumptions of +Theorem 1.5 (in fact, it only requires the regularity assumption (1.8) and not (1.9)). +Theorem 1.6 (Bound on η(Hh)). Under the assumptions of Theorem 1.5, there +exists C > 0 such that +(1.20) +η(Hh) ≤ C +� ⌊ℓ/2⌋−1 +� +j=0 +∥(I − Π)∥Z2(j+1)→H + ∥(I − Π)∥Zℓ+1→H +� +1 + ∥R∗∥H0→H +�� +. +Example 1.7. In §4 and §5 below we show how Helmholtz problems with the ra- +diation condition approximated by either a PML or an impedance boundary condition, +respectively, fit into the abstract framework of Theorems 1.5 and 1.6. In both these +cases, the norm of the adjoint solution operator, i.e., ∥R∗∥H0→H, is the same as the +norm of the solution operator of the original (non-adjoint) problem, which we denote +by Csol. +Furthermore, with {Hh}h>0 corresponding to the standard finite-element +spaces of piecewise degree-p polynomials on shape-regular simplicial triangulations, +indexed by the meshwidth h, +∥(I − Π)∥Zm+1→H ≤ C(hk)m +for 0 ≤ m ≤ p. +The meshthreshold (1.15) then becomes that (hk)2ℓCsol is sufficiently small. Recall +that ℓ is a parameter in the elliptic-regularity assumptions (Assumption 1.2). If the +polynomial degree p is taken to be ℓ then (1.15) becomes (1.1). The bounds (1.16) and +(1.17) then become (1.2) and (1.3), respectively. +1.3. Discussion of the context, novelty, and ideas behind Theorem 1.5. +The work of Ihlenburg and Babuˇska in 1-d. The celebrated work of [25, 26] studied +the h-FEM applied to the constant-coefficient Helmholtz equation in 1-d (a nontrap- +ping problem), and split the behaviour of the finite-element solutions as a function of +h into the so-called asymptotic and preasymptotic regimes. +The asymptotic regime is when h is small enough, as a function of k, for the +sequence of Galerkin solutions to be quasi-optimal uniformly in k, i.e., +∥u − uh∥H1 +k(Ω) ≤ C min +vh∈Hh ∥u − vh∥H1 +k(Ω) +with C > 0 independent of k. [26, Theorem 3.5] showed that a sufficient condition to +be in the asymptotic regime is “hk2/p sufficiently small”, with later work (discussed +below) then showing that a sufficient condition for nontrapping problems (when Csol ∼ +k) is “(hk)pk sufficiently small”, with this condition then indicated to be necessary +by numerical experiments. Therefore, the pollution effect for the h-FEM, i.e., the +fact that one needs h ≪ k−1 to maintain accuracy, becomes less pronounced as p +increases. +4 + +The preasymptotic regime is when the relative H1 +k error is controllably small, uni- +formly as k → ∞, provided that the data is k-oscillatory, in the sense that it satisfies +the bound (1.18) 1. [26, Corollary 3.2] used the explicit form of the Helmholtz Green’s +function in 1-d to prove that if (hk)2pk sufficiently small then the finite-element solu- +tion is in the preasymptotic regime, with the numerical experiments in [26, Table 2] +(for p = 1, . . . , 6) indicating that this condition is also necessary. [26] also studied the +phase difference between the exact and finite-element solutions (following [23, 43]), +with [26, Theorem 3.2] showing that the difference between the true wavenumber and +the numerical wavenumber is bounded by C(hk)2pk. Thus the condition “(hk)2pk +sufficiently small” also controls this phase difference; see also [1, Equation 3.5]. +Error bounds in the asymptotic regime using the Schatz argument.. We now out- +line the argument that gives the condition “(hk)pCsol sufficiently small” for quasiop- +timality, with this argument also used in the proof of Theorem 1.5. We work in the +setting of Examples 1.4 and 1.7; i.e., the PML approximation to the Helmholtz exte- +rior Dirichlet problem, so that H0 = L2(Ω) and H = H1 +0(Ω). The G˚arding inequality +(1.7) is then +ℜa(w, w) ≥ CG1 ∥w∥2 +H1 +k(Ω) − CG2 ∥w∥2 +L2(Ω) +for all w ∈ H1 +0(Ω) +for CG1, CG2 > 0 (see Corollary 4.6 below). Combining the G˚arding inequality with +the Galerkin orthogonality +(1.21) +a(u − uh, vh) = 0 +for all vh ∈ Hh, +we find that, for all vh ∈ Hh, +∥u − uh∥2 +H1 +k(Ω) ≤ C−1 +G1 +��a(u − uh, u − vh) +�� + C−1 +G1CG2 ∥u − uh∥2 +L2(Ω) +≤ C−1 +G1Ccont ∥u − uh∥H1 +k(Ω) ∥u − vh∥H1 +k(Ω) + C−1 +G1CG2 ∥u − uh∥2 +L2(Ω) , +(1.22) +where Ccont is the continuity constant of the sesquilinear form a. Therefore, (1.22) +implies that a sufficient condition for quasioptimality is that the L2 error is sufficiently +small relative to the H1 +k error. +By the definition of R∗ (1.13) (recalling that H = H1 +0(Ω) here) and Galerkin +orthogonality (1.21), for any vh ∈ Hh, +∥u − uh∥2 +L2(Ω) = a +� +u − uh, R∗(u − uh) +� += a +� +u − uh, R∗(u − uh) − vh +� +≤ Ccont ∥u − uh∥H1 +k(Ω) +��R∗(u − uh) − vh +�� +H1 +k(Ω), +(1.23) +and thus, by the definition of η(Hh) (1.14) (recalling that H0 = L2(Ω)), +(1.24) +∥u − uh∥L2(Ω) ≤ Ccontη(Hh) ∥u − uh∥H1 +k(Ω) . +Combining this last inequality with (1.22), we see that a sufficient condition for qua- +sioptimality is that η(Hh) is sufficiently small. Schatz [42] was the first to use the +Aubin-Nitsche-type bound (1.24) with the G˚arding inequality, and thus the argument +above is often called the Schatz argument. The “adjoint approximability” concept, +and associated definition of η(Hh), was introduced by Sauter in [41]. +1The relative error can only be small for a certain subclass of data, since, given a finite- +dimensional subspace Hh, one can choose data such that the solution v ∈ H is orthogonal to Hh. +Then ∥u − uh∥2 +H = ∥u∥2 +H + ∥uh∥2 +H ≥ ∥u∥2 +H. +5 + +The bound +(1.25) +η +� +Hh +� +≤ C +� +hk + (hk)pCsol +� +under sufficient regularity of the coefficients and obstacle has now been proved for a +wide variety of Helmholtz problems, with this bound sharp by the recent results of [17]. +The bound (1.25) therefore gives the sufficient condition “(hk)pCsol sufficiently small” +for quasioptimality, with this condition observed sharp for nontrapping problems in, +e.g., [6, Figures 3, 5, and 8] for p = 1, 2, 3, 4. +For p = 1, the bound (1.25) can be proved using only H2 regularity of the +Helmholtz solution, with the condition “hk2 sufficient small” for quasiopimality ob- +tained for 1-d problems in [2, Theorem 3.1], [11, Lemma 2.6], [27, Theorem 3], and +[33, Theorem 3.2], 2-d problems in [35, Proposition 8.2.7], and variable-coefficient +problems in 2- and 3-d in [22, 21]. +For p > 1 the bound (1.25) is proved by a judicious splitting of the solution in +[37, 38, 13, 36] for constant-coefficient problems and [6, 29, 19, 20, 3] for variable- +coefficient problems. All these papers apart from [6] make the constant C in (1.25) +explicit in p under suitably analyticity/smoothness assumptions on the obstacle and +coefficients, and thus give results about the hp-FEM (showing that quasioptimality +holds if hk/p is sufficiently small and p/ log k is sufficiently large). In addition, all these +papers apart from [6] split the solution into “high-” and “low-” frequency components. +In constrast, [6] instead expands the solution in a series whose terms increase with +regularity, and with only the remainder satisfying a bound involving Csol; see Lemma +2.2 below. +Bounds in the preasymptotic regime. Numerical experiments indicate that, at +least for nontrapping problems, the condition “(hk)2pCsol sufficiently small” for the +relative H1 +k error to be controllably small is necessary and sufficient for 2- and 3-d +Helmholtz problems; see, e.g., [12, Figure 3]. Nevertheless, despite the fact that sharp +asymptotic error bounds have now been obtained for a variety of Helmholtz problems +in 2- and 3-d and for arbitrary p ∈ Z+, until now the sharp preasymptotic error bounds +were obtained only in the following cases. +1. p = 1, the constant-coefficient Helmholtz equation with an impedance bound- +ary condition [44, Theorem 6.1] or PML (and no obstacle) [32, Theorem +4.4], the variable-coefficient Helmholtz equation with truncation via the ex- +act Dirichlet-to-Neumann map [28, Theorem 4.1]. +2. p ∈ Z+, the constant-coefficient Helmholtz equation with no obstacle and +an impedance boundary condition approximating the radiation condition [12, +Theorem 5.1], +3. p ∈ Z+, the variable-coefficient Helmholtz equation in the exterior of a Dirich- +let obstacle with an impedance boundary condition approximating the radi- +ation condition [40, Theorem 2.39]. +The bounds in Point 1 for p = 1 come from the so-called elliptic projection argument, +which proves error bounds under the condition “(hk)p+1Csol is sufficiently small”; i.e., +the sharp condition when p = 1, but not when p > 1. The initial ideas behind this +argument were introduced in the Helmholtz context in [15, 16] for interior-penalty +discontinuous Galerkin methods, and then further developed for the standard FEM +and continuous interior-penalty methods in [44, 45]. +The bounds in Point 2 used an error-splitting argument (with this idea called +“stability-error iterative improvement”, and used earlier in [16, 44]) together with the +idea of using discrete Sobolev norms in the duality argument. The bounds in Point 3 +6 + +for variable-coefficients were obtained by repeating the constant-coefficient arguments +in Point 2, but now keeping track of how the constants depend on the coefficients. +The elliptic-projection argument. Theorem 1.5 is proved by generalising the +elliptic-projection argument, allowing it to prove error bounds under the sharp condi- +tion “(hk)2pCsol sufficiently small” for p > 1. We therefore recap the main ideas of the +elliptic-projection argument here, and then we explain below how we generalise this +argument. Here, and in the rest of the paper, C is used for a constant, independent +of h and k, but dependent on p, whose value may change line by line. +The bounds (1.2) and (1.3) come from the bounds +(1.26) +∥u − uh∥H1 +k(Ω) ≤ C +� +1 + η(Hh) +� +min +vh∈Hh ∥u − vh∥H1 +k(Ω) +and +(1.27) +∥u − uh∥L2(Ω) ≤ Cη(Hh) min +vh∈Hh ∥u − vh∥H1 +k(Ω) +and the bound (1.25) on η(Hh). +Observe that, by the consequence (1.22) of the +G˚arding inequality, the bound (1.26) follows from the bound (1.27). +To prove (1.27), the elliptic-projection argument writes (1.23) as +∥u − uh∥2 +L2(Ω) = a +� +u − uh, R∗(u − uh) − vh +� += �a +� +u − uh, R∗(u − uh) − vh +� +− +� +(1 + c−2)(u − uh), R∗(u − uh) − vh +� +L2(Ω), +(1.28) +where +�a(u, v) := +� +Ω +k−2A∇u · ∇v + u v. +Let �Π : H1 +0(Ω) → Hh be the solution of the variational problem +�a(wh, �Πv) = �a(wh, v) +for all wh ∈ Hh. +Since �a is coercive on H1 +0(Ω) and the continuity and coercivity constants of �a in +∥ · ∥H1 +k(Ω) are independent of k, �Π is well-defined by the Lax–Milgram theorem and +(1.29) +��(I − �Π)v +�� +H1 +k(Ω) ≤ C min +wh∈Hh ∥v − wh∥H1 +k(Ω) +with C > 0 independent of k by C´ea’s lemma. The definition of �Π implies the Galerkin +orthogonality +(1.30) +�a +� +wh, (I − �Π)v +� += 0 +for all wh ∈ Hh. +We now choose vh = �ΠR∗(u − uh) in (1.28) so that, by (1.30), for all wh ∈ Hh, +∥u − uh∥2 +L2(Ω) = �a +� +v − wh, (I − �Π)R∗(u − uh) +� +− +� +(1 + c−2)(u − uh), (I − �Π)R∗(u − uh) +� +L2(Ω). +(1.31) +For the first term on the right-hand side of (1.31) we use the continuity of �a, (1.29), +and the definition of η(Hh) (1.14) to bound this term by +C ∥v − wh∥H1 +k(Ω) η(Hh) ∥u − uh∥L2(Ω) . +7 + +The second term on the right-hand side of (1.31) is bounded by +C ∥u − uh∥L2(Ω) +��(I − �Π)R∗(u − uh) +�� +L2(Ω). +Using the Schatz argument for �a, one can show that +(1.32) +��(I − �Π)R∗(u − uh) +�� +L2(Ω) ≤ Chk +��(I − �Π)R∗(u − uh) +�� +H1 +k(Ω) +and then (1.29) and the definition of η(Hh) (1.14) imply that the second term on the +right-hand side of (1.31) is bounded by +(1.33) +Chk η(Hh) ∥u − uh∥2 +L2(Ω) , +which can be absorbed into the left-hand side if hk η(Hh) is sufficiently small, giving +the result (1.27). +The ideas behind the proof of Theorem 1.5. We generalise the elliptic-projection +argument based on the observation that if �a(u, v) = a(u, v) + (Su, v)L2(Ω) with S a +self-adjoint smoothing operator, then the second term on the right-hand side of (1.31) +is replaced by +(1.34) +� +u − uh, S∗(I − �Π)R∗(u − uh) +� +L2(Ω) +(see (2.14) below). Using the Schatz argument for �a and the smoothing property of +S, the modulus of this term is bounded by +(1.35) +��S∗(I − �Π)R∗(u − uh) +�� +L2(Ω) ≤ C(hk)p��(I − �Π)R∗(u − uh) +�� +H1 +k(Ω) +(see (2.16) below). Provided that �Π still satisfies (1.29), the term (1.34) is therefore +bounded by +(1.36) +C(hk)pη(Hh) ∥u − uh∥2 +L2(Ω) . +Comparing (1.32) and (1.35), and also (1.33) and (1.36), we see that this new argument +replaces the condition “hkη(Hh) sufficiently small” in the standard elliptic-projection +argument by the condition “(hk)pη(Hh) sufficiently small”, which is the condition +(hk)2pCsol sufficiently small” after using the bound (1.25) on η(Hh). +The challenge now is to ensure that the smoothing operator S is such that the +projection �Π is well-defined and satisfies (1.29). This is achieved in Lemma 2.1 below, +where a suitable S such that �a(u, v) = a(u, v) + (Su, v)L2(Ω) is coercive is construc- +ted. S is defined by an expansion in terms of the eigenfunctions of the (self-adjoint) +operator associated with the real part of the sesquilinear form a (defined by (1.10)). +2. Proofs of the main results (Theorems 1.5 and 1.6). +2.1. Construction of a regularizing operator that produces coercivity +when added to a. +Lemma 2.1. Suppose that a : H × H → C satisfies (1.6), (1.7), and Assumption +1.2. Then there exists S : H0 → H0 self adjoint and c, C > 0 such that, with +(2.1) +�a(u, v) := a(u, v) + ⟨Su, v⟩H0, +(2.2) +ℜ�a(v, v) ≥ c ∥v∥2 +H +for all v ∈ H, +8 + +(2.3) +∥S∥H0→Zj ≤ C, +j = 0, . . . , ℓ + 1 +and �R : H∗ → H defined by +�a( �Rf, u) = ⟨f, u⟩ +for all u ∈ H, f ∈ H∗, +(2.4) +is well defined with +(2.5) +∥ �R∥Zj−2→Zj ≤ C, +2 ≤ j ≤ ℓ + 1. +The proof of Lemma 2.1 uses the spectral theorem for bounded self-adjoint op- +erators, B : H → H∗, which we recap here. With H0 and H as in §1.2, let b be a +sesquilinear form on H satisfying b(u, v) = b(v, u), with associated operator B; i.e., +b(u, v) = ⟨Bu, v⟩ for all u, v ∈ H. If b satisfies the G˚arding inequality (1.7) (with +a replaced by b) then there exist an orthonormal basis (in H0) of eigenfunctions of +B, {φj}∞ +j=1, with associated eigenvalues satisfying λ1 ≤ λ2 ≤ . . . with λj → ∞ as +j → ∞. Furthermore, for all u ∈ H, +(2.6) +Bu = +∞ +� +j=1 +λj⟨φj, u⟩φj +(where the sum converges in H∗); see, e.g., [34, Theorem 2.37]. Given a bounded +function f, we define f(B) : H0 → H0 by +(2.7) +f(B)u := +∞ +� +j=1 +f(λj)⟨φj, u⟩φj, +so that +∥f(B)∥H0→H0 ≤ +sup +λ∈[λ1,∞) +|f(λ)|. +Proof of Lemma 2.1. Let P : H → H∗ be the operator associated with the +sesquilinear form ℜa defined by (1.10), i.e., (ℜa)(u, v) = ⟨Pu, v⟩ for all u, v ∈ H; +observe that P is self-adjoint. Since (ℜa) also satisfies the G˚arding equality satis- +fied by a (1.7), the spectral theorem recapped above applies. Let {λj}∞ +j=1 be the +eigenvalues of P, let ψ ∈ C∞ +comp(R; [0, ∞)) be such that +(2.8) +x + ψ(x) ≥ 1 +for x ≥ −λ1, +and let S := ψ(P), in the sense of (2.7). +We now use (1.9) to prove that S : H0 → Zj satisfying (2.3). Since ψ has compact +support, the function t �→ tmψ(t) is bounded for any m ≥ 0. Thus (2.7) implies that, +for any m ≥ 0, +(2.9) +∥Pmψ(P)∥H0→H0 ≤ Cm. +By (1.9), +∥ψ(P)∥H0→Zj ≤ Cℓ +� +∥ψ(P)∥H0→H0 + ∥Pψ(P)∥H0→Zj−2 +� +, +j = 2, . . . , ℓ + 1, +so that, by induction and (2.9), +∥S∥H0→Zℓ+1 = ∥ψ(P)∥H0→Zℓ+1 ≤ Cℓ +⌈(ℓ+1)/2⌉ +� +j=0 +��Pjψ(P) +�� +H0→H0 ≤ Cℓ. +9 + +We now show that �a satisfies (2.2). By the definitions of P and S, (2.6), (2.7), +and the inequality (2.8), for all v ∈ H, +ℜ�a(v, v) = ℜa(v, v) + ⟨ψ(P)v, v⟩ = ⟨(P + ψ(P))v, v⟩ ≥ ∥v∥2 +H0 . +Since ψ ≥ 0, S is positive, and thus ℜ�a(v, v) ≥ ℜa(v, v) for all v ∈ H, for any ǫ > 0 +and for all v ∈ H, +ℜ�a(v, v) ≥ ǫℜa(v, v) + (1 − ǫ)ℜ�a(v, v) ≥ ǫCG1 ∥v∥2 +H − CG2ǫ ∥v∥2 +H0 + (1 − ǫ)∥v∥2 +H0, +so that, choosing ǫ = min( +1 +2CG2 , 1 +2), we have +ℜ�a(v, v) ≥ CG1 +2 +min +� 1 +CG2 +, 1 +� +∥v∥2 +H + 1 +2 ∥v∥2 +H0 ; +i.e., �a is coercive. The existence of �R : H∗ → H satisfying (2.4) and ∥ �R∥H∗→H ≤ C +then follows from the Lax–Milgram theorem. Finally, to see that +∥ �R∥Zj−2→Zj ≤ C, +2 ≤ j ≤ ℓ + 1, +observe that, since S is self-adjoint and satisfies (2.3), for v ∈ (Zj−2)∗, +|a( �Rg, v)| = |�a( �Rg, v) − ⟨S �Rg, v⟩| ≤ |�a( �Rg, v)| + |⟨S �Rg, v⟩| +≤ |⟨v, g⟩| + ∥v∥(Zj−2)∗∥S∥H→Zj−2∥( �R)∗∥H∗→H∥g∥H∗ +≤ ∥v∥(Zj−2)∗(∥g∥Zj−2 + C∥g∥H∗), +and the claim follows from (1.8). +2.2. Proof Theorem 1.5 using Lemma 2.1. We claim it is sufficient to prove +the bounds (1.16) and (1.17) under the assumption of existence. Indeed, by uniqueness +of the variational problem (1.11), either of the bounds (1.16) or (1.17) under the +assumption of existence implies uniqueness of uh, and uniqueness implies existence +for the finite-dimensional Galerkin linear system. +We next show that the bound (1.16) follows from (1.17). Now, by the G˚arding +inequality (1.7), Galerkin orthogonality (1.21), and (1.17), for any vh ∈ Hh, +∥u − uh∥2 +H ≤ C +���a(u − uh, u − vh) +�� + ∥u − uh∥2 +H0 +� +≤ C +� +∥u − uh∥H ∥u − vh∥H + +� +η(Hh) min +wh∈Hh ∥u − wh∥H +�2� +. +(2.10) +The bound (1.16) on the error in H then follows by using the inequality 2ab ≤ ǫa2 + +b2/ǫ for all a, b, ǫ > 0 in the first term on the right-hand side of (2.10), and then using +the inequality a2 + b2 ≤ (a + b)2 for a, b > 0. +We now prove (1.17) (using the ideas outlined in §1.3). By the definition of R∗, +Galerkin orthogonality (1.21), and the definition of �a (2.1) +∥u − uh∥2 +H0 = a +� +u − uh, R∗(u − uh) +� += a +� +u − uh, R∗(u − uh) − vh +� += �a +� +u − uh, R∗(u − uh) − vh +� +− +� +S(u − uh), R∗(u − uh) − vh +� +H0. +(2.11) +Let �Π : H → Hh be the solution of the variational problem +�a(wh, �Πv) = �a(wh, v) +for all wh ∈ Hh. +10 + +Since �a is continuous and coercive, with constants independent of k (see (2.2), (1.6), +and (2.3)), by the Lax–Milgram lemma and C´ea’s lemma given k0 > 0 there exists +C > 0 such that for all k ≥ k0 and v ∈ H, �Π is well-defined with +(2.12) +��(I − �Π)v +�� +H ≤ C min +wh∈Hh ∥v − wh∥H . +The definition of �Π implies the Galerkin orthogonality +(2.13) +�a +� +wh, (I − �Π)u +� += 0 +for all wh ∈ Hh. +We now choose vh = �ΠR∗(u − uh) in (2.11) so that, by (2.13), for all wh ∈ Hh, +(2.14) +∥u − uh∥2 +H0 += �a +� +u − wh, (I − �Π)R∗(u − uh) +� +− +� +u − uh, S∗(I − �Π)R∗(u − uh) +� +H0 +≤ C ∥u − wh∥H +��(I − �Π)R∗(u − uh) +�� +H + ∥u − uh∥H0 +��S∗(I − �Π)R∗(u − uh) +�� +H0. +By (2.12) and the definition of η(Hh) (1.14), +(2.15) +��(I − �Π)R∗(u − uh) +�� +H ≤ C min +wh∈Hh ∥R∗(u − uh) − wh∥H ≤ Cη(Hh) ∥u − uh∥H0 . +We now claim that the bound (1.17) follows if we can prove that, for all v ∈ H, +(2.16) +��S∗(I − �Π)v +�� +H0 ≤ C∥I − Π∥Zℓ+1→H +��(I − �Π)v +�� +H. +Indeed, we use (2.15) in the first term on the right-hand side of (2.14), and then (2.16) +with v = R∗(u − uh) in the second term on the right-hand side of (2.14) to obtain +∥u − uh∥2 +H0 ≤ Cη(Hh) ∥u − wh∥H ∥u − uh∥H0 ++ C∥I − Π∥Zℓ+1→H +��(I − �Π)R∗(u − uh) +�� +H ∥u − uh∥H0 . +By (2.15), the last term on the right-hand side is ≤ C∥I−Π∥Zℓ+1→H η(Hh)∥u−uh∥2 +H0 +and (1.17) follows. +We now prove (2.16) by using the duality argument described in §1.3 (as part of +the Schatz argument). By the definition of �R (2.4) and Galerkin orthogonality (2.13), +for all wh ∈ Hh, +��S∗(I − �Π)v +��2 +H0 = +� +SS∗(I − �Π)v, (I − �Π)v +� +H0 = �a +� �RSS∗(I − �Π)v − wh, (I − �Π)v +� +. +Then, by the bounds (2.5) and (2.3), +��S∗(I − �Π)v +��2 +H0 ≤ C min +wh∈Hh +�� �RSS∗(I − �Π)v − wh +�� +H +��(I − �Π)v +�� +H +≤ ∥I − Π∥Zℓ+1→H +�� �RSS∗(I − �Π)v +�� +Zℓ+1 +��(I − �Π)v +�� +H, +≤ C∥I − Π∥Zℓ+1→H +��SS∗(I − �Π)v +�� +Zℓ−1 +��(I − �Π)v +�� +H, +≤ C∥I − Π∥Zℓ+1→H +��S∗(I − �Π)v +�� +H0 +��(I − �Π)v +�� +H +which implies the bound (2.16), and hence (1.17). +11 + +Finally, we prove (1.19). By (1.11), (1.18), and the abstract elliptic-regularity +assumption (1.8), u ∈ Zℓ+1 with +∥u∥Zℓ+1 ≤ C +� +∥u∥H0 + ∥g∥Zℓ−1 +� +≤ C +� +∥u∥H0 + ∥g∥H∗ +� +. +The variational problem (1.11) implies that +∥g∥H∗ = +sup +v∈H∗,v̸=0 +|a(u, v)| +∥v∥H∗ +≤ C ∥u∥H , +and thus ∥u∥Zℓ+1 ≤ C ∥u∥H. The bound (1.16) then implies that +∥u − uh∥H ≤ C2 +� +1 + η(Hh) +� +∥I − Π∥Zℓ+1→H ∥u∥Zℓ+1 +and (1.19) follows. +2.3. Proof of Theorem 1.6. The following lemma is essentially [6, Theorem +2.6], rewritten in the abstract notation in §1.2. +Lemma 2.2. Under the assumptions of Theorem 1.5, let u = R∗g with R∗ defined +by (1.13) and g ∈ H0. Let um ∈ H, m = 0, . . . , ⌊ℓ/2⌋, be defined by +(2.17) +�a(v, u0) = ⟨v, g⟩ +for all v ∈ H, +and +(2.18) +�a(v, um) = ⟨Sv, um−1⟩ +for all v ∈ H, m = 1, . . . , ⌊ℓ/2⌋. +Then +(2.19) +um ∈ Z2(m+1) with +∥um∥Z2(m+1) ≤ C ∥g∥H0 +for m = 0, . . . , ⌊ℓ/2⌋ − 1, +and +(2.20) +u⌊ℓ/2⌋ ∈ Zℓ+1 with +��u⌊ℓ/2⌋ +�� +Zℓ+1 ≤ C ∥g∥H0 . +Furthermore, with +(2.21) +rm := u − +m−1 +� +j=0 +uj, +(2.22) +rm ∈ Z2(m+1) with +∥rm∥Z2(m+1) ≤ +� +1+∥R∗∥H0→H +� +∥g∥H0 +for m = 0, . . . , ⌊ℓ/2⌋−1, +and +(2.23) +r⌊ℓ/2⌋ ∈ Zℓ+1 with +��r⌊ℓ/2⌋ +�� +Zℓ+1 ≤ +� +1 + ∥R∗∥H0→H +� +∥g∥H0 . +Proof. We first prove (2.19) by induction. By the definition of u0 (2.17), conti- +nuity and coercivity of �a, and boundedness of S (2.3), ∥u0∥H ≤ C ∥g∥H0. Then, by +(1.8) with j = 2, +∥u0∥Z2 ≤ C +� +∥u0∥H0 + ∥g∥H0 +� +≤ C ∥g∥H0 , +12 + +which is (2.19) with m = 0. +Assume that (2.19) holds with m = q. By the definition of uq+1 (2.18), continuity +and coercivity of �a, and boundedness of S (2.3), +(2.24) +∥uq+1∥H ≤ C ∥uq∥H∗ . +By (1.8) with j = 2(q + 1) and the definition of uq+1 (2.18) +∥uq+1∥Z2(q+1) ≤ C +� +∥uq+1∥H0 + +sup +v∈H, ∥v∥(Z2q )∗ =1 +|⟨Sv, uq⟩| +� +. +(2.25) +By duality +∥S∥(Zj)∗→H0 ≤ C +j = 0, . . . , ℓ + 1, +and thus +(2.26) +sup +v∈H, ∥v∥(Z2q )∗ =1 +|⟨Sv, uq⟩| ≤ ∥S∥(Z2q)∗→H0 ∥uq∥H0 ≤ C ∥uq∥H0 . +Combining (2.25), (2.26), and (2.24), we find that +∥uq+1∥Z2(q+2) ≤ C +� +∥uq+1∥H0 + ∥uq∥H0 +� +≤ C ∥uq∥H . +Using (2.19) with m = q, we obtain (2.19) with m = q + 1, and the induction is +complete. +If ℓ is odd, i.e., ℓ + 1 is even, then this establishes both (2.19) and (2.20) since +2(⌊ℓ/2⌋ + 1) = ℓ + 1 (i.e., the highest-order case is even, and can be reached by +increasing the regularity at each step by two). If ℓ is even, i.e., ℓ + 1 is odd, then +the argument above establishes (2.19). The bound for u⌊ℓ/2⌋ (i.e., (2.20)) then follows +from elliptic regularity, using that u⌊ℓ/2⌋−1 = uℓ/2−1 ∈ Zℓ ⊂ Zℓ−1 (i.e., at the last +step, we only increase the regularity by one). +For the proof that rm ∈ Z2(m+1) and satisfies (2.22), observe that the definition +of rm (2.21) and the definition of um (2.18) implies that r0 = u and +�a(v, rm) = ⟨Sv, rm−1⟩ +for all v ∈ H, m = 1, . . . , ⌊ℓ/2⌋. +The proof of (2.22) is then very similar to the proof of (2.19), with the first step being +that, by (1.8), the fact that u = R∗g, and the definition of R∗ (1.13), +∥r0∥Z2 = ∥u∥Z2 ≤ C +� +∥u∥H0 + ∥g∥H0 +� +≤ C +� +1 + ∥R∗∥H0→H +� +∥g∥H0 . +Proof of Theorem 1.6 using Lemma 2.2. As in Lemma 2.2, given g ∈ H0, let u = +R∗g. By (2.21), +∥(I − Π)R∗g∥H ≤ +⌊ℓ/2⌋−1 +� +j=0 +∥(I − Π)∥Z2(j+1)→H ∥uj∥Z2(j+1) + ∥(I − Π)∥Zℓ+1 +��r⌊ℓ/2⌋ +�� +Zℓ+1 +so that, by the bounds (2.19), (2.20), and (2.23), +∥(I − Π)R∗g∥H ≤ C +� ⌊ℓ/2⌋−1 +� +j=0 +∥(I − Π)∥Z2(j+1)→H ++ ∥(I − Π)∥Zℓ+1→H +� +1 + ∥R∗∥H0→H +�� +∥g∥H0 ; +the result (1.20) then follows from the definition of η(Hh) (1.14). +13 + +3. Elliptic-regularity results. This section collects the elliptic-regularity re- +sults that are used to verify that Assumption 1.2 holds for Helmholtz problems with +truncation of the exterior domain either by a PML (in §4) or an impedance boundary +condition (in §5). Let +Lu = −k−2∇ · (A∇u) − c−2u, +with associated sesquilinear form +a(u, v) = +� +Ω +� +k−2(A∇u) · ∇v − c−2u v +� +, +where Ω be a bounded Lipschitz domain with outward-pointing unit normal vector +n. The conormal derivative ∂n,Au is defined for u ∈ H2(Ω) by ∂n,Au := n · (A∇u); +recall that ∂n,Au can be defined for u ∈ H1(Ω) with Lu ∈ L2(Ω) by Green’s identity; +see, e.g., [34, Lemma 4.3]. +Assumption 3.1. For all x ∈ Ω, Ajℓ(x) = Aℓj(x) and +ℜ +d +� +j=1 +d +� +ℓ=1 +Ajℓ(x)ξkξj ≥ c|ξ|2 +for all ξ ∈ Cd. +Theorem 3.2 (Local elliptic regularity near a Dirichlet or Neumann boundary). +Let Ω be a Lipschitz domain and let G1, G2 be open subsets of Rd with G1 ⋐ G2 and +G1 ∩ ∂Ω ̸= ∅. Let +(3.1) +Ωj := Gj ∩ Ω, j = 1, 2, +and +Γ2 := G2 ∩ ∂Ω. +Suppose that A satisfies Assumption 3.1, A, c ∈ Cm,1(Ω2), Γ2 ∈ Cm+1,1, u ∈ H1(Ω2), +and Lu ∈ Hm(Ω2) for some m ∈ N, and either u = 0 or ∂n,Au = 0 on Γ2. Then +(3.2) +∥u∥Hm+2 +k +(Ω1) ≤ C +� +∥u∥H1 +k(Ω2) + ∥Lu∥Hm +k (Ω2) +� +. +Proof. In unweighted norms, this follows from, e.g., [34, Theorems 4.7 and 4.16]; +the proof in the weighted norms (4.11) is very similar. +Theorem 3.3 (Local elliptic regularity for the transmission problem). +Let Ωin +be a Lipschitz domain, and let Ωout := Rd \ Ωin. Let G1, G2 be open subsets of Rd +with G1 ⋐ G2 and G1 ∩ ∂Ωin ̸= ∅. Let +Ωin/out,j := Gj ∩ Ωin/out, +j = 1, 2, +and Γ2 := G2 ∩ ∂Ωin. +Suppose that A satisfies Assumption 3.1, A|Ωin/out,2, c|Ωin/out,2 ∈ Cm,1(Ωin/out,2), Γ2 ∈ +Cm+1,1, uin/out ∈ H1(Ωin/out), and Lu ∈ Hm(Ωin/out,2) for some m ∈ N. Suppose +further that +uin = uout +and +∂n,Auin = β∂n,Auout +on Γ2 +for some β > 0. Then +∥uin∥Hm+2 +k +(Ωin,1) + ∥uout∥Hm+2 +k +(Ωout,1) +≤ C +� +∥uin∥H1 +k(Ωin,2) + ∥uout∥H1 +k(Ωout,2) + ∥Luin∥Hm +k (Ωin,2) + ∥Luout∥Hm +k (Ωout,2) +� +. +(3.3) +14 + +Proof. In unweighted norms, this is, e.g., [10, Theorem 5.2.1(i)] (and [34, The- +orems 4.7 and 4.16] when β = 1); the proof in the weighted norms (4.11) is very +similar. +Theorem 3.4 (Local elliptic regularity for the impedance problem). Let Ω be a +Lipschitz domain and let G1, G2 be open subsets of Rd with G1 ⋐ G2 and G1∩∂Ω ̸= ∅. +Let Ωj and Γ2 be defined by (3.1). Suppose that, for some m ∈ N, Γ2 ∈ Cm+1,1, +u ∈ H1(Ω2), and ∆u ∈ Hm(Ω2), and (k−1∂n − i)u = 0 on Γ2. Then +(3.4) +∥u∥Hm+2 +k +(Ω1) ≤ C +� +∥u∥H1 +k(Ω2) + +��k−2∆u +�� +Hm +k (Ω2) +� +. +Proof. When m = 0, the result can be obtained from [7, Lemma 4.1] by multiply- +ing by k−2 to switch to weighted norms, and using that the trace operator has norm +bounded by Ck1/2 from H1 +k to L2 (which can be obtained from, e.g., [39, Theorem +5.6.4] since the weighted norms there are, up to a constant, the weighted norms (1.5)). +The proof that (3.4) follow for m > 0 is then standard and can be found e.g. +in [14, §6.3.2, Theorem 5]. We repeat it here in the context of impedance boundary +conditions for completeness. +We now prove that if the bound holds for m = q, then it holds for m = q + 1 +(assuming the appropriate regularity of the coefficients and the domain). Without +loss of generality, we can change coordinates and work with U := B(0, s) ∩ {xd > 0} +and V := B(0, t) ∩ {xd > 0} for some 0 < t < s. In these coordinates +Lu := (−k−2aij∂xi∂xj −k−2(bi∂xi−c))u = f, +(−k−1∂xd−i)u = 0 on {xd = 0}∩U. +Suppose that for some q ≥ 0, for any 0 < t < s, +(3.5) +∥u∥Hq+2 +k +(V ) ≤ Ct +� +∥u∥L2(U) + ∥f∥Hq +k(U) +� +. +Now suppose that f ∈ Hq+1 +k +(U) and a, b, c ∈ Cq+1,1(U), and let W := B(0, r)∩{xd > +0} with t < r < s. By (3.5), +(3.6) +∥u∥Hq+2 +k +(W) ≤ C +� +∥u∥L2(U) + ∥f∥Hq +k(U) +� +, +and, by interior elliptic regularity, u ∈ Hq+3 +loc (U). +The next step is to bound tangential derivatives of u: let |α| = q + 1 with αd = 0 +(so that ∂α +x is a tangential derivative). Let +�f := L +� +k−|α|∂α +x u +� +so that +�f = [L, k−|α|∂α +x ]u + k−|α|∂α +x f +(where [A, B] := AB − BA) and, by (3.6) and the fact that the coefficients of L are +Cq+1,1(U), +(3.7) +∥ �f∥L2(W) ≤ C +� +∥u∥Hq+2(W) + ∥f∥Hq+1 +k +(W) +� +≤ C +� +∥u∥L2(U) + ∥f∥Hq+1 +k +(U) +� +. +Furthermore +(−k−1∂xd − i)k−|α|∂α +x u|xd=0 = k−|α|∂α +x +� +(−k−1∂xd − iu)|xd=0 +� += 0, +so that, by the analogue of (3.5) with q = 0 and U replaced by W, (3.6), and (3.7), +��k−|α|∂α +x u +�� +H2 +k(V ) ≤ C +���k−|α|∂α +x u +�� +L2(W) + +�� �f +�� +L2(W) +� +≤ C +� +∥u∥L2(U) + ∥f∥Hq+1 +k +(U) +� +. +15 + +Therefore, by the definition of α, +��k−|β|∂β +xu +�� +L2(V ) ≤ C +� +∥u∥L2(U) + ∥f∥Hq+1 +k +(U) +� +for all |β| = q + 3 with βd ∈ {0, 1, 2}. +(3.8) +To prove that the bound (3.5) holds with q replaced by q + 1, i.e., +∥u∥Hq+3 +k +(V ) ≤ C +� +∥u∥L2(U) + ∥f∥Hq+1 +k +(U) +� +, +it is sufficient to prove that +��k−|β|∂β +xu +�� +L2(V ) ≤ C +� +∥u∥L2(U) + ∥f∥Hq+1 +k +(U) +� +for all |β| = q + 3 with βd ∈ {0, . . . , q + 3}. +We therefore now prove by induction that if +(3.9) +��k−|β|∂β +xu +�� +L2(V ) ≤ C +� +∥u∥L2(U) + ∥f∥Hq+1 +k +(U) +� +for any |β| = q + 3 with βd ∈ {0, . . . , j} for some j ∈ {2, . . . , q + 2}, then (3.9) holds +for |β| = q + 3 with βd = j + 1. Combined with (3.8), this completes the proof. +We therefore assume that |β| = q + 3 with βd = j + 1. Then, putting β = γ + δ +with δ = (0, . . . , 0, 2) and |γ| = q + 1, and using that u ∈ Hq+3 +loc (U), we have +(3.10) +k−|γ|∂γLu = addk−|β|∂βu + Bu +in V, +where +Bu = +� +|α|≤q+3, αd≤j +aαk−|α|∂α +x u. +By the induction hypothesis (3.9), +∥Bu∥L2(V ) ≤ C +� +∥u∥L2(U) + ∥f∥Hq+1 +k +(U) +� +. +Dividing (3.10) by add, taking the L2(V ) norm, and using that 1/add is bounded, we +have +∥k−|β|∂βu∥L2(V ) ≤ C +� +∥u∥L2(U) + ∥f∥Hq+1 +k +(U) +� +; +i.e., we have proved that (3.9) holds for |β| = q + 3 with βd = j + 1, and the proof is +complete. +4. Theorem 1.5 applied to the PML problem. +4.1. Definition of the PML problem. +Obstacles and coefficients for Dirichlet/Neumann/penetrable obstacle problem. +Let Ωp, Ω− ⊂ BR0 := {x : |x| < R0} ⊂ Rd, d = 2, 3, be bounded open sets with +Lipschitz boundaries, Γp and Γ−, respectively, such that Γp ∩ Γ− = ∅, and Rd\Ω− is +connected. Let Ωout := Rd\Ω− ∪ Ωp and Ωin := (Rd\Ω−) ∩ Ωp. +Let Aout ∈ C0,1(Ωout, Rd×d) and Ain ∈ C0,1(Ωin, Rd×d) be symmetric positive +definite, let cout ∈ L∞(Ωout; R), cin ∈ L∞(Ωin; R) be strictly positive, and let Aout +and cout be such that there exists Rscat > R0 > 0 such that +Ω− ∪ supp(I − Aout) ∪ supp(1 − cout) ⋐ BRscat. +16 + +The obstacle Ω− is the impenetrable obstacle, on which we impose either a zero +Dirichlet or a zero Neumann condition, and the obstacle Ωin is the penetrable obstacle, +across whose boundary we impose transmission conditions. +For simplicity, we do not cover the case when Ω− is disconnected, with Dirichlet +boundary conditions on some connected components and Neumann boundary con- +ditions on others, but the main results hold for this problem too (at the cost of +introducing more notation). +Definition of the radial PML. Let Rtr > RPML,− > Rscat and let Ωtr ⊂ Rd be a +bounded Lipschitz open set with BRtr ⊂ Ωtr ⊂ BCRtr for some C > 0 (i.e., Ωtr has +characteristic length scale Rtr). Let Ω := Ωtr ∩ Ω+ and Γtr := ∂Ωtr. For 0 ≤ θ < π/2, +let the PML scaling function fθ ∈ C3([0, ∞); R) be defined by fθ(r) := f(r) tan θ for +some f satisfying +(4.1) +� +f(r) = 0 +� += +� +f ′(r) = 0 +� += +� +r ≤ RPML,− +� +, +f ′(r) ≥ 0, +f(r) ≡ r on r ≥ RPML,+; +i.e., the scaling “turns on” at r = RPML,−, and is linear when r ≥ RPML,+. We +emphasize that Rtr can be < RPML,+, i.e., we allow truncation before linear scaling +is reached. Indeed, RPML,+ > RPML,− can be arbitrarily large and therefore, given +any bounded interval [0, R] and any function �f ∈ C3([0, R]) satisfying +� �f(r) = 0 +� += +� �f ′(r) = 0 +� += +� +r ≤ RPML,− +� +, +�f ′(r) ≥ 0, +our results hold for an f with f|[0,R] = �f. Given fθ(r), let +(4.2) +α(r) := 1 + if ′ +θ(r) +and +β(r) := 1 + ifθ(r)/r. +and let +(4.3) +A := + + + + + +Ain +in Ωin, +Aout +in Ωout ∩ BRPML,−, +HDHT +in (BRPML,−)c +and 1 +c2 := + + + + + +c−2 +in +in Ωin, +c−2 +out +in Ωout ∩ BRPML,−, +α(r)β(r)d−1 +in (BRPML,−)c, +where, in polar coordinates, +(4.4) +D = +� +β(r)α(r)−1 +0 +0 +α(r)β(r)−1 +� +and +H = +� +cos θ +− sin θ +sin θ +cos θ +� +for d = 2, +and +(4.5) +D = + + +β(r)2α(r)−1 +0 +0 +0 +α(r) +0 +0 +0 +α(r) + + and H = + + +sin θ cos φ +cos θ cos φ +− sin φ +sin θ sin φ +cos θ sin φ +cos φ +cos θ +− sin θ +0 + + +for d = 3 (observe that then Aout = I and c−2 +out = 1 when r = RPML,− and thus A and +c−2 are continuous at r = RPML,−). +We highlight that, in other papers on PMLs, the scaled variable, which in our +case is r+ifθ(r), is often written as r(1+i�σ(r)) with �σ(r) = σ0 for r sufficiently large; +see, e.g., [24, §4], [4, §2]. Therefore, to convert from our notation, set �σ(r) = fθ(r)/r +and σ0 = tan θ. +Let +(4.6) +H := H1 +0(Ω) +or +{v ∈ H1(Ω) : v = 0 on Γtr}, +17 + +with the former corresponding to zero Dirichlet boundary conditions on Ω− and the +latter corresponding to zero Neumann boundary conditions on Ω−. +Definition 4.1 (A variational formulation of the PML problem). Given G ∈ +(H)∗ and β > 0, +(4.7) +find u ∈ H such that a(u, v) = G(v) for all v ∈ H, +where +(4.8) +a(u, v) := +�� +Ω∩Ωout ++ 1 +β +� +Ω∩Ωin +� � +k−2(A∇u) · ∇v − c−2uv +� +. +When +(4.9) +G(v) := +�� +BRPML,− ∩Ωout ++ 1 +β +� +Ω∩Ωin +� +c−2gv +for g ∈ L2(Ω+) with supp g ⊂ BRPML,−, the variational problem (4.7) is a weak form +of the problem +(4.10) +k−2c2 +out∇ · (Aout∇uout) + uout = −g +in Ωout, +k−2c2 +in∇ · (Ain∇uin) + uin = −g +in Ωin, +uin = uout +and +∂n,Ainuin = β∂n,Aoutuout +on ∂Ωin, +either +uin = 0 +or +∂n,Ainuin = 0 +on ∂Ω−, +and with the Sommerfeld radiation condition approximated by a radial PML ((4.7) is +obtained by multiplying the PDEs above by c−2 +in/outαβd−1 and integrating by parts). +Using the fact that the solution of the true scattering problem exists and is unique +with Aout, Ain, cout, cin, Ω−, and Ωin described above, the solution of (4.7) exists and +is unique (i) for fixed k and sufficiently large Rtr − R1 by [30, Theorem 2.1], [31, +Theorem A], [24, Theorem 5.8] and (ii) for fixed Rtr > R1 and sufficiently large k by +[18, Theorem 1.5]. +For the particular data G (4.9), it is well-known that, for fixed k, the error +∥u−v∥H1 +k(BRPML,− \Ω) decays exponentially in Rtr−RPML,− and tan θ; see [30, Theorem +2.1], [31, Theorem A], [24, Theorem 5.8]. It was recently proved in [18, Theorems 1.2 +and 1.5] that the error ∥u − v∥H1 +k(BRPML,− \Ω) also decreases exponentially in k. +4.2. Showing that the PML problem fits in the abstract framework +used in Theorem 1.5. Recall that H is defined by (4.6) and let H0 = L2(Ω). We +work with the norm ∥ · ∥H1 +k(Ω) (1.5) on H, and use below the higher-order norms +(4.11) +∥v∥2 +Hm +k (Ω) := +� +0≤|α|≤m +k−2|α| ∥∂αv∥2 +L2(Ω) . +The rationale for using these norms is that if a function v oscillates with frequency k, +then |(k−1∂)αv| ∼ |v| for all α; this is true, e.g., if v(x) = exp(ikx · a). We highlight +that many papers on the FEM applied to the Helmholtz equation use the weighted H1 +norm |||v|||2 := ∥∇v∥2 +L2(Ω)+k2 ∥v∥2 +L2(Ω); we work with (1.5) instead, because weighting +the jth derivative with k−j is easier to keep track of than weighting the jth derivative +with k−j+1. +We first check that the sesquilinear form a (4.8) is continuous and satisfies a +G˚arding inequality, with constants uniform for ǫ ≤ θ ≤ π/2 − ǫ. +18 + +Lemma 4.2 (Bounds on the coefficients A and c). +Given A and c as in (4.3), a +scaling function f(r) satisfying (4.1), and ǫ > 0 there exist A+ and c− such that, for +all ǫ ≤ θ ≤ π/2 − ǫ, x ∈ Ω, and ξ, ζ ∈ Cd, +|(A(x)ξ, ζ)2| ≤ A+∥ξ∥2∥ζ∥2 +and +1 +|c(x)|2 ≥ 1 +c2 +− +. +Proof. This follows from the definitions of A and c in (4.3), the definitions of α +and β in (4.2), and the fact that fθ(r) := f(r) tan θ. +Continuity of a (1.6) with Ccont := max{A+, c−2 +− } then follows from the Cauchy- +Schwarz inequality and the definition of ∥ · ∥H1 +k(Ω) (1.5). +Assumption 4.3. When d = 3, fθ(r)/r is nondecreasing. +Assumption 4.3 is standard in the literature; e.g., in the alternative notation +described above it is that �σ is non-decreasing – see [4, §2]. +Remark 4.4. As noted above, the variational problem (4.7) is obtained by multi- +plying the PDEs in (4.10) by c−2 +in/outαβd−1 and integrating by parts (as in [9, §3]). If +one integrates by parts the PDEs directly (as in, e.g., [24, Lemma 4.2 and Equation +4.8]), the resulting sesquilinear form satisfies Assumption 1.2 after multiplication by +eiω, for some suitable ω (see Remark 1.3), without the need for Assumption 4.3. +Lemma 4.5. Suppose that fθ satisfies Assumption 4.3. With A defined by (4.3), +given ǫ > 0 there exists A− > 0 such that, for all ǫ ≤ θ ≤ π/2 − ǫ, +ℜ +� +A(x)ξ, ξ +� +2 ≥ A−∥ξ∥2 +2 +for all ξ ∈ Cd and x ∈ Ω+. +Reference for the proof. See, e.g., [20, Lemma 2.3]. +Corollary 4.6. If fθ satisfies Assumption 4.3 then +ℜa(w, w) ≥ A−∥w∥2 +H1 +k(Ω) − +� +A− + c−2 +min +� +∥w∥2 +L2(Ω) +for all w ∈ H. +Let R : L2(Ω) → H be defined by a(Rg, v) = (g, v)L2(Ω) for all v ∈ H; i.e., R +is the solution operator of the PML problem. The definition of a and the facts that +(with the matrices H and D defined by (4.4), (4.5)) H is real and the matrix D is +diagonal (and hence symmetric) imply that a(u, v) = a(v, u) for all u, v ∈ H, and thus +Rg = R∗g. We therefore let +(4.12) +Csol := ∥R∥L2(Ω)→H = ∥R∗∥L2(Ω)→H . +We highlight that (i) Csol is bounded by the norm of the solution operator of the true +scattering problem (i.e., with the Sommerfeld radiation condition) by [18, Theorem +1.6], (ii) Csol ∼ k when the problem is nontrapping (with this the slowest-possible +growth in k), and (iii) an advantage of working with the weighted norms (4.11) is that +Csol in fact describes the k-dependence of the Helmholtz solution operator between +Hm +k and Hm+2 +k +for any m. +Lemma 4.7 (The PML problem satisfies Assumption 1.2). +Suppose that, for +some ℓ ∈ Z+, Aout, Ain, cout, cin ∈ Cℓ−1,1 and fθ ∈ Cℓ,1 on the closures of the domains +on which they are defined, ∂Ω is Cℓ,1, and fθ satisfies Assumption 4.3. Let +(4.13) +Zj = +� +v : vout ∈ Hj(Ω ∩ Ωout), vin ∈ Hj(Ωin) +� +∩ H +19 + +with norm +(4.14) +∥v∥2 +Zj := ∥vout∥2 +Hj +k(Ωout∩Ω) + ∥vin∥2 +Hj +k(Ωin) . +where the “out” and “in” subscripts denote restriction to Ωout∩Ω and Ωin, respectively. +Then a defined by (4.8) satisfies Assumption 1.2 and given ǫ > 0 and k0 > 0 there +exists C > 0 such the bounds (1.8) and (1.9) hold for all k ≥ k0 and ǫ ≤ θ ≤ π/2 − ǫ. +Proof. First observe that Assumption 3.1 is satisfied by the definition (4.3) of A. +Since +sup +v∈H, ∥v∥(Zj−2 )∗=1 +|a(u, v)| = ∥Lu∥Zj−2 , +the bound (1.9) holds by combining Theorem 3.2 (used near Γ− and Γtr) and Theorem +3.3 (used near Γp) and using the fact that, by Green’s identity, for u ∈ H1 +0(Ω) with +Lu ∈ L2(Ω) and ∂n,Ainuin = β∂n,Aoutuout on ∂Ωin, +∥uin∥H1 +k(Ωin) + ∥uout∥H1 +k(Ωout) +≤ C +� +∥uin∥L2(Ωin) + ∥uout∥L2(Ωout) + ∥Luin∥L2(Ωin) + ∥Luout∥L2(Ωout) +� +(so that the H1 +k norms on the right-hand sides of (3.2) and (3.3) can be replaced by +L2 norms). Since the operator associated with the sesquilinear form ℜa is +�L + L∗ +2 +� +u = −k−2∇ · +�A + A +2 +∇u +� +− +�c−2 + c−2 +2 +� +u +and the matrix A is symmetric, this operator also satisfies Assumption 3.1. +The +bound (1.8) then holds by a very similar argument. +4.3. Theorem 1.5 applied to the PML problem. +Assumption 4.8. Given p ∈ Z+, (Hh)h>0 are such that the following holds. +There exists C > 0 such that, for all h > 0, 0 ≤ j ≤ m+1 ≤ p+1, and v ∈ H∩Hℓ+1(Ω) +there exists Ih,pv ∈ Hh such that +��vout − (Ih,pv)out +�� +Hj(Ωout∩Ω) + +��vin − (Ih,pv)in +�� +Hj(Ωin) +≤ Chm+1−j� +∥vout∥Hm+1(Ωout∩Ω) + ∥vin∥Hm+1(Ωin) +� +. +(4.15) +where the “out” and “in” subscripts denote restriction to Ωout∩Ω and Ωin, respectively. +Assumption 4.8 holds when (Hh)h>0 consists of piecewise degree-p polynomials +on shape-regular simplicial triangulations, indexed by the meshwidth; see, e.g., [8, +Theorem 17.1], [5, Proposition 3.3.17]. +Theorem 4.9 (Existence, uniqueness, and error bound in the preasymptotic +regime for the PML problem). +Suppose that, for some ℓ ∈ Z+, Aout, Ain, cout, cin ∈ +Cℓ−1,1 and fθ ∈ Cℓ,1 on the closures of the domains where they are defined, ∂Ω is +Cℓ,1, fθ satisfies Assumption 4.3, and β > 0. Let Csol be defined by (4.12), and as- +sume that {Hh}h>0 satisfy Assumption 4.8. Given ǫ > 0 and p ∈ Z+ with p ≥ ℓ, +there exists k0 > 0 and Cj, j = 1, 2, 3, 4, such that the following is true for all k ≥ k0, +ǫ ≤ θ ≤ π/2 − ǫ, and Rtr > R1 + ǫ. +The solution u of the PML problem (4.7) exists and is unique, and if +(4.16) +(hk)2ℓCsol ≤ C1 +20 + +then the Galerkin solution uh, exists, is unique, and satisfies +∥u − uh∥H1 +k(Ω) ≤ C2 +� +1 + hk + (hk)ℓCsol +� +min +wh∈Hh ∥u − vh∥H1 +k(Ω) , +(4.17) +∥u − uh∥L2(Ω) ≤ C3 +� +hk + (hk)ℓCsol +� +min +wh∈Hh ∥u − vh∥H1 +k(Ω) . +(4.18) +If, in addition, g ∈ Hp−1(Ω) ∩ H (with H defined by (4.6)) with +(4.19) +∥g∥Hp−1 +k +(Ω) ≤ C ∥g∥H∗ +for some C > 0, then there exists C4 > 0 such that if h satisfies (4.16) then +(4.20) +∥u − uh∥H1 +k(Ω) +∥u∥H1 +k(Ω) +≤ C4 +� +hk + (hk)ℓCsol +� +(hk)ℓ. +Theorem 4.9 is most interesting when p = ℓ, i.e., the polynomial degree is the +smallest possible covered by the theorem. In this case, (4.16) becomes the condi- +tion (1.1), and the bounds (4.17), (4.18), and (4.20) become (1.2), (1.3), and (1.4), +respectively. +Proof of Theorem 4.9. By the results in §4.2, a defined by (4.8) satisfies the as- +sumptions of Theorem 1.5. By (4.15), the definition of ∥·∥Zj (4.14), and the definition +(4.11) of the weighted norms, ∥I − Π∥Zm+1→H ≤ C(hk)m. This bound along with +Theorem 1.6 and (4.12) imply that +η(Hh) ≤ C +� ⌊ℓ/2⌋−1 +� +j=0 +(hk)2j+1 + (hk)ℓCsol +� +. +If hk ≤ C, then η(Hh) ≤ C(hk + (hk)ℓCsol); the result then follows from Theorem +1.5 and the fact that if the condition (4.16) holds, then hk ≤ C (since Csol ≥ Ck). +5. Theorem 1.5 applied to the impedance problem. +5.1. Definition of the impedance problem. Let Aout, Ain, cout, cin, Ω−, Ωin, +and Ωtr be as in §4.1. Let +A := +� +Ain +in Ωin, +Aout +in Ωout ∩ Ω, +and +1 +c2 := +� +c−2 +in +in Ωin, +c−2 +out +in Ωout ∩ Ω. +Let +(5.1) +H := {v ∈ H1(Ω) : v = 0 on ∂Ω−} +or +H1(Ω), +with the former corresponding to zero Dirichlet boundary conditions on Ω− and the +latter corresponding to zero Neumann boundary conditions on Ω−. +Definition 5.1 (Variational formulation of the impedance problem). Given G ∈ +(H)∗ and β > 0, +(5.2) +find u ∈ H such that a(u, v) = G(v) for all v ∈ H, +where +(5.3) +a(u, v) := +�� +Ω∩Ωout ++ 1 +β +� +Ω∩Ωin +� � +k−2(A∇u) · ∇v − c−2uv +� +− ik−1 +� +Γtr +uv. +The solution of this variational problem exists and is unique by, e.g., [22, Theorem +2.4]. +21 + +5.2. Showing that the impedance problem fits in the abstract frame- +work used in Theorem 1.5. The proofs that the sesquilinear form a is continuous +and satisfies a G˚arding inequality are very similar to those for the PML problem in +§4.2 (in fact, they are simpler because there is no PML scaling parameter in which +the bounds need to be uniform). +Lemma 5.2 (The impedance problem satisfies Assumption 1.2). +Suppose that, +for some ℓ ∈ Z+, Aout, Ain, cout, cin ∈ Cℓ−1,1 on the closures of the domains on which +they are defined, and ∂Ω is Cℓ,1. With Zj and its norm defined by (4.13) and (4.14), +a defined by (5.3) satisfies Assumption 1.2 and given k0 > 0 there exists C > 0 such +the bounds (1.8) and (1.9) hold for all k ≥ k0. +Proof. This is very similar to the proof of Lemma 4.7. The regularity assumption +(1.8) follows by combining Theorem 3.2 used near ∂Ω−, Theorem 3.3 used near ∂Ωin, +and Theorem 3.4 used near Γtr. The regularity assumption (1.9) follows by combining +Theorem 3.2 used near ∂Ω−, Theorem 3.3 used near ∂Ωin, and now Theorem 3.2 +(with Neumann boundary condition) used near Γtr. Indeed, near Γtr, the operator +associated with (ℜa) is −k−2∆−1 with Neumann boundary conditions (coming from +Aout = I and cout = 1 near Γtr and the fact that no boundary condition is imposed +on Γtr in H (5.1)). +5.3. Theorem 1.5 applied to the impedance problem. +Theorem 5.3 (Existence, +uniqueness, +and error bound in the preasymp- +totic regime for the impedance problem). +Suppose that, for some ℓ ∈ Z+, +Aout, Ain, cout, cin ∈ Cℓ−1,1 on the closures of the domains where they are defined, ∂Ω +is Cℓ,1, and β > 0. Let Csol be defined by (4.12), and assume that {Hh}h>0 satisfy +Assumption 4.8. Given p ∈ Z+ with p ≥ ℓ, there exists k0 > 0 and Cj, j = 1, 2, 3, 4, +such that the following is true for all k ≥ k0. +The solution u of the impedance problem (5.2) exists and is unique, and if (4.16) +holds then the Galerkin solution uh, exists, is unique, and satisfies the bounds (4.17) +and (4.18). If, in addition, g ∈ Hp−1(Ω) ∩ H (with H defined by (5.1)) with (4.19) +for some C > 0, then there exists C4 > 0 such that if h satisfies (4.16) then the bound +(4.20) holds. +Given Lemma 5.2, the proof of Theorem 5.3 is very similar to the proof of Theorem +4.9. +Remark 5.4 (Imposing the exact Dirichlet-to-Neumann map on Γtr). +With the +exact Dirichlet-to-Neumann map imposed on Γtr, the Helmholtz sesquilinear form is +continuous and satisfies a G˚arding inequality (see, e.g., [37, Lemma 3.3 and Corollary +3.4]). To apply Theorem 1.5 to this problem, one therefore only needs to check the +elliptic-regularity assumptions of Assumption 1.2. Using Theorems 3.2 and 3.3, this +boils down to knowing the analogue of Theorem 3.4 with the impedance boundary +condition replaced by k−1∂nu = DtNu (for (1.8)) and also k−1∂nu = (DtN+DtN∗)u/2 +(for (1.9)). When m = 0 (i.e., the lowest-order regularity shift covered in Theorem +3.4), the first of these regularity results is given by [28, Theorem 6.1]. To prove this +result for m > 1 one would need to make an argument similar to that in the proof +of Theorem 3.4 except that, because DtN and DtN∗ do not commute with tangential +derivatives, one would need to obtain two additional estimates: 1) estimates on u +with nontrivial boundary data, e.g., when k−1∂nu − (DtN)u = g ∈ Hs +k and 2) trace +estimates for u that are needed to bound, e.g., [T, DtN]u where T is a vector field +tangent to the boundary. The same strategy could also be used to handle higher-order +22 + +impedance boundary conditions. +Acknowledgements. EAS was supported by EPSRC grant EP/R005591/1 and +JG was supported by EPSRC grants EP/V001760/1 and EP/V051636/1. +REFERENCES +[1] M. Ainsworth, Discrete dispersion relation for hp-version finite element approximation at +high wave number, SIAM Journal on Numerical Analysis, 42 (2004), pp. 553–575. +[2] A. K. Aziz, R. B. Kellogg, and A. B. Stephens, A two point boundary value problem with +a rapidly oscillating solution, Numer. Math., 53 (1988), pp. 107–121. +[3] M. Bernkopf, T. Chaumont-Frelet, and J. M. Melenk, Stability and convergence of +Galerkin discretizations of the Helmholtz equation in piecewise smooth media, arXiv pre- +print arXiv:2209.03601, (2022). +[4] J. H. Bramble and J. Pasciak, Analysis of a finite PML approximation for the three dimen- +sional time-harmonic Maxwell and acoustic scattering problems, Mathematics of Compu- +tation, 76 (2007), pp. 597–614. +[5] S. C. Brenner and L. R. 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Anal., 51 (2013), +pp. 1828–1852. +24 + diff --git a/F9E1T4oBgHgl3EQf-wbm/content/tmp_files/load_file.txt b/F9E1T4oBgHgl3EQf-wbm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f6b0efa6f36a9416c8ca0a5cd7003c650b9b7898 --- /dev/null +++ b/F9E1T4oBgHgl3EQf-wbm/content/tmp_files/load_file.txt @@ -0,0 +1,1349 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf,len=1348 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='03574v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='NA] 9 Jan 2023 SHARP PREASYMPTOTIC ERROR BOUNDS FOR THE HELMHOLTZ h-FEM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' GALKOWSKI∗ AND E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' SPENCE† Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' In the analysis of the h-version of the finite-element method (FEM), with fixed polynomial degree p, applied to the Helmholtz equation with wavenumber k ≫ 1, the asymptotic regime is when (hk)pCsol is sufficiently small and the sequence of Galerkin solutions are quasioptimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' here Csol is the norm of the Helmholtz solution operator, normalised so that Csol ∼ k for nontrapping problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The preasymptotic regime is when (hk)2pCsol is sufficiently small, and (for physical data) one expects the relative error of the Galerkin solution to be controllably small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' In this paper, we prove the natural error bounds in the preasymptotic regime for the variable- coefficient Helmholtz equation in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or combinations of these) and with the radiation condition approximated either by a radial perfectly- matched layer (PML) or an impedance boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Previously, such bounds for p > 1 were only available for Dirichlet obstacles with the radiation condition approximated by an impedance boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Our result is obtained via a novel generalisation of the “elliptic-projection” argument (the argument used to obtain the result for p = 1) which can be applied to a wide variety of abstract Helmholtz-type problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' AMS subject classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 35J05, 65N15, 65N30, 78A45 Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Helmholtz, FEM, high order, pollution effect, preasymptotic, perfectly-matched layer, elliptic projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Informal statement of the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We consider the h-version of the finite-element method (h-FEM), where accuracy is increased by decreasing the meshwidth h while keeping the polynomial degree p constant, applied to the Helmholtz equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1 (Informal statement of the main result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let u be the solution to the variable-coefficient Helmholtz equation, with wavenumber k > 0, in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or combinations of these) and with the radiation condition approximated either by a perfectly-matched layer (PML) or an impedance boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Csol be the norm of the solution operator, normalised so that Csol ∼ k for nontrapping problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Under the natural regularity assumptions on the domain and coefficients, if (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1) (hk)2pCsol is sufficiently small then the Galerkin solution uh exists, is unique, and satisfies ∥u − uh∥H1 k(Ω) ≤ C � 1 + hk + (hk)pCsol � min vh∈Hh ∥u − vh∥H1 k(Ω) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2) ∥u − uh∥L2(Ω) ≤ C � hk + (hk)pCsol � min vh∈Hh ∥u − vh∥H1 k(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3) Furthermore, if the data is k-oscillatory (in a sense made precise below), then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4) ∥u − uh∥H1 k(Ω) ∥u∥H1 k(Ω) ≤ C � 1 + hk + (hk)pCsol � (hk)p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' ∗Department of Mathematics, University College London, 25 Gordon Street, London, WC1H 0AY, UK, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='Galkowski@ucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='uk †Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='Spence@bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='uk 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', the relative H1 k error can be made controllably small by making (hk)2pCsol suffi- ciently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The norm ∥ · ∥H1 k(Ω) in the bounds above is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5) ∥v∥2 H1 k(Ω) := k−2 ∥∇v∥2 L2(Ω) + ∥v∥2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The fact that, for oscillatory data, the relative H1 k error for the Helmholtz h-FEM is controllably small if (hk)2pCsol is sufficiently small was famously identified for 1-d nontrapping problems by the work of Ihlenburg and Babuˇska [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The bounds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3) have previously been obtained (i) for the Dirichlet obstacle problem with impedance boundary conditions approximating the radiation condition [12, 40] and (ii) for PML with constant-coefficients, no obstacle, and p = 1 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The present paper proves the bounds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3), and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4) assuming only that the sesquilinear form is continuous, satisfies a G˚arding inequality, and satisfies certain standard elliptic-regularity assumptions, therefore covering a variety of scatterers and methods for truncating the exterior domain (to approximate the radiation condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Regarding the latter: in this paper we consider truncating with a PML or an imped- ance boundary condition, but truncating with the exact Dirichlet-to-Neumann map is also, in principle, covered by the abstract framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Statement of the main abstract result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let H ⊂ H0 ⊂ H∗ be Hilbert spaces with H0 identified with its dual and H ⊂ H0 compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let a : H × H → C be a continuous sesquilinear form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6) |a(u, v)| ≤ Ccont ∥u∥H ∥v∥H and a(λu, µv) = λ¯µa(u, v) for all u, v ∈ H, satisfying the G˚arding inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7) ℜa(v, v) ≥ CG1 ∥v∥2 H − CG2 ∥v∥2 H0 for all v ∈ H for some CG1, CG2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We assume further that Ccont, c, C and all the other constants in this section are independent of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 (“Elliptic regularity” assumptions on a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Z0 = H0, Z1 = H, and Zj ⊂ Zj−1 for j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , ℓ + 1 such that Zj is dense in Zj−1, and assume that for all u ∈ H with sup v∈H, ∥v∥(Zj−2)∗=1 |a(u, v)| < ∞, u ∈ Zj and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8) ∥u∥Zj ≤ C � ∥u∥H0 + sup v∈H, ∥v∥(Zj−2)∗=1 |a(u, v)| � , j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Assume further that for any w ∈ H such that sup w∈H, ∥v∥(Zj−2)∗=1 |(ℜa)(u, v)| < ∞, w ∈ Zj with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9) ∥w∥Zj ≤ C � ∥u∥H0 + sup v∈H, ∥v∥(Zj−2)∗=1 |(ℜa)(u, v)| � , j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , ℓ + 1, 2 where the sesquilinear form ℜa is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='10) (ℜa)(u, v) := 1 2 � a(u, v) + a(v, u) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Note that ℜa in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='10) could be replaced by ℜ(eiωa), so long as one uses the same value of ω in both conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4 below describes a situation where this is useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Given g ∈ H∗, suppose that u ∈ H satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='11) a(u, v) = ⟨g, v⟩ for all v ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Given a sequence of finite dimensional subspace {Hh}h>0 with Hh ⊂ H, the sequence of Galerkin approximations of u, {uh}h>0, are defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='12) a(uh, vh) = ⟨g, vh⟩ for all vh ∈ Hh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' For the Helmholtz equation outside a Dirichlet obstacle with PML truncation and Ω the truncated exterior domain, H0 = L2(Ω), H = H1 0(Ω), and Zj = Hj(Ω) ∩ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 is then elliptic regularity for the Helmholtz PML operator and its real part, which both hold if the coefficients of the Helmholtz equation are in Cℓ−1,1, the PML scaling function is Cℓ,1, and ∂Ω is Cℓ,1 (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5 (Abstract generalisation of the elliptic-projection argument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let a : H × H → C satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7), and Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Suppose that R∗ : H∗ → H defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='13) a(w, R∗v) = ⟨w, v⟩ for all w ∈ H, v ∈ H∗, is well defined and let (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='14) η(Hh) := sup g∈H0,g̸=0 ∥(I − Π)R∗g∥H ∥g∥H0 , where Π : H → Hh is the orthogonal projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Then the solution, u, to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='11) exists and is unique and there exist C1, C2, C3 > 0 such that if h satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='15) η(Hh)∥I − Π∥Zℓ+1→H ≤ C1, then the solution uh to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='12) exists, is unique, and satisfies ∥u − uh∥H ≤ C2 � 1 + η(Hh) � min wh∈Hh ∥u − vh∥H , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) ∥u − uh∥H0 ≤ C3 η(Hh) min wh∈Hh ∥u − vh∥H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17) If, in addition, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='18) ∥g∥Zℓ−1 ≤ C ∥g∥H∗ for some C > 0, then there exists C4 > 0 such that if h satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='15) then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19) ∥u − uh∥H ∥u∥H ≤ C4 � 1 + η(Hh) � ∥I − Π∥Zℓ+1→H ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', the relative error in H can be made controllably small by making η(Hh) ∥I − Π∥Zℓ+1→H sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5 includes the result that the sequence of Galerkin solutions are qua- sioptimal with constant independent of k if η(Hh) is sufficiently small – with this the so-called asymptotic regime (see the discussion in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The bounds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17), and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19) and the meshthreshold (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='15) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5 all involve the quantity η(Hh), which measures how well solutions of the adjoint problem are approximated in the space Hh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Bounds on η(Hh) are given in [37, 38, 36, 13, 6, 29, 19, 20, 3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' see the discussion in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The following bound on η(Hh) is proved using the ideas in [6] (although the end result is phrased in a different way there);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' we include it here both for completeness, and because it holds under the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5 (in fact, it only requires the regularity assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8) and not (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6 (Bound on η(Hh)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Under the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5, there exists C > 0 such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='20) η(Hh) ≤ C � ⌊ℓ/2⌋−1 � j=0 ∥(I − Π)∥Z2(j+1)→H + ∥(I − Π)∥Zℓ+1→H � 1 + ∥R∗∥H0→H �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' In §4 and §5 below we show how Helmholtz problems with the ra- diation condition approximated by either a PML or an impedance boundary condition, respectively, fit into the abstract framework of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' In both these cases, the norm of the adjoint solution operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', ∥R∗∥H0→H, is the same as the norm of the solution operator of the original (non-adjoint) problem, which we denote by Csol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Furthermore, with {Hh}h>0 corresponding to the standard finite-element spaces of piecewise degree-p polynomials on shape-regular simplicial triangulations, indexed by the meshwidth h, ∥(I − Π)∥Zm+1→H ≤ C(hk)m for 0 ≤ m ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The meshthreshold (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='15) then becomes that (hk)2ℓCsol is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Recall that ℓ is a parameter in the elliptic-regularity assumptions (Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' If the polynomial degree p is taken to be ℓ then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='15) becomes (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The bounds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17) then become (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Discussion of the context, novelty, and ideas behind Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The work of Ihlenburg and Babuˇska in 1-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The celebrated work of [25, 26] studied the h-FEM applied to the constant-coefficient Helmholtz equation in 1-d (a nontrap- ping problem), and split the behaviour of the finite-element solutions as a function of h into the so-called asymptotic and preasymptotic regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The asymptotic regime is when h is small enough, as a function of k, for the sequence of Galerkin solutions to be quasi-optimal uniformly in k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', ∥u − uh∥H1 k(Ω) ≤ C min vh∈Hh ∥u − vh∥H1 k(Ω) with C > 0 independent of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' [26, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5] showed that a sufficient condition to be in the asymptotic regime is “hk2/p sufficiently small”, with later work (discussed below) then showing that a sufficient condition for nontrapping problems (when Csol ∼ k) is “(hk)pk sufficiently small”, with this condition then indicated to be necessary by numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Therefore, the pollution effect for the h-FEM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', the fact that one needs h ≪ k−1 to maintain accuracy, becomes less pronounced as p increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 4 The preasymptotic regime is when the relative H1 k error is controllably small, uni- formly as k → ∞, provided that the data is k-oscillatory, in the sense that it satisfies the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='18) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' [26, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2] used the explicit form of the Helmholtz Green’s function in 1-d to prove that if (hk)2pk sufficiently small then the finite-element solu- tion is in the preasymptotic regime, with the numerical experiments in [26, Table 2] (for p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , 6) indicating that this condition is also necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' [26] also studied the phase difference between the exact and finite-element solutions (following [23, 43]), with [26, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2] showing that the difference between the true wavenumber and the numerical wavenumber is bounded by C(hk)2pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Thus the condition “(hk)2pk sufficiently small” also controls this phase difference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' see also [1, Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Error bounds in the asymptotic regime using the Schatz argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='. We now out- line the argument that gives the condition “(hk)pCsol sufficiently small” for quasiop- timality, with this argument also used in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We work in the setting of Examples 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', the PML approximation to the Helmholtz exte- rior Dirichlet problem, so that H0 = L2(Ω) and H = H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The G˚arding inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7) is then ℜa(w, w) ≥ CG1 ∥w∥2 H1 k(Ω) − CG2 ∥w∥2 L2(Ω) for all w ∈ H1 0(Ω) for CG1, CG2 > 0 (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Combining the G˚arding inequality with the Galerkin orthogonality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='21) a(u − uh, vh) = 0 for all vh ∈ Hh, we find that, for all vh ∈ Hh, ∥u − uh∥2 H1 k(Ω) ≤ C−1 G1 ��a(u − uh, u − vh) �� + C−1 G1CG2 ∥u − uh∥2 L2(Ω) ≤ C−1 G1Ccont ∥u − uh∥H1 k(Ω) ∥u − vh∥H1 k(Ω) + C−1 G1CG2 ∥u − uh∥2 L2(Ω) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='22) where Ccont is the continuity constant of the sesquilinear form a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Therefore, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='22) implies that a sufficient condition for quasioptimality is that the L2 error is sufficiently small relative to the H1 k error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By the definition of R∗ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='13) (recalling that H = H1 0(Ω) here) and Galerkin orthogonality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='21), for any vh ∈ Hh, ∥u − uh∥2 L2(Ω) = a � u − uh, R∗(u − uh) � = a � u − uh, R∗(u − uh) − vh � ≤ Ccont ∥u − uh∥H1 k(Ω) ��R∗(u − uh) − vh �� H1 k(Ω), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='23) and thus, by the definition of η(Hh) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='14) (recalling that H0 = L2(Ω)), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='24) ∥u − uh∥L2(Ω) ≤ Ccontη(Hh) ∥u − uh∥H1 k(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Combining this last inequality with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='22), we see that a sufficient condition for qua- sioptimality is that η(Hh) is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Schatz [42] was the first to use the Aubin-Nitsche-type bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='24) with the G˚arding inequality, and thus the argument above is often called the Schatz argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The “adjoint approximability” concept, and associated definition of η(Hh), was introduced by Sauter in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 1The relative error can only be small for a certain subclass of data, since, given a finite- dimensional subspace Hh, one can choose data such that the solution v ∈ H is orthogonal to Hh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Then ∥u − uh∥2 H = ∥u∥2 H + ∥uh∥2 H ≥ ∥u∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 5 The bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='25) η � Hh � ≤ C � hk + (hk)pCsol � under sufficient regularity of the coefficients and obstacle has now been proved for a wide variety of Helmholtz problems, with this bound sharp by the recent results of [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='25) therefore gives the sufficient condition “(hk)pCsol sufficiently small” for quasioptimality, with this condition observed sharp for nontrapping problems in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', [6, Figures 3, 5, and 8] for p = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' For p = 1, the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='25) can be proved using only H2 regularity of the Helmholtz solution, with the condition “hk2 sufficient small” for quasiopimality ob- tained for 1-d problems in [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1], [11, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6], [27, Theorem 3], and [33, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2], 2-d problems in [35, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7], and variable-coefficient problems in 2- and 3-d in [22, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' For p > 1 the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='25) is proved by a judicious splitting of the solution in [37, 38, 13, 36] for constant-coefficient problems and [6, 29, 19, 20, 3] for variable- coefficient problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' All these papers apart from [6] make the constant C in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='25) explicit in p under suitably analyticity/smoothness assumptions on the obstacle and coefficients, and thus give results about the hp-FEM (showing that quasioptimality holds if hk/p is sufficiently small and p/ log k is sufficiently large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' In addition, all these papers apart from [6] split the solution into “high-” and “low-” frequency components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' In constrast, [6] instead expands the solution in a series whose terms increase with regularity, and with only the remainder satisfying a bound involving Csol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Bounds in the preasymptotic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Numerical experiments indicate that, at least for nontrapping problems, the condition “(hk)2pCsol sufficiently small” for the relative H1 k error to be controllably small is necessary and sufficient for 2- and 3-d Helmholtz problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', [12, Figure 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Nevertheless, despite the fact that sharp asymptotic error bounds have now been obtained for a variety of Helmholtz problems in 2- and 3-d and for arbitrary p ∈ Z+, until now the sharp preasymptotic error bounds were obtained only in the following cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' p = 1, the constant-coefficient Helmholtz equation with an impedance bound- ary condition [44, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1] or PML (and no obstacle) [32, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4], the variable-coefficient Helmholtz equation with truncation via the ex- act Dirichlet-to-Neumann map [28, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' p ∈ Z+, the constant-coefficient Helmholtz equation with no obstacle and an impedance boundary condition approximating the radiation condition [12, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1], 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' p ∈ Z+, the variable-coefficient Helmholtz equation in the exterior of a Dirich- let obstacle with an impedance boundary condition approximating the radi- ation condition [40, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The bounds in Point 1 for p = 1 come from the so-called elliptic projection argument, which proves error bounds under the condition “(hk)p+1Csol is sufficiently small”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', the sharp condition when p = 1, but not when p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The initial ideas behind this argument were introduced in the Helmholtz context in [15, 16] for interior-penalty discontinuous Galerkin methods, and then further developed for the standard FEM and continuous interior-penalty methods in [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The bounds in Point 2 used an error-splitting argument (with this idea called “stability-error iterative improvement”, and used earlier in [16, 44]) together with the idea of using discrete Sobolev norms in the duality argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The bounds in Point 3 6 for variable-coefficients were obtained by repeating the constant-coefficient arguments in Point 2, but now keeping track of how the constants depend on the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The elliptic-projection argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5 is proved by generalising the elliptic-projection argument, allowing it to prove error bounds under the sharp condi- tion “(hk)2pCsol sufficiently small” for p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We therefore recap the main ideas of the elliptic-projection argument here, and then we explain below how we generalise this argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Here, and in the rest of the paper, C is used for a constant, independent of h and k, but dependent on p, whose value may change line by line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The bounds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3) come from the bounds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='26) ∥u − uh∥H1 k(Ω) ≤ C � 1 + η(Hh) � min vh∈Hh ∥u − vh∥H1 k(Ω) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='27) ∥u − uh∥L2(Ω) ≤ Cη(Hh) min vh∈Hh ∥u − vh∥H1 k(Ω) and the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='25) on η(Hh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Observe that, by the consequence (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='22) of the G˚arding inequality, the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='26) follows from the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' To prove (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='27), the elliptic-projection argument writes (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='23) as ∥u − uh∥2 L2(Ω) = a � u − uh, R∗(u − uh) − vh � = �a � u − uh, R∗(u − uh) − vh � − � (1 + c−2)(u − uh), R∗(u − uh) − vh � L2(Ω), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='28) where �a(u, v) := � Ω k−2A∇u · ∇v + u v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let �Π : H1 0(Ω) → Hh be the solution of the variational problem �a(wh, �Πv) = �a(wh, v) for all wh ∈ Hh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Since �a is coercive on H1 0(Ω) and the continuity and coercivity constants of �a in ∥ · ∥H1 k(Ω) are independent of k, �Π is well-defined by the Lax–Milgram theorem and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='29) ��(I − �Π)v �� H1 k(Ω) ≤ C min wh∈Hh ∥v − wh∥H1 k(Ω) with C > 0 independent of k by C´ea’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The definition of �Π implies the Galerkin orthogonality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='30) �a � wh, (I − �Π)v � = 0 for all wh ∈ Hh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We now choose vh = �ΠR∗(u − uh) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='28) so that, by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='30), for all wh ∈ Hh, ∥u − uh∥2 L2(Ω) = �a � v − wh, (I − �Π)R∗(u − uh) � − � (1 + c−2)(u − uh), (I − �Π)R∗(u − uh) � L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='31) For the first term on the right-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='31) we use the continuity of �a, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='29), and the definition of η(Hh) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='14) to bound this term by C ∥v − wh∥H1 k(Ω) η(Hh) ∥u − uh∥L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 7 The second term on the right-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='31) is bounded by C ∥u − uh∥L2(Ω) ��(I − �Π)R∗(u − uh) �� L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Using the Schatz argument for �a, one can show that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='32) ��(I − �Π)R∗(u − uh) �� L2(Ω) ≤ Chk ��(I − �Π)R∗(u − uh) �� H1 k(Ω) and then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='29) and the definition of η(Hh) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='14) imply that the second term on the right-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='31) is bounded by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='33) Chk η(Hh) ∥u − uh∥2 L2(Ω) , which can be absorbed into the left-hand side if hk η(Hh) is sufficiently small, giving the result (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The ideas behind the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We generalise the elliptic-projection argument based on the observation that if �a(u, v) = a(u, v) + (Su, v)L2(Ω) with S a self-adjoint smoothing operator, then the second term on the right-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='31) is replaced by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='34) � u − uh, S∗(I − �Π)R∗(u − uh) � L2(Ω) (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='14) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Using the Schatz argument for �a and the smoothing property of S, the modulus of this term is bounded by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='35) ��S∗(I − �Π)R∗(u − uh) �� L2(Ω) ≤ C(hk)p��(I − �Π)R∗(u − uh) �� H1 k(Ω) (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Provided that �Π still satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='29), the term (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='34) is therefore bounded by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='36) C(hk)pη(Hh) ∥u − uh∥2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Comparing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='32) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='35), and also (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='33) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='36), we see that this new argument replaces the condition “hkη(Hh) sufficiently small” in the standard elliptic-projection argument by the condition “(hk)pη(Hh) sufficiently small”, which is the condition (hk)2pCsol sufficiently small” after using the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='25) on η(Hh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The challenge now is to ensure that the smoothing operator S is such that the projection �Π is well-defined and satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' This is achieved in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1 below, where a suitable S such that �a(u, v) = a(u, v) + (Su, v)L2(Ω) is coercive is construc- ted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' S is defined by an expansion in terms of the eigenfunctions of the (self-adjoint) operator associated with the real part of the sesquilinear form a (defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Proofs of the main results (Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Construction of a regularizing operator that produces coercivity when added to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Suppose that a : H × H → C satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7), and Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Then there exists S : H0 → H0 self adjoint and c, C > 0 such that, with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1) �a(u, v) := a(u, v) + ⟨Su, v⟩H0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2) ℜ�a(v, v) ≥ c ∥v∥2 H for all v ∈ H, 8 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3) ∥S∥H0→Zj ≤ C, j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , ℓ + 1 and �R : H∗ → H defined by �a( �Rf, u) = ⟨f, u⟩ for all u ∈ H, f ∈ H∗, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4) is well defined with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5) ∥ �R∥Zj−2→Zj ≤ C, 2 ≤ j ≤ ℓ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1 uses the spectral theorem for bounded self-adjoint op- erators, B : H → H∗, which we recap here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' With H0 and H as in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2, let b be a sesquilinear form on H satisfying b(u, v) = b(v, u), with associated operator B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', b(u, v) = ⟨Bu, v⟩ for all u, v ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' If b satisfies the G˚arding inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7) (with a replaced by b) then there exist an orthonormal basis (in H0) of eigenfunctions of B, {φj}∞ j=1, with associated eigenvalues satisfying λ1 ≤ λ2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' with λj → ∞ as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Furthermore, for all u ∈ H, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6) Bu = ∞ � j=1 λj⟨φj, u⟩φj (where the sum converges in H∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', [34, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Given a bounded function f, we define f(B) : H0 → H0 by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7) f(B)u := ∞ � j=1 f(λj)⟨φj, u⟩φj, so that ∥f(B)∥H0→H0 ≤ sup λ∈[λ1,∞) |f(λ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let P : H → H∗ be the operator associated with the sesquilinear form ℜa defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='10), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', (ℜa)(u, v) = ⟨Pu, v⟩ for all u, v ∈ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' observe that P is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Since (ℜa) also satisfies the G˚arding equality satis- fied by a (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7), the spectral theorem recapped above applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let {λj}∞ j=1 be the eigenvalues of P, let ψ ∈ C∞ comp(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' [0, ∞)) be such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8) x + ψ(x) ≥ 1 for x ≥ −λ1, and let S := ψ(P), in the sense of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We now use (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9) to prove that S : H0 → Zj satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Since ψ has compact support, the function t �→ tmψ(t) is bounded for any m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7) implies that, for any m ≥ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9) ∥Pmψ(P)∥H0→H0 ≤ Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9), ∥ψ(P)∥H0→Zj ≤ Cℓ � ∥ψ(P)∥H0→H0 + ∥Pψ(P)∥H0→Zj−2 � , j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , ℓ + 1, so that, by induction and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9), ∥S∥H0→Zℓ+1 = ∥ψ(P)∥H0→Zℓ+1 ≤ Cℓ ⌈(ℓ+1)/2⌉ � j=0 ��Pjψ(P) �� H0→H0 ≤ Cℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 9 We now show that �a satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By the definitions of P and S, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7), and the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8), for all v ∈ H, ℜ�a(v, v) = ℜa(v, v) + ⟨ψ(P)v, v⟩ = ⟨(P + ψ(P))v, v⟩ ≥ ∥v∥2 H0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Since ψ ≥ 0, S is positive, and thus ℜ�a(v, v) ≥ ℜa(v, v) for all v ∈ H, for any ǫ > 0 and for all v ∈ H, ℜ�a(v, v) ≥ ǫℜa(v, v) + (1 − ǫ)ℜ�a(v, v) ≥ ǫCG1 ∥v∥2 H − CG2ǫ ∥v∥2 H0 + (1 − ǫ)∥v∥2 H0, so that, choosing ǫ = min( 1 2CG2 , 1 2), we have ℜ�a(v, v) ≥ CG1 2 min � 1 CG2 , 1 � ∥v∥2 H + 1 2 ∥v∥2 H0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', �a is coercive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The existence of �R : H∗ → H satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4) and ∥ �R∥H∗→H ≤ C then follows from the Lax–Milgram theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Finally, to see that ∥ �R∥Zj−2→Zj ≤ C, 2 ≤ j ≤ ℓ + 1, observe that, since S is self-adjoint and satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3), for v ∈ (Zj−2)∗, |a( �Rg, v)| = |�a( �Rg, v) − ⟨S �Rg, v⟩| ≤ |�a( �Rg, v)| + |⟨S �Rg, v⟩| ≤ |⟨v, g⟩| + ∥v∥(Zj−2)∗∥S∥H→Zj−2∥( �R)∗∥H∗→H∥g∥H∗ ≤ ∥v∥(Zj−2)∗(∥g∥Zj−2 + C∥g∥H∗), and the claim follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Proof Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5 using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We claim it is sufficient to prove the bounds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17) under the assumption of existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Indeed, by uniqueness of the variational problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='11), either of the bounds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17) under the assumption of existence implies uniqueness of uh, and uniqueness implies existence for the finite-dimensional Galerkin linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We next show that the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Now, by the G˚arding inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7), Galerkin orthogonality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='21), and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17), for any vh ∈ Hh, ∥u − uh∥2 H ≤ C ���a(u − uh, u − vh) �� + ∥u − uh∥2 H0 � ≤ C � ∥u − uh∥H ∥u − vh∥H + � η(Hh) min wh∈Hh ∥u − wh∥H �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='10) The bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) on the error in H then follows by using the inequality 2ab ≤ ǫa2 + b2/ǫ for all a, b, ǫ > 0 in the first term on the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='10), and then using the inequality a2 + b2 ≤ (a + b)2 for a, b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We now prove (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17) (using the ideas outlined in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By the definition of R∗, Galerkin orthogonality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='21), and the definition of �a (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1) ∥u − uh∥2 H0 = a � u − uh, R∗(u − uh) � = a � u − uh, R∗(u − uh) − vh � = �a � u − uh, R∗(u − uh) − vh � − � S(u − uh), R∗(u − uh) − vh � H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='11) Let �Π : H → Hh be the solution of the variational problem �a(wh, �Πv) = �a(wh, v) for all wh ∈ Hh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 10 Since �a is continuous and coercive, with constants independent of k (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3)), by the Lax–Milgram lemma and C´ea’s lemma given k0 > 0 there exists C > 0 such that for all k ≥ k0 and v ∈ H, �Π is well-defined with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='12) ��(I − �Π)v �� H ≤ C min wh∈Hh ∥v − wh∥H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The definition of �Π implies the Galerkin orthogonality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='13) �a � wh, (I − �Π)u � = 0 for all wh ∈ Hh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We now choose vh = �ΠR∗(u − uh) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='11) so that, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='13), for all wh ∈ Hh, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='14) ∥u − uh∥2 H0 = �a � u − wh, (I − �Π)R∗(u − uh) � − � u − uh, S∗(I − �Π)R∗(u − uh) � H0 ≤ C ∥u − wh∥H ��(I − �Π)R∗(u − uh) �� H + ∥u − uh∥H0 ��S∗(I − �Π)R∗(u − uh) �� H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='12) and the definition of η(Hh) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='14), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='15) ��(I − �Π)R∗(u − uh) �� H ≤ C min wh∈Hh ∥R∗(u − uh) − wh∥H ≤ Cη(Hh) ∥u − uh∥H0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We now claim that the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17) follows if we can prove that, for all v ∈ H, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) ��S∗(I − �Π)v �� H0 ≤ C∥I − Π∥Zℓ+1→H ��(I − �Π)v �� H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Indeed, we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='15) in the first term on the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='14), and then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) with v = R∗(u − uh) in the second term on the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='14) to obtain ∥u − uh∥2 H0 ≤ Cη(Hh) ∥u − wh∥H ∥u − uh∥H0 + C∥I − Π∥Zℓ+1→H ��(I − �Π)R∗(u − uh) �� H ∥u − uh∥H0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='15), the last term on the right-hand side is ≤ C∥I−Π∥Zℓ+1→H η(Hh)∥u−uh∥2 H0 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We now prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) by using the duality argument described in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3 (as part of the Schatz argument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By the definition of �R (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4) and Galerkin orthogonality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='13), for all wh ∈ Hh, ��S∗(I − �Π)v ��2 H0 = � SS∗(I − �Π)v, (I − �Π)v � H0 = �a � �RSS∗(I − �Π)v − wh, (I − �Π)v � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Then, by the bounds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3), ��S∗(I − �Π)v ��2 H0 ≤ C min wh∈Hh �� �RSS∗(I − �Π)v − wh �� H ��(I − �Π)v �� H ≤ ∥I − Π∥Zℓ+1→H �� �RSS∗(I − �Π)v �� Zℓ+1 ��(I − �Π)v �� H, ≤ C∥I − Π∥Zℓ+1→H ��SS∗(I − �Π)v �� Zℓ−1 ��(I − �Π)v �� H, ≤ C∥I − Π∥Zℓ+1→H ��S∗(I − �Π)v �� H0 ��(I − �Π)v �� H which implies the bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16), and hence (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 11 Finally, we prove (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='11), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='18), and the abstract elliptic-regularity assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8), u ∈ Zℓ+1 with ∥u∥Zℓ+1 ≤ C � ∥u∥H0 + ∥g∥Zℓ−1 � ≤ C � ∥u∥H0 + ∥g∥H∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The variational problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='11) implies that ∥g∥H∗ = sup v∈H∗,v̸=0 |a(u, v)| ∥v∥H∗ ≤ C ∥u∥H , and thus ∥u∥Zℓ+1 ≤ C ∥u∥H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) then implies that ∥u − uh∥H ≤ C2 � 1 + η(Hh) � ∥I − Π∥Zℓ+1→H ∥u∥Zℓ+1 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The following lemma is essentially [6, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6], rewritten in the abstract notation in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Under the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5, let u = R∗g with R∗ defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='13) and g ∈ H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let um ∈ H, m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , ⌊ℓ/2⌋, be defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17) �a(v, u0) = ⟨v, g⟩ for all v ∈ H, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='18) �a(v, um) = ⟨Sv, um−1⟩ for all v ∈ H, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , ⌊ℓ/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19) um ∈ Z2(m+1) with ∥um∥Z2(m+1) ≤ C ∥g∥H0 for m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , ⌊ℓ/2⌋ − 1, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='20) u⌊ℓ/2⌋ ∈ Zℓ+1 with ��u⌊ℓ/2⌋ �� Zℓ+1 ≤ C ∥g∥H0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Furthermore, with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='21) rm := u − m−1 � j=0 uj, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='22) rm ∈ Z2(m+1) with ∥rm∥Z2(m+1) ≤ � 1+∥R∗∥H0→H � ∥g∥H0 for m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , ⌊ℓ/2⌋−1, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='23) r⌊ℓ/2⌋ ∈ Zℓ+1 with ��r⌊ℓ/2⌋ �� Zℓ+1 ≤ � 1 + ∥R∗∥H0→H � ∥g∥H0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We first prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19) by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By the definition of u0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17), conti- nuity and coercivity of �a, and boundedness of S (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3), ∥u0∥H ≤ C ∥g∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Then, by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8) with j = 2, ∥u0∥Z2 ≤ C � ∥u0∥H0 + ∥g∥H0 � ≤ C ∥g∥H0 , 12 which is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19) with m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Assume that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19) holds with m = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By the definition of uq+1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='18), continuity and coercivity of �a, and boundedness of S (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='24) ∥uq+1∥H ≤ C ∥uq∥H∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8) with j = 2(q + 1) and the definition of uq+1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='18) ∥uq+1∥Z2(q+1) ≤ C � ∥uq+1∥H0 + sup v∈H, ∥v∥(Z2q )∗ =1 |⟨Sv, uq⟩| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='25) By duality ∥S∥(Zj)∗→H0 ≤ C j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , ℓ + 1, and thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='26) sup v∈H, ∥v∥(Z2q )∗ =1 |⟨Sv, uq⟩| ≤ ∥S∥(Z2q)∗→H0 ∥uq∥H0 ≤ C ∥uq∥H0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='25), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='26), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='24), we find that ∥uq+1∥Z2(q+2) ≤ C � ∥uq+1∥H0 + ∥uq∥H0 � ≤ C ∥uq∥H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19) with m = q, we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19) with m = q + 1, and the induction is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' If ℓ is odd, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', ℓ + 1 is even, then this establishes both (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='20) since 2(⌊ℓ/2⌋ + 1) = ℓ + 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', the highest-order case is even, and can be reached by increasing the regularity at each step by two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' If ℓ is even, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', ℓ + 1 is odd, then the argument above establishes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The bound for u⌊ℓ/2⌋ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='20)) then follows from elliptic regularity, using that u⌊ℓ/2⌋−1 = uℓ/2−1 ∈ Zℓ ⊂ Zℓ−1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', at the last step, we only increase the regularity by one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' For the proof that rm ∈ Z2(m+1) and satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='22), observe that the definition of rm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='21) and the definition of um (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='18) implies that r0 = u and �a(v, rm) = ⟨Sv, rm−1⟩ for all v ∈ H, m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , ⌊ℓ/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='22) is then very similar to the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19), with the first step being that, by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8), the fact that u = R∗g, and the definition of R∗ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='13), ∥r0∥Z2 = ∥u∥Z2 ≤ C � ∥u∥H0 + ∥g∥H0 � ≤ C � 1 + ∥R∗∥H0→H � ∥g∥H0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6 using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' As in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2, given g ∈ H0, let u = R∗g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='21), ∥(I − Π)R∗g∥H ≤ ⌊ℓ/2⌋−1 � j=0 ∥(I − Π)∥Z2(j+1)→H ∥uj∥Z2(j+1) + ∥(I − Π)∥Zℓ+1 ��r⌊ℓ/2⌋ �� Zℓ+1 so that, by the bounds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='20), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='23), ∥(I − Π)R∗g∥H ≤ C � ⌊ℓ/2⌋−1 � j=0 ∥(I − Π)∥Z2(j+1)→H + ∥(I − Π)∥Zℓ+1→H � 1 + ∥R∗∥H0→H �� ∥g∥H0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' the result (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='20) then follows from the definition of η(Hh) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Elliptic-regularity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' This section collects the elliptic-regularity re- sults that are used to verify that Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 holds for Helmholtz problems with truncation of the exterior domain either by a PML (in §4) or an impedance boundary condition (in §5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Lu = −k−2∇ · (A∇u) − c−2u, with associated sesquilinear form a(u, v) = � Ω � k−2(A∇u) · ∇v − c−2u v � , where Ω be a bounded Lipschitz domain with outward-pointing unit normal vector n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The conormal derivative ∂n,Au is defined for u ∈ H2(Ω) by ∂n,Au := n · (A∇u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' recall that ∂n,Au can be defined for u ∈ H1(Ω) with Lu ∈ L2(Ω) by Green’s identity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', [34, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' For all x ∈ Ω, Ajℓ(x) = Aℓj(x) and ℜ d � j=1 d � ℓ=1 Ajℓ(x)ξkξj ≥ c|ξ|2 for all ξ ∈ Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 (Local elliptic regularity near a Dirichlet or Neumann boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Ω be a Lipschitz domain and let G1, G2 be open subsets of Rd with G1 ⋐ G2 and G1 ∩ ∂Ω ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1) Ωj := Gj ∩ Ω, j = 1, 2, and Γ2 := G2 ∩ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Suppose that A satisfies Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1, A, c ∈ Cm,1(Ω2), Γ2 ∈ Cm+1,1, u ∈ H1(Ω2), and Lu ∈ Hm(Ω2) for some m ∈ N, and either u = 0 or ∂n,Au = 0 on Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2) ∥u∥Hm+2 k (Ω1) ≤ C � ∥u∥H1 k(Ω2) + ∥Lu∥Hm k (Ω2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' In unweighted norms, this follows from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', [34, Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' the proof in the weighted norms (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='11) is very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3 (Local elliptic regularity for the transmission problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Ωin be a Lipschitz domain, and let Ωout := Rd \\ Ωin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let G1, G2 be open subsets of Rd with G1 ⋐ G2 and G1 ∩ ∂Ωin ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Ωin/out,j := Gj ∩ Ωin/out, j = 1, 2, and Γ2 := G2 ∩ ∂Ωin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Suppose that A satisfies Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1, A|Ωin/out,2, c|Ωin/out,2 ∈ Cm,1(Ωin/out,2), Γ2 ∈ Cm+1,1, uin/out ∈ H1(Ωin/out), and Lu ∈ Hm(Ωin/out,2) for some m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Suppose further that uin = uout and ∂n,Auin = β∂n,Auout on Γ2 for some β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Then ∥uin∥Hm+2 k (Ωin,1) + ∥uout∥Hm+2 k (Ωout,1) ≤ C � ∥uin∥H1 k(Ωin,2) + ∥uout∥H1 k(Ωout,2) + ∥Luin∥Hm k (Ωin,2) + ∥Luout∥Hm k (Ωout,2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3) 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' In unweighted norms, this is, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', [10, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1(i)] (and [34, The- orems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16] when β = 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' the proof in the weighted norms (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='11) is very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4 (Local elliptic regularity for the impedance problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Ω be a Lipschitz domain and let G1, G2 be open subsets of Rd with G1 ⋐ G2 and G1∩∂Ω ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Ωj and Γ2 be defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Suppose that, for some m ∈ N, Γ2 ∈ Cm+1,1, u ∈ H1(Ω2), and ∆u ∈ Hm(Ω2), and (k−1∂n − i)u = 0 on Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4) ∥u∥Hm+2 k (Ω1) ≤ C � ∥u∥H1 k(Ω2) + ��k−2∆u �� Hm k (Ω2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' When m = 0, the result can be obtained from [7, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1] by multiply- ing by k−2 to switch to weighted norms, and using that the trace operator has norm bounded by Ck1/2 from H1 k to L2 (which can be obtained from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', [39, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4] since the weighted norms there are, up to a constant, the weighted norms (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The proof that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4) follow for m > 0 is then standard and can be found e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' in [14, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2, Theorem 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We repeat it here in the context of impedance boundary conditions for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We now prove that if the bound holds for m = q, then it holds for m = q + 1 (assuming the appropriate regularity of the coefficients and the domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Without loss of generality, we can change coordinates and work with U := B(0, s) ∩ {xd > 0} and V := B(0, t) ∩ {xd > 0} for some 0 < t < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' In these coordinates Lu := (−k−2aij∂xi∂xj −k−2(bi∂xi−c))u = f, (−k−1∂xd−i)u = 0 on {xd = 0}∩U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Suppose that for some q ≥ 0, for any 0 < t < s, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5) ∥u∥Hq+2 k (V ) ≤ Ct � ∥u∥L2(U) + ∥f∥Hq k(U) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Now suppose that f ∈ Hq+1 k (U) and a, b, c ∈ Cq+1,1(U), and let W := B(0, r)∩{xd > 0} with t < r < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6) ∥u∥Hq+2 k (W) ≤ C � ∥u∥L2(U) + ∥f∥Hq k(U) � , and, by interior elliptic regularity, u ∈ Hq+3 loc (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The next step is to bound tangential derivatives of u: let |α| = q + 1 with αd = 0 (so that ∂α x is a tangential derivative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let �f := L � k−|α|∂α x u � so that �f = [L, k−|α|∂α x ]u + k−|α|∂α x f (where [A, B] := AB − BA) and, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6) and the fact that the coefficients of L are Cq+1,1(U), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7) ∥ �f∥L2(W) ≤ C � ∥u∥Hq+2(W) + ∥f∥Hq+1 k (W) � ≤ C � ∥u∥L2(U) + ∥f∥Hq+1 k (U) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Furthermore (−k−1∂xd − i)k−|α|∂α x u|xd=0 = k−|α|∂α x � (−k−1∂xd − iu)|xd=0 � = 0, so that, by the analogue of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5) with q = 0 and U replaced by W, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7), ��k−|α|∂α x u �� H2 k(V ) ≤ C ���k−|α|∂α x u �� L2(W) + �� �f �� L2(W) � ≤ C � ∥u∥L2(U) + ∥f∥Hq+1 k (U) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 15 Therefore, by the definition of α, ��k−|β|∂β xu �� L2(V ) ≤ C � ∥u∥L2(U) + ∥f∥Hq+1 k (U) � for all |β| = q + 3 with βd ∈ {0, 1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8) To prove that the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5) holds with q replaced by q + 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', ∥u∥Hq+3 k (V ) ≤ C � ∥u∥L2(U) + ∥f∥Hq+1 k (U) � , it is sufficient to prove that ��k−|β|∂β xu �� L2(V ) ≤ C � ∥u∥L2(U) + ∥f∥Hq+1 k (U) � for all |β| = q + 3 with βd ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , q + 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We therefore now prove by induction that if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9) ��k−|β|∂β xu �� L2(V ) ≤ C � ∥u∥L2(U) + ∥f∥Hq+1 k (U) � for any |β| = q + 3 with βd ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , j} for some j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , q + 2}, then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9) holds for |β| = q + 3 with βd = j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Combined with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8), this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We therefore assume that |β| = q + 3 with βd = j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Then, putting β = γ + δ with δ = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' , 0, 2) and |γ| = q + 1, and using that u ∈ Hq+3 loc (U), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='10) k−|γ|∂γLu = addk−|β|∂βu + Bu in V, where Bu = � |α|≤q+3, αd≤j aαk−|α|∂α x u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By the induction hypothesis (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9), ∥Bu∥L2(V ) ≤ C � ∥u∥L2(U) + ∥f∥Hq+1 k (U) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Dividing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='10) by add, taking the L2(V ) norm, and using that 1/add is bounded, we have ∥k−|β|∂βu∥L2(V ) ≤ C � ∥u∥L2(U) + ∥f∥Hq+1 k (U) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', we have proved that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9) holds for |β| = q + 3 with βd = j + 1, and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5 applied to the PML problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Definition of the PML problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Obstacles and coefficients for Dirichlet/Neumann/penetrable obstacle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Ωp, Ω− ⊂ BR0 := {x : |x| < R0} ⊂ Rd, d = 2, 3, be bounded open sets with Lipschitz boundaries, Γp and Γ−, respectively, such that Γp ∩ Γ− = ∅, and Rd\\Ω− is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Ωout := Rd\\Ω− ∪ Ωp and Ωin := (Rd\\Ω−) ∩ Ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Aout ∈ C0,1(Ωout, Rd×d) and Ain ∈ C0,1(Ωin, Rd×d) be symmetric positive definite, let cout ∈ L∞(Ωout;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' R), cin ∈ L∞(Ωin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' R) be strictly positive, and let Aout and cout be such that there exists Rscat > R0 > 0 such that Ω− ∪ supp(I − Aout) ∪ supp(1 − cout) ⋐ BRscat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 16 The obstacle Ω− is the impenetrable obstacle, on which we impose either a zero Dirichlet or a zero Neumann condition, and the obstacle Ωin is the penetrable obstacle, across whose boundary we impose transmission conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' For simplicity, we do not cover the case when Ω− is disconnected, with Dirichlet boundary conditions on some connected components and Neumann boundary con- ditions on others, but the main results hold for this problem too (at the cost of introducing more notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Definition of the radial PML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Rtr > RPML,− > Rscat and let Ωtr ⊂ Rd be a bounded Lipschitz open set with BRtr ⊂ Ωtr ⊂ BCRtr for some C > 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', Ωtr has characteristic length scale Rtr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Ω := Ωtr ∩ Ω+ and Γtr := ∂Ωtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' For 0 ≤ θ < π/2, let the PML scaling function fθ ∈ C3([0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' R) be defined by fθ(r) := f(r) tan θ for some f satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1) � f(r) = 0 � = � f ′(r) = 0 � = � r ≤ RPML,− � , f ′(r) ≥ 0, f(r) ≡ r on r ≥ RPML,+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', the scaling “turns on” at r = RPML,−, and is linear when r ≥ RPML,+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We emphasize that Rtr can be < RPML,+, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', we allow truncation before linear scaling is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Indeed, RPML,+ > RPML,− can be arbitrarily large and therefore, given any bounded interval [0, R] and any function �f ∈ C3([0, R]) satisfying � �f(r) = 0 � = � �f ′(r) = 0 � = � r ≤ RPML,− � , �f ′(r) ≥ 0, our results hold for an f with f|[0,R] = �f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Given fθ(r), let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2) α(r) := 1 + if ′ θ(r) and β(r) := 1 + ifθ(r)/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' and let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3) A := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Ain in Ωin, Aout in Ωout ∩ BRPML,−, HDHT in (BRPML,−)c and 1 c2 := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 c−2 in in Ωin, c−2 out in Ωout ∩ BRPML,−, α(r)β(r)d−1 in (BRPML,−)c, where, in polar coordinates, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4) D = � β(r)α(r)−1 0 0 α(r)β(r)−1 � and H = � cos θ − sin θ sin θ cos θ � for d = 2, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5) D = \uf8eb \uf8ed β(r)2α(r)−1 0 0 0 α(r) 0 0 0 α(r) \uf8f6 \uf8f8 and H = \uf8eb \uf8ed sin θ cos φ cos θ cos φ − sin φ sin θ sin φ cos θ sin φ cos φ cos θ − sin θ 0 \uf8f6 \uf8f8 for d = 3 (observe that then Aout = I and c−2 out = 1 when r = RPML,− and thus A and c−2 are continuous at r = RPML,−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We highlight that, in other papers on PMLs, the scaled variable, which in our case is r+ifθ(r), is often written as r(1+i�σ(r)) with �σ(r) = σ0 for r sufficiently large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', [24, §4], [4, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Therefore, to convert from our notation, set �σ(r) = fθ(r)/r and σ0 = tan θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6) H := H1 0(Ω) or {v ∈ H1(Ω) : v = 0 on Γtr}, 17 with the former corresponding to zero Dirichlet boundary conditions on Ω− and the latter corresponding to zero Neumann boundary conditions on Ω−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1 (A variational formulation of the PML problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Given G ∈ (H)∗ and β > 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7) find u ∈ H such that a(u, v) = G(v) for all v ∈ H, where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8) a(u, v) := �� Ω∩Ωout + 1 β � Ω∩Ωin � � k−2(A∇u) · ∇v − c−2uv � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' When (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9) G(v) := �� BRPML,− ∩Ωout + 1 β � Ω∩Ωin � c−2gv for g ∈ L2(Ω+) with supp g ⊂ BRPML,−, the variational problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7) is a weak form of the problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='10) k−2c2 out∇ · (Aout∇uout) + uout = −g in Ωout, k−2c2 in∇ · (Ain∇uin) + uin = −g in Ωin, uin = uout and ∂n,Ainuin = β∂n,Aoutuout on ∂Ωin, either uin = 0 or ∂n,Ainuin = 0 on ∂Ω−, and with the Sommerfeld radiation condition approximated by a radial PML ((4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7) is obtained by multiplying the PDEs above by c−2 in/outαβd−1 and integrating by parts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Using the fact that the solution of the true scattering problem exists and is unique with Aout, Ain, cout, cin, Ω−, and Ωin described above, the solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7) exists and is unique (i) for fixed k and sufficiently large Rtr − R1 by [30, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1], [31, Theorem A], [24, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8] and (ii) for fixed Rtr > R1 and sufficiently large k by [18, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' For the particular data G (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9), it is well-known that, for fixed k, the error ∥u−v∥H1 k(BRPML,− \\Ω) decays exponentially in Rtr−RPML,− and tan θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' see [30, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1], [31, Theorem A], [24, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' It was recently proved in [18, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5] that the error ∥u − v∥H1 k(BRPML,− \\Ω) also decreases exponentially in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Showing that the PML problem fits in the abstract framework used in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Recall that H is defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6) and let H0 = L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We work with the norm ∥ · ∥H1 k(Ω) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5) on H, and use below the higher-order norms (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='11) ∥v∥2 Hm k (Ω) := � 0≤|α|≤m k−2|α| ∥∂αv∥2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The rationale for using these norms is that if a function v oscillates with frequency k, then |(k−1∂)αv| ∼ |v| for all α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' this is true, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', if v(x) = exp(ikx · a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We highlight that many papers on the FEM applied to the Helmholtz equation use the weighted H1 norm |||v|||2 := ∥∇v∥2 L2(Ω)+k2 ∥v∥2 L2(Ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' we work with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5) instead, because weighting the jth derivative with k−j is easier to keep track of than weighting the jth derivative with k−j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We first check that the sesquilinear form a (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8) is continuous and satisfies a G˚arding inequality, with constants uniform for ǫ ≤ θ ≤ π/2 − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 18 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 (Bounds on the coefficients A and c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Given A and c as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3), a scaling function f(r) satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1), and ǫ > 0 there exist A+ and c− such that, for all ǫ ≤ θ ≤ π/2 − ǫ, x ∈ Ω, and ξ, ζ ∈ Cd, |(A(x)ξ, ζ)2| ≤ A+∥ξ∥2∥ζ∥2 and 1 |c(x)|2 ≥ 1 c2 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' This follows from the definitions of A and c in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3), the definitions of α and β in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2), and the fact that fθ(r) := f(r) tan θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Continuity of a (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6) with Ccont := max{A+, c−2 − } then follows from the Cauchy- Schwarz inequality and the definition of ∥ · ∥H1 k(Ω) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' When d = 3, fθ(r)/r is nondecreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3 is standard in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', in the alternative notation described above it is that �σ is non-decreasing – see [4, §2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' As noted above, the variational problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7) is obtained by multi- plying the PDEs in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='10) by c−2 in/outαβd−1 and integrating by parts (as in [9, §3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' If one integrates by parts the PDEs directly (as in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', [24, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 and Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8]), the resulting sesquilinear form satisfies Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 after multiplication by eiω, for some suitable ω (see Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3), without the need for Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Suppose that fθ satisfies Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' With A defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3), given ǫ > 0 there exists A− > 0 such that, for all ǫ ≤ θ ≤ π/2 − ǫ, ℜ � A(x)ξ, ξ � 2 ≥ A−∥ξ∥2 2 for all ξ ∈ Cd and x ∈ Ω+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Reference for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' See, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', [20, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' If fθ satisfies Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3 then ℜa(w, w) ≥ A−∥w∥2 H1 k(Ω) − � A− + c−2 min � ∥w∥2 L2(Ω) for all w ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let R : L2(Ω) → H be defined by a(Rg, v) = (g, v)L2(Ω) for all v ∈ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', R is the solution operator of the PML problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The definition of a and the facts that (with the matrices H and D defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5)) H is real and the matrix D is diagonal (and hence symmetric) imply that a(u, v) = a(v, u) for all u, v ∈ H, and thus Rg = R∗g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We therefore let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='12) Csol := ∥R∥L2(Ω)→H = ∥R∗∥L2(Ω)→H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' We highlight that (i) Csol is bounded by the norm of the solution operator of the true scattering problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', with the Sommerfeld radiation condition) by [18, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6], (ii) Csol ∼ k when the problem is nontrapping (with this the slowest-possible growth in k), and (iii) an advantage of working with the weighted norms (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='11) is that Csol in fact describes the k-dependence of the Helmholtz solution operator between Hm k and Hm+2 k for any m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7 (The PML problem satisfies Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Suppose that, for some ℓ ∈ Z+, Aout, Ain, cout, cin ∈ Cℓ−1,1 and fθ ∈ Cℓ,1 on the closures of the domains on which they are defined, ∂Ω is Cℓ,1, and fθ satisfies Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='13) Zj = � v : vout ∈ Hj(Ω ∩ Ωout), vin ∈ Hj(Ωin) � ∩ H 19 with norm (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='14) ∥v∥2 Zj := ∥vout∥2 Hj k(Ωout∩Ω) + ∥vin∥2 Hj k(Ωin) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' where the “out” and “in” subscripts denote restriction to Ωout∩Ω and Ωin, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Then a defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8) satisfies Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 and given ǫ > 0 and k0 > 0 there exists C > 0 such the bounds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9) hold for all k ≥ k0 and ǫ ≤ θ ≤ π/2 − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' First observe that Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1 is satisfied by the definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3) of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Since sup v∈H, ∥v∥(Zj−2 )∗=1 |a(u, v)| = ∥Lu∥Zj−2 , the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9) holds by combining Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 (used near Γ− and Γtr) and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3 (used near Γp) and using the fact that, by Green’s identity, for u ∈ H1 0(Ω) with Lu ∈ L2(Ω) and ∂n,Ainuin = β∂n,Aoutuout on ∂Ωin, ∥uin∥H1 k(Ωin) + ∥uout∥H1 k(Ωout) ≤ C � ∥uin∥L2(Ωin) + ∥uout∥L2(Ωout) + ∥Luin∥L2(Ωin) + ∥Luout∥L2(Ωout) � (so that the H1 k norms on the right-hand sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3) can be replaced by L2 norms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Since the operator associated with the sesquilinear form ℜa is �L + L∗ 2 � u = −k−2∇ · �A + A 2 ∇u � − �c−2 + c−2 2 � u and the matrix A is symmetric, this operator also satisfies Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8) then holds by a very similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5 applied to the PML problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Given p ∈ Z+, (Hh)h>0 are such that the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' There exists C > 0 such that, for all h > 0, 0 ≤ j ≤ m+1 ≤ p+1, and v ∈ H∩Hℓ+1(Ω) there exists Ih,pv ∈ Hh such that ��vout − (Ih,pv)out �� Hj(Ωout∩Ω) + ��vin − (Ih,pv)in �� Hj(Ωin) ≤ Chm+1−j� ∥vout∥Hm+1(Ωout∩Ω) + ∥vin∥Hm+1(Ωin) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='15) where the “out” and “in” subscripts denote restriction to Ωout∩Ω and Ωin, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8 holds when (Hh)h>0 consists of piecewise degree-p polynomials on shape-regular simplicial triangulations, indexed by the meshwidth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', [8, Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1], [5, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9 (Existence, uniqueness, and error bound in the preasymptotic regime for the PML problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Suppose that, for some ℓ ∈ Z+, Aout, Ain, cout, cin ∈ Cℓ−1,1 and fθ ∈ Cℓ,1 on the closures of the domains where they are defined, ∂Ω is Cℓ,1, fθ satisfies Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3, and β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Csol be defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='12), and as- sume that {Hh}h>0 satisfy Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Given ǫ > 0 and p ∈ Z+ with p ≥ ℓ, there exists k0 > 0 and Cj, j = 1, 2, 3, 4, such that the following is true for all k ≥ k0, ǫ ≤ θ ≤ π/2 − ǫ, and Rtr > R1 + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The solution u of the PML problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7) exists and is unique, and if (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) (hk)2ℓCsol ≤ C1 20 then the Galerkin solution uh, exists, is unique, and satisfies ∥u − uh∥H1 k(Ω) ≤ C2 � 1 + hk + (hk)ℓCsol � min wh∈Hh ∥u − vh∥H1 k(Ω) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17) ∥u − uh∥L2(Ω) ≤ C3 � hk + (hk)ℓCsol � min wh∈Hh ∥u − vh∥H1 k(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='18) If, in addition, g ∈ Hp−1(Ω) ∩ H (with H defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6)) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19) ∥g∥Hp−1 k (Ω) ≤ C ∥g∥H∗ for some C > 0, then there exists C4 > 0 such that if h satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='20) ∥u − uh∥H1 k(Ω) ∥u∥H1 k(Ω) ≤ C4 � hk + (hk)ℓCsol � (hk)ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9 is most interesting when p = ℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', the polynomial degree is the smallest possible covered by the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' In this case, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) becomes the condi- tion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1), and the bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='18), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='20) become (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3), and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By the results in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2, a defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8) satisfies the as- sumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='15), the definition of ∥·∥Zj (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='14), and the definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='11) of the weighted norms, ∥I − Π∥Zm+1→H ≤ C(hk)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' This bound along with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='6 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='12) imply that η(Hh) ≤ C � ⌊ℓ/2⌋−1 � j=0 (hk)2j+1 + (hk)ℓCsol � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' If hk ≤ C, then η(Hh) ≤ C(hk + (hk)ℓCsol);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' the result then follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5 and the fact that if the condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) holds, then hk ≤ C (since Csol ≥ Ck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5 applied to the impedance problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Definition of the impedance problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Aout, Ain, cout, cin, Ω−, Ωin, and Ωtr be as in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let A := � Ain in Ωin, Aout in Ωout ∩ Ω, and 1 c2 := � c−2 in in Ωin, c−2 out in Ωout ∩ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1) H := {v ∈ H1(Ω) : v = 0 on ∂Ω−} or H1(Ω), with the former corresponding to zero Dirichlet boundary conditions on Ω− and the latter corresponding to zero Neumann boundary conditions on Ω−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1 (Variational formulation of the impedance problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Given G ∈ (H)∗ and β > 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2) find u ∈ H such that a(u, v) = G(v) for all v ∈ H, where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3) a(u, v) := �� Ω∩Ωout + 1 β � Ω∩Ωin � � k−2(A∇u) · ∇v − c−2uv � − ik−1 � Γtr uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The solution of this variational problem exists and is unique by, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', [22, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Showing that the impedance problem fits in the abstract frame- work used in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The proofs that the sesquilinear form a is continuous and satisfies a G˚arding inequality are very similar to those for the PML problem in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 (in fact, they are simpler because there is no PML scaling parameter in which the bounds need to be uniform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 (The impedance problem satisfies Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Suppose that, for some ℓ ∈ Z+, Aout, Ain, cout, cin ∈ Cℓ−1,1 on the closures of the domains on which they are defined, and ∂Ω is Cℓ,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' With Zj and its norm defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='14), a defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3) satisfies Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 and given k0 > 0 there exists C > 0 such the bounds (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9) hold for all k ≥ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' This is very similar to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The regularity assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8) follows by combining Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 used near ∂Ω−, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3 used near ∂Ωin, and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4 used near Γtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The regularity assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9) follows by combining Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 used near ∂Ω−, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3 used near ∂Ωin, and now Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 (with Neumann boundary condition) used near Γtr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Indeed, near Γtr, the operator associated with (ℜa) is −k−2∆−1 with Neumann boundary conditions (coming from Aout = I and cout = 1 near Γtr and the fact that no boundary condition is imposed on Γtr in H (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5 applied to the impedance problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3 (Existence, uniqueness, and error bound in the preasymp- totic regime for the impedance problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Suppose that, for some ℓ ∈ Z+, Aout, Ain, cout, cin ∈ Cℓ−1,1 on the closures of the domains where they are defined, ∂Ω is Cℓ,1, and β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Let Csol be defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='12), and assume that {Hh}h>0 satisfy Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Given p ∈ Z+ with p ≥ ℓ, there exists k0 > 0 and Cj, j = 1, 2, 3, 4, such that the following is true for all k ≥ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The solution u of the impedance problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2) exists and is unique, and if (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) holds then the Galerkin solution uh, exists, is unique, and satisfies the bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='17) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' If, in addition, g ∈ Hp−1(Ω) ∩ H (with H defined by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1)) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='19) for some C > 0, then there exists C4 > 0 such that if h satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='16) then the bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='20) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Given Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2, the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3 is very similar to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4 (Imposing the exact Dirichlet-to-Neumann map on Γtr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' With the exact Dirichlet-to-Neumann map imposed on Γtr, the Helmholtz sesquilinear form is continuous and satisfies a G˚arding inequality (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', [37, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' To apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='5 to this problem, one therefore only needs to check the elliptic-regularity assumptions of Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Using Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='3, this boils down to knowing the analogue of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4 with the impedance boundary condition replaced by k−1∂nu = DtNu (for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='8)) and also k−1∂nu = (DtN+DtN∗)u/2 (for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' When m = 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', the lowest-order regularity shift covered in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4), the first of these regularity results is given by [28, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' To prove this result for m > 1 one would need to make an argument similar to that in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='4 except that, because DtN and DtN∗ do not commute with tangential derivatives, one would need to obtain two additional estimates: 1) estimates on u with nontrivial boundary data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', when k−1∂nu − (DtN)u = g ∈ Hs k and 2) trace estimates for u that are needed to bound, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=', [T, DtN]u where T is a vector field tangent to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' The same strategy could also be used to handle higher-order 22 impedance boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' EAS was supported by EPSRC grant EP/R005591/1 and JG was supported by EPSRC grants EP/V001760/1 and EP/V051636/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' REFERENCES [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E1T4oBgHgl3EQf-wbm/content/2301.03574v1.pdf'} +page_content=' 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b/I9AyT4oBgHgl3EQffvhT/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a588906877a180ebdfc01e80892b7d1ee7831f328500aea77b41d6a5bb1d2312 +size 5505069 diff --git a/IdE2T4oBgHgl3EQfUQce/content/tmp_files/2301.03810v1.pdf.txt b/IdE2T4oBgHgl3EQfUQce/content/tmp_files/2301.03810v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4d82d58306f3050233039c3d31502300cec948b --- /dev/null +++ b/IdE2T4oBgHgl3EQfUQce/content/tmp_files/2301.03810v1.pdf.txt @@ -0,0 +1,5549 @@ +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +Algebraic approach and exact solutions of superintegrable +systems in 2D Darboux spaces +Ian Marquette ∗, Junze Zhang †and Yao-Zhong Zhang ‡ +School of Mathematics and Physics, The University of Queensland +Brisbane, QLD 4072, Australia +January 11, 2023 +Abstract +Superintegrable systems in 2D Darboux spaces were classified and it was found that there exist 12 +distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry +algebras generated by the integrals) in the Darboux spaces. In this paper, we obtain exact solutions via +purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four +different 2D Darboux spaces. This is achieved by constructing the deformed oscillator realization and +finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated +by quadratic integrals respectively for each of the 12 superintegrable systems. We also introduce generic +cubic and quintic algebras, generated respectively by linear and quadratic integrals and linear and +cubic integrals, and obtain their Casimir operators and deformed oscillator realizations. As examples +of applications, we present three classes of new superintegrable systems with cubic symmetry algebras +in 2D Darboux spaces. +1 +Introduction +Superintegrable systems of different orders have been attracting a large amount of international research +activities, see e.g. [1], [2], [3], [4], [5], [6] , [7], [8] and [9]. This paper is a contribution to the underlying +algebraic structures and exact solutions of superintegrable systems in 2-dimensional (2D) curved spaces. +Superintegrable systems in 2D spaces with constant or non-constant curvatures have been widely +studied by means of separation of variables and St¨ackel transforms [10], [11], [12], [13], [14] and [15]. The +St¨ackel transforms have been widely studied [16], [14], [17] provide useful tools in the classification of 2D +superintegrable systems. Through the method of the so-called coupling constant metamorphosis, St¨ackel +transforms [18] enable one to establish the relationship between different superintegrable systems: they +provide equivalence classes at the level of integrable and superintegrable Hamiltonians. However, even if +such Hamiltonians are connected via the St¨ackel transformations, they are distinct as Sturm-Liouville and +spectral problem, and their exact solvability (with possibly different boundary conditions) and algebraic +solutions need to be investigated separately. For a given superintegrable Hamiltonian which is separable +in various coordinates, its solvability would in general depend on the coordinates used in the separation +of variables, e.g. it is exactly solvable in one coordinate system but only quasi-exactly solvable in another +coordinate system. +It is well known that symmetry algebra structures play an important role in the analytic analysis of +physical systems. In the context of superintegrable models, the underlying symmetry algebra structures +are usually polynomial algebras such as quadratic and cubic algebras. In [19], [2], [10], rank-1 quadratic +algebra structures underlying certain 2D superintegrable systems, generated by integrals of motion of the +∗i.marquette@uq.edu.au +†junze.zhang@uqconnect.edu.au +‡yzz@maths.uq.edu.au +1 +arXiv:2301.03810v1 [nlin.SI] 10 Jan 2023 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +systems, were exploited. The authors in these references obtained the Casimir operator and deformed +oscillator algebra realization of a generic quadratic algebra, and applied the relates to study the energy +spectrum of the superintegrable systems. In [14] and [11], examples of rank-1 cubic and quintic algebras +in Darboux spaces were given. Higher order or higher rank polynomial algebras generated by integrals +and their deformed oscillator algebra realizations were studied in [5], [20], [21], [22], [23], [24] , [25], [26], +[27] and [28]. More recently, by extending the Wigner-In¨on¨u method of Lie algebra contraction, the +authors in [29], [30] showed that quadratic algebras from certain second-order superintegrable systems in +2D spaces are contractions of those with general 3-parameter potentials on S2. +Superintegrable systems in 2D Darboux spaces were classified in [14][15]. In 2 dimensions, there exist +4 possible Darboux spaces with metrics given by [31] +I. +d1(x, y) = (x + y) dxdy +II. +d2(x, y) = +� +Ω +(x − y)2 + Λ +� +dxdy +III. +d3(x, y) = +� +Ω exp +� +−x + y +2 +� ++ Λ exp(−x − y) +� +dxdy +IV. +d4(x, y) = +Ω +� +exp +� +x−y +2 +� ++ exp +� +y−x +2 +�� ++ Λ +exp +� +x−y +2 ++ exp +� +y−x +2 +��2 +dxdy +Here Ω, Λ ∈ R are constants. According to the classification in [14][15], there exist 12 distinct classes of +superintegrable systems with non-trivial potentials in the 2D Darboux spaces. In each case, quadratic +integrals of motion of the system were determined and were found to form a quadratic algebra. The +wave functions and energy spectra of the systems were obtained by means of separation of variables. +Superintegrable systems in Darboux spaces were also studied in [11] [32] [33]. It was shown there that +free superintegrable systems (i.e. systems without potentials) in 2D and 3D flat conformal spaces are +equivalent to systems in 2D and 3D Darboux spaces, respectively. However, as far as we know, finite +dimensional representations of the polynomial algebras and algebraic derivations of the energy spectrum +of the superintegrable systems have remained an open problem. +In this paper we present a genuine algebraic approach to superintegrable systems in the 2D Darboux +spaces. The purpose of this paper is twofold. One is to give algebraic solutions to the existing 12 distinct +classes of superintegrable systems in the four 2D Darboux spaces. This is achieved by constructing the +finite dimensional irreducible representation of the quadratic algebras underlying the 12 superintegrable +systems via the deformed oscillator algebra techniques in [1] and [5]. As one will see, energy spectrum for +superintegrable systems in Darboux spaces are often determined by very complicated algebraic equations +whose analytic and closed-form solutions can only be obtained by restricting the model parameter spaces. +The second purpose is to investigate superintegrable systems in 2D Darboux spaces with linear, quadratic +or cubic integrals of motion. It was found in [11] that the free systems with linear and quadratic integrals +in 2D Darboux spaces have cubic algebras as their underlying symmetry algebras. We will introduce +generic cubic and quintic algebras, generated by linear and quadratic integrals and linear and cubic +integrals, respectively, and construct their Casimir operators and deformed oscillator algebra realizations. +We also present three classes of new superintegrable systems with non-trivial potentials in 2D Darboux +spaces which have cubic algebras as their symmetry algebras. These superintegrable systems do not seem +to belong to the families classified in [14][15] for systems with quadratic integrals in 2D Darboux spaces. +This paper is organised as follows. +In Section 2, we obtain the Casimir operators, the deformed +oscillator algebra realizations and finite-dimensional irreducible representations for the quadratic algebras +generated by the quadratic integrals of motion of the 12 superintegrable systems in 2D Darboux spaces. +This enables us to give an algebraic derivation for the energy spectra of all the 12 classes of superintegrable +systems. In Section 3, we introduce generic cubic and quintic algebras generated by linear and higher order +integrals of motion. We construct their Casimir operators and deformed oscillator algebra realizations. +We also present examples of new superintegrable systems with linear and quadratic integrals in the 2D +Darboux spaces. In Section 4, we provide a summary of the main results of our work. +2 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +2 +Solutions of the 12 distinct classes of superintegrable systems in 2D +Darboux spaces +Consider a superintegrable system in a 2D Darboux space with coordinates (x, y) and metric gij(x, y). +The Hamiltonian of the system with potential V (x, y) is given by +ˆH = +2 +� +j,k=1 +1 +� +det(gjk) +∂ +∂xk +�� +det(gjk)gjk +∂ +∂xk +� ++ V (x, y). +Let ˆX be an integral of motion (aka, constant of motion) of the system which commute with the Hamil- +tonian, i.e. [ ˆX, ˆH] = 0. An integral of motion is said to be a polynomial in momenta of degree p, denoted +by deg ˆX = p, if it has the form +ˆX = +p +� +j=0 +rj(x, y) ∂p−j +x +∂j +y + s(x, y), +where rj(x, y), s(x, y) are smooth functions in the coordinates x, y. In particular, integrals of motion +of degree 1, 2 or 3 are usually called linear, quadratic or cubic integrals, respectively. Note that the +Hamiltonian has degree 2, i.e. deg ˆH = 2. +As mentioned in the Introduction, superintegrable systems in the four 2D Darboux spaces with +quadratic integrals of motion were classified in [14][15], and 12 distinct classes of potentials which pre- +serve superintegrability were found. In this section, we present algebraic solutions to all the 12 existing +superintegrable systems. +Note that in the following we will use the so-called separable coordinates in [14][15] for each case. As +indicated in [14][15], in such coordinates the parameters Ω, Λ in the metrics of the Darboux spaces can +be conveniently absorbed into the model parameters of the systems by redefinition so that they do not +appear explicitly in the expressions of Hamiltonians and integrals. +2.1 +Darboux Space I +According to the classification in [14][15], in the Darboux space I, there are two possible superintegrable +systems with potentials given by +V1(x, y) = b1(4x2 + y2) +4x ++ b2 +x + b3 +xy2 , +V2(x, y) = a1 +x + a2y +x + a3(x2 + y2) +x +, +respectively, where bi, ai are real constants. +2.1.1 +Potential V1(x, y) +For superintegrable system in Darboux space I with the Hamiltonian ˆH = +1 +4x +� +∂2 +x + ∂2 +y +� ++ V1(x, y) asso- +ciated to V1, the constants of motion are given by [15] +A = ∂2 +y + 4b3 +y2 + b1y2, +B = y∂y∂x − x∂2 +y + ∂x +2 − y2 +4x +� +∂2 +x + ∂2 +y +� ++ b1y4 +4x + b2y2 +x ++ b3(4x2 + y2) +y2x +. +These integrals satisfy the following quadratic algebra relations +[A, B] = C, +[A, C] = −8 ˆHA − 16b1B, +[B, C] = 6A2 + 8 ˆHB + 16b2A − 2b1(3 + 16b3). +3 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +This is the symmetry algebra of the superintegrable system The Casimir of this algebra is given by +K1 = C2 − 4A3 + 8 ˆH{A, B} − 16b2A2 − 16b1B2 + 4b1(11 + 16b3)A. +We can show that with the differential realization of A, B the Casimir K1 has the following form in terms +of the Hamiltonian ˆH, +K1 = −4(3 + 16b3) ˆH2 + 16b1b2(3 + 16b3). +In order to obtain the energy spectrum of the system via algebraic means, we now construct realization +of the quadratic algebra in terms of the deformed oscillator algebra of the form +[N, b†] = b†, +[N, b] = −b, +bb† = Φ(N + 1), +b†b = Φ(N), +(1) +where N is the number operator and Φ(z) is a well-defined real function satisfying +Φ(0) = 0, +Φ(z) > 0, ∀z > 0. +(2) +Φ(x) is called the structure function of the deformed oscillator algebra. +It is non-trivial to obtain such a realization and the corresponding structure function Φ(z). After a +long computation, we find in the present case that +A = 4 +� +−b1 (N + η), +B = +2 ˆH +√−b1 +(N + η) + b† + b +map the quadratic algebra to the deformed oscillator algebra with structure function given by +Φ(I) +1 (N, η) = − +1 +16b1 +� +−4(N + η)16b1b2 + (−b1)3/2(16b3 + 11) − 4 ˆH2 + 2b3/2 +1 +(16b3 + 3) ++64 +� +−b1b1(N + η)3 + 16(N + η)2(4b1b2 − ˆH2) − (16b3 + 3)(2b1 +� +−b1 − 4b1b2 + ˆH2) +� +. +Here η is a constant to be determined from the constraints on the structure function Φ. +We now obtain the finite-dimensional unitary irreducible representations (unirreps) of the deformed +oscillator algebra in the Fock space. Let |z, E⟩, denote the Fock basis states labelled by the eigenvalues +z and E of N and ˆH, respectively. Acting the structure function on the Fock states, we find that it is +factorized to the following form +ΦI +1(z, η) = +� +z + η − 1 +4 +� +2 − +� +1 − 16b3 +�� � +z + η − 1 +4 +� +2 + +� +1 − 16b3 +�� +� +z + η + 2b1 +�√−b1 − 2b2 +� + E2 +4(−b1)3/2 +� +. +For the unirreps to be finite dimensional, we impose the following constraints on the structure function, +Φ(0, η) = 0, +Φ(p + 1, η) = 0, +(3) +where p is a positive integer, p = 0, 1, 2, · · · . These constraints give (p+1)-dimensional unirreps in the Fock +space and their solutions give the constant η and energy spectrum E of the underlying superintegrable +system. +There are two sets of solutions from the constraints on the structure function: +η = 1 +4 +� +2 + ϵ√1 − 16a2 +� +, +Eim = ±2 +√ +−1 (−b1)3/4 +� +p + 1 − ϵ +4 +� +1 − 16b3 + +b2 +√−b1 +, +4 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +and +η = −2b1 +�√−b1 − 2b2 +� + E2 +4(−b1)3/2 +, +Eϵ = ±2(−b1)3/4 +� +p + 1 + ϵ +4 +� +1 − 16b3 − +b2 +√−b1 +, +where ϵ = ±1. The first set of solutions give complex energies which are not physical and thus will be +discarded. So the energy spectrum of the system is given by the second set of solutions which are real +for ϵ = +1, b1 < 0, b2 ≤ 0, b3 < 1/16. The structure function for the corresponding (p + 1)-dimensional +unirreps is +Φ(I) +E+(z) = z(z − p − 1) +� +z − 2b1 +�√−b1 − 2b2 +� + E2 ++ +4(−b1)3/2 +− 1 +4 +� +2 + +� +1 − 16b3 +�� +. +In the following subsections, we would only give the values of parameter η which can lead to real +energies E. +2.1.2 +Potential V2(x, y) +Constants of motion for the superintegrable system in Darboux space I with the Hamiltonian ˆH = +1 +4x +� +∂2 +x + ∂2 +y +� ++ V2(x, y) corresponding to the potential V2 are given by [15] +A = y∂y∂x − x∂2 +y + ∂x +2 − y2 +4x +� +∂2 +x + ∂2 +y +� +− 2a2y +x ++ 2a2(x2 − y2) +x ++ 2a2y(x2 − y2) +x +, +B = ∂2 +y + 4a2y + 4a3y2. +They satisfy the following quadratic algebra relations +[A, B] = C, +[A, C] = 16a2 ˆH − 16a3B, +[B, C] = 16a3A + 8(a2 +2 + 4a1a3) − 8 ˆH2. +The Casimir operator of the algebra is given by +K2 = C2 + 16a3A2 + 16a3B2 − 32a2 ˆHB + 16 +� +(a2 +2 + 4a1a3) − ˆH2� +A, +which in terms of the differential realization of A, B takes the constant value K2 = 64(a2 +3 − a1a2 +2). +We then determine the realization of above quadratic algebra in terms of the deformed oscillator +algebra (1) and apply its finite dimensional unirreps to obtain the energy spectrum of the system. After +computations, we find that +A = 4√−a3(N + η), +B = a2 ˆH +a3 ++ b† + b. +transform the quadratic algebra to the deformed oscillator algebra with structure function +Φ(I) +2 (N, η) = +� +4a1a3 + a2 +2 − ˆH2�2 +16a2 +3 +− a1a2 +2 +a3 +− 1 +12(N + η) +� +24a2 ˆH +√−a3 ++ 48a3 +� ++ a2 ˆH +√−a3 ++ 4a3(N + η)2 + a3. +Moreover, the action of this structure function on the Fock states |z, E⟩ is factorized as +Φ(I) +2 (z, η) = +� +z + η − m+(E) + 2a3 +4a3 +� � +z + η − m−(E) + 2a3 +4a3 +� +, +5 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +where η is a constant to be determined and +m±(E) = a2E +√−a3 +± +� +64a2 +1a2 +3 + 4a1 +�a2 +2(8a3 − 1) − 8a3E2� + 4a4 +2 − a2 +2(8a3 + 1)E2 +a3 ++ 4E4. +We now impose the constraints (3) to obtain finite-dimensional unirreps of the algebra. We find that +for p = 0, 1, 2, · · · , we have +Case 1: η−(E) = +1 +4a3 (m−(E) + 2a3) and +� +64a2 +1a2 +3 + 4a1 +�a2 +2(8a3 − 1) − 8a3E2� + 4a4 +2 − a2 +2(8a3 + 1)E2 +a3 ++ 4E4 = 2a3(p + 1), +(4) +which has solutions only for a3 > 0 and the energy spectrum of the system is given by +E+a3 = ± +1 +√8a3 +�� +128a1a2 +2a2 +3 + a4 +2(16a3 + 1) + 64a4 +3(p + 1)2 + 32a1a2 +3 + a2 +2(8a3 + 1), +Notice that E+a3 is real for a1 > 0, a3 > 0. +Case 2 η+(E) = +1 +4a3 (m+(E) + 2a3) and +� +64a2 +1a2 +3 + 4a1 +�a2 +2(8a3 − 1) − 8a3E2� + 4a4 +2 − a2 +2(8a3 + 1)E2 +a3 ++ 4E4 = −2a3(p + 1), +which has solutions only for a3 < 0 and the energies of the system are +E−a3 = ± +1 +√−8a3 +�� +128a1a2 +2a2 +3 + a4 +2(16a3 + 1) + 64a4 +3(p + 1)2 − 32a1a2 +3 − a2 +2(8a3 + 1). +(5) +Obviously for a3 < 0 there exist ranges of model parameters a1, a2 such that the eneries E−a3 of the +system are real. +The structure function for both cases 1 and 2 corresponding to the (p + 1)-dimensional unirreps of +the algebra is given by Φ(I) +E±a3(z) = z(z − p − 1). +2.2 +Darboux Space II +In the Darboux space II, there are three superintegrable systems with potentials given by [14] +V1(x, y) = +x2 +x2 + 1 +� +a1 +� +x2 +4 + y2 +� ++ a2y + a3 +x2 +� +, +V2(x, y) = +x2 +x2 + 1 +� +b1(x2 + y2) + b2 +x2 + b3 +y2 +� +V3(x, y) = +c1 + c2 +x2 + c3 +y2 +x2 + y2 + 1 +x2 + 1 +y2 +, +respectively, where aj, bj, cj are real constants. +2.2.1 +Potential V1(x, y) +The constants of motion of the superintegrable system in Darboux space II with the Hamiltonian ˆH = +x2 +x2+1 +� +∂2 +x + ∂2 +y +� ++ V1(x, y) associated to the potential V1 are +A = ∂2 +y + a1y2 + a2y, +B = +2y +x2 + 1 +� +∂2 +y − x2∂2 +x +� ++ 2x∂x∂y + ∂y + a1 +2 y +� +x2 + x2 + 4y2 +x2 + 1 +� ++ a2 +2 +� +x2 + +4y2 +x2 + 1 +� +− 2a3y +x2 + 1. +6 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +They satisfy the following quadratic algebra relations [14] +[A, B] = C, +[A, C] = −4a1B − 4a2A, +[B, C] = −24A2 + 4a2B + 32 ˆHA − 8 ˆH2 − 8a1 ˆH + 6a1 + 8a1a3. +Its Casimir operator can be shown to be given by +K1 = C2 − 16A3 + 4a1B2 + 4a2{A, B} + +� +4a1(4a3 − 11) − (16a1 ˆH + 16 ˆH2) +� +A + 32 ˆHA2. +In term of the differential realization of A, B, the Casimir K1 takes the simple form K1 = (32a1+4a2 +2) ˆH− +a2 +2(3 + 4a3). +By computations similar to those in the previous subsection, we find that +A = 2√−a1(N + η), +B = +2a2 +√−a1 +(N + η) + a2 ˆH +a1 ++ b† + b +map the quadratic algebra to the deformed oscillator algebra (1) with structure function given by +Φ(II) +1 +(N, η) = − 12a3 +1 − 3√−a1a1a2 +2 − 16a3 +1a3 − 4√−a1a1a2 +2a3 + 32√−a1a2 +1 ˆH + 16a3 +1 ˆH +− 8a1a2 +2 ˆH + 4√−a1a1a2 +2 ˆH + 16a2 +1H2 + 4√−a1a2 +2H2 ++ (N + η) +� +88a3 +1 + 16√−a1a1a2 +2 + 32a3 +1a3 − 128√−a1a2 +1 ˆH − 32a3 +1 ˆH + 16a1a2 +2 ˆH − 32a2 +1 ˆH2� ++ (N + η)2 � +−192a3 +1 − 16√−a1a1a2 +2 + 128√−a1a2 +1 ˆH +� ++ 128a3 +1(N + η)3. +Here η is a constant to be determined from the constraints of the structure function. Acting on the Fock +basis states |z, E⟩, the structure function Φ(II) +1 +becomes factorized +Φ(II) +1 +(z, η) = +� +z + η − f1(E) + ω(E) + f2(E) +24a3 +1 +� +� +z + η − +1 +96a3 +1 +� +4f1(E) − 2 +� +1 − i +√ +3 +� +ω(E) + +� +1 + i +√ +3 +� +f2(E) +�� +� +z + η − +1 +96a3 +1 +� +4f1(E) − 2 +� +1 + i +√ +3 +� +ω(E) + +� +1 − i +√ +3 +� +f2(E) +�� +, +where +f1(E) =a1 +� +12a2 +1 + √−a1a2 +2 + 8(−a1)3/2E +� +, +f2(E) = +1 +ω(E) +� +a3 +1(12a3 +1(4E − 4a3 + 1) − a4 +2 − 16a2 +1E2 − 8a1a2 +2E) +� +, +ω(E) = 3� +τ1(E) + τ2(E), +τ1(E) =6a6 +1 +� +a4 +2 + 8a1Ea2 +2 + 16a2 +1E2 + a3 +1(−16a3 + 16E + 4) +� � +3(4a3 − 4E − 1), +τ2(E) =a1a6 +2(−a1)7/2 + 12a4 +2E(−a1)11/2 − 48a2 +2E2(−a1)13/2 +− 4 +� +9a2 +2(4a3 − 4E − 1) − 16E3� +(−a1)15/2 − 144E(−4a3 + 4E + 1)(−a1)17/2. +To determine the constant η and energy spectrum E of the superintegrable system, we impose the +constraints (3) which give (p + 1)-dimensional unirreps of the algebra. We find +Case 1: The constant η is given by +η1(E) = f1(E) + ω(E) + f2(E) +24a3 +1 +7 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +and the energy E satisfies the algebraic equation, +ω(E) + f2(E) + 1 +2 +� +1 − ϵ i +√ +3 +� +ω(E) − 1 +4 +� +1 + ϵ i +√ +3 +� +f2(E) = −24(p + 1)a3 +1. +(6) +Case 2: +η2(E) = +1 +96a3 +1 +� +4f1(E) − 2 +� +1 − ϵ i +√ +3 +� +ω(E) + +� +1 + ϵ i +√ +3 +� +f2(E) +� +and the energy is determined by +ω(E) + f2(E) + 1 +2 +� +1 − ϵ i +√ +3 +� +ω(E) − 1 +4 +� +1 + ϵ +√ +3i +� +f2(E) = 24(p + 1)a3 +1. +(7) +In both cases above, ϵ = ±1. +The energy spectrum E of the system are obtained by solving the algebraic equations (6) and (7). +However, it is in general very difficult to obtain analytical solutions of these equations, due to their +complicated form. To demonstrate that these equations have real solutions, we have a closer look at +restricted model parameter spaces. Without the loss of generality, we consider the case where −a1 = +a2 = a3 = a for any a ∈ R. For such model parameters, the structure function has the simple form, +Φ(II) +1 +(z, η) = +� +z + η − 1 +8 +�√a + 4 +�� � +z + η − 1 +4a +� +2√aE + 2a − a +√ +4E + 1 − 4a +�� +� +z + η − 1 +4a +� +2√aE + 2a + a +√ +4E + 1 − 4a +�� +. +Imposing the constraints (3) on the structure function lead to the determination of constant η and energy +E of the superintegrable system for the model parameters −a1 = a2 = a3 = a. There are two sets of +solutions: One is that η = 1 +8 (√a + 4) and +Eϵ = 1 +4 +� +8√a(p + 1) + 3a + 2ϵ +� +8a3/2(p + 1) − 2a2 + a +� +, +where ϵ = ±1, with the associated structure function Φ(II) +Eϵ (z) = z(p + 1 − z)2. The energy spectrum Eϵ +is real for 0 < a ≤ 1/2. +The second set of solutions is given by +η(E) = 1 +4a +� +2√aE + 2a − a +√ +4E + 1 − 4a +� +and the corresponding energy spectrum of the system and structure function for the (p + 1)-dimensional +unirreps of the deformed oscillator algebra are given by +E = p(p + 2) + a + 3 +4, +Φ(II) +E +(z) = z(z − p − 1) +� +z + 1 +8a +� +3a3/2 − 4a(p + 1) + √a(4p2 + 8p + 3) +�� +. +Thus we have demonstrated that there exist indeed non-trivial model parameters which give real +energies of the superintegrable system in both Case 1 and Case 2 above. +2.2.2 +Potential V2(x, y) +The superintegrable system in Darboux II with potential V2(x, y) has Hamiltonian ˆH = +x2 +x2+1 +� +∂2 +x + ∂2 +y +� ++ +V2(x, y). This system possesses the following integrals of motion [14] +A = ∂2 +y + b1y2 + b3 +y2 , +B = (y2 − x4)∂2 +y + x2(1 − y2)∂2 +x +x2 + 1 ++ 2xy∂x∂y + x∂x + y∂y − 1 +4 + x2 + y2 +x2 + 1 +� +b1(x2 + y2) − b2 − b3 +x2 +y2 +� +, +8 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +which form the quadratic algebra relations +[A, B] = C, +[A, C] = 8A2 − 16b1B + 16b1 ˆH − 16b1(b2 + b3 + 3 +4), +[B, C] = −8{A, B} + 8 ˆHB + 12A − 8 ˆH2 + 8(b2 − b3 − 3 +4) ˆH. +By a direct calculation, we find the Casimir operator of the algebra +K2 =C2 − 8{A2, B} + 8 ˆH{A, B} + 16b1B2 + 76A2 + +� +16(b3 − b2 + 19 +4 ) ˆH − 16 ˆH2 +� +A ++ +� +8b1(4(b2 + b3) + 3) − 32b1 ˆH +� +B. +This Casimir operator can be expressed in terms of Hamiltonian as +K2 = −16 +� +b1 + b3 + 3 +4 +� +ˆH2 − 8b1(4b3 − 4b2 + 3) ˆH + b1 +� +36 + 48b3 − (4b3 − 4b2 + 3)2� +. +It can be shown that after the change of basis +A = 4 +� +−b1(N + η), +B = 8(N + η)2 − +2 ˆH +√−b1 +− 16(b2 + b3 + 3 +4)(N + η) − b1 ˆH +b1 ++ b† + b, +the quadratic algebra becomes the deformed oscillator algebra with structure function +Φ(II) +2 +(N, η) = 1 +16 +� +4b3 + 16N 2 + 16N(2η − 1) + 16η2 − 16η + 3 +� +� +4b2 + 1 − 4 ˆH +�√−b1 + 2N + 2η − 1 +� +√−b1 ++ 16N 2 + 32Nη − 16N + 16η2 − 16η + 3 +� +. +On the Fock states |z, E⟩, the structure function is factorized as follows +Φ(II) +2 +(z, η) = +� +z + η − 1 +4 +� +2 − +� +1 − 4b3 +�� � +z + η − 1 +4 +� +2 + +� +1 − 4b3 +�� +� +z + η − 2b1 − γ+(E) +4b1 +� � +z + η − 2b1 − γ−(E) +4b1 +� +, +where +γ±(E) = +� +b2 +1(4E − 4b2 + 1) ± +� +−b1E. +Imposing the constraints (3), for any p ∈ N+ we get the following values for the parameter η and +energy E: +Case 1. η+(E) = +1 +4b1 (2b1 − γ+(E)). This η value gives the following energy spectrum of the system +and the corresponding structure function of the deformed oscillator algebra +E− = −(p + 2) +� +−b1, +Φ(II) +E− (z) = z(z − p − 1) +� +z + 1 +4b1 +� +b1 +� +1 − 4b3 − γ+(E−) +�� � +z − 1 +4b1 +� +b1 +� +1 − 4b3 + γ+(E−) +�� +. +The enegry E− is real for b1 < 0. +Case 2. η−(E) = +1 +4b1 (2b1 − γ−(E)). This η value gives two sets of energies of the system, +E+ =(p + 2) +� +−b1, +(8) +Eϵ = − 2b1 + 4 +� +−b1 +� +p + 1 + ϵ +4 +� +1 − 4b3 +� +± +� +4b2 +1 − 16b1 +� +−b1 +� +p + 1 + ϵ +4 +� +1 − 4b3 +� ++ 4b1b2 − b1. +(9) +9 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +Eϵ above is obtained by solving the algebraic equation γ−(E) = 4b1 +�p + 1 + ϵ +4 +√1 − 4b3 +� from the con- +straints. (Notice that the other algebraic equation γ+(E) = 4b1 +�p + 1 + ϵ +4 +√1 − 4b3 +� lead to complex +solutions and its solutions are not shown here.) +Obviously E+ is real for b1 < 0 and Eϵ is real for +ϵ = +1, b1 < 0, b2 < 1 +4, b3 < 1 +4. The structure functions corresponding to E+, Eϵ in the Case 2 above are +given by +Φ(II) +E+ (z) = z(z − p − 1) +� +z + 1 +4b1 +� +b1 +� +1 − 4b3 − γ−(E+) +�� � +z − 1 +4b1 +� +b1 +� +1 − 4b3 + γ−(E+) +�� +, +Φ(II) +Eϵ (z) = z(z − p − 1) +� +z + 1 +2b1 +� +−b1 Eϵ +� � +z − 1 +4b1 +� +γ−(Eϵ) + ϵb1 +� +1 − 4b3 +�� +, +respectively. +Case 3: η = 1 +4 +�2 + ϵ√1 − 4b3 +�. The corresponding energies are given the same expression as Eϵ +above (and are obtained from solving the algebraic equation γ+(E) = −4b1 +�p + 1 + ϵ +4 +√1 − 4b3 +�). +2.2.3 +Potential V3(x, y) +The constants of motion for the superintegrable system in Darboux space II with the Hamiltonian ˆH = +x2 +x2+1 +� +∂2 +x + ∂2 +y +� ++ V3(x, y) associated to the potential V3 are given by +A = +� +y2 + 1 +y2 +� +∂2 +x − +� +x2 + 1 +x2 +� +∂2 +y +x2 + y2 + 1 +x2 + 1 +y2 ++ c1x2(y4 + 1) + c2(y4 + 1) − c3(x4 + 1) +(x2y2 + 1)(x2 + y2) +, +B =c1(x2 + y2) − c2(y4 − 1) − c3(x4 − 1) +4(x2y2 + 1) ++ xy(x2 − y2) +� +xy∂2 +x − xy∂2 +y + (x2 − y2)∂x∂y +� ++ +1 +x2y2 + 1 +�� +x2 − y2 +4 ++ y4 +� +x2∂2 +x + +� +x2 − y2 +4 ++ x4 +� +y2∂2 +y + 2xy +� +x2 − y2 +2 +− x2y2 +� +∂x∂y +� +. +They form the following quadratic algebra relations +[A, B] = C, +[A, C] = 2A2 + 2c1A + 16 ˆHB + 6 ˆH − 8 ˆH2, +[B, C] = −2{A, B} + (c2 + c3)A − c1c3. +The Casimir operator of this algebra is +K3 = C2 − 2{A2, B} − 16 ˆHB2 + (c2 + c3 + 4)A2 + 2c1{A, B}a − 2c1(c3 + 2)A + (16 ˆH2 − 12 ˆH)B. +With the differential realization of A, B, the Casimir operator can be expressed in terms of the Hamilto- +nian as +K3 = 4(c2 + c3) ˆH2 + (c2 +1 − 4c2c3 − 3(c2 + c3)) ˆH − 3 + 4c3 +4 +c2 +1. +We can convert the quadratic algebra into the deformed oscillator algebra by using the realization +A = 4 +� +ˆH(N + η), +B = −2(N + η)2 + +c1 +2 +� ˆH +(N + η) − 3 ˆH − 4 ˆH2 +8 ˆH ++ b† + b +with the corresponding structure function given by +Φ(II) +3 +(N, η) = − +1 +256 ˆH +� +4c3 − 4 ˆH + 16N 2 + 32Nη − 16N + 16η2 − 16η + 3 +� +× +� +−c2 +1 + 4c1 +� +ˆH(2N + 2η − 1) + ˆH +� +−4c2 + 4 ˆH − 16N 2 − 32Nη + 16N − 16η2 + 16η − 3 +�� +. +10 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +By acting Φ(II) +3 +on the Fock basis states |z, E⟩, we find that the structure function is factorised as +Φ(II) +3 +(z, η) = +� +z + η − 1 +4 +� +2 − +� +−4c3 + 4E + 1 +�� � +z + η − 1 +4 +� +2 + +� +−4c3 + 4E + 1 +�� +× +� +z + η − 1 +4E +� +−E +� +−4c2 + 4E + 1 + c1 +√ +E + 2E +�� +× +� +z + η − 1 +4E +� +E +� +−4c2 + 4E + 1 + c1 +√ +E + 2E +�� +. +We now obtain the energy spectrum of the system from the finite-dimensional unirreps of the deformed +oscillator algebra. Imposing the constraints (3) which give (p + 1)-dimensional unirreps for any p ∈ N+, +we determine the parameter η and the energy E of the system. There are two sets of solutions; +Case 1: η(E) = 1 +4 +�2 − √−4c3 + 4E + 1 +� and the energies are determined by either +� +−4c3 + 4E + 1 − 2(p + 1) = 0 +(10) +or +c1 +√ +E ++ +� +−4c3 + 4E + 1 + +� +−4c2 + 4E + 1 = 4(p + 1). +(11) +Solution to the algebraic equation (10) gives the energies +Ec3 = p(p + 2) + c3 + 3 +4. +The structure function of the corresponding (p + 1)-dimensional unirreps is +Φ(II) +Ec3 (z) =z(z − p − 1) +� +� +� +�z − 1 +2 +� +p + 1 − +� +(p + 1)2 + c3 − c2 +� +− +c1 +4 +�� +p + 1 +2 +� � +p + 3 +2 +� ++ c3 +� +� +� +� +� +� +� +�z − 1 +2 +� +p + 1 + +� +(p + 1)2 + c3 − c2 +� +− +c1 +4 +�� +p + 1 +2 +� � +p + 3 +2 +� ++ c3 +� +� +� +� . +Other possible energies of the system are given by solutions to the algebraic equation (11), which read +E± = 1 +4 +(p + 1 + c1)2 � +2c2 + 2c3 − 1 ± +� +(p + 1 + c1)2 + 4c2c3 − (c2 + c3) + 1 +4 +� +(p + 1 + c1)2 − (c2 − c3)2 +. +These energies are real for the model parameters satisfying 4c2c3 + 1 +4 > c2 + c3. The corresponding +structure functions for the (p+1)-dimensional unirreps of the algebra are +Φ(II) +E± (z) =z(z − p − 1) +� +z − 1 +2 +� +−4c3 + 4E± + 1 +� +� +z − 1 +4 +�� +−4c3 + 4E± + 1 − +� +−4c2 + 4E± + 1 + +c1 +√E± +�� +. +Case 2: η(E) = +1 +4E +� +c1 +√ +E + 2E − E√−4c2 + 4E + 1 +� +. +This η value gives the following energy +spectrum of the system and the corresponding structure function of the unirreps, +Ec2 = p(p + 2) + c2 + 3 +4, +Φ(II) +Ec2 (z) =z(z − p − 1) +� +� +� +�z + 1 +2 +� +p + 1 − +� +(p + 1)2 + c2 − c3 +� ++ +c1 +4 +�� +p + 1 +2 +� � +p + 3 +2 +� ++ c2 +� +� +� +� +� +� +� +�z − 1 +2 +� +p + 1 + +� +(p + 1)2 + c2 − c3 +� ++ +c1 +4 +�� +p + 1 +2 +� � +p + 3 +2 +� ++ c2 +� +� +� +� . +11 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +2.3 +Darboux Space III +In Darboux space III, there exist 4 different potentials. In terms of the separable coordinates (u, v) and +(µ, ν), they are given by +V1(u, v) = a1u + a2v + a3 +4 + u2 + v2 +, +V2(u, v) = +b1 +u2 + b2 +v2 + b3 +4 + u2 + v2 , +V3(µ, ν) = +c1(µ + ν) + c2 +µ+ν +µν + c3 +ν2−µ2 +ν2µ2 +(µ + ν)(2 + µ − ν) +, +V4(µ, ν) = d1µ + d2ν + d3ν2 + µ2 +(µ + ν)(2 + µ − ν) +, +where ai, bi, ci, di are real constants. +2.3.1 +Potential V1(u, v) +The constants of motion of the superintegrable system in Darboux space III with the Hamiltonian ˆH = +exp(2u) +4(exp(u))+1 +�∂2 +u + ∂2 +v +� + V1(u, v) associated to the potential V2 are given by [14] +A = (2 + v2)∂2 +u − (2 + u2)∂2 +v +2(4 + u2 + v2) ++ a1u(2 + v2) − 2a2v(2 + u2) + a3(v2 − u2) +4(4 + u2 + v2) +, +B = 2uv +(∂2 +u + ∂2 +v) +2(4 + u2 + v2) − 2∂u∂v + a1v(v2 − u2 + 4) + a2u(u2 − v2 + 4) − 2a3vu +4(4 + u2 + v2) +. +They form the quadratic algebra with the commutation relations +[A, B] = C, +[A, C] = ˆHB − a2a1 +8 +, +[B, C] = − ˆHA − a2 +2 − a2 +1 +16 +, +which is the symmetry algebra of the superintegrable system. +By a direct computation, we obtain the Casimir operator of this algebra +K1 = − ˆHA2 − ˆHB2 − a2 +2 − a2 +1 +8 +A + a1a2 +4 +B. +We can show that in terms of the Hamiltonian this Casimir operator takes the form +K1 = − ˆH3 + 1 +2(a3 + 1 +2) ˆH2 + 1 +16(2a2 +1 + 2a2 +2 − a2 +3) ˆH − a3(a2 +1 + a2 +2) +32 +. +To determine the energy spectrum of the system, we now construct the deformed oscillator algebra +realization of the quadratic algebra. We find that +A = +� +ˆH(N + η), +B = a1a2 +8 ˆH ++ b† + b, +trsansform the quadratic algebra into the deformed oscillator algebra with the structure function +Φ(III) +1 +(N, η) = +1 +256 ˆH +� +a2 +1 + 2a3 ˆH − 4 ˆH3/2 +� +2 +� +ˆH + 2N + 2η − 1 +�� +× +� +a2 +2 + 2a3 ˆH + 4 ˆH3/2 +� +−2 +� +ˆH + 2N + 2η − 1 +�� +. +Here η is a constant parameter to be determined from the constraints (1). Acting on the Fock basis states +|z, E⟩, the structure function Φ(III) +1 +becomes +Φ(III) +1 +(z, η) = +� +z + η − +1 +8E3/2 +� +a2 +1 + 2E +� +a3 − 4E + 2 +√ +E +��� +× +� +z + η + +1 +8E3/2 +� +a2 +2 + 2E(a3 − 4E − 2 +√ +E) +�� +. +12 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +The constraints (3) give the (p + 1)-dimensional unirreps of the deformed oscillaor algebra and their +solutions determine the constant η and energy spectrum of the superintegrable system. There are two +sets of solutions: +Case 1: η(E) = +1 +8E3/2 +� +a2 +1 + 2E +� +a3 − 4E + 2 +√ +E +�� +and energies E are determined by the algebraic +equation +8E3/2 (p + 1) + 4a3E + a2 +1 + a2 +2 = 16E2. +(12) +Case 2: η(E) = − +1 +8E3/2 +� +a2 +2 + 2E +� +a3 − 4E − 2 +√ +E) +�� +and energies E satisfy +16E2 + 8E3/2 (p + 1) = 4a3E + a2 +1 + a2 +2. +(13) +The algebraic equations (12) and (13) can be solved by using symbolic computation packages. +It can be shown that there exist model parameters ai such that solutions to these algebraic equations +for energies are real. To demonstrate this, we consider the case in which the model parameters satisfy +a1 = 0 and a2 = a3 = 1. In this case we find that the structure function reduces to +Φ(III) +1 +(z, η) = +� +z + η − +�1 +2 − +√ +E + +1 +4 +√ +E +�� � +z + η − +�1 +2 + +1 +8E3/2 (8E2 − 2E − 1) +�� +. +Imposing the constraints (3) gives the constant η and energies as follows. +a. η(E) = 1 +2 − +√ +E + +1 +4 +√ +E which leads to the algebraic equation 4E(1 − 4E) + 1 + 8E3/2(p + 1) = 0 +for E. This equation has real solution given by +E± = 1 +48 +� +3p2 + +√ +3 g(p) + 6p + 9 ± +� +6 f(p) +� +, +where +e(p) = +3 +� +27p4 + 108p3 + 252p2 + 3 +√ +3 +� +(p + 1)4 (27p4 + 108p3 + 310p2 + 404p + 575) + 288p + 367 +g(p) = +� +3 (p2 + 2p + 3)2 + 2 × 22/3e(p) + +1 +e(p) 4 +3√ +2 (6p2 + 12p + 31) + 8 +f(p) = 3 +� +p2 + 2p + 3 +�2 − 22/3e(p) − +1 +e(p)2 +3√ +2 +� +6p2 + 12p + 31 +� ++ +1 +g(p) 3 +√ +3(p + 1)2 � +p4 + 4p3 + 12p2 + 16p + 23 +� ++ 8. +It is clear that g(p) is real for all p ∈ N+. We now show that f(p) > 0 for all p ∈ N+. Let +f0(p) = 3 +� +p2 + 2p + 3 +�2 − 22/3e(p) − +1 +e(p) 2 +3√ +2 +� +6p2 + 12p + 31 +� ++ 8. +By using symbolic computation package, we found that df0(p) +dp +> 0 for all p ∈ N+. Hence f0(p) is strictly +increasing. Moreover, f0(0) = −62 3� +2 +15 +√ +69+367 − 22/3 3� +15 +√ +69 + 367 + 35 ∼= 12.5741 > 0. It follows that +f(p) > 0 for all p ∈ N+ and the energy E given above is real. +b. η(E) = 1 +2 + +1 +8E3/2 (8E2−2E−1). This leads to the algebraic equation 4E(4E−1)−1+8E3/2(p+1) = +0. It gives the same energy expression as in Case a above. +For both case a and case b above, the structure function corresponding to the (p + 1)-dimensional +unirreps of the deformed oscillator algebra is simply Φ(III +a,b (z) = z(z − p − 1). +13 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +2.3.2 +Potential V2(u, v) +The integrals of motion of the superintegrable system associated to the potential V2 with Hamiltonian +ˆH = +exp(2u) +4(exp(u))+1 +�∂2 +u + ∂2 +v +� + V2(u, v) in Darboux space III are given by [14], +A = u2∂2 +v − 2uv∂u∂v + v2∂2 +u + b1v2 +4u2 + b2u2 +4v2 , +B = (2 + v2)∂2 +u − (2 + u2)∂2 +v +2(4 + u2 + v2) ++ 2b1v2(v2 + 2) − 2b2u2(u2 + 2) + b3(v2 − u2) +4(4 + u2 + v2) +. +These integrals form the quadratic algebra of the form +[A, B] = C, +[A, C] = −2{A, B} − (b1 + b2 + 1)B + (b1 − b2) ˆH + (b2 − b1)b3 +4 +, +[B, C] = −2B2 − (b1 + b2 + 1)B + (b1 − b2) ˆH + (b2 − b1)b3 +4 +. +By a direct calculation, we find the Casimir operator of the algebra +K2 = C2 + 2{A, B2} + (b1 + b2 + 5)B2 − 4 ˆHA2 − 2(b1 − b2) ˆHB +− b3(b2 − b1)B − 4 ˆHA + (2b3 − 1) ˆHA − b2 +3 +4 A. +With the differential realization of A, B and in terms of ˆH, the Casimir K2 takes the simple form +K2 = −(b1 + b2 − 2) ˆH2 + +� +(b3 + 3 +2)(b1 + b2) +2 +− b3 − b1b2 − 1 +2 +� +ˆH − b2 +3(b1 + b2 − 2) +16 +. +The quadratic algebra can be transformed into the deformed oscillator algebra via the realization +(i.e., change of basis) +A = − +� +(N + η)2 − 1 +4 + b1 + b2 + 1 +4 +� +, +B = −(b1 − b2) ˆH + (b2−b1)b3 +4 +16 +� +(N + η)2 − 1 +4 +� ++ b†ρ(N) + ρ(N)b, +where +ρ(N) = +1 +3 · 212 · (−2)8(N + η)(1 + N + η)(1 + 2(N + η))2 . +The structure function is given by +Φ(III) +2 +(N, η) = 4096((2N + 2η − 1)2 � +b2 +2(3b1 + 3b2 + 7) − 4 ˆH +� +3b2 +1 + b1(12b2 − 12 ˆH + 11) ++9b2 +2 − 12b2 ˆH + 25b2 − 28 ˆH + 4 +�� +− 48(1 − 2(N + η))2 +� +− 1 +16b2 +2(b1 + b2 − 2) ++1 +2 +ˆH +�� +b2 + 3 +2 +� +(b1 + b2) − 2b1b2 − 2b2 − 1 +� +− ˆH2(b1 + b2 − 2) +� ++ (2N + 2η − 1)2 � +12N 2 + 12N(2η − 1) + 12η2 − 12η − 1 +� +× +� +b2 +2 − 4 ˆH(2b1 + 4b2 − 4 ˆH + 1) +� ++ 12(b1 − b2)2(b2 − 2 ˆH)2 − 12 ˆH(2(N + η) − 3)(2(N + η) + 1)(1 − 2(N + η))4). +Here η is a constant to be determined from the constraints on the structure function. +14 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +By acting on Fock basis states |z, E⟩, we can show that the structure function Φ(III) +2 +is factorized as +Φ(III) +2 +(z, η) = +� +z + η − 1 +12 +� +6 − +√ +3 +� +δ1(E) + (b3 − 4E)2 +E +− 8(b1 + b2) + +g(E) +Eδ1(E) + 12 +�� +� +z + η − 1 +12 +� +6 + +√ +3 +� +δ1(E) + (b3 − 4E)2 +E +− 8(b1 + b2) + +g(E) +Eδ1(E) + 12 +�� +� +�z + η − 1 +24 +� +�12 − +√ +6 +� +f(E) +E ++ (−1 + i +√ +3)δ1(E) +E +− (1 + i +√ +3)g(E) +Eδ1(E) ++ 24 +� +� +� +� +� +�z + η − 1 +24 +� +�12 + +√ +6 +� +f(E) +E ++ (−1 + i +√ +3)δ1(E) +E +− (1 + i +√ +3)g(E) +Eδ1(E) ++ 24 +� +� +� +� +� +�z + η − 1 +24 +� +�12 − +√ +6 +� +f(E) +E +− (1 + i +√ +3)δ1(E) +E ++ (−1 + i +√ +3)g(E) +Eδ1(E) ++ 24 +� +� +� +� +� +�z + η − 1 +24 +� +�12 + +√ +6 +� +f(E) +E +− (1 + i +√ +3)δ1(E) +E ++ (−1 + i +√ +3)g(E) +Eδ1(E) ++ 24 +� +� +� +� , +where +f(E) = 2 +� +b2 +3 − 8Eb3 − 8(b1 + b2 − 2E)E +� +, +g(E) = b4 +3 − 16Eb3 +3 + 4E(2b1 + 2b2 + 24E + 3)b2 +3 − 32E2(2b1 + 2b2 + 8E + 3)b3, ++ 16E2(b2 +1 + 14b2b1 + 8(E − 3)b1 + b2 +2 + 16E2 + 8b2(E − 3) + 12E + 15), +δ1(E) = +3� +ρ1(E) + ρ2(E), +with +ρ1(E) = b6 +3 − 24Eb5 +3 + 240E2b4 +3 + 12b1Eb4 +3 + 12b2Eb4 +3 + 18Eb4 +3 − 1280E3b3 +3 +− 192b1E2b3 +3 − 192b2E2b3 +3 − 288E2b3 +3 + 3840E4b2 +3 + 1152b1E3b2 +3 + 1152b2E3b2 +3 ++ 1728E3b2 +3 + 48b2 +1E2b2 +3 + 48b2 +2E2b2 +3 − 288b1E2b2 +3 − 480b1b2E2b2 +3 +− 288b2E2b2 +3 + 360E2b2 +3 − 6144E5b3 − 3072b1E4b3 − 3072b2E4b3 − 4608E4b3 +− 384b2 +1E3b3 − 384b2 +2E3b3 + 2304b1E3b3 + 3840b1b2E3b3 + 2304b2E3b3 − 2880E3b3 ++ 4096E6 + 3072b1E5 + 3072b2E5 + 4608E5 + 768b2 +1E4 + 768b2 +2E4 − 4608b1E4 +− 7680b1b2E4 − 4608b2E4 + 5760E4 + 64b3 +1E3 + 64b3 +2E3 + 3744b2 +1E3 − 2112b1b2 +2E3 ++ 3744b2 +2E3 − 8064b1E3 − 2112b2 +1b2E3 + 10944b1b2E3 − 8064b2E3 + 3456E3; +ρ2(E) = 128 +2043 +�� +− +� +b2 +3 − 8Eb3 − 8(b1 + b2 − 2E)E +�2 − 12E +� +(2b1 + 2b2 + 1)b2 +3 +−8(2b1 + 2b2 + 1)Eb3 − 4E +� +b2 +1 − 2(b2 + 4E − 4)b1 + b2 +2 + 8b2 − 8b2E − 4E − 5 +���3 ++ 262144 +� +b6 +3 − 24Eb5 +3 + 6E(2b1 + 2b2 + 40E + 3)b4 +3 − 32E2(6b1 + 6b2 + 40E + 9)b3 +3 ++ 24E2 � +2b2 +1 − 4(5b2 − 12E + 3)b1 + 2b2 +2 + 160E2 + 72E + 12b2(4E − 1) + 15 +� +b2 +3 +− 192E3 � +2b2 +1 − 4(5b2 − 4E + 3)b1 + 2b2 +2 + 32E2 + 24E + 4b2(4E − 3) + 15 +� +b3 ++ 32E3 � +2b3 +1 + (−66b2 + 24E + 117)b2 +1 − 6 +� +11b2 +2 + (40E − 57)b2 − 16E2 + 24E + 42 +� +b1 ++2b3 +2 + 3b2 +2(8E + 39) + 12b2 +� +8E2 − 12E − 21 +� ++ 4 +� +32E3 + 36E2 + 45E + 27 +���2� 1 +2 . +Imposing the constraints (3) which give the (p + 1)-dimensional unirreps of thedeformed oscillator +algebra, we determine the constant η and obtain the following algebraic equations for the energies E: +15 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +1. η1(E) = +1 +12 +� +6 − +√ +3 +� +δ1(E)+(b3−4E)2 +E +− 8(b1 + b2) + +g(E) +Eδ1(E) + 12 +� +. This η value gives five sets of +algebraic equations, +δ1(E) + (b3 − 4E)2 + g(E) +δ1(E) = (12p(p + 2) + 8(b1 + b2)) E, +η1(E) − 1 +24 +� +�12 + ϵ +√ +6 +� +f(E) +E ++ (−1 + i +√ +3)δ1(E) +E +− (1 + i +√ +3)g(E) +Eδ1(E) ++ 24 +� +� + p + 1 = 0, +η1(E) − 1 +24 +� +�12 + ϵ +√ +6 +� +f(E) +E +− (1 + i +√ +3)δ1(E) +E ++ (−1 + i +√ +3)g(E) +Eδ1(E) ++ 24 +� +� + p + 1 = 0, +where ϵ = ±1. Real solutions to each algebraic equation above give the energies of the system. +2. η2(E) = 1 +24 +� +12 − +√ +6 +� +f(E) +E ++ +i(i+ +√ +3)δ1(E) +E +− +i(−i+ +√ +3)g(E) +Eδ1 ++ 24 +� +and energy spectra from the real +solutions of the three sets of algebraic equations +f(E) + i(i + +√ +3)δ1(E) − +i +� +−i + +√ +3 +� +g(E) +δ1(E) += 24p(p + 2)E, +η2(E) − 1 +24 +� +� +�12 + ϵ +√ +6 +� +� +� +�f(E) +E +− +i +� +−i + +√ +3 +� +δ1(E) +E ++ +i +� +i + +√ +3 +� +g(E) +Eδ1(E) ++ 24 +� +� +� + p + 1 = 0, +where again ϵ = ±1. +3. η3(E) = 1 +24 +� +12 − +√ +6 +� +f(E) +E +− (1+i +√ +3)δ1(E) +E ++ (−1+i +√ +3)g(E) +Eδ1(E) ++ 24 +� +. This η yields the algebraic equa- +tion whose real solutions gives other possible energies of the system, +f(E) − (1 + i +√ +3)δ1(E) + (−1 + i +√ +3)g(E) +δ1(E) += 24p(p + 2)E. +It is in general very difficult to solve the above algebraic equations for E analytically due to their +complicated forms. To demonstrate the existence of real solutions to the above algebraic equations, we +have a closer look at cases of restricted model parameter space. As an example, we consider b1 = b2 = +b3 = h for any h ∈ R. In this case the structure function reduces to +Φ(III) +2 +(z, η) = +� +z + η − 1 +2 +�2 +� +�z + η − 1 +8 +� +�4 − +� +2h2(E) − 2h1(E) +E +� +� +� +� +� +�z + η − 1 +8 +� +�4 + +� +2h2(E) − 2h1(E) +E +� +� +� +� +� +�z + η − 1 +8 +� +�4 − +� +2h2(E) + 2h1(E) +E +� +� +� +� +� +�z + η − 1 +8 +� +�4 + +� +2h2(E) + 2h1(E) +E +� +� +� +� , +where +h1(E) = +� +h4 + 16h3E + 8h2E(12E + 1) + 64hE2(4E − 15) + 16E2 (16E2 + 8E + 17), +h2(E) = h2 − 24hE + 16E2 + 12E. +Imposing the constraints (3), we determine the constant η and the corresponding energies for the model +parameters b1 = b2 = b3 = h as follows. +16 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +a. η1 = 1 +2 and +E1,± = 4h2 + 12hp2 + 24hp + 15h + 8p4 + 32p3 + 42p2 + 20p + 1 ± m(h, p) +4 (4h + 4p2 + 8p + 3) +, +where +m(h, p) = +� +(4h2 + 3h (4p2 + 8p + 5) + 8p4 + 32p3 + 42p2 + 20p + 1)2 − h2 (4h + 4p2 + 8p + 3)2. +It is easy to check that m(p) is real for any p ∈ N+ if h > 0 and so E1,± give the energies of the system +for the model parameters b1 = b2 = b3 = h > 0. +b. η2(E) = 1 +8 +� +4 + ϵ +� +2h2(E)−2h1(E) +E +� +with ϵ = ±1. For this η value, the energies are +E2,± = 8h2 + 6hp2 + 12hp + 12h + p4 + 4p3 + 3p2 − 2p − 4 ± n(h, p) +8(4h + p(p + 2)) +, +where +n(h, p) = +� +(8h2 + 6h (p2 + 2p + 2) + p4 + 4p3 + 3p2 − 2p − 4)2 − 4h2(4h + p(p + 2))2. +It can be checked that n(p) is real for h > 1 and so E2,± give the energies of the system for the model +parameters b1 = b2 = b3 = h > 1. +Other possible energies corresponding to η2(E) are +E3,± = 1 +8 +� +4 +� +(1 − 4h)p(p + 2) − 2h + 4p2 + 8p + 3 ± l(h, p) +� +, +where +l(p, h) = 8p2� +4z(h, p)(16p + 7) − 4h (4z(h, p) + 20p2 + 40p + 3) + 16p4 + 64p3 + 80p + 22, +z(p, h) = +� +(1 − 4h)p(p + 2). +It is easily seen that l(p) and z(p) are real for h < 0. Hence, E3,± are real for h < 0 and give the energies +of the system for model parameters b1 = b2 = b3 = h < 0. +2.3.3 +Potential V3(µ, ν) +The constants of motion of the superintegrable system with potential V3 and the Hamiltonian ˆH = +µ2∂2 +µ−ν2∂2 +ν +(µ+ν)(2+µ−ν) + V3(µ, ν) are given by [14], +A = −4µ2ν2 (∂µ + ∂ν)2 +(µ + ν)2 +− c2 +µ − ν +µν +− c3 +(µ − ν)2 +µ2ν2 +, +B = ν2(µ + 2)µ∂2 +ν − µ2(ν − 2)ν∂2 +µ +(µ + ν)(2 + µ − ν) +− 4µ2ν2 (∂µ + ∂ν)2 +(µ + ν)2 +− c1µ2ν2 + c2µν + 2c3(1 + µ − ν) +µν(2 + µ − ν) +. +These integrals form the quadratic algebra with the commutation relations +[A, B] = C, +[A, C] = −2{A, B} − B − 2c1c2 + 4c2 ˆH, +[B, C] = 2B2 − 8c3 ˆH. +The Casimir operator for the quadratic algebra is given by +K3 = C2 + 2{A, B2} − 16c3 ˆHA + 5B2 + 4c2 +� +c1 − 2 ˆH +� +B, +17 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +which can be expressed in terms of the Hamiltonian as +K3 = 16c3 ˆH2 + 4(c2 +2 − 4c1c3) ˆH + 4c2 +1c3. +It can be shown that +A = (N + η)2 − 1 +2, +B = − +c1c2 − 2c2 ˆH +2 +� +(N + η)2 − 1 +4 +� + b†ρ(N) + ρ(N)b, +where η is a constant to be determined and +ρ(N) = +1 +3 · 220(N + η)(1 + N + η)(1 + 2(N + η))2 , +convert the quadratic algebra into the deformed oscillator algebra with the structure function +Φ(III) +3 +(N, η) = −786432 +� +−c2 +1 + 4c1 ˆH + ˆH (2N + 2η − 1)2 − 4 ˆH2� � +c2 +2 − c3(2N + 2η − 1)2� +. +By acting it on a Fock basis |z, E⟩, the structure function becomes +Φ(III) +3 +(z, η) = 12582912 c3 E +� +z + η − +� +1 +2 − +c2 +2√c3 +�� � +z + η − +� +1 +2 + +c2 +2√c3 +�� +� +z + η − E − +√ +E(c1 − 2E) +2E +� � +z + η − E + +√ +E(c1 − 2E) +2E +� +. +Imposing the constraint conditions (3), we obtain +1. η(E) = E− +√ +E(c1−2E) +2E +or η(E) = E+ +√ +E(c1−2E) +2E +. For both cases, we have +E = 1 +8 +� +4c1 + (p + 1)2 ± (p + 1) +� +8c1 + (p + 1)2 +� +, +which is real for c1 > 0. This gives the energy spectrum of the system for any model parameters c1, c2, c3 +with c1 > 0. +2. ηϵ = 1 +2 +� +1 + ϵ +c2 +√c3 +� +, where ϵ = ±1. The corresponding energies are given by +Eϵ = 1 +8 +� +��4c1 + +� +2(p + 1) + ϵ c2 +√c3 +�2 +± +� +2(p + 1) + ϵ c2 +√c3 +� � +� +� +�8c1 + +� +2(p + 1) + ϵ c2 +√c3 +�2 +� +�� , +which is real for c1 > 0 and c3 > 0. +2.3.4 +Potential V4(µ, ν) +For a superintegrable system with the Hamiltonian ˆH = +µ2∂2 +µ−ν2∂2 +ν +(2+µ−ν)(µ+ν)+V4(µ, ν) associated to the potential +V4, the constants of motion are given by [14] +A = ν2(µ + 2)µ∂2 +ν − µ2(ν − 2)ν∂2 +µ +(µ + ν)(2 + µ − ν) +− µν (d1(ν − 2) + d2(µ + 2) + 2d3(ν − µ + µν)) +(µ + ν)(2 + µ − ν) +, +B = +1 +4µν(µ − ν + 2)(µ + ν)2 +�� +µ4(12ν3 − 12ν2 + ν + 1) + 2µ3ν − (ν − 1)µ2ν2� +∂2 +µ ++ µν(µ − ν + 2) +� +µ2(12ν2 + 1) + 2µν + ν2� +∂µ∂ν ++ν2 � +µ3(12ν2 − 1) + µ2(12ν2 − 1) + µ(ν − 2)ν − ν2� +∂2 +ν +� +− (µ − ν) +�(µ − ν)(d1µ + d2ν) − 2d3(µ2 + ν2 + µν(2 + µ − ν)) +� +4µν(µ + ν)(2 + µ − ν) +. +18 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +They satisfy the quadratic algebra relations +[A, B] = C, +[B, C] = −2B2 + 2 ˆHB − d2 +3 +2 , +[A, C] = 2{A, B} − 2 ˆHA − B + (d1 + d2 + 1 +2) ˆH − d1d2 +2 +− 2 ˆH2. +By a direct calculation, we find the Casimir operator of this algebra +K4 = C2 − 2{A, B2} + 5B2 + 2 ˆH{A, B} − d2 +3A + +� +4 ˆH − (2d1 + 2d2 + 5) ˆH + d1d2 +� +B. +By means of the differential operator representation of A and B above, the Casimir operator K4 can be +expressed in terms of ˆH as +K4 = 4 ˆH3 − (2d1 + 2d2 + 1) ˆH2 + +� +(d1 + d2)2 +4 ++ d3(d2 − d1) +� +ˆH − d3(d3 − d2 +1 + d2 +2) +4 +. +We can show that +A =(N + η)2, +B = +ˆH +2 − −4 ˆH + 8(d1 + d2 + 1 +2) ˆH − 4d1d2 +32 +� +(N + η)2 − 1 +4 +� ++ ρ(N)b† + bρ(N) +with +ρ(N) = +1 +3 · 220(N + η)(1 + N + η)(1 + 2(N + η))2 , +give a realization of the quadratic algebra in terms of the deformed oscillator algebra, with the structure +function given by +Φ(III) +4 +(N, η) =16384(1 − 2(N + η))2 � +3 +� +d3 +� +−d2 +1 + d2 +2 + d3 +� ++ 4d3 ˆH(d1 − d2) ++4 ˆH2(2d1 + 2d2 + 1) − ˆH(d1 + d2)2 − 16 ˆH3� ++ 6d1 ˆH(d2 − 2 ˆH) +−4 ˆH2(3d2 − 6 ˆH + 2) + +� ˆH2 − d2 +3 +� � +12(N + η)2 − 12(N + η) − 1 +� +− 7d2 +3 +� +. +Here η is a constant parameter to be determined. Acting on the Fock states |z, E⟩, the structure function +is factorized as +Φ(III) +3 +(z, η) = +� +z + η − 1 +2 +�2 � +z + η − +� +1 +2 − +γ1(E) +2 +�d2 +3 − E2� +�� � +z + η − +� +1 +2 + +γ1(E) +2 +�d2 +3 − E2� +�� +, +where +γ1(E) = +� +d2 +3 − E2 × +� +−d2 +1(d3 + E) + 4d1E(d3 + E) + d2 +2(d3 − E) + 4d2E(E − d3) − 8E3. +From the constraints (3), we determine the constant η and the energy spectrum E of the system. We list +the results as follows. +Case 1: η(E) = 1 +2 − +γ1(E) +2(d2 +3−E2). Corresponding to this η value, we have either +� +−d2 +1(d3 + E) + 4d1E(d3 + E) + d2 +2(d3 − E) + 4d2E(E − d3) − 8E3 = (p + 1) +��d2 +3 − E2� +(14) +or +� +−d2 +1(d3 + E) + 4d1E(d3 + E) + d2 +2(d3 − E) + 4d2E(E − d3) − 8E3 = 2(p + 1) +��d2 +3 − E2�. +(15) +19 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +Notice that obviously the solution space of the algebraic equation (15) is subspace of that of (14) and so +the energy spectrum of the system corresponding to η1(E) is given by solutions to (14). +Case 2: η = 1 +2. In this case, E satisfies the same algebraic equation as (15) and so do not give new +energies of the system. +Case 3: η(E) = +� +1 +2 + +γ1(E) +2(d2 +3−E2) +� +. This η value give the same equations for E as those in Case 1 +above. +Due to the complexity of the algebraic equations, it is hard to see whether or not they lead to real +energies E for general model parameters. However, we can show that when the model parameter d3 = 0, +the structure function reduces to +Φ(III) +3 +(z, η) = +� +z + η − 1 +2 +�2 � +z + η − 1 +2E +� +E − +� +E +�d2 +1 − 4d1E + d2 +2 − 4d2E + 8E2��� +� +z + η − 1 +2E +� +E + +� +E +�d2 +1 − 4d1E + d2 +2 − 4d2E + 8E2��� +. +In this case, by imposing the constraints (3) we obtain the parameter η and the energies of the system +with model parameter d3 = 0, +η−(E) = 1 +2E +� +E − +� +E +�d2 +1 − 4d1E + d2 +2 − 4d2E + 8E2�� +, +E± = 1 +16 +� +4d1 + 4d2 + (p + 1)2 ± +� +(p + 1)2 ((p + 1)2 + 8(d1 + d2)) − 16 (d1 − d2)2 +� +. +Other η values from the constraints give rise to same energies as E± above. +It is clear that both E± are real for d1 = d2 > 0. So E± give the energy spectrum of the system for +model parameters d1 = d2 > 0, d3 = 0. The corresponding structure function of the p + 1)-dimensional +unirreps is +Φ(III)E±(z) = z(z − p − 1) +� +z − +1 +2E± +� +E± +�d2 +1 − 4d1E± + d2 +2 − 4d2E± + 8E2 +± +� �2 +. +2.4 +Darboux Space IV +In Darboux space IV, there are 3 different potentials in the separable coordinates (µ, ν), (u, v) and (ω, ϕ): +V1(µ, ν) = −sin2(2µ)(4a1 exp(2ν) + 4a2 csc2(2µ) + 4a3 exp(4ν)) +2 cos 2µ + a4 +, +V2(u, v) = − +sin2(2u)( +b2 +sinh2 v + +b3 +cosh2 v) + b1 +2 cos 2u + b4 +, +V3(ω, ϕ) = +c1 +cos2 ϕ + +c2 +cosh2 ω + c3 +� +1 +sin2 ϕ − +1 +sinh2 ω +� +c4+2 +sinh2(2ω) + +c4−2 +sin2(2ϕ) +, +where ai, bi ci are real model parameters. +2.4.1 +Potential V1(µ, ν) +The integrals of motion of the superintegrable system in Darboux space IV with potential V1 and the +Hamiltonian ˆH = − +4µ2ν2 +(a4+2)µ2+(a4−2)ν2 + V1(µ, ν) are +A =µ2∂2 +µ + 2µν∂µ∂ν + ν2∂2 +ν + µ∂µ + ν∂ν + a1(µ2 + ν2) + a3(µ2 + ν2)2; +B = 4(a4 + 2)µ2∂2 +µ − 4(a4 − 2)ν2∂2 +ν +(a4 + 2)µ2 + (a4 − 2)ν2 ++ 2a1 +�(a4 + 2)µ2 − (a4 − 2)ν2� + 4a3 +�(a4 + 2)µ4 − (a4 − 2)ν4� + 16a2 +(a4 + 2)µ2 + (a4 − 2)ν2 +. +20 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +They form the quadratic algebra with the commutation relations given by [14] +[A, B] = C, +[A, C] = 8{A, B}a − 16B + 32a1 ˆH, +[B, C] = −8B2 + 256a3A + 128 a3a4 ˆH + 32(a2 +1 + 4a3 + 16). +By a direct calculation, we find that the Casimir operator is +K1 = C2 − 8{A, B2} + 256 a3A2 + 80B2 + +� +256 a3a4 ˆH + 64(16a2a3 + a2 +1 + 4a3) +� +A − 64 a1 ˆHB. +With the differential operator representation of A and B, the Casimir operator can be expressed in terms +of ˆH as +K1 = −256 a3 ˆH2 + 64 a4(4a3 − a2 +1) ˆH + 128(a2 +1 + 4a3 + 8a2a3 − 2a2 +1a2). +After a long calculation, we find that the change of basis +A = 4(N + η)2, +B = − +128a1 ˆH +256 +� +(N + η)2 − 1 +4 +� + ρ(N)b† + bρ(N), +where +ρ(N) = +1 +3 · 215(N + η)(1 + N + η)(1 + 2(N + η))2 +maps the quadratic algebra to the deformed oscillator algebra with the structure function +Φ(IV ) +1 +(N, η) = − 805306368 (2(N + η) − 1)2 +× +� +128(−2a2 +1a2 + a2 +1 + 8a2a3 + 4a3) + 64 ˆHa4(4a3 − a2 +1) − 256a3 ˆH2� ++ 131072 +� +12(N + η)2 − 12(N + η) − 1 +� +(2(N + η) − 1)2 +× +� +131072(a2 +1 + 16a2a3 + 4a3) + 524288 a3a4 ˆH + 1048576 a3 +� +− 16384 (2(N + η) − 1)2 � +−7340032(a2 +1 + 16a2a3 + 4a3) + 29360128 a3a4 ˆH − 46137344 a3 +� ++ 51539607552 a2 +1 ˆH2 + 206158430208 a3 (2(N + η) − 3) (2(N + η) + 1) (2(N + η) − 1)4. +Acting on the Fock basis state |z, E⟩, we find that the structure function has the factorization +Φ(IV ) +1 +(z, η) = +� +z + η − +� +1 +2 − +ia1 +4√a3 +�� � +z + η − +� +1 +2 + +ia1 +4√a3 +�� +� +z + η − +1 +2 +√ +2 +�√ +2 − +� +1 − 4a2 − Ea4 − m1(E) +�� +� +z + η − +1 +2 +√ +2 +�√ +2 + +� +1 − 4a2 − Ea4 − m1(E) +�� +� +z + η − +1 +2 +√ +2 +�√ +2 − +� +1 − 4a2 − Ea4 + m1(E) +�� +� +z + η − +1 +2 +√ +2 +�√ +2 + +� +1 − 4a2 − Ea4 + m1(E) +�� +, +where +m1(E) = +� +(4a2 + Ea4 + 1)2 − 4(4a2 + E(E + a4)). +Imposing the constraints (3), we have +21 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +1. +η1(E) = +1 +2 +√ +2 +�√ +2 − +� +1 − 4a2 − Ea4 − m1(E) +� +. +This η value gives the following two sets of +energies and corresponding structure functions +E1,ϵ =1 +2(p + 1) +� +−(p + 1) a4 + ϵ +� +(a2 +4 − 4)(p + 1)2 + 16(1 − a2) +� +, +Φ(IV ) +E1,ϵ (z) =z(z − p − 1) +� +z − +1 +2 +√ +2 +� +1 − 4a2 − E1,ϵa4 − m1E1,ϵ) − +ia1 +4√a3 +� +� +z − +1 +2 +√ +2 +� +1 − 4a2 − E1,ϵa4 − m1(E1,ϵ) + +ia1 +4√a3 +� +� +z − +1 +2 +√ +2 +�� +1 − 4a2 − E1,ϵa4 − m1(E1,ϵ) − +� +1 − 4a2 − E1,ϵa4 + m1(E1,ϵ +�� +� +z − +1 +2 +√ +2 +�� +1 − 4a2 + E1,ϵa4 − m1(E1,ϵ) + +� +1 − 4a2 − E1,ϵa4 + m1(E1,ϵ) +�� +, +E2,ϵ = − +1 +a4 + ϵ 4 +� +4a2 + 4p2 + 8p + 3 +� +, +a4 ̸= ±4, +ΦE2,ϵ(z) =z(z − p − 1) +� +z − +1 +2 +√ +2 +� +1 − 4a2 − E2,ϵa4 − m1E2,ϵ) − +ia1 +4√a3 +� +� +z − +1 +2 +√ +2 +� +1 − 4a2 − E2,ϵa4 − m1(E1,ϵ) + +ia1 +4√a3 +� +� +z − 1 +√ +2 +�� +1 − 4a2 − E2,ϵa4 − m1(E2,ϵ) +�� +� +z − +1 +2 +√ +2 +�� +1 − 4a2 + E2,ϵa4 − m1(E2,ϵ) + ϵ +� +1 − 4a2 − E2,ϵa4 + m1(E2,ϵ) +�� +, +where ϵ = ±1. Notice that the energies E1,ϵ are real for a2 +4 > 4, a2 < 1. +2. η2(E) = +1 +2 +√ +2 +�√ +2 − +� +1 − 4a2 − Ea4 + m1(E) +� +. The energies are the same as those given in case +1 above +2.4.2 +Potential V2(u, v) +The constants of motion of the superintegrable system in Darboux space IV with the Hamiltonian ˆH = +− +sin2(2u)(∂2 +v+∂2 +u) +2 cos(2u)+b4 ++ V2(u, v) are [14], +A = e−2v +��e4v + 1 +� (2b4 cos(2u) + 3 cos(4u) + 1) +2b4 + 4 cos(2u) +∂2 +v − sin(2u) +�e4v + 1 +� sin(2u) +b4 + 2 cos(2u) ∂2 +u +� ++ e−2v � +sin(2u) +� +e4v + 1 +� +∂u + sin(2u) +� +e4v − 1 +� +∂u∂v + cos(2u) +� +e4v − 1 +� +∂v +� ++ +1 +2 cos 2u + b4 +� +2b1 cosh 2v + (b2 + b3)(4 − b2 +4) + (cos 4u + 2b4 cos 2u + 3) +� +b2 +sinh2 v − +b3 +cosh2 v +�� +, +B = ∂2 +v + +b2 +sinh2 v + +b3 +cosh2 v. +These integrals generates the quadratic algebra the commutation relations as follows +[A, B] =C, +[A, C] =8{A, B} + 16b4(b2 + b3)A − 16B + 32(b1 + b3) ˆH − 16b4(b2 + b3), +[B, C] =8B2 + 96A2 + +� +64b4 ˆH + (2b2 − 2b3 + b1 + 3) +� +A + 32 ˆH2 + 32b4(2b2 − 2b3 + 1) ˆH ++ 64b1(b2 − b3) − 8(b2 +4 − 4)(b2 + b3)2 + 32(b1 + 2b2 − 2b3). +22 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +By a direct computation, we find the Casimir operator of the algbebra, +K2 = C2 + 64A3 − 8{A, B2} − 16b4(b2 + b3){A, B} + 64 +� +b4 ˆH + 2b2 − 2b3 + b1 + 7 +� +A2 ++ +� +160b4(b2 + b3) − 64(b2 + b3) ˆH +� +B − 64b4(2b3 − 2b2 − 1) ˆHA +− 16 +� +(b2 +4 − 4)(b2 + b3)2 + 8(b1 + 1)(b3 − b2) − 4b1 + 32 +� +A + 64 ˆH2A. +With the differential realization of A and B, the Casimir K2 is expressible in terms of ˆH as follows +K2 =128(b3 − b2 + 1) ˆH2 + 128b4(b2 − b3 + 1) ˆH ++ (128 − 80b2 +4 − 64b1)(b2 + b3)2 − 128(b1 + 2)(b3 − b2 − 1) − 256. +After a long computation, we find that the realization +A = −4 +� +(N + η)2 − 1 +2 +� +, +B = b4(b2 + b3) +8 +− −32 · 16b4(b2 + b3) − 256 · (b1 + 2b2 − 2b3) +4γ3 +� +(N + η)2 − 1 +4 +� ++ ρ(N)b† + bρ(N), +where +ρ(N) = +1 +3 · 212 · (−8)8(N + η)(1 + N + η)(1 + 2(N + η))2 , +converts the quadratic algebra into the deformed oscillator algebra with the structure function +Φ(IV ) +2 +(N, η) =268435456 +� +(2N + 2η − 1)2 � +48(5a2 + 4b1 − 8)(b2 + b3)2 ++384 +� +(b1 + 2)(b2 − b3 + 1) + ˆH2(−b2 + b3 + 1) + ˆHb4(b2 − b3 + 1) − 2 +�� ++ 64 (2N + 2η − 1)2 � +12(N + η)2 − 12(N + η−) − 1 +� +× +� +b1(2b2 − 2b3 + 3) + b2 +2 + 2b2(b3 + ˆHb4 + 3) + b2 +3 − 2b3 ˆHb4 − 6b3 + ˆH2 + 3 ˆHb4 + 9 +� ++ (2N + 2η + 1)(2N + 2η − 1)6(b1 + 2b2 − 2b3 + ˆHb4 + 6) +× +� +986 b1(b2 − b3) + 448 +� +b1 + 2b2 − 2b3 + ˆHb4(2b2 − 2b3 + 1) +� +− 112 +� +b2 +4 − 4 +� +(b2 + b3)2 + 448 ˆH2 + 96 b4(b2 + b3) +� +2 ˆH(b1 + b3) − b4(b2 + b3) +� ++704 +� +b1 + 2b2 − 2b3 + ˆHb4 + 3 +� +− 32b2 +4(b2 + b3)2 + 192 +� ++192 +� +2N + 2η − 3 + ˆH2(b1 + b3)2 − (2N + 2η − 1)4 � +4(N + η)2 − 4(N + η) − 3 +�2�� +. +23 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +Acting on Fock basis states |z, E⟩, the structure function is factorized as follows +Φ(IV ) +2 +(z, η) = +� +z + η − +1 +2 +√ +2 +�√ +2 − +� +1 − 2b2 + 2b3 − +� +(1 − 4b2)(1 + 4b3) +�� +� +z + η − +1 +2 +√ +2 +�√ +2 + +� +1 − 2b2 + 2b3 − +� +(1 − 4b2)(1 + 4b3) +�� +� +z + η − +1 +2 +√ +2 +�√ +2 − +� +1 − 2b2 + 2b3 + +� +(1 − 4b2)(1 + 4b3) +�� +� +z + η − +1 +2 +√ +2 +�√ +2 + +� +1 − 2b2 + 2b3 + +� +(1 − 4b2)(1 + 4b3) +�� +� +z + η − +1 +2 +√ +2 +�√ +2 − +� +1 − b1 − Eb4 − m2(E) +�� +� +z + η − +1 +2 +√ +2 +�√ +2 + +� +1 − b1 − Eb4 − m2(E) +�� +� +z + η − +1 +2 +√ +2 +�√ +2 − +� +1 − b1 − Eb4 + m2(E) +�� +� +z + η − +1 +2 +√ +2 +�√ +2 + +� +1 − b1 − Eb4 + m2(E) +�� +, +where +m2(E) = +� +(1 + b1 + Eb4)2 − 4 (b1 + E2 + Eb4). +Imposing the constraint conditions (3), we determine the parameter η and the corresponding energies +of the system. We find +Case 1: η1,± = +1 +2 +√ +2 +�√ +2 − +� +1 − 2b2 + 2b3 ± +� +(1 − 4b2)(1 + 4b3) +� +and the energies E satisfy +2 +√ +2(p + 1) − n2,± = +� +1 − b1 − Eb4 + m2(E), +where +n2,± = +� +1 − 2b2 + 2b3 ± +� +(1 − 4b2)(1 + 4b3). +Noticing (1 − 4b2)(1 + 4b3) = (1 − 2b2 + 2b3)2 − (2b2 + 2b3)2 ≤ (1 − 2b2 + 2b3)2, we conclude that both +n2,± are real if b2 < 1 +4, b3 > −1 +4. +Solving the algebraic equations give the energies of the system and the corresponding structure func- +tions of the (p + 1)-dimensional unirreps of the algebra. We have +E1±,ϵ = − b4 +4 +� +2 +√ +2(p + 1) − n2,± +�2 ++ ϵ +� +2 +√ +2(p + 1) − n2,± +� � +8(1 − b1) + (b2 +4 − 4) +� +2 +√ +2(p + 1) − n2,± +�2, +Φ(IV ) +E1±,ϵ(z, η) =z (z − p − 1) +� +z − 1 +√ +2n± +� � +z − +1 +2 +√ +2(n± − n∓) +� � +z − +1 +2 +√ +2(n± + n∓) +� +� +z − +1 +2 +√ +2 +� +n± − +� +1 − b1 − E1±,ϵb4 − m2(E1±,ϵ) +�� +� +z − +1 +2 +√ +2 +� +n± + +� +1 − b1 − E1±,ϵb4 − m2(E1±,ϵ) +�� +� +z − +1 +2 +√ +2 +� +n± − +� +1 − b1 − E1±,ϵb4 + m2(E1±,ϵ) +�� +, +where ϵ = ±1. The energies E1±,ϵ are real for the model parameters b2 < 1 +4, b3 > −1 +4, b1 < 1, b2 +4 > 4. +24 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +Case 2: η1,± = +1 +2 +√ +2 +�√ +2 + +� +1 − 2b2 + 2b3 ± +� +(1 − 4b2)(1 + 4b3) +� +and the energies E satisfy +2 +√ +2(p + 1) + n2,± = +� +1 − b1 − Eb4 + m2(E). +Solutions of the equations give the energies of the system and the corresponding structre functions of the +(p + 1)-dimensional unirreps of the algebra. We have +E2±,ϵ = − b4 +4 +� +2 +√ +2(p + 1) + n2,± +�2 ++ ϵ +� +2 +√ +2(p + 1) + n2,± +� � +8(1 − b1) + (b2 +4 − 4) +� +2 +√ +2(p + 1) + n2,± +�2, +Φ(IV ) +E2±,ϵ(z, η) =z (z − p − 1) +� +z + 1 +√ +2n± +� � +z + +1 +2 +√ +2(n± − n∓) +� � +z + +1 +2 +√ +2(n± + n∓) +� +� +z + +1 +2 +√ +2 +� +n± − +� +1 − b1 − E2±,ϵb4 − m2(E2±,ϵ) +�� +� +z + +1 +2 +√ +2 +� +n± + +� +1 − b1 − E2±,ϵb4 − m2(E2±,ϵ) +�� +� +z + +1 +2 +√ +2 +� +n± + +� +1 − b1 − E2±,ϵb4 + m2(E2±,ϵ) +�� +, +where ϵ = ±1. The energies E2±,ϵ are real for the model parameters b2 < 1 +4, b3 > −1 +4, b1 < 1, b2 +4 > 4. +Case 3: η3,−(E) = +1 +2 +√ +2 +�√ +2 − +� +1 − b1 − Eb4 + m2(E) +� +and +√ +2(p + 1) = +� +1 − b1 − Eb4 + m2(E) +or +2 +√ +2(p + 1) = +� +1 − b1 − Eb4 + m2(E) + +� +1 − b1 − Eb4 − m2(E). +The first algebraic equation gives the energies +E3,1 = p + 1 +2 +� +−(p + 1)b4 ± +� +4(1 − b1) + (b2 +4 − 4)(p + 1)2 +� +, +which is real for b1 < 1, b2 +4 > 4. The corresponding structure function is +Φ(IV ) +E3,1 (z, η) =z (z − p − 1) +� +z − +1 +2 +√ +2 +�� +1 − b1 − E3,1b4 + m2(E3,1) − n− +�� +� +z − +1 +2 +√ +2 +�� +1 − b1 − E3,1b4 + m2(E3,1) + n− +�� +� +z − +1 +2 +√ +2 +�� +1 − b1 − E3,1b4 + m2(E3,1) − n+ +�� +� +z − +1 +2 +√ +2 +�� +1 − b1 − E3,1b4 + m2(E3,1) + n+ +�� +� +z − +1 +2 +√ +2 +�� +1 − b1 − E3,1b4 + m2(E3,1) − +� +1 − b1 − E3,1b4 − m2(E3,1) +�� +� +z − +1 +2 +√ +2 +�� +1 − b1 − E3,1b4 + m2(E3,1) + +� +1 − b1 − E3,1b4 − m2(E3,1) +�� +. +The second algebraic equation yields the energies +E3,2 = +1 +2 − b4 +� +4(p + 1)2 + b1 − 1 +� +, +25 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +which is well defined for the model parameter b4 ̸= 2. The associated structure function is given by +Φ(IV ) +E3,2 (z, η) =z (z − p − 1) +� +z − +1 +2 +√ +2 +�� +1 − b1 − E3,2b4 + m2(E3,2) − n− +�� +� +z − +1 +2 +√ +2 +�� +1 − b1 − E3,2b4 + m2(E3,2) + n− +�� +� +z − +1 +2 +√ +2 +�� +1 − b1 − E3,2b4 + m2(E3,2) − n+ +�� +� +z − +1 +2 +√ +2 +�� +1 − b1 − E3,2b4 + m2(E3,2) + n+ +�� +� +z − +1 +2 +√ +2 +�� +1 − b1 − E3,2b4 + m2(E3,2) − +� +1 − b1 − E3,2b4 − m2(E3,2) +�� +� +z − 1 +√ +2 +� +1 − b1 − E3,2b4 + m2(E3,2) +� +. +Case 4: η4,+ = +1 +2 +√ +2 +�√ +2 + +� +1 − b1 − Eb4 − m2(E) +� +. We have the algebraic equation +2 +√ +2(p + 1) = +� +1 − b1 − Eb4 + m2(E) − +� +1 − b1 − Eb4 − m2(E). +Solving, we obtain +E4 = +1 +2 + b4 +� +4(p + 1)2 + b1 − 1 +� +. +This gives the energy spectrum of the system for the model parameter b4 ̸= −2. The corresponding +structure function is +Φ(IV ) +E4 (z, η) =z (z − p − 1) +� +z + +1 +2 +√ +2 +�� +1 − b1 − E4b4 − m2(E4) − n− +�� +� +z + +1 +2 +√ +2 +�� +1 − b1 − E4b4 − m2(E4) + n− +�� +� +z + +1 +2 +√ +2 +�� +1 − b1 − E4b4 − m2(E4) − n+ +�� +� +z + +1 +2 +√ +2 +�� +1 − b1 − E4b4 − m2(E4) + n+ +�� +� +z + 1 +√ +2 +� +1 − b1 − E4b4 − m2(E4) +� +� +z + +1 +2 +√ +2 +�� +1 − b1 − E4b4 − m2(E4) + +� +1 − b1 − E4b4 + m2(E4) +�� +. +26 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +2.4.3 +Potential V3(ω, ϕ) +The constants of motion of the superintegrable system in Darboux space IV with the potential V3(ω, ϕ) +are [14] +A = − 2c4 +∂2 +ϕ + ∂2 +ω +c4+2 +sinh2(2ω) + +c4−2 +sin2(2ϕ) ++ (c4 + 2) sin2(2ϕ)∂2 +ϕ − (c4 − 2) sinh2(2ω)∂2 +ω +(c4 + 2) sin2(2ϕ) + (c4 − 2) sinh2(2ω) ++ +1 +c4+2 +sinh2(2ω) + c4−2 +sin2 ω +� +c4 + 2 +sinh2(2ω) +� +c3 +sin2 ϕ + +c1 +cos2 ϕ +� ++ +c4 − 2 +sin2(2ω) +� +c3 +sinh2 ω − +c2 +cosh2 ω +�� +, +B =1 +2 sin(2ϕ) sinh(2ω) tan(ϕ − iω) tan(ϕ + iω) +� +cot(2ϕ) ∂2 +ω + coth(2ω) ∂2 +ϕ +� ++ +� +−i cos(2ϕ) sinh(2ω) sinh +� +log +�tan(ϕ − iω) +tan(ϕ + iω) +�� +∂ω ++i cosh(2ω) sin(2ϕ) sinh +� +log +�tan(ϕ − iω) +tan(ϕ + iω) +�� +∂ϕ + 2 cosh +� +log +�tan(ϕ − iω) +tan(ϕ + iω) +��� ++ +1 +c4+2 +sinh2 2ω + c4−2 +sin2 ω +� +� c4 + 2 +sinh2 2ω +� +�c1 cosh 2ω tan2 ϕ − c2 cos 2ϕ − +c3 +� +2 cos2 ϕ(sinh2 ω − sin2 ϕ) +� ++ 1 +sin2 ϕ +� +� ++ c4 − 2 +sin2 2ϕ +� +�c2 cos 2ϕ tanh2 ω + c1 cosh 2ω − +c3 +� +2 cosh2 ω(sinh2 ω − sin2 ϕ) + 1 +� +sinh2 ω +� +� +� +� . +These integrals form the quadratic algebra with the following commutation relations +[A, B] =C, +[A, C] = − 8{A, B} − 16B − 16(c1 − c3)(c2 − c3), +[B, C] = − 24A2 + 8B2 + 16 +� +2c4 ˆH − 2c1 + 2c2 + 3 +� +A − 16 [(c4 + 2)c1 + (c4 − 2)c2 − c4 + 64c3] ˆH +− 8(c2 +4 − 4) ˆH2 − 8c2 +1 − 8c2 +2 + 16c2 +3 + 32c1c2 + 48c3(c1 + c2) − 16(c1 − c2), +which is the symmetry algebra of the superintegrable system. We can calculate the Casimir operator of +the algebra +K3 =C2 − 16A3 + 8{A, B2} − 16(2c2 − 2c1 − 7)A2 + 80B2 +− 16 +� +c2 +4 − 4 + 2(c4 + 2)c1 − 2(c4 − 2)c2 + 8c3 + 2c4 +� ˆH ++ 16 +� +c2 +1 + c2 +2 − 2c2 +3 − 6c3(c1 + c2) − 4c1c2 + 2c1 − 2c2 − 8 +� +A + 32(c2 − c3)(c1 − c3)B. +We can show that in terms of the Hamiltonian the Casimir K3 takes the simple form +K3 =16(c2 +4 − 4) ˆH2 − 16 +� +(c4 + 2)((c1 − c3)2 − 2c1) + (c4 − 2)((c2 − c3)2 + 2c2) − 8c3 − 4c4 +� ˆH +− 32(c1 − c2)(3c2 +3 − c1c2 − c3(c1 + c2)) + 32(c2 +1 + c2 +2 − 4c3(c1 + c2) − 2c1c2 + 2c1 − 2c2). +After a long computation, we find that the realization +A(N) = −4(N + η)2, +B = −(c1 − c3)(c2 − c3) +4 +� +(N + η)2 − 1 +4 +� + ρ(N)b + b†ρ(N) +with +ρ(N) = +1 +3 · 212 · (−8)8(N + η)(1 + N + η)(1 + 2(N + η))2 , +27 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +changes the quadratic algebra to the deformed oscillator algebra with the structure function +Φ(IV ) +3 +(N, η) =268435456 +� +16 +� +12N 2 + 12N(2η − 1) + 12η2 − 12η − 1 +� +(2N + 2η − 1)2 +× +� +c2 +1 + c1 +� +−4c2 − 6c3 + 2 ˆHc4 + 4 ˆH − 2 +� ++ c2 +2 + c2 +� +−6c3 − 2 ˆHc4 + 4 ˆH + 2 +� +−2c2 +3 + 128c3 ˆH + ˆH2c2 +4 − 4 ˆH2 + 6 ˆHc4 + 9 +� ++ 16 (2N + 2η − 1)2 � +7c2 +1 − 2c1 +� +14c2 + 21c3 − 7 ˆHc4 − 14 ˆH + 4 +� ++ 7c2 +2 ++c2 +� +−42c3 − 14 ˆH(c4 − 2) + 8 +� +− 14c2 +3 + 896c3 ˆH + 7 ˆH2c2 +4 − 28 ˆH2 + 36 ˆHc4 + 36 +� +− 3 (2N + 2η − 1)2 � +352 ˆH +� +−c1 + (c4 − 2)(c2 − c3)2 + 2c2(c4 − 2) − c2 +3 − 8c3 − 4c4 +� ++32(c1 − c2) (c1(c2 + c3) + c3(c2 − 3c3)) − 16 ˆH2 � +c2 +4 − 4 +�� ++ 48(c1 − c3)2(c2 − c3)2 +− 96 (2N + 2η − 3)(2N + 2η + 1)(2N + 2η − 1)4(c1 − c2 − ˆHc4 − 3) ++48 (2N + 2η − 1)4 � +(2N + 2η − 1)4 − 8(2N + 2η − 1)2 + 16 +�� +. +The structure function is a polynomial of degree 8 in N. Acting on the Fock basis states |z, E⟩, it becomes a +polynomial of degree 8 in z. In order to determine the energy spectrum of the superintegrable system, we have +to find the finite-dimensional unirreps of the deformed oscillator algebra by solving the constraints. This requires +the factorization of the structure function. However, it turns out to be very difficult to factorize the structure +function for general model parameters ci (even using symbolic computation softwares such as Mathematica). In +the following we restrict our attention to special model parameters and present analytic and closed-form results +for c1 = c2 = 1, c3 = c4 = 0. +In this case, we find that the structure function factorizes as +Φ(IV ) +3 +(z, η) = +� +z + η − +1 +2 +√ +2 +�√ +2 − +� +− +� +4E2 − 12E + 5 − 2E + 3 +�� +� +z + η − +1 +2 +√ +2 +�√ +2 + +� +− +� +4E2 − 12E + 5 − 2E + 3 +�� +� +z + η − +1 +2 +√ +2 +�√ +2 − +�� +4E2 − 12E + 5 − 2E + 3 +�� +� +z + η − +1 +2 +√ +2 +�√ +2 + +�� +4E2 − 12E + 5 − 2E + 3 +�� +� +z + η − +1 +2 +√ +2 +�√ +2 − +� +− +� +4E2 − 4E − 3 + 2E − 1 +�� +� +z + η − +1 +2 +√ +2 +�√ +2 + +� +− +� +4E2 − 4E − 3 + 2E − 1 +�� +� +z + η − +1 +2 +√ +2 +�√ +2 − +�� +4E2 − 4E − 3 + 2E − 1 +�� +� +z + η − +1 +2 +√ +2 +�√ +2 + +�� +4E2 − 4E − 3 + 2E − 1 +�� +. +Imposing the constraints (3), we determine the constant η and obtain the energies of the system and the structure +function of the symmetry algebra for the model parameters c1 = c2 = 1, c3 = c4 = 0 as follows. +a. The constant η is ηa(E) = +1 +2 +√ +2 +�√ +2 − +� +− +√ +4E2 − 12E + 5 − 2E + 3 +� +and the energies are given by the +equation +2 +√ +2(p + 1) = +�� +4E2 − 12E + 5 − 2E + 3 + +� +− +� +4E2 − 12E + 5 − 2E + 3 +=⇒ +Ea = −2(p + 1)2 + 5 +2. +28 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +The associated structure function is +Φ(IV ) +Ea (z, η) = z (z − p − 1) +� +z − 1 +√ +2 +� +− +� +4E2a − 12E2a + 5 − 2Ea + 3 +� +� +z − +1 +2 +√ +2 +�� +− +� +4E2a − 12Ea + 5 − 2Ea + 3 − +�� +4E2a − 12Ea + 5 − 2Ea + 3 +�� +� +z − +1 +2 +√ +2 +�� +− +� +4E2a − 12Ea + 5 − 2Ea + 3 − +� +− +� +4E2a − 4Ea − 3 + 2Ea − 1 +�� +� +z − +1 +2 +√ +2 +�� +− +� +4E2a − 12Ea + 5 − 2Ea + 3 + +� +− +� +4E2a − 4Ea − 3 + 2Ea − 1 +�� +� +z − +1 +2 +√ +2 +�� +− +� +4E2a − 12Ea + 5 − 2Ea + 3 − +�� +4E2a − 4Ea − 3 + 2Ea − 1 +�� +� +z − +1 +2 +√ +2 +�� +− +� +4E2a − 12Ea + 5 − 2Ea + 3 + +�� +4E2a − 4Ea − 3 + 2Ea − 1 +�� +. +b. The constant η is given by ηb(E) = +1 +2 +√ +2 +�√ +2 − +�√ +4E2 − 12E + 5 − 2E + 3 +� +and the energies are +√ +2(p + 1) = +�� +4E2 − 12E + 5 − 2E + 3 +=⇒ +Eb = −1 +2 +� +(p + 1)2 − 3 + +1 +(p + 1)2 +� +. +The corresponding structure function of the (p + 1)-dimensional unirreps is +Φ(IV ) +Eb +(z, η) = z (z − p − 1) +� +z − 1 +√ +2 +�� +4E2 +b − 12Eb + 5 − 2Eb + 3 +� +� +z − +1 +2 +√ +2 +��� +4E2 +b − 12Eb + 5 − 2Eb + 3 − +� +− +� +4E2 +b − 12Eb + 5 − 2Eb + 3 +�� +� +z − +1 +2 +√ +2 +��� +4E2 +b − 12Eb + 5 − 2Eb + 3 − +� +− +� +4E2 +b − 4Eb − 3 + 2Eb − 1 +�� +� +z − +1 +2 +√ +2 +��� +4E2 +b − 12Eb + 5 − 2Eb + 3 + +� +− +� +4E2 +b − 4Eb − 3 + 2Eb − 1 +�� +� +z − +1 +2 +√ +2 +��� +4E2 +b − 12Eb + 5 − 2Eb + 3 − +�� +4E2 +b − 4Eb − 3 + 2Eb − 1 +�� +� +z − +1 +2 +√ +2 +��� +4E2 +b − 12Eb + 5 − 2Eb + 3 + +�� +4E2 +b − 4Eb − 3 + 2Eb − 1 +�� +. +c. The constant η is η2c(E) = +1 +2 +√ +2 +�√ +2 + +� +− +√ +4E2 − 12E + 5 − 2E + 3 +� +and the energy E satisfies +2 +√ +2(p + 1) = +�� +4E2 − 12E + 5 − 2E + 3 − +� +− +� +4E2 − 12E + 5 − 2E + 3 +=⇒ +Ec = −2(p + 1)2 + 1 +2. +The structure function is given by +Φ(IV ) +Ec +(z, η) = z (z − p − 1) +� +z + 1 +√ +2 +� +− +� +4E2c − 12Ec + 5 − 2Ec + 3 +� +� +z + +1 +2 +√ +2 +�� +− +� +4E2c − 12Ec + 5 − 2Ec + 3 − +�� +4E2c − 12Ec + 5 − 2Ec + 3 +�� +� +z + +1 +2 +√ +2 +�� +− +� +4E2c − 12Ec + 5 − 2Ec + 3 − +� +− +� +4E2c − 4Ec − 3 + 2Ec − 1 +�� +� +z + +1 +2 +√ +2 +�� +− +� +4E2c − 12Ec + 5 − 2Ec + 3 + +� +− +� +4E2c − 4Ec − 3 + 2Ec − 1 +�� +� +z + +1 +2 +√ +2 +�� +− +� +4E2c − 12Ec + 5 − 2Ec + 3 − +�� +4E2c − 4Ec − 3 + 2Ec − 1 +�� +� +z + +1 +2 +√ +2 +�� +− +� +4E2c − 12Ec + 5 − 2Ec + 3 + +�� +4E2c − 4Ec − 3 + 2Ec − 1 +�� +. +29 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +d. The constant η is ηd(E) = +1 +2 +√ +2 +�√ +2 − +� +− +√ +4E2 − 4E − 3 + 2E − 1 +� +and the energies are +2 +√ +2(p + 1) = +�� +4E2 − 4E − 3 + 2E − 1 + +� +− +� +4E2 − 4E − 3 + 2E − 1 +=⇒ +Ed = 2(p + 1)2 − 1 +2. +The corresponding structure function reads +Φ(IV ) +d +(z, η) = z (z − p − 1) +� +z − +1 +2 +√ +2 +�� +− +� +4E2 +d − 4Ed − 3 + 2Ed − 1 − +� +− +� +4E2 +d − 12Ed + 5 − 2Ed + 3 +�� +� +z − +1 +2 +√ +2 +�� +− +� +4E2 +d − 4Ed − 3 + 2Ed − 1 + +� +− +� +4E2 +d − 12Ed + 5 − 2Ed + 3 +�� +� +z − +1 +2 +√ +2 +�� +− +� +4E2 +d − 4Ed − 3 + 2Ed − 1 − +�� +4E2 +d − 12Ed + 5 − 2Ed + 3 +�� +� +z − +1 +2 +√ +2 +�� +− +� +4E2 +d − 4Ed − 3 + 2Ed − 1 + +�� +4E2 +d − 12Ed + 5 − 2Ed + 3 +�� +� +z − 1 +√ +2 +� +− +� +4E2 +d − 4Ed − 3 + 2Ed − 1 +� +� +z − +1 +2 +√ +2 +�� +− +� +4E2 +d − 4Ed − 3 + 2Ed − 1 − +�� +4E2 +d − 4Ed − 3 + 2Ed − 1 +�� +. +e. The constant η is ηe(E) = +1 +2 +√ +2 +�√ +2 − +�√ +4E2 − 4E − 3 + 2E − 1 +� +and the energies and structure functions +are given by +√ +2(p + 1) = +�� +4E2 − 4E − 3 + 2E − 1 +=⇒ +Ee = 1 +2 +� +(p + 1)2 + 1 + +1 +(p + 1)2 +� +, +Φ(IV ) +e +(z, η) = z (z − p − 1) +� +z − +1 +2 +√ +2 +��� +4E2e − 4Ee − 3 + 2Ee − 1 − +� +− +� +4E2e − 12Ee + 5 − 2Ee + 3 +�� +� +z − +1 +2 +√ +2 +��� +4E2e − 4Ee − 3 + 2Ee − 1 + +� +− +� +4E2e − 12Ee + 5 − 2Ee + 3 +�� +� +z − +1 +2 +√ +2 +��� +4E2e − 4Ee − 3 + 2Ee − 1 − +�� +4E2e − 12Ee + 5 − 2Ee + 3 +�� +� +z − +1 +2 +√ +2 +��� +4E2e − 4Ee − 3 + 2Ee − 1 + +�� +4E2e − 12Ee + 5 − 2Ee + 3 +�� +� +z − 1 +√ +2 +�� +4E2e − 4Ee − 3 + 2Ee − 1 +� +� +z − +1 +2 +√ +2 +��� +4E2e − 4Ee − 3 + 2Ee − 1 − +� +− +� +4E2e − 4Ee − 3 + 2Ee − 1 +�� +. +f. The constant η is ηf(E) = +1 +2 +√ +2 +�√ +2 + +� +− +√ +4E2 − 4E − 3 + 2E − 1 +� +and the energies are +2 +√ +2(p + 1) = +�� +4E2 − 4E − 3 + 2E − 1 − +� +− +� +4E2 − 4E − 3 + 2E − 1 +=⇒ +Ef = 2(p + 1)2 + 3 +2. +30 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +The structure function of the (p + 1)-dimensional unirreps has the form +Φ(IV ) +f +(z, η) = z (z − p − 1) +� +z + +1 +2 +√ +2 +�� +− +� +4E2 +f − 4Ef − 3 + 2Ef − 1 − +� +− +� +4E2 +f − 12Ef + 5 − 2Ef + 3 +�� +� +z + +1 +2 +√ +2 +�� +− +� +4E2 +f − 4Ef − 3 + 2Ef − 1 + +� +− +� +4E2 +f − 12Ef + 5 − 2Ef + 3 +�� +� +z + +1 +2 +√ +2 +�� +− +� +4E2 +f − 4Ef − 3 + 2Ef − 1 − +�� +4E2 +f − 12Ef + 5 − 2Ef + 3 +�� +� +z + +1 +2 +√ +2 +�� +− +� +4E2 +f − 4Ef − 3 + 2Ef − 1 + +�� +4E2 +f − 12Ef + 5 − 2Ef + 3 +�� +� +z + 1 +√ +2 +� +− +� +4E2 +f − 4Ef − 3 + 2Ef − 1 +� +� +z + +1 +2 +√ +2 +�� +− +� +4E2 +f − 4Ef − 3 + 2Ef − 1 − +�� +4E2 +f − 4Ef − 3 + 2Ef − 1 +�� +. +3 +New superintegrable systems in 2D Darboux spaces +In this section, we investigate superintegrable systems in 2D Darboux spaces with linear and quadratic or quintic +integrals of motion. We will first construct generic cubic and quintic algebras and derive their Casimir operators +and realizations in terms of the deformed oscillator algebras. We will then present examples of new superintegrable +systems in 2D Darboux spaces with cubic symmetry algebras. +3.1 +Generic cubic and quintic algebras generated by linear, quadratic or quintic +integrals +We start with the construction of generic cubic and quintic algebras with structure coefficients involving the +Hamiltonians. +Let ˆX1, ˆY1 be linear integrals, and let ˆX2, ˆY2 be quadratic and cubic integrals, respectively. That is, deg ˆX1 = +1 = deg ˆY1, deg ˆX2 = 2, deg ˆY2 = 3. We define the operators ˆF and ˆG by ˆF = [ ˆX1, ˆX2] and ˆG = [ ˆY1, ˆY2]. Then +deg ˆF = deg ˆX1 + deg ˆX2 − 1 = 2 and deg ˆG = deg ˆY1 + deg ˆY2 − 1 = 3. By analysing the degrees of the integrals +and applying the Jacobi identity constraint [34], we obtain the following generic cubic and quintic algebras +Proposition 3.1. Integrals { ˆX1, ˆX2, ˆF} satisfy the cubic commutation relations, +[ ˆX1, ˆX2] = ˆF, +[ ˆX1, ˆF] =u1 ˆX2 +1 + u2 ˆX1 + u3 ˆX2 + u, +[ ˆX2, ˆF] =v1 ˆX3 +1 + v2 ˆX2 +1 + v3 ˆX1 − u2 ˆX2 − u1{ ˆX1, ˆX2} + v, +(16) +and integrals { ˆY1, ˆY2, ˆG} form the following quintic commutation relations, +[ ˆY1, ˆY2] = ˆG, +[ ˆY1, ˆK] =α ˆY 3 +1 + β ˆY 2 +1 + δ ˆY1 + ϵ ˆY2 + ζ, +[ ˆY2, ˆK] =a ˆY 5 +1 + b ˆY 4 +1 + c ˆY 3 +1 + d ˆY 2 +1 + e ˆY1 + 1 +2 (α ϵ − 2 δ) ˆY2 − 3 +2α{ ˆY 2 +1 , ˆY2} − β{ ˆY1, ˆY2} + z, +(17) +where uj, vj, . . . , α, . . . , z are polynomials of the Hamiltonian ˆH. Moreover, the coefficients v1 in (16)and a in (17) +are not zero polynomials of ˆH. +The proof of this proposition is a short and straightforward computation from the Jacobi identity requirement. +Remark 3.2. For the polynomials on both sides of the commutation relations (16) and (17) to have the same degree, +we must have that v1, v2, u1, u2, u3, α, β, ϵ, a, b are constants and +u = u(0) + u(1) ˆH, +v3 = v(0) +3 ++ v(1) +3 +ˆH, +v = v(0) + v(1) ˆH, +δ = δ(0) + δ(1) ˆH, +ζ = ζ(0) + ζ(1) ˆH, +c = c(0) + c(1) ˆH, +d = d(0) + d(1) ˆH, +e = e(0) + e(1) ˆH + e(2) ˆH2, +z = z(0) + z(1) ˆH + z(2) ˆH2, +31 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +where u(0), u(1), . . . , are constants. +We now construct the Casimir operators for both polynomial algebras. We have +Proposition 3.3. The Casimir operators C(3) and C(5) for the cubic and quintic algebras are respectively given by +C(3) = ˆF 2 − u1{ ˆX2 +1, ˆX2} − u2{ ˆX1, ˆX2} + v1 +2 +ˆX4 +1 + 2 +3v2 ˆX3 +1 + +� +v3 + u2 +1 +� ˆX2 +1 + (u1u2 + 2v) ˆX1 − 2u ˆX2 − u3 ˆX2 +2, +C(5) = ˆG2 − α{ ˆY 3 +1 , ˆY2} + β{ ˆY 2 +1 , ˆY2} − δ{ ˆY1, ˆY2} − ϵ ˆY 2 +2 − 2ζ ˆY2 + a +3 +ˆY 6 +1 + 2 +5b ˆY 5 +1 ++ 1 +2 +� +c + 5 +3aϵ + 3αδ +� +ˆY 4 +1 + +� +2β(δ + 3α) + 2 +5ϵb − 2d +� +ˆY 3 +1 ++ +�1 +6 (5 − 6a) ϵ2 + e + 1 +2ϵc + β2 − 3 +4α (αϵ − 2δ) +� +ˆY 2 +1 + +� +2z + βδ + βϵ(α + δ) − 1 +5bϵ2 − ϵd +� +ˆY1. +Proof. By analysinf the degrees of the integrals in the algebras, we see that the Casimir operators C(3) and C(5) of +the cubic and quintic algebras have the following general form +C(3) = ˆF 2 + w1{ ˆX2 +1, ˆX2} + w2{ ˆX1, ˆX2 +2} + w3{ ˆX1, ˆX2} + w4 ˆX4 +1 ++ w5 ˆX3 +1 + w6 ˆX2 +1 + w7 ˆX1 + w8 ˆX2 + w9 ˆX2 +2, +C(5) = ˆG2 + ω1{ ˆY 3 +1 , ˆY2} + ω2{ ˆY 2 +1 , ˆY2} + ω3{ ˆY1, ˆY 2 +2 } + ω4{ ˆY1, ˆY2} ++ ω5 ˆY 2 +2 + ω6 ˆY2 + ω7 ˆY 6 +1 + ω8 ˆY 5 +1 + ω9 ˆY 4 +1 + ω10 ˆY 3 +1 + ω11 ˆY 2 +1 + ω12 ˆY1, +where wj and ωj are coefficients. +Now using the quadratic commutation relations (16), we have +[C(3), ˆX1] = − (u1 + w1){ ˆF, ˆX2 +1} − (w3 + u2 + w2u1){ ˆF, ˆX1} − (w9 + u3){ ˆF, ˆX2} +− (2u + w8 + w2u2) ˆF − w2{ ˆF, { ˆX1, ˆX2}} + ˆX1(w1){ ˆX2 +1, ˆX2} + . . . + ˆX1(w9) ˆX2 +2. +Setting the coefficients to be zero gives +w1 = −u1, +w2 = 0, +w3 = −u2, +w8 = −2u, +w9 = −u3. +Similarly, from [C(3), ˆX2] = 0,, we have +0 =(2w4 − v1){ ˆF, ˆX3 +1} + (3w5 +2 +− v2){ ˆF, ˆX2 +1} + (w6 − v3 − u2 +1){ ˆF, ˆX1} ++ (w7 − u1u2 − 2v) ˆF. +This gives +w4 = v1 +2 , +w5 = 2v2 +3 , +w6 = v3 + u2 +1, +w7 = u1u2 + 2v. +This finishes the proof for C(3). +The derivation of C(5) is slightly more complicated. We express [C(5), ˆY1] and [C(5), ˆY2] in terms of { ˆG, ˆY n +1 } +and { ˆG, { ˆY1, ˆY2}}. By [34, Lemma 2] and quintic commutation relations, we have +[C(5), ˆY1] = − (α + ω1){ ˆG, ˆY 3 +1 } − +� +β + ω2 − 3αω3 +2 +� +{ ˆG, ˆY 2 +1 } − (δ + ω3β + ω4){ ˆG, ˆY1} +− (ϵ + ω5){ ˆG, ˆY2} − ω3{ ˆG, { ˆY1, ˆY2}} + +��αϵ +2 − δ +� +ω3 − 2ζ − ω6 +� +ˆG. +Setting the coefficients of { ˆK, { ˆY1, ˆY2}} and { ˆK, ˆY2} to be zero, we obtain that ω3 = 0 and ω5 = −ϵ. Then [C(5), ˆY1] +is reduced to the form +[C(5), ˆY1] =(α − ω1){ ˆG, ˆY 3 +1 } + (β − ω2){ ˆG, ˆY 2 +1 } + [δ − ω4)]{ ˆG, ˆY1} + (2ζ − ω6) ˆG. +From [C(5), ˆY1] = 0 it follows that the coefficients of { ˆG, ˆY l +1} are zero for all 1 ≤ n ≤ 3. Thus +ω1 = −α, ω2 = −β, ω4 = −δ and ω6 = −2ζ. +32 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +Similarly, after some manipulations we find +[C(5), ˆY2] =(3ω7 − a){ ˆG, ˆY 5 +1 } + +�5ω8 +2 +− b +� +{ ˆG, ˆY 4 +1 } − (c + 3αδ + 5ϵω7 − 2ω9){ ˆG, ˆY 3 +1 } ++ +�3α +2 +�αϵ +2 − δ +� +− β2 + 3αδϵ +2 ++ 3ϵ2ω7 − e − ϵω9 + ω11 +� +{ ˆG, ˆY1} +− +� +βδ + ϵω8 +2 +− 1 +2ω10 − 3αβ + d +� +{ ˆG, ˆY 2 +1 } ++ +� +β +�αϵ +2 − δ +� +− ϵω10 +2 ++ ϵ2ω8 + 3αβϵ +2 ++ (ω12 − 2z) +� +ˆG. +It follows from [C(5), ˆY2] = 0 that +ω7 = a +3, +ω8 = 2b +5 , +ω9 = 1 +2 +� +c + 5ϵ +3 + 3αδ +� +, +ω10 = 2(β(δ + 3α) + ϵb +5 − d) +ω11 = +�5 +6 − a +� +ϵ2 + e + ϵc +2 + β2 − 3α +2 +�αϵ +2 − δ +� +, +ω12 = 2z + βδ + βϵ(α + δ) − bϵ2 +5 − ϵd +as required. +we now construct realizations of these algebras in terms of the deformed oscillator algebras (1) and determine +their structure functions. After long computations, we obtain the following results. +Proposition 3.4. The realization +ˆX1 =√u3 (N + η), +ˆX2 = − u3 (N + η)2 − u2 +√u3 +(N + η) + b† + b − u +u3 +, +(18) +where η is a constant parameter to be determined, changes the cubic algebra (16) to the deformed oscillator algebra +(1) with the structure function given by +Φ(N, η) = +1 +1 − 2u3 +� +C(3) − u2 +u3 ++ uu2 +√u3 ++ √u3v + (N + η)2 � +−2uu1 + 2u1u2 +√u3 − u2 +2 + (u2 + v2)u3/2 +3 +− u3 +� ++ (N + η) +� +2uu1 − 2uu2 +√u3 +− √u3(u1u2 + 2v) + u2 +2 + u3v3 +� ++(N + η)3 +� +−2u1u2 +√u3 + 2u1u2 +3 − 2 +3v2u3/2 +3 ++ v1u2 +3 +� ++ (N + η)4 +� +−2u1u2 +3 + u3 +3 − 1 +2v1u2 +3 +�� +. +Note that Φ is a quartic polynomial of the number operator N. +Proposition 3.5. The transformation +ˆY1 =√ϵ(N + η), +ˆY2 = − α√ϵ(N + η)3 − β(N + η)2 − δ +√ϵ(N + η) + b† + b − ζ +ϵ , +(19) +where η is a constant parameter to be determined, maps the quintic algebra (17) to the deformed oscillator algebra +with the structure function +Φ(N, η) = 1 +4ϵC(5) + (N + η)6 +�aϵ4 +12 − 3α2ϵ3 +4 +� ++ (N + η)5 +�3α2ϵ +4 ++ 1 +2αβϵ5/2 − aϵ2 +4 + 1 +10bϵ7/2 +� ++ (N + η)4 +� +−1 +4αβ√ϵ + 1 +8ϵ3 +� +3αδ + 5aϵ +3 ++ c +� ++ 1 +2αδϵ2 + β2ϵ2 +4 +− 1 +4bϵ3/2 +� ++ (N + η)3 +� +−1 +4α(2αϵ − δ) − 3αδ +4 ++ 1 +2αζϵ3/2 − β2 +2 + 1 +2βδϵ3/2 − cϵ +4 + 1 +4ω10ϵ5/2 +� ++ (N + η)2 +�β(2αϵ − δ) +4√ϵ +− 3αζ +4√ϵ − βδ +2√ϵ + βζϵ +2 ++ δ2ϵ +4 − d√ϵ +4 ++ ω11ϵ2 +4 +� ++ (N + η) +�δ(2αϵ − δ) +4ϵ +− βζ +2ϵ + 1 +2δζ√ϵ − e +4 + 1 +4ω12ϵ3/2 +� ++ ζ2 +4 + ζ(2αϵ − δ) +4ϵ3/2 +. +The structure function Φ(N) is a polynomial of N of degree 6. +In the next subsection, we will present new superintegrable systems in 2D Darboux spaces with cubic symmetry +algebras. +33 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +3.2 +Superintegrable systems in 2D Darboux spaces with cubic symmetry algebras +In this subsection we obtain potentials in the 2D Darboux spaces which can be added to the Hamiltonians of +the free superintegrable systems studied in [11] and preserve their superintegrability. The free systems have only +kinetic terms and possess linear and quadratic integrals of motion. We will determine the integrals corresponding +to the superintegrable systems with potnetials. +3.2.1 +Darboux space I +The Hamiltonian of the free system in Darboux space I with separable local coordinates (x, y) studied in [11] has +the form H1 = ϕ1(x)(∂2 +x + ∂2 +y), where ϕ1(x) = +1 +αx+β . This system has linear integral X1 = ∂y and quadratic +integral given by +X2 = y∂x∂y − x∂2 +y + 1 +2∂x − 1 +4αy2ϕ1(x)(∂2 +x + ∂2 +y), +where α is a constant. +We seek new superintegrable system in Darboux space I with Hamiltonian +ˆH1 = H1 + V1(x, y), +where V1(x, y) is potential function, which preserves the separability of the coordinates and the superintegrability +of the original system. Without loss of generality, we assume that the local separable coordinates (x, y) is an +orthogonal system. After some computations, we find the allowed potential V1 and the corresponding integrals +ˆX1, ˆX2. The results are as follows. +ˆH1 = ϕ1(x)(∂2 +x + ∂2 +y) + c1ϕ1(x), +ˆX1 = ∂y, +ˆX2 = y∂x∂y − x∂2 +y + 1 +2∂x − 1 +4αy2ϕ1(x)(∂2 +x + ∂2 +y) − 1 +4c1αϕ1(x)y2 +where c1 is a constant. +By a direct calculation, we can show that the integrals ˆX1, ˆX2 form the cubic algebra, +[ ˆX1, ˆX2] = ˆF, +[ ˆX1, ˆF] = α +2 +ˆH1, +[ ˆX2, ˆF] = −2X3 +1 + α ˆH1X1 − c1X1, +(20) +where explicitly ˆF = ∂x∂y − 1 +2αyϕ1(x) +� +∂2 +x + ∂2 +y +� ++ 1 +2c1αϕ1(x)y. This cubic algebra is a special case of (16) in +Proposition 3.1 with +v1 = −2, +u1 = u2 = u3 = v2 = v = 0, +u = α +2 +ˆH1, +v3 = β ˆH1 − c1. +Then it follows that its Casimir operator is +C(3) = ˆF 2 − X4 +1 − α ˆH1 ˆX2 + (β ˆH1 − c1)X2 +1. +Since u3 = 0 it follows from Proposition 3.4 that this cubic algebra does not have realization in terms of the +deformed oscillator algebra. +3.2.2 +Darboux space II +The Hamiltonian of the free superintegrable system in 2D Darboux space II is H2 = ϕ2(x) +� +∂2 +x + ∂2 +y +� +, where +ϕ2(x) = +x2 +a2−a1x2 , a1, a2 ∈ R. The system possesses the following linear and quadratic integrals of motion, +X1 = ∂y, +X2 = 2xy∂x∂y + (y2 − x2)∂2 +y + x∂x + y∂y + a1y2H2. +It can be shown that we can add the potential V2(x, y) = c2 ϕ2(x), where c2 is a real constant, to the free +Hamiltonian such that +ˆH2 = ϕ2(x) +� +∂2 +x + ∂2 +y +� ++ c2 ϕ2(x) +is separable and superintegrable in the 2D Darboux space II, with integrals of motion given by +ˆX1 = ∂y, +ˆX2 = 2xy∂x∂y + (y2 − x2 + 1)∂2 +y + x∂x + y∂y + a1y2H2 + +a2c2y2 +a2 − a1x2 . +34 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +By a direct computation, we find that these integrals obey the cubic commutation relations +[ ˆX1, ˆX2] = ˆF, +[ ˆX1, ˆF] = 2a1 ˆH2 + 2 ˆX2 +1 + 2c2, +[ ˆX2, ˆF] = 4 ˆX3 +1 − 2{ ˆX1, ˆX2} + (2c2 + 1 − 2a2 ˆH2)X1. +(21) +The Casimir operator of this cubic algeba is given by +C(3) = ˆF 2 − 2{X2 +1, ˆX2}a + 2 ˆX4 +1 + (c2 + 5 − 2a2 ˆH2)X2 +1 − 4(a1 ˆH2 + c2) ˆX2. +By Proposition 3.1 and Proposition 3.4, we again find that the cubic algebra has no realization in terms of the +deformed oscillator algebra. +3.2.3 +Darboux space III +In the 2D Darboux space III, the free superintegrable system Hamiltonian and its constants of motion in the +separable local coordinates (u, v) are given by +H3 = ϕ3(v)(∂2 +u + ∂2 +v), +X1 = ∂u, +X2 = 1 +2e−v � +cos u(2∂2 +u + ∂v) + sin u (2∂u∂v − ∂u) +� ++ α cos uH3, +where ϕ3(v) = +e−v +βev−2α with α, β being real constants. +We seek potential of the form V3(u, v) = ϕ3(v)(f3(u) + g3(v)) such that system in Darboux space III with +this potential is superintegrable. We thus expect that ˆH3 = H3 + V3(u, v) possesses linear and quadratic integrals +of the form, ˆX1 = X1, +ˆX2 = X2 + f3(u, v). After some manipulations, we find that V3(u, v) = c3 ϕ3(v) and +f3(u, v) = c3 βev cos(u) +2βev−4α , where c3 is a constant. That is, we obtain the superintegrable system in Darboux space III +with Hamiltonian and integrals given by +ˆH3 = ϕ3(v)(∂2 +u + ∂2 +v) + +c3 +βev − 2α, +ˆX1 = ∂u, +ˆX2 = 1 +2 exp(−v) +� +cos u(2∂2 +u + ∂v) + sin u(2∂u∂v − ∂u) +� ++ α cos uH3 + c3 βev cos(u) +2βev − 4α . +These integrals form the following algebra, +[ ˆX1, ˆX2] = ˆF, +[ ˆX1, ˆF] = − ˆX2, +[ ˆX2, ˆF] = −β ˆH3 ˆX1. +(22) +The Casimir operator of this algebra is given by C(3) = ˆF 2−β ˆH3X2 +1+ ˆX2 +2. It is interesting that the algebra generated +by the above linear and quadratic integrals in the Darboux space III is “linear” in the generators (though with +coefficient involving the Hamiltonian ˆH3). +3.2.4 +Darboux space IV +In terms of separable local coordinates (u, v), the Hamiltonian of the free superintegrable system in 2D Darboux +space IV is H4 = ϕ4(u) +� +∂2 +u + ∂2 +v +� +, where ϕ4(u) = +sin2 u +β−2α cos u α, β ∈ R. The system possesses the following linear +and quadratic integrals of motion, +X1 = ∂v, +X2 = exp(v) +2 +� +cos u(2∂2 +v − ∂v) − sin u(2∂u∂v − ∂v) − 2αH4 +� +. +By analysis similar to previous cases, we find that the system with the Hamiltonian +ˆH4 = ϕ4(u) +� +∂2 +u + ∂2 +v +� ++ +c4 +β − 2α cos u, +where c4 is a constant, is superintegrable with linear and quadratic integrals given by +ˆX1 = ∂v, +ˆX2 = exp(v) +2 +� +cos u(2∂2 +v − ∂v) − sin u(2∂u∂v − ∂v) − 2αH4 +� ++ +4c4e−v +β − 2α cos u. +These integrals form the cubic algebra, +[X1, ˆX2] = ˆF, +[X1, ˆF] = − ˆX2, +[ ˆX2, ˆF] = 4X3 +1 − 2β ˆH4X1 + 1 +2 +ˆX1 − 2αc4X1. +(23) +35 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +The Casimir operator of the algebra is +C(3) = −1 +2{ ˆX2, ˆF}a + β ˆH4X2 +1 + X4 +1 + (5 + 4c4)X2 +1, +which can be expressed as C(3) = ˆH2 +4 + β ˆH4 + 4c4 in terms of the Hamiltonian ˆH4. Through the change of basis, +X1 = (N + η), +X2 = −(N + η)2 + b† + b +the cubic algebra relations become those of the deformed oscillator algebra with structure function +Φ(N, η) = (N + η)4 − 4(+N + η)3 + (N + η)2 − (N + η) +� +−2αc4 − 2βE + 1 +2 +� +− 4c4 − E2 − βE. +Here η is a constant which can be determined from the constraints on the structure function. +4 +Conclusions +We have presented a genuine algebraic analysis for the superintegrable systems in 2D Darboux spaces. The main +results in this paper are following. +The first main result is the construction of the Casimir operators, deformed oscillator algebra realizations +and finite-dimensional unirreps for all the 12 distinct quadratic algebras underlying the 12 superintegrable systems +found in the classification of [14][15]. This allows us to give an algebraic derivation for the energy spectrum of the 12 +existing classes of superintegrable systems with quadratic integrals in the 2D Darboux spaces and the determination +for the structure functions of the finite-dimensional unitary irreducible representations of the deformed oscillator +algebras (corresponding to the quadratic algebras). As our results demonstrate, superintegrable systems in curved +(Darboux) spaces have much richer structures than those in flat spaces. For instance, the structures of energies of +the systems and structure functions of the associated deformed oscillator algebras can be very complicated in the +Darboux spaces, and in some cases we have to restrict the model parameter spaces in order to find explicit analytic +and closed form solutions. +Another main result of the paper is the construction of generic cubic and quintic algebras, generated by first, +quadratic and cubic integrals, their Casimir operators and deformed oscillator algebra realizations. As examples +of applications, we obtain four classes of new superintegrable systems with non-trivial potentials and with linear +and quadratic integrals in the 2D Darboux spaces, three of which have cubic algebras as their symmetry algebras. +Acknowledgement +IM and YZZ were supported by Australian Research Council Future Fellowship FT180100099 and Discovery Project +DP190101529, respectively. +References +[1] C. 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Marquette, S. Post, and Y.-Z. Zhang. Algebraic calculations for spectrum of superintegrable +system from exceptional orthogonal polynomials. Ann. Physics, 391:203–215, 2018. +[27] M. F. Hoque, I. Marquette, and Y.-Z. Zhang. Recurrence approach and higher order polynomial algebras for +superintegrable monopole systems. J. Math. Phys., 59(5):052101, 10, 2018. +[28] F. Correa, M. F. Hoque, I. Marquette, and Y.-Z. Zhang. N-dimensional Smorodinsky-Winternitz model and +related higher rank quadratic algebra sw(n). J. Phys. A, 54(39):Paper No. 395201, 19, 2021. +[29] J. A. Calzada, J. Negro, M. A. del Olmo, and M. A. Rodr´ıguez. Contraction of superintegrable Hamiltonian +systems. J. Math. Phys., 41(1):317–336, 2000. +[30] E. G. Kalnins, W. Miller Jr, and S. Post. Contractions of 2D 2nd order quantum superintegrable systems and +the Askey scheme for hypergeometric orthogonal polynomials. SIGMA Symmetry Integrability Geom. Methods +Appl., 9:Paper 057, 28, 2013. +[31] G. Darboux. Le¸cons sur la th´eorie g´en´erale des surfaces et les applications g´eom´etriques du calcul infinit´esimal. +Troisi`eme partie. Chelsea Publishing Co., Bronx, N.Y., 1972. +37 + +ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES +[32] A. P. Fordy and Q. Huang. Superintegrable systems on 3 dimensional conformally flat spaces. J. Geom. Phys., +153:103687, 27, 2020. +[33] A. P. Fordy and Q. Huang. Adding potentials to superintegrable systems with symmetry. Proc. R. Soc. +London A, 477(2248):Paper No. 20200800, 21, 2021. +[34] P. S. Isaac and I. Marquette. +On realizations of polynomial algebras with three generators via deformed +oscillator algebras. J. Phys. A, 47(20):205203, 26, 2014. +38 + diff --git a/IdE2T4oBgHgl3EQfUQce/content/tmp_files/load_file.txt b/IdE2T4oBgHgl3EQfUQce/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cbc8aae8ef8a767518b3b8d20875d5c03b1f58e0 --- /dev/null +++ b/IdE2T4oBgHgl3EQfUQce/content/tmp_files/load_file.txt @@ -0,0 +1,2098 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf,len=2097 +page_content='ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces Ian Marquette ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Junze Zhang †and Yao-Zhong Zhang ‡ School of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The University of Queensland Brisbane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' QLD 4072,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Australia January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2023 Abstract Superintegrable systems in 2D Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in the Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this paper, we obtain exact solutions via purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four different 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This is achieved by constructing the deformed oscillator realization and finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated by quadratic integrals respectively for each of the 12 superintegrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We also introduce generic cubic and quintic algebras, generated respectively by linear and quadratic integrals and linear and cubic integrals, and obtain their Casimir operators and deformed oscillator realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' As examples of applications, we present three classes of new superintegrable systems with cubic symmetry algebras in 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 1 Introduction Superintegrable systems of different orders have been attracting a large amount of international research activities, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [1], [2], [3], [4], [5], [6] , [7], [8] and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This paper is a contribution to the underlying algebraic structures and exact solutions of superintegrable systems in 2-dimensional (2D) curved spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Superintegrable systems in 2D spaces with constant or non-constant curvatures have been widely studied by means of separation of variables and St¨ackel transforms [10], [11], [12], [13], [14] and [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The St¨ackel transforms have been widely studied [16], [14], [17] provide useful tools in the classification of 2D superintegrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Through the method of the so-called coupling constant metamorphosis, St¨ackel transforms [18] enable one to establish the relationship between different superintegrable systems: they provide equivalence classes at the level of integrable and superintegrable Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' However, even if such Hamiltonians are connected via the St¨ackel transformations, they are distinct as Sturm-Liouville and spectral problem, and their exact solvability (with possibly different boundary conditions) and algebraic solutions need to be investigated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For a given superintegrable Hamiltonian which is separable in various coordinates, its solvability would in general depend on the coordinates used in the separation of variables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' it is exactly solvable in one coordinate system but only quasi-exactly solvable in another coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is well known that symmetry algebra structures play an important role in the analytic analysis of physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In the context of superintegrable models, the underlying symmetry algebra structures are usually polynomial algebras such as quadratic and cubic algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In [19], [2], [10], rank-1 quadratic algebra structures underlying certain 2D superintegrable systems, generated by integrals of motion of the ∗i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='marquette@uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='au †junze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='zhang@uqconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='au ‡yzz@maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='au 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='03810v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='SI] 10 Jan 2023 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES systems, were exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The authors in these references obtained the Casimir operator and deformed oscillator algebra realization of a generic quadratic algebra, and applied the relates to study the energy spectrum of the superintegrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In [14] and [11], examples of rank-1 cubic and quintic algebras in Darboux spaces were given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Higher order or higher rank polynomial algebras generated by integrals and their deformed oscillator algebra realizations were studied in [5], [20], [21], [22], [23], [24] , [25], [26], [27] and [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' More recently, by extending the Wigner-In¨on¨u method of Lie algebra contraction, the authors in [29], [30] showed that quadratic algebras from certain second-order superintegrable systems in 2D spaces are contractions of those with general 3-parameter potentials on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Superintegrable systems in 2D Darboux spaces were classified in [14][15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In 2 dimensions, there exist 4 possible Darboux spaces with metrics given by [31] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' d1(x, y) = (x + y) dxdy II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' d2(x, y) = � Ω (x − y)2 + Λ � dxdy III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' d3(x, y) = � Ω exp � −x + y 2 � + Λ exp(−x − y) � dxdy IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' d4(x, y) = Ω � exp � x−y 2 � + exp � y−x 2 �� + Λ exp � x−y 2 + exp � y−x 2 ��2 dxdy Here Ω, Λ ∈ R are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' According to the classification in [14][15], there exist 12 distinct classes of superintegrable systems with non-trivial potentials in the 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In each case, quadratic integrals of motion of the system were determined and were found to form a quadratic algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The wave functions and energy spectra of the systems were obtained by means of separation of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Superintegrable systems in Darboux spaces were also studied in [11] [32] [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It was shown there that free superintegrable systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' systems without potentials) in 2D and 3D flat conformal spaces are equivalent to systems in 2D and 3D Darboux spaces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' However, as far as we know, finite dimensional representations of the polynomial algebras and algebraic derivations of the energy spectrum of the superintegrable systems have remained an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this paper we present a genuine algebraic approach to superintegrable systems in the 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The purpose of this paper is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' One is to give algebraic solutions to the existing 12 distinct classes of superintegrable systems in the four 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This is achieved by constructing the finite dimensional irreducible representation of the quadratic algebras underlying the 12 superintegrable systems via the deformed oscillator algebra techniques in [1] and [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' As one will see, energy spectrum for superintegrable systems in Darboux spaces are often determined by very complicated algebraic equations whose analytic and closed-form solutions can only be obtained by restricting the model parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The second purpose is to investigate superintegrable systems in 2D Darboux spaces with linear, quadratic or cubic integrals of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It was found in [11] that the free systems with linear and quadratic integrals in 2D Darboux spaces have cubic algebras as their underlying symmetry algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We will introduce generic cubic and quintic algebras, generated by linear and quadratic integrals and linear and cubic integrals, respectively, and construct their Casimir operators and deformed oscillator algebra realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We also present three classes of new superintegrable systems with non-trivial potentials in 2D Darboux spaces which have cubic algebras as their symmetry algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These superintegrable systems do not seem to belong to the families classified in [14][15] for systems with quadratic integrals in 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In Section 2, we obtain the Casimir operators, the deformed oscillator algebra realizations and finite-dimensional irreducible representations for the quadratic algebras generated by the quadratic integrals of motion of the 12 superintegrable systems in 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This enables us to give an algebraic derivation for the energy spectra of all the 12 classes of superintegrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In Section 3, we introduce generic cubic and quintic algebras generated by linear and higher order integrals of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We construct their Casimir operators and deformed oscillator algebra realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We also present examples of new superintegrable systems with linear and quadratic integrals in the 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In Section 4, we provide a summary of the main results of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES 2 Solutions of the 12 distinct classes of superintegrable systems in 2D Darboux spaces Consider a superintegrable system in a 2D Darboux space with coordinates (x, y) and metric gij(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The Hamiltonian of the system with potential V (x, y) is given by ˆH = 2 � j,k=1 1 � det(gjk) ∂ ∂xk �� det(gjk)gjk ∂ ∂xk � + V (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Let ˆX be an integral of motion (aka, constant of motion) of the system which commute with the Hamil- tonian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [ ˆX, ˆH] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' An integral of motion is said to be a polynomial in momenta of degree p, denoted by deg ˆX = p, if it has the form ˆX = p � j=0 rj(x, y) ∂p−j x ∂j y + s(x, y), where rj(x, y), s(x, y) are smooth functions in the coordinates x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In particular, integrals of motion of degree 1, 2 or 3 are usually called linear, quadratic or cubic integrals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Note that the Hamiltonian has degree 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' deg ˆH = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' As mentioned in the Introduction, superintegrable systems in the four 2D Darboux spaces with quadratic integrals of motion were classified in [14][15], and 12 distinct classes of potentials which pre- serve superintegrability were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this section, we present algebraic solutions to all the 12 existing superintegrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Note that in the following we will use the so-called separable coordinates in [14][15] for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' As indicated in [14][15], in such coordinates the parameters Ω, Λ in the metrics of the Darboux spaces can be conveniently absorbed into the model parameters of the systems by redefinition so that they do not appear explicitly in the expressions of Hamiltonians and integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 Darboux Space I According to the classification in [14][15], in the Darboux space I, there are two possible superintegrable systems with potentials given by V1(x, y) = b1(4x2 + y2) 4x + b2 x + b3 xy2 , V2(x, y) = a1 x + a2y x + a3(x2 + y2) x , respectively, where bi, ai are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 Potential V1(x, y) For superintegrable system in Darboux space I with the Hamiltonian ˆH = 1 4x � ∂2 x + ∂2 y � + V1(x, y) asso- ciated to V1, the constants of motion are given by [15] A = ∂2 y + 4b3 y2 + b1y2, B = y∂y∂x − x∂2 y + ∂x 2 − y2 4x � ∂2 x + ∂2 y � + b1y4 4x + b2y2 x + b3(4x2 + y2) y2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These integrals satisfy the following quadratic algebra relations [A, B] = C, [A, C] = −8 ˆHA − 16b1B, [B, C] = 6A2 + 8 ˆHB + 16b2A − 2b1(3 + 16b3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES This is the symmetry algebra of the superintegrable system The Casimir of this algebra is given by K1 = C2 − 4A3 + 8 ˆH{A, B} − 16b2A2 − 16b1B2 + 4b1(11 + 16b3)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We can show that with the differential realization of A, B the Casimir K1 has the following form in terms of the Hamiltonian ˆH, K1 = −4(3 + 16b3) ˆH2 + 16b1b2(3 + 16b3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In order to obtain the energy spectrum of the system via algebraic means, we now construct realization of the quadratic algebra in terms of the deformed oscillator algebra of the form [N, b†] = b†, [N, b] = −b, bb† = Φ(N + 1), b†b = Φ(N), (1) where N is the number operator and Φ(z) is a well-defined real function satisfying Φ(0) = 0, Φ(z) > 0, ∀z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (2) Φ(x) is called the structure function of the deformed oscillator algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is non-trivial to obtain such a realization and the corresponding structure function Φ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After a long computation, we find in the present case that A = 4 � −b1 (N + η), B = 2 ˆH √−b1 (N + η) + b† + b map the quadratic algebra to the deformed oscillator algebra with structure function given by Φ(I) 1 (N, η) = − 1 16b1 � −4(N + η)16b1b2 + (−b1)3/2(16b3 + 11) − 4 ˆH2 + 2b3/2 1 (16b3 + 3) +64 � −b1b1(N + η)3 + 16(N + η)2(4b1b2 − ˆH2) − (16b3 + 3)(2b1 � −b1 − 4b1b2 + ˆH2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Here η is a constant to be determined from the constraints on the structure function Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We now obtain the finite-dimensional unitary irreducible representations (unirreps) of the deformed oscillator algebra in the Fock space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Let |z, E⟩, denote the Fock basis states labelled by the eigenvalues z and E of N and ˆH, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Acting the structure function on the Fock states, we find that it is factorized to the following form ΦI 1(z, η) = � z + η − 1 4 � 2 − � 1 − 16b3 �� � z + η − 1 4 � 2 + � 1 − 16b3 �� � z + η + 2b1 �√−b1 − 2b2 � + E2 4(−b1)3/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For the unirreps to be finite dimensional, we impose the following constraints on the structure function, Φ(0, η) = 0, Φ(p + 1, η) = 0, (3) where p is a positive integer, p = 0, 1, 2, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These constraints give (p+1)-dimensional unirreps in the Fock space and their solutions give the constant η and energy spectrum E of the underlying superintegrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' There are two sets of solutions from the constraints on the structure function: η = 1 4 � 2 + ϵ√1 − 16a2 � , Eim = ±2 √ −1 (−b1)3/4 � p + 1 − ϵ 4 � 1 − 16b3 + b2 √−b1 , 4 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES and η = −2b1 �√−b1 − 2b2 � + E2 4(−b1)3/2 , Eϵ = ±2(−b1)3/4 � p + 1 + ϵ 4 � 1 − 16b3 − b2 √−b1 , where ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The first set of solutions give complex energies which are not physical and thus will be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' So the energy spectrum of the system is given by the second set of solutions which are real for ϵ = +1, b1 < 0, b2 ≤ 0, b3 < 1/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure function for the corresponding (p + 1)-dimensional unirreps is Φ(I) E+(z) = z(z − p − 1) � z − 2b1 �√−b1 − 2b2 � + E2 + 4(−b1)3/2 − 1 4 � 2 + � 1 − 16b3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In the following subsections, we would only give the values of parameter η which can lead to real energies E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 Potential V2(x, y) Constants of motion for the superintegrable system in Darboux space I with the Hamiltonian ˆH = 1 4x � ∂2 x + ∂2 y � + V2(x, y) corresponding to the potential V2 are given by [15] A = y∂y∂x − x∂2 y + ∂x 2 − y2 4x � ∂2 x + ∂2 y � − 2a2y x + 2a2(x2 − y2) x + 2a2y(x2 − y2) x , B = ∂2 y + 4a2y + 4a3y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' They satisfy the following quadratic algebra relations [A, B] = C, [A, C] = 16a2 ˆH − 16a3B, [B, C] = 16a3A + 8(a2 2 + 4a1a3) − 8 ˆH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The Casimir operator of the algebra is given by K2 = C2 + 16a3A2 + 16a3B2 − 32a2 ˆHB + 16 � (a2 2 + 4a1a3) − ˆH2� A, which in terms of the differential realization of A, B takes the constant value K2 = 64(a2 3 − a1a2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We then determine the realization of above quadratic algebra in terms of the deformed oscillator algebra (1) and apply its finite dimensional unirreps to obtain the energy spectrum of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After computations, we find that A = 4√−a3(N + η), B = a2 ˆH a3 + b† + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' transform the quadratic algebra to the deformed oscillator algebra with structure function Φ(I) 2 (N, η) = � 4a1a3 + a2 2 − ˆH2�2 16a2 3 − a1a2 2 a3 − 1 12(N + η) � 24a2 ˆH √−a3 + 48a3 � + a2 ˆH √−a3 + 4a3(N + η)2 + a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Moreover, the action of this structure function on the Fock states |z, E⟩ is factorized as Φ(I) 2 (z, η) = � z + η − m+(E) + 2a3 4a3 � � z + η − m−(E) + 2a3 4a3 � , 5 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES where η is a constant to be determined and m±(E) = a2E √−a3 ± � 64a2 1a2 3 + 4a1 �a2 2(8a3 − 1) − 8a3E2� + 4a4 2 − a2 2(8a3 + 1)E2 a3 + 4E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We now impose the constraints (3) to obtain finite-dimensional unirreps of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We find that for p = 0, 1, 2, · · · , we have Case 1: η−(E) = 1 4a3 (m−(E) + 2a3) and � 64a2 1a2 3 + 4a1 �a2 2(8a3 − 1) − 8a3E2� + 4a4 2 − a2 2(8a3 + 1)E2 a3 + 4E4 = 2a3(p + 1), (4) which has solutions only for a3 > 0 and the energy spectrum of the system is given by E+a3 = ± 1 √8a3 �� 128a1a2 2a2 3 + a4 2(16a3 + 1) + 64a4 3(p + 1)2 + 32a1a2 3 + a2 2(8a3 + 1), Notice that E+a3 is real for a1 > 0, a3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 2 η+(E) = 1 4a3 (m+(E) + 2a3) and � 64a2 1a2 3 + 4a1 �a2 2(8a3 − 1) − 8a3E2� + 4a4 2 − a2 2(8a3 + 1)E2 a3 + 4E4 = −2a3(p + 1), which has solutions only for a3 < 0 and the energies of the system are E−a3 = ± 1 √−8a3 �� 128a1a2 2a2 3 + a4 2(16a3 + 1) + 64a4 3(p + 1)2 − 32a1a2 3 − a2 2(8a3 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (5) Obviously for a3 < 0 there exist ranges of model parameters a1, a2 such that the eneries E−a3 of the system are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure function for both cases 1 and 2 corresponding to the (p + 1)-dimensional unirreps of the algebra is given by Φ(I) E±a3(z) = z(z − p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 Darboux Space II In the Darboux space II, there are three superintegrable systems with potentials given by [14] V1(x, y) = x2 x2 + 1 � a1 � x2 4 + y2 � + a2y + a3 x2 � , V2(x, y) = x2 x2 + 1 � b1(x2 + y2) + b2 x2 + b3 y2 � V3(x, y) = c1 + c2 x2 + c3 y2 x2 + y2 + 1 x2 + 1 y2 , respectively, where aj, bj, cj are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 Potential V1(x, y) The constants of motion of the superintegrable system in Darboux space II with the Hamiltonian ˆH = x2 x2+1 � ∂2 x + ∂2 y � + V1(x, y) associated to the potential V1 are A = ∂2 y + a1y2 + a2y, B = 2y x2 + 1 � ∂2 y − x2∂2 x � + 2x∂x∂y + ∂y + a1 2 y � x2 + x2 + 4y2 x2 + 1 � + a2 2 � x2 + 4y2 x2 + 1 � − 2a3y x2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 6 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES They satisfy the following quadratic algebra relations [14] [A, B] = C, [A, C] = −4a1B − 4a2A, [B, C] = −24A2 + 4a2B + 32 ˆHA − 8 ˆH2 − 8a1 ˆH + 6a1 + 8a1a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Its Casimir operator can be shown to be given by K1 = C2 − 16A3 + 4a1B2 + 4a2{A, B} + � 4a1(4a3 − 11) − (16a1 ˆH + 16 ˆH2) � A + 32 ˆHA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In term of the differential realization of A, B, the Casimir K1 takes the simple form K1 = (32a1+4a2 2) ˆH− a2 2(3 + 4a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By computations similar to those in the previous subsection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we find that A = 2√−a1(N + η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = 2a2 √−a1 (N + η) + a2 ˆH a1 + b† + b map the quadratic algebra to the deformed oscillator algebra (1) with structure function given by Φ(II) 1 (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = − 12a3 1 − 3√−a1a1a2 2 − 16a3 1a3 − 4√−a1a1a2 2a3 + 32√−a1a2 1 ˆH + 16a3 1 ˆH − 8a1a2 2 ˆH + 4√−a1a1a2 2 ˆH + 16a2 1H2 + 4√−a1a2 2H2 + (N + η) � 88a3 1 + 16√−a1a1a2 2 + 32a3 1a3 − 128√−a1a2 1 ˆH − 32a3 1 ˆH + 16a1a2 2 ˆH − 32a2 1 ˆH2� + (N + η)2 � −192a3 1 − 16√−a1a1a2 2 + 128√−a1a2 1 ˆH � + 128a3 1(N + η)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Here η is a constant to be determined from the constraints of the structure function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Acting on the Fock basis states |z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' E⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' the structure function Φ(II) 1 becomes factorized Φ(II) 1 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = � z + η − f1(E) + ω(E) + f2(E) 24a3 1 � � z + η − 1 96a3 1 � 4f1(E) − 2 � 1 − i √ 3 � ω(E) + � 1 + i √ 3 � f2(E) �� � z + η − 1 96a3 1 � 4f1(E) − 2 � 1 + i √ 3 � ω(E) + � 1 − i √ 3 � f2(E) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where f1(E) =a1 � 12a2 1 + √−a1a2 2 + 8(−a1)3/2E � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' f2(E) = 1 ω(E) � a3 1(12a3 1(4E − 4a3 + 1) − a4 2 − 16a2 1E2 − 8a1a2 2E) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ω(E) = 3� τ1(E) + τ2(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' τ1(E) =6a6 1 � a4 2 + 8a1Ea2 2 + 16a2 1E2 + a3 1(−16a3 + 16E + 4) � � 3(4a3 − 4E − 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' τ2(E) =a1a6 2(−a1)7/2 + 12a4 2E(−a1)11/2 − 48a2 2E2(−a1)13/2 − 4 � 9a2 2(4a3 − 4E − 1) − 16E3� (−a1)15/2 − 144E(−4a3 + 4E + 1)(−a1)17/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' To determine the constant η and energy spectrum E of the superintegrable system, we impose the constraints (3) which give (p + 1)-dimensional unirreps of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We find Case 1: The constant η is given by η1(E) = f1(E) + ω(E) + f2(E) 24a3 1 7 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES and the energy E satisfies the algebraic equation, ω(E) + f2(E) + 1 2 � 1 − ϵ i √ 3 � ω(E) − 1 4 � 1 + ϵ i √ 3 � f2(E) = −24(p + 1)a3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (6) Case 2: η2(E) = 1 96a3 1 � 4f1(E) − 2 � 1 − ϵ i √ 3 � ω(E) + � 1 + ϵ i √ 3 � f2(E) � and the energy is determined by ω(E) + f2(E) + 1 2 � 1 − ϵ i √ 3 � ω(E) − 1 4 � 1 + ϵ √ 3i � f2(E) = 24(p + 1)a3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (7) In both cases above, ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The energy spectrum E of the system are obtained by solving the algebraic equations (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' However, it is in general very difficult to obtain analytical solutions of these equations, due to their complicated form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' To demonstrate that these equations have real solutions, we have a closer look at restricted model parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Without the loss of generality, we consider the case where −a1 = a2 = a3 = a for any a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For such model parameters, the structure function has the simple form, Φ(II) 1 (z, η) = � z + η − 1 8 �√a + 4 �� � z + η − 1 4a � 2√aE + 2a − a √ 4E + 1 − 4a �� � z + η − 1 4a � 2√aE + 2a + a √ 4E + 1 − 4a �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3) on the structure function lead to the determination of constant η and energy E of the superintegrable system for the model parameters −a1 = a2 = a3 = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' There are two sets of solutions: One is that η = 1 8 (√a + 4) and Eϵ = 1 4 � 8√a(p + 1) + 3a + 2ϵ � 8a3/2(p + 1) − 2a2 + a � , where ϵ = ±1, with the associated structure function Φ(II) Eϵ (z) = z(p + 1 − z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The energy spectrum Eϵ is real for 0 < a ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The second set of solutions is given by η(E) = 1 4a � 2√aE + 2a − a √ 4E + 1 − 4a � and the corresponding energy spectrum of the system and structure function for the (p + 1)-dimensional unirreps of the deformed oscillator algebra are given by E = p(p + 2) + a + 3 4, Φ(II) E (z) = z(z − p − 1) � z + 1 8a � 3a3/2 − 4a(p + 1) + √a(4p2 + 8p + 3) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Thus we have demonstrated that there exist indeed non-trivial model parameters which give real energies of the superintegrable system in both Case 1 and Case 2 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 Potential V2(x, y) The superintegrable system in Darboux II with potential V2(x, y) has Hamiltonian ˆH = x2 x2+1 � ∂2 x + ∂2 y � + V2(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This system possesses the following integrals of motion [14] A = ∂2 y + b1y2 + b3 y2 , B = (y2 − x4)∂2 y + x2(1 − y2)∂2 x x2 + 1 + 2xy∂x∂y + x∂x + y∂y − 1 4 + x2 + y2 x2 + 1 � b1(x2 + y2) − b2 − b3 x2 y2 � , 8 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES which form the quadratic algebra relations [A, B] = C, [A, C] = 8A2 − 16b1B + 16b1 ˆH − 16b1(b2 + b3 + 3 4), [B, C] = −8{A, B} + 8 ˆHB + 12A − 8 ˆH2 + 8(b2 − b3 − 3 4) ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By a direct calculation, we find the Casimir operator of the algebra K2 =C2 − 8{A2, B} + 8 ˆH{A, B} + 16b1B2 + 76A2 + � 16(b3 − b2 + 19 4 ) ˆH − 16 ˆH2 � A + � 8b1(4(b2 + b3) + 3) − 32b1 ˆH � B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This Casimir operator can be expressed in terms of Hamiltonian as K2 = −16 � b1 + b3 + 3 4 � ˆH2 − 8b1(4b3 − 4b2 + 3) ˆH + b1 � 36 + 48b3 − (4b3 − 4b2 + 3)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It can be shown that after the change of basis A = 4 � −b1(N + η), B = 8(N + η)2 − 2 ˆH √−b1 − 16(b2 + b3 + 3 4)(N + η) − b1 ˆH b1 + b† + b, the quadratic algebra becomes the deformed oscillator algebra with structure function Φ(II) 2 (N, η) = 1 16 � 4b3 + 16N 2 + 16N(2η − 1) + 16η2 − 16η + 3 � � 4b2 + 1 − 4 ˆH �√−b1 + 2N + 2η − 1 � √−b1 + 16N 2 + 32Nη − 16N + 16η2 − 16η + 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' On the Fock states |z, E⟩, the structure function is factorized as follows Φ(II) 2 (z, η) = � z + η − 1 4 � 2 − � 1 − 4b3 �� � z + η − 1 4 � 2 + � 1 − 4b3 �� � z + η − 2b1 − γ+(E) 4b1 � � z + η − 2b1 − γ−(E) 4b1 � , where γ±(E) = � b2 1(4E − 4b2 + 1) ± � −b1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3), for any p ∈ N+ we get the following values for the parameter η and energy E: Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η+(E) = 1 4b1 (2b1 − γ+(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This η value gives the following energy spectrum of the system and the corresponding structure function of the deformed oscillator algebra E− = −(p + 2) � −b1, Φ(II) E− (z) = z(z − p − 1) � z + 1 4b1 � b1 � 1 − 4b3 − γ+(E−) �� � z − 1 4b1 � b1 � 1 − 4b3 + γ+(E−) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The enegry E− is real for b1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η−(E) = 1 4b1 (2b1 − γ−(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This η value gives two sets of energies of the system, E+ =(p + 2) � −b1, (8) Eϵ = − 2b1 + 4 � −b1 � p + 1 + ϵ 4 � 1 − 4b3 � ± � 4b2 1 − 16b1 � −b1 � p + 1 + ϵ 4 � 1 − 4b3 � + 4b1b2 − b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (9) 9 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES Eϵ above is obtained by solving the algebraic equation γ−(E) = 4b1 �p + 1 + ϵ 4 √1 − 4b3 � from the con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (Notice that the other algebraic equation γ+(E) = 4b1 �p + 1 + ϵ 4 √1 − 4b3 � lead to complex solutions and its solutions are not shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=') Obviously E+ is real for b1 < 0 and Eϵ is real for ϵ = +1, b1 < 0, b2 < 1 4, b3 < 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure functions corresponding to E+, Eϵ in the Case 2 above are given by Φ(II) E+ (z) = z(z − p − 1) � z + 1 4b1 � b1 � 1 − 4b3 − γ−(E+) �� � z − 1 4b1 � b1 � 1 − 4b3 + γ−(E+) �� , Φ(II) Eϵ (z) = z(z − p − 1) � z + 1 2b1 � −b1 Eϵ � � z − 1 4b1 � γ−(Eϵ) + ϵb1 � 1 − 4b3 �� , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 3: η = 1 4 �2 + ϵ√1 − 4b3 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The corresponding energies are given the same expression as Eϵ above (and are obtained from solving the algebraic equation γ+(E) = −4b1 �p + 1 + ϵ 4 √1 − 4b3 �).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 Potential V3(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' y) The constants of motion for the superintegrable system in Darboux space II with the Hamiltonian ˆH = x2 x2+1 � ∂2 x + ∂2 y � + V3(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' y) associated to the potential V3 are given by A = � y2 + 1 y2 � ∂2 x − � x2 + 1 x2 � ∂2 y x2 + y2 + 1 x2 + 1 y2 + c1x2(y4 + 1) + c2(y4 + 1) − c3(x4 + 1) (x2y2 + 1)(x2 + y2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B =c1(x2 + y2) − c2(y4 − 1) − c3(x4 − 1) 4(x2y2 + 1) + xy(x2 − y2) � xy∂2 x − xy∂2 y + (x2 − y2)∂x∂y � + 1 x2y2 + 1 �� x2 − y2 4 + y4 � x2∂2 x + � x2 − y2 4 + x4 � y2∂2 y + 2xy � x2 − y2 2 − x2y2 � ∂x∂y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' They form the following quadratic algebra relations [A, B] = C, [A, C] = 2A2 + 2c1A + 16 ˆHB + 6 ˆH − 8 ˆH2, [B, C] = −2{A, B} + (c2 + c3)A − c1c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The Casimir operator of this algebra is K3 = C2 − 2{A2, B} − 16 ˆHB2 + (c2 + c3 + 4)A2 + 2c1{A, B}a − 2c1(c3 + 2)A + (16 ˆH2 − 12 ˆH)B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' With the differential realization of A, B, the Casimir operator can be expressed in terms of the Hamilto- nian as K3 = 4(c2 + c3) ˆH2 + (c2 1 − 4c2c3 − 3(c2 + c3)) ˆH − 3 + 4c3 4 c2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We can convert the quadratic algebra into the deformed oscillator algebra by using the realization A = 4 � ˆH(N + η), B = −2(N + η)2 + c1 2 � ˆH (N + η) − 3 ˆH − 4 ˆH2 8 ˆH + b† + b with the corresponding structure function given by Φ(II) 3 (N, η) = − 1 256 ˆH � 4c3 − 4 ˆH + 16N 2 + 32Nη − 16N + 16η2 − 16η + 3 � × � −c2 1 + 4c1 � ˆH(2N + 2η − 1) + ˆH � −4c2 + 4 ˆH − 16N 2 − 32Nη + 16N − 16η2 + 16η − 3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 10 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES By acting Φ(II) 3 on the Fock basis states |z, E⟩, we find that the structure function is factorised as Φ(II) 3 (z, η) = � z + η − 1 4 � 2 − � −4c3 + 4E + 1 �� � z + η − 1 4 � 2 + � −4c3 + 4E + 1 �� × � z + η − 1 4E � −E � −4c2 + 4E + 1 + c1 √ E + 2E �� × � z + η − 1 4E � E � −4c2 + 4E + 1 + c1 √ E + 2E �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We now obtain the energy spectrum of the system from the finite-dimensional unirreps of the deformed oscillator algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3) which give (p + 1)-dimensional unirreps for any p ∈ N+, we determine the parameter η and the energy E of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' There are two sets of solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 1: η(E) = 1 4 �2 − √−4c3 + 4E + 1 � and the energies are determined by either � −4c3 + 4E + 1 − 2(p + 1) = 0 (10) or c1 √ E + � −4c3 + 4E + 1 + � −4c2 + 4E + 1 = 4(p + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (11) Solution to the algebraic equation (10) gives the energies Ec3 = p(p + 2) + c3 + 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure function of the corresponding (p + 1)-dimensional unirreps is Φ(II) Ec3 (z) =z(z − p − 1) � � � �z − 1 2 � p + 1 − � (p + 1)2 + c3 − c2 � − c1 4 �� p + 1 2 � � p + 3 2 � + c3 � � � � � � � �z − 1 2 � p + 1 + � (p + 1)2 + c3 − c2 � − c1 4 �� p + 1 2 � � p + 3 2 � + c3 � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Other possible energies of the system are given by solutions to the algebraic equation (11), which read E± = 1 4 (p + 1 + c1)2 � 2c2 + 2c3 − 1 ± � (p + 1 + c1)2 + 4c2c3 − (c2 + c3) + 1 4 � (p + 1 + c1)2 − (c2 − c3)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These energies are real for the model parameters satisfying 4c2c3 + 1 4 > c2 + c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The corresponding structure functions for the (p+1)-dimensional unirreps of the algebra are Φ(II) E± (z) =z(z − p − 1) � z − 1 2 � −4c3 + 4E± + 1 � � z − 1 4 �� −4c3 + 4E± + 1 − � −4c2 + 4E± + 1 + c1 √E± �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 2: η(E) = 1 4E � c1 √ E + 2E − E√−4c2 + 4E + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This η value gives the following energy spectrum of the system and the corresponding structure function of the unirreps, Ec2 = p(p + 2) + c2 + 3 4, Φ(II) Ec2 (z) =z(z − p − 1) � � � �z + 1 2 � p + 1 − � (p + 1)2 + c2 − c3 � + c1 4 �� p + 1 2 � � p + 3 2 � + c2 � � � � � � � �z − 1 2 � p + 1 + � (p + 1)2 + c2 − c3 � + c1 4 �� p + 1 2 � � p + 3 2 � + c2 � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 11 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 Darboux Space III In Darboux space III, there exist 4 different potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In terms of the separable coordinates (u, v) and (µ, ν), they are given by V1(u, v) = a1u + a2v + a3 4 + u2 + v2 , V2(u, v) = b1 u2 + b2 v2 + b3 4 + u2 + v2 , V3(µ, ν) = c1(µ + ν) + c2 µ+ν µν + c3 ν2−µ2 ν2µ2 (µ + ν)(2 + µ − ν) , V4(µ, ν) = d1µ + d2ν + d3ν2 + µ2 (µ + ν)(2 + µ − ν) , where ai, bi, ci, di are real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 Potential V1(u, v) The constants of motion of the superintegrable system in Darboux space III with the Hamiltonian ˆH = exp(2u) 4(exp(u))+1 �∂2 u + ∂2 v � + V1(u, v) associated to the potential V2 are given by [14] A = (2 + v2)∂2 u − (2 + u2)∂2 v 2(4 + u2 + v2) + a1u(2 + v2) − 2a2v(2 + u2) + a3(v2 − u2) 4(4 + u2 + v2) , B = 2uv (∂2 u + ∂2 v) 2(4 + u2 + v2) − 2∂u∂v + a1v(v2 − u2 + 4) + a2u(u2 − v2 + 4) − 2a3vu 4(4 + u2 + v2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' They form the quadratic algebra with the commutation relations [A, B] = C, [A, C] = ˆHB − a2a1 8 , [B, C] = − ˆHA − a2 2 − a2 1 16 , which is the symmetry algebra of the superintegrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By a direct computation, we obtain the Casimir operator of this algebra K1 = − ˆHA2 − ˆHB2 − a2 2 − a2 1 8 A + a1a2 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We can show that in terms of the Hamiltonian this Casimir operator takes the form K1 = − ˆH3 + 1 2(a3 + 1 2) ˆH2 + 1 16(2a2 1 + 2a2 2 − a2 3) ˆH − a3(a2 1 + a2 2) 32 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' To determine the energy spectrum of the system, we now construct the deformed oscillator algebra realization of the quadratic algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We find that A = � ˆH(N + η), B = a1a2 8 ˆH + b† + b, trsansform the quadratic algebra into the deformed oscillator algebra with the structure function Φ(III) 1 (N, η) = 1 256 ˆH � a2 1 + 2a3 ˆH − 4 ˆH3/2 � 2 � ˆH + 2N + 2η − 1 �� × � a2 2 + 2a3 ˆH + 4 ˆH3/2 � −2 � ˆH + 2N + 2η − 1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Here η is a constant parameter to be determined from the constraints (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Acting on the Fock basis states |z, E⟩, the structure function Φ(III) 1 becomes Φ(III) 1 (z, η) = � z + η − 1 8E3/2 � a2 1 + 2E � a3 − 4E + 2 √ E ��� × � z + η + 1 8E3/2 � a2 2 + 2E(a3 − 4E − 2 √ E) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 12 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES The constraints (3) give the (p + 1)-dimensional unirreps of the deformed oscillaor algebra and their solutions determine the constant η and energy spectrum of the superintegrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' There are two sets of solutions: Case 1: η(E) = 1 8E3/2 � a2 1 + 2E � a3 − 4E + 2 √ E �� and energies E are determined by the algebraic equation 8E3/2 (p + 1) + 4a3E + a2 1 + a2 2 = 16E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (12) Case 2: η(E) = − 1 8E3/2 � a2 2 + 2E � a3 − 4E − 2 √ E) �� and energies E satisfy 16E2 + 8E3/2 (p + 1) = 4a3E + a2 1 + a2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (13) The algebraic equations (12) and (13) can be solved by using symbolic computation packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It can be shown that there exist model parameters ai such that solutions to these algebraic equations for energies are real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' To demonstrate this, we consider the case in which the model parameters satisfy a1 = 0 and a2 = a3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this case we find that the structure function reduces to Φ(III) 1 (z, η) = � z + η − �1 2 − √ E + 1 4 √ E �� � z + η − �1 2 + 1 8E3/2 (8E2 − 2E − 1) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3) gives the constant η and energies as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η(E) = 1 2 − √ E + 1 4 √ E which leads to the algebraic equation 4E(1 − 4E) + 1 + 8E3/2(p + 1) = 0 for E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This equation has real solution given by E± = 1 48 � 3p2 + √ 3 g(p) + 6p + 9 ± � 6 f(p) � , where e(p) = 3 � 27p4 + 108p3 + 252p2 + 3 √ 3 � (p + 1)4 (27p4 + 108p3 + 310p2 + 404p + 575) + 288p + 367 g(p) = � 3 (p2 + 2p + 3)2 + 2 × 22/3e(p) + 1 e(p) 4 3√ 2 (6p2 + 12p + 31) + 8 f(p) = 3 � p2 + 2p + 3 �2 − 22/3e(p) − 1 e(p)2 3√ 2 � 6p2 + 12p + 31 � + 1 g(p) 3 √ 3(p + 1)2 � p4 + 4p3 + 12p2 + 16p + 23 � + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is clear that g(p) is real for all p ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We now show that f(p) > 0 for all p ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Let f0(p) = 3 � p2 + 2p + 3 �2 − 22/3e(p) − 1 e(p) 2 3√ 2 � 6p2 + 12p + 31 � + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By using symbolic computation package, we found that df0(p) dp > 0 for all p ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Hence f0(p) is strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Moreover, f0(0) = −62 3� 2 15 √ 69+367 − 22/3 3� 15 √ 69 + 367 + 35 ∼= 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='5741 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It follows that f(p) > 0 for all p ∈ N+ and the energy E given above is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η(E) = 1 2 + 1 8E3/2 (8E2−2E−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This leads to the algebraic equation 4E(4E−1)−1+8E3/2(p+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It gives the same energy expression as in Case a above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For both case a and case b above, the structure function corresponding to the (p + 1)-dimensional unirreps of the deformed oscillator algebra is simply Φ(III a,b (z) = z(z − p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 13 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 Potential V2(u, v) The integrals of motion of the superintegrable system associated to the potential V2 with Hamiltonian ˆH = exp(2u) 4(exp(u))+1 �∂2 u + ∂2 v � + V2(u, v) in Darboux space III are given by [14], A = u2∂2 v − 2uv∂u∂v + v2∂2 u + b1v2 4u2 + b2u2 4v2 , B = (2 + v2)∂2 u − (2 + u2)∂2 v 2(4 + u2 + v2) + 2b1v2(v2 + 2) − 2b2u2(u2 + 2) + b3(v2 − u2) 4(4 + u2 + v2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These integrals form the quadratic algebra of the form [A, B] = C, [A, C] = −2{A, B} − (b1 + b2 + 1)B + (b1 − b2) ˆH + (b2 − b1)b3 4 , [B, C] = −2B2 − (b1 + b2 + 1)B + (b1 − b2) ˆH + (b2 − b1)b3 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By a direct calculation, we find the Casimir operator of the algebra K2 = C2 + 2{A, B2} + (b1 + b2 + 5)B2 − 4 ˆHA2 − 2(b1 − b2) ˆHB − b3(b2 − b1)B − 4 ˆHA + (2b3 − 1) ˆHA − b2 3 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' With the differential realization of A, B and in terms of ˆH, the Casimir K2 takes the simple form K2 = −(b1 + b2 − 2) ˆH2 + � (b3 + 3 2)(b1 + b2) 2 − b3 − b1b2 − 1 2 � ˆH − b2 3(b1 + b2 − 2) 16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The quadratic algebra can be transformed into the deformed oscillator algebra via the realization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=', change of basis) A = − � (N + η)2 − 1 4 + b1 + b2 + 1 4 � , B = −(b1 − b2) ˆH + (b2−b1)b3 4 16 � (N + η)2 − 1 4 � + b†ρ(N) + ρ(N)b, where ρ(N) = 1 3 · 212 · (−2)8(N + η)(1 + N + η)(1 + 2(N + η))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure function is given by Φ(III) 2 (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = 4096((2N + 2η − 1)2 � b2 2(3b1 + 3b2 + 7) − 4 ˆH � 3b2 1 + b1(12b2 − 12 ˆH + 11) +9b2 2 − 12b2 ˆH + 25b2 − 28 ˆH + 4 �� − 48(1 − 2(N + η))2 � − 1 16b2 2(b1 + b2 − 2) +1 2 ˆH �� b2 + 3 2 � (b1 + b2) − 2b1b2 − 2b2 − 1 � − ˆH2(b1 + b2 − 2) � + (2N + 2η − 1)2 � 12N 2 + 12N(2η − 1) + 12η2 − 12η − 1 � × � b2 2 − 4 ˆH(2b1 + 4b2 − 4 ˆH + 1) � + 12(b1 − b2)2(b2 − 2 ˆH)2 − 12 ˆH(2(N + η) − 3)(2(N + η) + 1)(1 − 2(N + η))4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Here η is a constant to be determined from the constraints on the structure function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 14 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES By acting on Fock basis states |z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' E⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we can show that the structure function Φ(III) 2 is factorized as Φ(III) 2 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='6 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='Eδ1(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�z + η − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�12 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='f(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− (1 + i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3)δ1(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ (−1 + i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3)g(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='Eδ1(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where f(E) = 2 � b2 3 − 8Eb3 − 8(b1 + b2 − 2E)E � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' g(E) = b4 3 − 16Eb3 3 + 4E(2b1 + 2b2 + 24E + 3)b2 3 − 32E2(2b1 + 2b2 + 8E + 3)b3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' + 16E2(b2 1 + 14b2b1 + 8(E − 3)b1 + b2 2 + 16E2 + 8b2(E − 3) + 12E + 15),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' δ1(E) = 3� ρ1(E) + ρ2(E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ρ1(E) = b6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 24Eb5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 240E2b4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 12b1Eb4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 12b2Eb4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 18Eb4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 1280E3b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 192b1E2b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 192b2E2b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 288E2b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 3840E4b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 1152b1E3b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 1152b2E3b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 1728E3b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 48b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1E2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 48b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2E2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 288b1E2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 480b1b2E2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 288b2E2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 360E2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 6144E5b3 − 3072b1E4b3 − 3072b2E4b3 − 4608E4b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 384b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1E3b3 − 384b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2E3b3 + 2304b1E3b3 + 3840b1b2E3b3 + 2304b2E3b3 − 2880E3b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 4096E6 + 3072b1E5 + 3072b2E5 + 4608E5 + 768b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1E4 + 768b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2E4 − 4608b1E4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 7680b1b2E4 − 4608b2E4 + 5760E4 + 64b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1E3 + 64b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2E3 + 3744b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1E3 − 2112b1b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2E3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 3744b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2E3 − 8064b1E3 − 2112b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b2E3 + 10944b1b2E3 − 8064b2E3 + 3456E3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ρ2(E) = 128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2043 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 8Eb3 − 8(b1 + b2 − 2E)E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�2 − 12E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(2b1 + 2b2 + 1)b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−8(2b1 + 2b2 + 1)Eb3 − 4E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 2(b2 + 4E − 4)b1 + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + 8b2 − 8b2E − 4E − 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='���3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 262144 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 24Eb5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 6E(2b1 + 2b2 + 40E + 3)b4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 32E2(6b1 + 6b2 + 40E + 9)b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 24E2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 4(5b2 − 12E + 3)b1 + 2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + 160E2 + 72E + 12b2(4E − 1) + 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 192E3 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 4(5b2 − 4E + 3)b1 + 2b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + 32E2 + 24E + 4b2(4E − 3) + 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 32E3 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 + (−66b2 + 24E + 117)b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='11b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + (40E − 57)b2 − 16E2 + 24E + 42 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+2b3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + 3b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2(8E + 39) + 12b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='8E2 − 12E − 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='32E3 + 36E2 + 45E + 27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='���2� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3) which give the (p + 1)-dimensional unirreps of thedeformed oscillator algebra, we determine the constant η and obtain the following algebraic equations for the energies E: 15 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η1(E) = 1 12 � 6 − √ 3 � δ1(E)+(b3−4E)2 E − 8(b1 + b2) + g(E) Eδ1(E) + 12 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This η value gives five sets of algebraic equations, δ1(E) + (b3 − 4E)2 + g(E) δ1(E) = (12p(p + 2) + 8(b1 + b2)) E, η1(E) − 1 24 � �12 + ϵ √ 6 � f(E) E + (−1 + i √ 3)δ1(E) E − (1 + i √ 3)g(E) Eδ1(E) + 24 � � + p + 1 = 0, η1(E) − 1 24 � �12 + ϵ √ 6 � f(E) E − (1 + i √ 3)δ1(E) E + (−1 + i √ 3)g(E) Eδ1(E) + 24 � � + p + 1 = 0, where ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Real solutions to each algebraic equation above give the energies of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η2(E) = 1 24 � 12 − √ 6 � f(E) E + i(i+ √ 3)δ1(E) E − i(−i+ √ 3)g(E) Eδ1 + 24 � and energy spectra from the real solutions of the three sets of algebraic equations f(E) + i(i + √ 3)δ1(E) − i � −i + √ 3 � g(E) δ1(E) = 24p(p + 2)E, η2(E) − 1 24 � � �12 + ϵ √ 6 � � � �f(E) E − i � −i + √ 3 � δ1(E) E + i � i + √ 3 � g(E) Eδ1(E) + 24 � � � + p + 1 = 0, where again ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η3(E) = 1 24 � 12 − √ 6 � f(E) E − (1+i √ 3)δ1(E) E + (−1+i √ 3)g(E) Eδ1(E) + 24 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This η yields the algebraic equa- tion whose real solutions gives other possible energies of the system, f(E) − (1 + i √ 3)δ1(E) + (−1 + i √ 3)g(E) δ1(E) = 24p(p + 2)E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is in general very difficult to solve the above algebraic equations for E analytically due to their complicated forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' To demonstrate the existence of real solutions to the above algebraic equations, we have a closer look at cases of restricted model parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' As an example, we consider b1 = b2 = b3 = h for any h ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this case the structure function reduces to Φ(III) 2 (z, η) = � z + η − 1 2 �2 � �z + η − 1 8 � �4 − � 2h2(E) − 2h1(E) E � � � � � �z + η − 1 8 � �4 + � 2h2(E) − 2h1(E) E � � � � � �z + η − 1 8 � �4 − � 2h2(E) + 2h1(E) E � � � � � �z + η − 1 8 � �4 + � 2h2(E) + 2h1(E) E � � � � , where h1(E) = � h4 + 16h3E + 8h2E(12E + 1) + 64hE2(4E − 15) + 16E2 (16E2 + 8E + 17), h2(E) = h2 − 24hE + 16E2 + 12E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3), we determine the constant η and the corresponding energies for the model parameters b1 = b2 = b3 = h as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 16 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η1 = 1 2 and E1,± = 4h2 + 12hp2 + 24hp + 15h + 8p4 + 32p3 + 42p2 + 20p + 1 ± m(h, p) 4 (4h + 4p2 + 8p + 3) , where m(h, p) = � (4h2 + 3h (4p2 + 8p + 5) + 8p4 + 32p3 + 42p2 + 20p + 1)2 − h2 (4h + 4p2 + 8p + 3)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is easy to check that m(p) is real for any p ∈ N+ if h > 0 and so E1,± give the energies of the system for the model parameters b1 = b2 = b3 = h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η2(E) = 1 8 � 4 + ϵ � 2h2(E)−2h1(E) E � with ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For this η value, the energies are E2,± = 8h2 + 6hp2 + 12hp + 12h + p4 + 4p3 + 3p2 − 2p − 4 ± n(h, p) 8(4h + p(p + 2)) , where n(h, p) = � (8h2 + 6h (p2 + 2p + 2) + p4 + 4p3 + 3p2 − 2p − 4)2 − 4h2(4h + p(p + 2))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It can be checked that n(p) is real for h > 1 and so E2,± give the energies of the system for the model parameters b1 = b2 = b3 = h > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Other possible energies corresponding to η2(E) are E3,± = 1 8 � 4 � (1 − 4h)p(p + 2) − 2h + 4p2 + 8p + 3 ± l(h, p) � , where l(p, h) = 8p2� 4z(h, p)(16p + 7) − 4h (4z(h, p) + 20p2 + 40p + 3) + 16p4 + 64p3 + 80p + 22, z(p, h) = � (1 − 4h)p(p + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is easily seen that l(p) and z(p) are real for h < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Hence, E3,± are real for h < 0 and give the energies of the system for model parameters b1 = b2 = b3 = h < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 Potential V3(µ, ν) The constants of motion of the superintegrable system with potential V3 and the Hamiltonian ˆH = µ2∂2 µ−ν2∂2 ν (µ+ν)(2+µ−ν) + V3(µ, ν) are given by [14], A = −4µ2ν2 (∂µ + ∂ν)2 (µ + ν)2 − c2 µ − ν µν − c3 (µ − ν)2 µ2ν2 , B = ν2(µ + 2)µ∂2 ν − µ2(ν − 2)ν∂2 µ (µ + ν)(2 + µ − ν) − 4µ2ν2 (∂µ + ∂ν)2 (µ + ν)2 − c1µ2ν2 + c2µν + 2c3(1 + µ − ν) µν(2 + µ − ν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These integrals form the quadratic algebra with the commutation relations [A, B] = C, [A, C] = −2{A, B} − B − 2c1c2 + 4c2 ˆH, [B, C] = 2B2 − 8c3 ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The Casimir operator for the quadratic algebra is given by K3 = C2 + 2{A, B2} − 16c3 ˆHA + 5B2 + 4c2 � c1 − 2 ˆH � B, 17 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES which can be expressed in terms of the Hamiltonian as K3 = 16c3 ˆH2 + 4(c2 2 − 4c1c3) ˆH + 4c2 1c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It can be shown that A = (N + η)2 − 1 2, B = − c1c2 − 2c2 ˆH 2 � (N + η)2 − 1 4 � + b†ρ(N) + ρ(N)b, where η is a constant to be determined and ρ(N) = 1 3 · 220(N + η)(1 + N + η)(1 + 2(N + η))2 , convert the quadratic algebra into the deformed oscillator algebra with the structure function Φ(III) 3 (N, η) = −786432 � −c2 1 + 4c1 ˆH + ˆH (2N + 2η − 1)2 − 4 ˆH2� � c2 2 − c3(2N + 2η − 1)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By acting it on a Fock basis |z, E⟩, the structure function becomes Φ(III) 3 (z, η) = 12582912 c3 E � z + η − � 1 2 − c2 2√c3 �� � z + η − � 1 2 + c2 2√c3 �� � z + η − E − √ E(c1 − 2E) 2E � � z + η − E + √ E(c1 − 2E) 2E � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraint conditions (3), we obtain 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η(E) = E− √ E(c1−2E) 2E or η(E) = E+ √ E(c1−2E) 2E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For both cases, we have E = 1 8 � 4c1 + (p + 1)2 ± (p + 1) � 8c1 + (p + 1)2 � , which is real for c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This gives the energy spectrum of the system for any model parameters c1, c2, c3 with c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ηϵ = 1 2 � 1 + ϵ c2 √c3 � , where ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The corresponding energies are given by Eϵ = 1 8 � ��4c1 + � 2(p + 1) + ϵ c2 √c3 �2 ± � 2(p + 1) + ϵ c2 √c3 � � � � �8c1 + � 2(p + 1) + ϵ c2 √c3 �2 � �� , which is real for c1 > 0 and c3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 Potential V4(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ν) For a superintegrable system with the Hamiltonian ˆH = µ2∂2 µ−ν2∂2 ν (2+µ−ν)(µ+ν)+V4(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ν) associated to the potential V4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' the constants of motion are given by [14] A = ν2(µ + 2)µ∂2 ν − µ2(ν − 2)ν∂2 µ (µ + ν)(2 + µ − ν) − µν (d1(ν − 2) + d2(µ + 2) + 2d3(ν − µ + µν)) (µ + ν)(2 + µ − ν) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = 1 4µν(µ − ν + 2)(µ + ν)2 �� µ4(12ν3 − 12ν2 + ν + 1) + 2µ3ν − (ν − 1)µ2ν2� ∂2 µ + µν(µ − ν + 2) � µ2(12ν2 + 1) + 2µν + ν2� ∂µ∂ν +ν2 � µ3(12ν2 − 1) + µ2(12ν2 − 1) + µ(ν − 2)ν − ν2� ∂2 ν � − (µ − ν) �(µ − ν)(d1µ + d2ν) − 2d3(µ2 + ν2 + µν(2 + µ − ν)) � 4µν(µ + ν)(2 + µ − ν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 18 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES They satisfy the quadratic algebra relations [A, B] = C, [B, C] = −2B2 + 2 ˆHB − d2 3 2 , [A, C] = 2{A, B} − 2 ˆHA − B + (d1 + d2 + 1 2) ˆH − d1d2 2 − 2 ˆH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By a direct calculation, we find the Casimir operator of this algebra K4 = C2 − 2{A, B2} + 5B2 + 2 ˆH{A, B} − d2 3A + � 4 ˆH − (2d1 + 2d2 + 5) ˆH + d1d2 � B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By means of the differential operator representation of A and B above, the Casimir operator K4 can be expressed in terms of ˆH as K4 = 4 ˆH3 − (2d1 + 2d2 + 1) ˆH2 + � (d1 + d2)2 4 + d3(d2 − d1) � ˆH − d3(d3 − d2 1 + d2 2) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We can show that A =(N + η)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = ˆH 2 − −4 ˆH + 8(d1 + d2 + 1 2) ˆH − 4d1d2 32 � (N + η)2 − 1 4 � + ρ(N)b† + bρ(N) with ρ(N) = 1 3 · 220(N + η)(1 + N + η)(1 + 2(N + η))2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' give a realization of the quadratic algebra in terms of the deformed oscillator algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' with the structure function given by Φ(III) 4 (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) =16384(1 − 2(N + η))2 � 3 � d3 � −d2 1 + d2 2 + d3 � + 4d3 ˆH(d1 − d2) +4 ˆH2(2d1 + 2d2 + 1) − ˆH(d1 + d2)2 − 16 ˆH3� + 6d1 ˆH(d2 − 2 ˆH) −4 ˆH2(3d2 − 6 ˆH + 2) + � ˆH2 − d2 3 � � 12(N + η)2 − 12(N + η) − 1 � − 7d2 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Here η is a constant parameter to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Acting on the Fock states |z, E⟩, the structure function is factorized as Φ(III) 3 (z, η) = � z + η − 1 2 �2 � z + η − � 1 2 − γ1(E) 2 �d2 3 − E2� �� � z + η − � 1 2 + γ1(E) 2 �d2 3 − E2� �� , where γ1(E) = � d2 3 − E2 × � −d2 1(d3 + E) + 4d1E(d3 + E) + d2 2(d3 − E) + 4d2E(E − d3) − 8E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' From the constraints (3), we determine the constant η and the energy spectrum E of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We list the results as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 1: η(E) = 1 2 − γ1(E) 2(d2 3−E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Corresponding to this η value, we have either � −d2 1(d3 + E) + 4d1E(d3 + E) + d2 2(d3 − E) + 4d2E(E − d3) − 8E3 = (p + 1) ��d2 3 − E2� (14) or � −d2 1(d3 + E) + 4d1E(d3 + E) + d2 2(d3 − E) + 4d2E(E − d3) − 8E3 = 2(p + 1) ��d2 3 − E2�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (15) 19 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES Notice that obviously the solution space of the algebraic equation (15) is subspace of that of (14) and so the energy spectrum of the system corresponding to η1(E) is given by solutions to (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 2: η = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this case, E satisfies the same algebraic equation as (15) and so do not give new energies of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 3: η(E) = � 1 2 + γ1(E) 2(d2 3−E2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This η value give the same equations for E as those in Case 1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Due to the complexity of the algebraic equations, it is hard to see whether or not they lead to real energies E for general model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' However, we can show that when the model parameter d3 = 0, the structure function reduces to Φ(III) 3 (z, η) = � z + η − 1 2 �2 � z + η − 1 2E � E − � E �d2 1 − 4d1E + d2 2 − 4d2E + 8E2��� � z + η − 1 2E � E + � E �d2 1 − 4d1E + d2 2 − 4d2E + 8E2��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this case, by imposing the constraints (3) we obtain the parameter η and the energies of the system with model parameter d3 = 0, η−(E) = 1 2E � E − � E �d2 1 − 4d1E + d2 2 − 4d2E + 8E2�� , E± = 1 16 � 4d1 + 4d2 + (p + 1)2 ± � (p + 1)2 ((p + 1)2 + 8(d1 + d2)) − 16 (d1 − d2)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Other η values from the constraints give rise to same energies as E± above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is clear that both E± are real for d1 = d2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' So E± give the energy spectrum of the system for model parameters d1 = d2 > 0, d3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The corresponding structure function of the p + 1)-dimensional unirreps is Φ(III)E±(z) = z(z − p − 1) � z − 1 2E± � E± �d2 1 − 4d1E± + d2 2 − 4d2E± + 8E2 ± � �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 Darboux Space IV In Darboux space IV, there are 3 different potentials in the separable coordinates (µ, ν), (u, v) and (ω, ϕ): V1(µ, ν) = −sin2(2µ)(4a1 exp(2ν) + 4a2 csc2(2µ) + 4a3 exp(4ν)) 2 cos 2µ + a4 , V2(u, v) = − sin2(2u)( b2 sinh2 v + b3 cosh2 v) + b1 2 cos 2u + b4 , V3(ω, ϕ) = c1 cos2 ϕ + c2 cosh2 ω + c3 � 1 sin2 ϕ − 1 sinh2 ω � c4+2 sinh2(2ω) + c4−2 sin2(2ϕ) , where ai, bi ci are real model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 Potential V1(µ, ν) The integrals of motion of the superintegrable system in Darboux space IV with potential V1 and the Hamiltonian ˆH = − 4µ2ν2 (a4+2)µ2+(a4−2)ν2 + V1(µ, ν) are A =µ2∂2 µ + 2µν∂µ∂ν + ν2∂2 ν + µ∂µ + ν∂ν + a1(µ2 + ν2) + a3(µ2 + ν2)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = 4(a4 + 2)µ2∂2 µ − 4(a4 − 2)ν2∂2 ν (a4 + 2)µ2 + (a4 − 2)ν2 + 2a1 �(a4 + 2)µ2 − (a4 − 2)ν2� + 4a3 �(a4 + 2)µ4 − (a4 − 2)ν4� + 16a2 (a4 + 2)µ2 + (a4 − 2)ν2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 20 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES They form the quadratic algebra with the commutation relations given by [14] [A, B] = C, [A, C] = 8{A, B}a − 16B + 32a1 ˆH, [B, C] = −8B2 + 256a3A + 128 a3a4 ˆH + 32(a2 1 + 4a3 + 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By a direct calculation, we find that the Casimir operator is K1 = C2 − 8{A, B2} + 256 a3A2 + 80B2 + � 256 a3a4 ˆH + 64(16a2a3 + a2 1 + 4a3) � A − 64 a1 ˆHB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' With the differential operator representation of A and B, the Casimir operator can be expressed in terms of ˆH as K1 = −256 a3 ˆH2 + 64 a4(4a3 − a2 1) ˆH + 128(a2 1 + 4a3 + 8a2a3 − 2a2 1a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After a long calculation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we find that the change of basis A = 4(N + η)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = − 128a1 ˆH 256 � (N + η)2 − 1 4 � + ρ(N)b† + bρ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where ρ(N) = 1 3 · 215(N + η)(1 + N + η)(1 + 2(N + η))2 maps the quadratic algebra to the deformed oscillator algebra with the structure function Φ(IV ) 1 (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = − 805306368 (2(N + η) − 1)2 × � 128(−2a2 1a2 + a2 1 + 8a2a3 + 4a3) + 64 ˆHa4(4a3 − a2 1) − 256a3 ˆH2� + 131072 � 12(N + η)2 − 12(N + η) − 1 � (2(N + η) − 1)2 × � 131072(a2 1 + 16a2a3 + 4a3) + 524288 a3a4 ˆH + 1048576 a3 � − 16384 (2(N + η) − 1)2 � −7340032(a2 1 + 16a2a3 + 4a3) + 29360128 a3a4 ˆH − 46137344 a3 � + 51539607552 a2 1 ˆH2 + 206158430208 a3 (2(N + η) − 3) (2(N + η) + 1) (2(N + η) − 1)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Acting on the Fock basis state |z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' E⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we find that the structure function has the factorization Φ(IV ) 1 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = � z + η − � 1 2 − ia1 4√a3 �� � z + η − � 1 2 + ia1 4√a3 �� � z + η − 1 2 √ 2 �√ 2 − � 1 − 4a2 − Ea4 − m1(E) �� � z + η − 1 2 √ 2 �√ 2 + � 1 − 4a2 − Ea4 − m1(E) �� � z + η − 1 2 √ 2 �√ 2 − � 1 − 4a2 − Ea4 + m1(E) �� � z + η − 1 2 √ 2 �√ 2 + � 1 − 4a2 − Ea4 + m1(E) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where m1(E) = � (4a2 + Ea4 + 1)2 − 4(4a2 + E(E + a4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3), we have 21 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η1(E) = 1 2 √ 2 �√ 2 − � 1 − 4a2 − Ea4 − m1(E) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This η value gives the following two sets of energies and corresponding structure functions E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ =1 2(p + 1) � −(p + 1) a4 + ϵ � (a2 4 − 4)(p + 1)2 + 16(1 − a2) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Φ(IV ) E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ (z) =z(z − p − 1) � z − 1 2 √ 2 � 1 − 4a2 − E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) − ia1 4√a3 � � z − 1 2 √ 2 � 1 − 4a2 − E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1(E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) + ia1 4√a3 � � z − 1 2 √ 2 �� 1 − 4a2 − E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1(E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) − � 1 − 4a2 − E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 + m1(E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ �� � z − 1 2 √ 2 �� 1 − 4a2 + E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1(E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) + � 1 − 4a2 − E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 + m1(E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ = − 1 a4 + ϵ 4 � 4a2 + 4p2 + 8p + 3 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' a4 ̸= ±4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ΦE2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ(z) =z(z − p − 1) � z − 1 2 √ 2 � 1 − 4a2 − E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) − ia1 4√a3 � � z − 1 2 √ 2 � 1 − 4a2 − E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1(E1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) + ia1 4√a3 � � z − 1 √ 2 �� 1 − 4a2 − E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1(E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) �� � z − 1 2 √ 2 �� 1 − 4a2 + E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 − m1(E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) + ϵ � 1 − 4a2 − E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵa4 + m1(E2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϵ) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Notice that the energies E1,ϵ are real for a2 4 > 4, a2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η2(E) = 1 2 √ 2 �√ 2 − � 1 − 4a2 − Ea4 + m1(E) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The energies are the same as those given in case 1 above 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 Potential V2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' v) The constants of motion of the superintegrable system in Darboux space IV with the Hamiltonian ˆH = − sin2(2u)(∂2 v+∂2 u) 2 cos(2u)+b4 + V2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' v) are [14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' A = e−2v ��e4v + 1 � (2b4 cos(2u) + 3 cos(4u) + 1) 2b4 + 4 cos(2u) ∂2 v − sin(2u) �e4v + 1 � sin(2u) b4 + 2 cos(2u) ∂2 u � + e−2v � sin(2u) � e4v + 1 � ∂u + sin(2u) � e4v − 1 � ∂u∂v + cos(2u) � e4v − 1 � ∂v � + 1 2 cos 2u + b4 � 2b1 cosh 2v + (b2 + b3)(4 − b2 4) + (cos 4u + 2b4 cos 2u + 3) � b2 sinh2 v − b3 cosh2 v �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = ∂2 v + b2 sinh2 v + b3 cosh2 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These integrals generates the quadratic algebra the commutation relations as follows [A, B] =C, [A, C] =8{A, B} + 16b4(b2 + b3)A − 16B + 32(b1 + b3) ˆH − 16b4(b2 + b3), [B, C] =8B2 + 96A2 + � 64b4 ˆH + (2b2 − 2b3 + b1 + 3) � A + 32 ˆH2 + 32b4(2b2 − 2b3 + 1) ˆH + 64b1(b2 − b3) − 8(b2 4 − 4)(b2 + b3)2 + 32(b1 + 2b2 − 2b3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 22 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES By a direct computation, we find the Casimir operator of the algbebra, K2 = C2 + 64A3 − 8{A, B2} − 16b4(b2 + b3){A, B} + 64 � b4 ˆH + 2b2 − 2b3 + b1 + 7 � A2 + � 160b4(b2 + b3) − 64(b2 + b3) ˆH � B − 64b4(2b3 − 2b2 − 1) ˆHA − 16 � (b2 4 − 4)(b2 + b3)2 + 8(b1 + 1)(b3 − b2) − 4b1 + 32 � A + 64 ˆH2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' With the differential realization of A and B, the Casimir K2 is expressible in terms of ˆH as follows K2 =128(b3 − b2 + 1) ˆH2 + 128b4(b2 − b3 + 1) ˆH + (128 − 80b2 4 − 64b1)(b2 + b3)2 − 128(b1 + 2)(b3 − b2 − 1) − 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After a long computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we find that the realization A = −4 � (N + η)2 − 1 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = b4(b2 + b3) 8 − −32 · 16b4(b2 + b3) − 256 · (b1 + 2b2 − 2b3) 4γ3 � (N + η)2 − 1 4 � + ρ(N)b† + bρ(N),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where ρ(N) = 1 3 · 212 · (−8)8(N + η)(1 + N + η)(1 + 2(N + η))2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' converts the quadratic algebra into the deformed oscillator algebra with the structure function Φ(IV ) 2 (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) =268435456 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(2N + 2η − 1)2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='48(5a2 + 4b1 − 8)(b2 + b3)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+384 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(b1 + 2)(b2 − b3 + 1) + ˆH2(−b2 + b3 + 1) + ˆHb4(b2 − b3 + 1) − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 64 (2N + 2η − 1)2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='12(N + η)2 − 12(N + η−) − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b1(2b2 − 2b3 + 3) + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + 2b2(b3 + ˆHb4 + 3) + b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 2b3 ˆHb4 − 6b3 + ˆH2 + 3 ˆHb4 + 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ (2N + 2η + 1)(2N + 2η − 1)6(b1 + 2b2 − 2b3 + ˆHb4 + 6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='986 b1(b2 − b3) + 448 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b1 + 2b2 − 2b3 + ˆHb4(2b2 − 2b3 + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 112 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(b2 + b3)2 + 448 ˆH2 + 96 b4(b2 + b3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ˆH(b1 + b3) − b4(b2 + b3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+704 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='b1 + 2b2 − 2b3 + ˆHb4 + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 32b2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4(b2 + b3)2 + 192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2N + 2η − 3 + ˆH2(b1 + b3)2 − (2N + 2η − 1)4 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4(N + η)2 − 4(N + η) − 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 23 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES Acting on Fock basis states |z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' E⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' the structure function is factorized as follows Φ(IV ) 2 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 2b2 + 2b3 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(1 − 4b2)(1 + 4b3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 2b2 + 2b3 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(1 − 4b2)(1 + 4b3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 2b2 + 2b3 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(1 − 4b2)(1 + 4b3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − b1 − Eb4 − m2(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − b1 − Eb4 − m2(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − b1 − Eb4 + m2(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − b1 − Eb4 + m2(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where m2(E) = � (1 + b1 + Eb4)2 − 4 (b1 + E2 + Eb4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraint conditions (3), we determine the parameter η and the corresponding energies of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We find Case 1: η1,± = 1 2 √ 2 �√ 2 − � 1 − 2b2 + 2b3 ± � (1 − 4b2)(1 + 4b3) � and the energies E satisfy 2 √ 2(p + 1) − n2,± = � 1 − b1 − Eb4 + m2(E), where n2,± = � 1 − 2b2 + 2b3 ± � (1 − 4b2)(1 + 4b3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Noticing (1 − 4b2)(1 + 4b3) = (1 − 2b2 + 2b3)2 − (2b2 + 2b3)2 ≤ (1 − 2b2 + 2b3)2, we conclude that both n2,± are real if b2 < 1 4, b3 > −1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Solving the algebraic equations give the energies of the system and the corresponding structure func- tions of the (p + 1)-dimensional unirreps of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We have E1±,ϵ = − b4 4 � 2 √ 2(p + 1) − n2,± �2 + ϵ � 2 √ 2(p + 1) − n2,± � � 8(1 − b1) + (b2 4 − 4) � 2 √ 2(p + 1) − n2,± �2, Φ(IV ) E1±,ϵ(z, η) =z (z − p − 1) � z − 1 √ 2n± � � z − 1 2 √ 2(n± − n∓) � � z − 1 2 √ 2(n± + n∓) � � z − 1 2 √ 2 � n± − � 1 − b1 − E1±,ϵb4 − m2(E1±,ϵ) �� � z − 1 2 √ 2 � n± + � 1 − b1 − E1±,ϵb4 − m2(E1±,ϵ) �� � z − 1 2 √ 2 � n± − � 1 − b1 − E1±,ϵb4 + m2(E1±,ϵ) �� , where ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The energies E1±,ϵ are real for the model parameters b2 < 1 4, b3 > −1 4, b1 < 1, b2 4 > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 24 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES Case 2: η1,± = 1 2 √ 2 �√ 2 + � 1 − 2b2 + 2b3 ± � (1 − 4b2)(1 + 4b3) � and the energies E satisfy 2 √ 2(p + 1) + n2,± = � 1 − b1 − Eb4 + m2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Solutions of the equations give the energies of the system and the corresponding structre functions of the (p + 1)-dimensional unirreps of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We have E2±,ϵ = − b4 4 � 2 √ 2(p + 1) + n2,± �2 + ϵ � 2 √ 2(p + 1) + n2,± � � 8(1 − b1) + (b2 4 − 4) � 2 √ 2(p + 1) + n2,± �2, Φ(IV ) E2±,ϵ(z, η) =z (z − p − 1) � z + 1 √ 2n± � � z + 1 2 √ 2(n± − n∓) � � z + 1 2 √ 2(n± + n∓) � � z + 1 2 √ 2 � n± − � 1 − b1 − E2±,ϵb4 − m2(E2±,ϵ) �� � z + 1 2 √ 2 � n± + � 1 − b1 − E2±,ϵb4 − m2(E2±,ϵ) �� � z + 1 2 √ 2 � n± + � 1 − b1 − E2±,ϵb4 + m2(E2±,ϵ) �� , where ϵ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The energies E2±,ϵ are real for the model parameters b2 < 1 4, b3 > −1 4, b1 < 1, b2 4 > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 3: η3,−(E) = 1 2 √ 2 �√ 2 − � 1 − b1 − Eb4 + m2(E) � and √ 2(p + 1) = � 1 − b1 − Eb4 + m2(E) or 2 √ 2(p + 1) = � 1 − b1 − Eb4 + m2(E) + � 1 − b1 − Eb4 − m2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The first algebraic equation gives the energies E3,1 = p + 1 2 � −(p + 1)b4 ± � 4(1 − b1) + (b2 4 − 4)(p + 1)2 � , which is real for b1 < 1, b2 4 > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The corresponding structure function is Φ(IV ) E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) =z (z − p − 1) � z − 1 2 √ 2 �� 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 + m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) − n− �� � z − 1 2 √ 2 �� 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 + m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) + n− �� � z − 1 2 √ 2 �� 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 + m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) − n+ �� � z − 1 2 √ 2 �� 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 + m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) + n+ �� � z − 1 2 √ 2 �� 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 + m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) − � 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 − m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) �� � z − 1 2 √ 2 �� 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 + m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) + � 1 − b1 − E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1b4 − m2(E3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The second algebraic equation yields the energies E3,2 = 1 2 − b4 � 4(p + 1)2 + b1 − 1 � , 25 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES which is well defined for the model parameter b4 ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The associated structure function is given by Φ(IV ) E3,2 (z, η) =z (z − p − 1) � z − 1 2 √ 2 �� 1 − b1 − E3,2b4 + m2(E3,2) − n− �� � z − 1 2 √ 2 �� 1 − b1 − E3,2b4 + m2(E3,2) + n− �� � z − 1 2 √ 2 �� 1 − b1 − E3,2b4 + m2(E3,2) − n+ �� � z − 1 2 √ 2 �� 1 − b1 − E3,2b4 + m2(E3,2) + n+ �� � z − 1 2 √ 2 �� 1 − b1 − E3,2b4 + m2(E3,2) − � 1 − b1 − E3,2b4 − m2(E3,2) �� � z − 1 √ 2 � 1 − b1 − E3,2b4 + m2(E3,2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Case 4: η4,+ = 1 2 √ 2 �√ 2 + � 1 − b1 − Eb4 − m2(E) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We have the algebraic equation 2 √ 2(p + 1) = � 1 − b1 − Eb4 + m2(E) − � 1 − b1 − Eb4 − m2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Solving, we obtain E4 = 1 2 + b4 � 4(p + 1)2 + b1 − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This gives the energy spectrum of the system for the model parameter b4 ̸= −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The corresponding structure function is Φ(IV ) E4 (z, η) =z (z − p − 1) � z + 1 2 √ 2 �� 1 − b1 − E4b4 − m2(E4) − n− �� � z + 1 2 √ 2 �� 1 − b1 − E4b4 − m2(E4) + n− �� � z + 1 2 √ 2 �� 1 − b1 − E4b4 − m2(E4) − n+ �� � z + 1 2 √ 2 �� 1 − b1 − E4b4 − m2(E4) + n+ �� � z + 1 √ 2 � 1 − b1 − E4b4 − m2(E4) � � z + 1 2 √ 2 �� 1 − b1 − E4b4 − m2(E4) + � 1 − b1 − E4b4 + m2(E4) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 26 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 Potential V3(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ϕ) The constants of motion of the superintegrable system in Darboux space IV with the potential V3(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ϕ) are [14] A = − 2c4 ∂2 ϕ + ∂2 ω c4+2 sinh2(2ω) + c4−2 sin2(2ϕ) + (c4 + 2) sin2(2ϕ)∂2 ϕ − (c4 − 2) sinh2(2ω)∂2 ω (c4 + 2) sin2(2ϕ) + (c4 − 2) sinh2(2ω) + 1 c4+2 sinh2(2ω) + c4−2 sin2 ω � c4 + 2 sinh2(2ω) � c3 sin2 ϕ + c1 cos2 ϕ � + c4 − 2 sin2(2ω) � c3 sinh2 ω − c2 cosh2 ω �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='B =1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 sin(2ϕ) sinh(2ω) tan(ϕ − iω) tan(ϕ + iω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='cot(2ϕ) ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ω + coth(2ω) ∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−i cos(2ϕ) sinh(2ω) sinh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�tan(ϕ − iω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='tan(ϕ + iω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='∂ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+i cosh(2ω) sin(2ϕ) sinh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�tan(ϕ − iω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='tan(ϕ + iω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='∂ϕ + 2 cosh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�tan(ϕ − iω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='tan(ϕ + iω) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='c4+2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='sinh2 2ω + c4−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='sin2 ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� c4 + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='sinh2 2ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�c1 cosh 2ω tan2 ϕ − c2 cos 2ϕ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 cos2 ϕ(sinh2 ω − sin2 ϕ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='sin2 ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ c4 − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='sin2 2ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�c2 cos 2ϕ tanh2 ω + c1 cosh 2ω − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 cosh2 ω(sinh2 ω − sin2 ϕ) + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='sinh2 ω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These integrals form the quadratic algebra with the following commutation relations [A, B] =C, [A, C] = − 8{A, B} − 16B − 16(c1 − c3)(c2 − c3), [B, C] = − 24A2 + 8B2 + 16 � 2c4 ˆH − 2c1 + 2c2 + 3 � A − 16 [(c4 + 2)c1 + (c4 − 2)c2 − c4 + 64c3] ˆH − 8(c2 4 − 4) ˆH2 − 8c2 1 − 8c2 2 + 16c2 3 + 32c1c2 + 48c3(c1 + c2) − 16(c1 − c2), which is the symmetry algebra of the superintegrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We can calculate the Casimir operator of the algebra K3 =C2 − 16A3 + 8{A, B2} − 16(2c2 − 2c1 − 7)A2 + 80B2 − 16 � c2 4 − 4 + 2(c4 + 2)c1 − 2(c4 − 2)c2 + 8c3 + 2c4 � ˆH + 16 � c2 1 + c2 2 − 2c2 3 − 6c3(c1 + c2) − 4c1c2 + 2c1 − 2c2 − 8 � A + 32(c2 − c3)(c1 − c3)B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We can show that in terms of the Hamiltonian the Casimir K3 takes the simple form K3 =16(c2 4 − 4) ˆH2 − 16 � (c4 + 2)((c1 − c3)2 − 2c1) + (c4 − 2)((c2 − c3)2 + 2c2) − 8c3 − 4c4 � ˆH − 32(c1 − c2)(3c2 3 − c1c2 − c3(c1 + c2)) + 32(c2 1 + c2 2 − 4c3(c1 + c2) − 2c1c2 + 2c1 − 2c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After a long computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we find that the realization A(N) = −4(N + η)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' B = −(c1 − c3)(c2 − c3) 4 � (N + η)2 − 1 4 � + ρ(N)b + b†ρ(N) with ρ(N) = 1 3 · 212 · (−8)8(N + η)(1 + N + η)(1 + 2(N + η))2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 27 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES changes the quadratic algebra to the deformed oscillator algebra with the structure function Φ(IV ) 3 (N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) =268435456 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='12N 2 + 12N(2η − 1) + 12η2 − 12η − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(2N + 2η − 1)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 + c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−4c2 − 6c3 + 2 ˆHc4 + 4 ˆH − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−6c3 − 2 ˆHc4 + 4 ˆH + 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 128c3 ˆH + ˆH2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 − 4 ˆH2 + 6 ˆHc4 + 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 16 (2N + 2η − 1)2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='7c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 − 2c1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='14c2 + 21c3 − 7 ˆHc4 − 14 ˆH + 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 7c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−42c3 − 14 ˆH(c4 − 2) + 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 14c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 + 896c3 ˆH + 7 ˆH2c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 − 28 ˆH2 + 36 ˆHc4 + 36 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 3 (2N + 2η − 1)2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='352 ˆH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−c1 + (c4 − 2)(c2 − c3)2 + 2c2(c4 − 2) − c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 − 8c3 − 4c4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+32(c1 − c2) (c1(c2 + c3) + c3(c2 − 3c3)) − 16 ˆH2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 48(c1 − c3)2(c2 − c3)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 96 (2N + 2η − 3)(2N + 2η + 1)(2N + 2η − 1)4(c1 − c2 − ˆHc4 − 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+48 (2N + 2η − 1)4 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='(2N + 2η − 1)4 − 8(2N + 2η − 1)2 + 16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure function is a polynomial of degree 8 in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Acting on the Fock basis states |z, E⟩, it becomes a polynomial of degree 8 in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In order to determine the energy spectrum of the superintegrable system, we have to find the finite-dimensional unirreps of the deformed oscillator algebra by solving the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This requires the factorization of the structure function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' However, it turns out to be very difficult to factorize the structure function for general model parameters ci (even using symbolic computation softwares such as Mathematica).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In the following we restrict our attention to special model parameters and present analytic and closed-form results for c1 = c2 = 1, c3 = c4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we find that the structure function factorizes as Φ(IV ) 3 (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 − 12E + 5 − 2E + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 − 12E + 5 − 2E + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 − 12E + 5 − 2E + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 − 12E + 5 − 2E + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 − 4E − 3 + 2E − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + η − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 − 4E − 3 + 2E − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Imposing the constraints (3), we determine the constant η and obtain the energies of the system and the structure function of the symmetry algebra for the model parameters c1 = c2 = 1, c3 = c4 = 0 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The constant η is ηa(E) = 1 2 √ 2 �√ 2 − � − √ 4E2 − 12E + 5 − 2E + 3 � and the energies are given by the equation 2 √ 2(p + 1) = �� 4E2 − 12E + 5 − 2E + 3 + � − � 4E2 − 12E + 5 − 2E + 3 =⇒ Ea = −2(p + 1)2 + 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 28 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES The associated structure function is Φ(IV ) Ea (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = z (z − p − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2a − 12E2a + 5 − 2Ea + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The constant η is η2c(E) = 1 2 √ 2 �√ 2 + � − √ 4E2 − 12E + 5 − 2E + 3 � and the energy E satisfies 2 √ 2(p + 1) = �� 4E2 − 12E + 5 − 2E + 3 − � − � 4E2 − 12E + 5 − 2E + 3 =⇒ Ec = −2(p + 1)2 + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure function is given by Φ(IV ) Ec (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = z (z − p − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z + 1 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 29 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The constant η is ηd(E) = 1 2 √ 2 �√ 2 − � − √ 4E2 − 4E − 3 + 2E − 1 � and the energies are 2 √ 2(p + 1) = �� 4E2 − 4E − 3 + 2E − 1 + � − � 4E2 − 4E − 3 + 2E − 1 =⇒ Ed = 2(p + 1)2 − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The corresponding structure function reads Φ(IV ) d (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = z (z − p − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='d − 4Ed − 3 + 2Ed − 1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='d − 12Ed + 5 − 2Ed + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='d − 4Ed − 3 + 2Ed − 1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='d − 12Ed + 5 − 2Ed + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='d − 4Ed − 3 + 2Ed − 1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='d − 12Ed + 5 − 2Ed + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='d − 4Ed − 3 + 2Ed − 1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='d − 12Ed + 5 − 2Ed + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='d − 4Ed − 3 + 2Ed − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='d − 4Ed − 3 + 2Ed − 1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='d − 4Ed − 3 + 2Ed − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The constant η is ηe(E) = 1 2 √ 2 �√ 2 − �√ 4E2 − 4E − 3 + 2E − 1 � and the energies and structure functions are given by √ 2(p + 1) = �� 4E2 − 4E − 3 + 2E − 1 =⇒ Ee = 1 2 � (p + 1)2 + 1 + 1 (p + 1)2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Φ(IV ) e (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = z (z − p − 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='z − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2e − 4Ee − 3 + 2Ee − 1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2e − 4Ee − 3 + 2Ee − 1 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4E2e − 4Ee − 3 + 2Ee − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The constant η is ηf(E) = 1 2 √ 2 �√ 2 + � − √ 4E2 − 4E − 3 + 2E − 1 � and the energies are 2 √ 2(p + 1) = �� 4E2 − 4E − 3 + 2E − 1 − � − � 4E2 − 4E − 3 + 2E − 1 =⇒ Ef = 2(p + 1)2 + 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 30 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES The structure function of the (p + 1)-dimensional unirreps has the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3 New superintegrable systems in 2D Darboux spaces In this section, we investigate superintegrable systems in 2D Darboux spaces with linear and quadratic or quintic integrals of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We will first construct generic cubic and quintic algebras and derive their Casimir operators and realizations in terms of the deformed oscillator algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We will then present examples of new superintegrable systems in 2D Darboux spaces with cubic symmetry algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 Generic cubic and quintic algebras generated by linear, quadratic or quintic integrals We start with the construction of generic cubic and quintic algebras with structure coefficients involving the Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Let ˆX1, ˆY1 be linear integrals, and let ˆX2, ˆY2 be quadratic and cubic integrals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' That is, deg ˆX1 = 1 = deg ˆY1, deg ˆX2 = 2, deg ˆY2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We define the operators ˆF and ˆG by ˆF = [ ˆX1, ˆX2] and ˆG = [ ˆY1, ˆY2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Then deg ˆF = deg ˆX1 + deg ˆX2 − 1 = 2 and deg ˆG = deg ˆY1 + deg ˆY2 − 1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By analysing the degrees of the integrals and applying the Jacobi identity constraint [34], we obtain the following generic cubic and quintic algebras Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Integrals { ˆX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆF} satisfy the cubic commutation relations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [ ˆX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2] = ˆF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [ ˆX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆF] =u1 ˆX2 1 + u2 ˆX1 + u3 ˆX2 + u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [ ˆX2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆF] =v1 ˆX3 1 + v2 ˆX2 1 + v3 ˆX1 − u2 ˆX2 − u1{ ˆX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2} + v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (16) and integrals { ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆG} form the following quintic commutation relations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [ ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2] = ˆG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [ ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆK] =α ˆY 3 1 + β ˆY 2 1 + δ ˆY1 + ϵ ˆY2 + ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [ ˆY2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆK] =a ˆY 5 1 + b ˆY 4 1 + c ˆY 3 1 + d ˆY 2 1 + e ˆY1 + 1 2 (α ϵ − 2 δ) ˆY2 − 3 2α{ ˆY 2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} − β{ ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} + z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (17) where uj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' vj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' , α, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' , z are polynomials of the Hamiltonian ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Moreover, the coefficients v1 in (16)and a in (17) are not zero polynomials of ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The proof of this proposition is a short and straightforward computation from the Jacobi identity requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For the polynomials on both sides of the commutation relations (16) and (17) to have the same degree, we must have that v1, v2, u1, u2, u3, α, β, ϵ, a, b are constants and u = u(0) + u(1) ˆH, v3 = v(0) 3 + v(1) 3 ˆH, v = v(0) + v(1) ˆH, δ = δ(0) + δ(1) ˆH, ζ = ζ(0) + ζ(1) ˆH, c = c(0) + c(1) ˆH, d = d(0) + d(1) ˆH, e = e(0) + e(1) ˆH + e(2) ˆH2, z = z(0) + z(1) ˆH + z(2) ˆH2, 31 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES where u(0), u(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' , are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We now construct the Casimir operators for both polynomial algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We have Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The Casimir operators C(3) and C(5) for the cubic and quintic algebras are respectively given by C(3) = ˆF 2 − u1{ ˆX2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2} − u2{ ˆX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2} + v1 2 ˆX4 1 + 2 3v2 ˆX3 1 + � v3 + u2 1 � ˆX2 1 + (u1u2 + 2v) ˆX1 − 2u ˆX2 − u3 ˆX2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' C(5) = ˆG2 − α{ ˆY 3 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} + β{ ˆY 2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} − δ{ ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} − ϵ ˆY 2 2 − 2ζ ˆY2 + a 3 ˆY 6 1 + 2 5b ˆY 5 1 + 1 2 � c + 5 3aϵ + 3αδ � ˆY 4 1 + � 2β(δ + 3α) + 2 5ϵb − 2d � ˆY 3 1 + �1 6 (5 − 6a) ϵ2 + e + 1 2ϵc + β2 − 3 4α (αϵ − 2δ) � ˆY 2 1 + � 2z + βδ + βϵ(α + δ) − 1 5bϵ2 − ϵd � ˆY1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By analysinf the degrees of the integrals in the algebras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we see that the Casimir operators C(3) and C(5) of the cubic and quintic algebras have the following general form C(3) = ˆF 2 + w1{ ˆX2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2} + w2{ ˆX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2 2} + w3{ ˆX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2} + w4 ˆX4 1 + w5 ˆX3 1 + w6 ˆX2 1 + w7 ˆX1 + w8 ˆX2 + w9 ˆX2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' C(5) = ˆG2 + ω1{ ˆY 3 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} + ω2{ ˆY 2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} + ω3{ ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY 2 2 } + ω4{ ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2} + ω5 ˆY 2 2 + ω6 ˆY2 + ω7 ˆY 6 1 + ω8 ˆY 5 1 + ω9 ˆY 4 1 + ω10 ˆY 3 1 + ω11 ˆY 2 1 + ω12 ˆY1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' where wj and ωj are coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Now using the quadratic commutation relations (16), we have [C(3), ˆX1] = − (u1 + w1){ ˆF, ˆX2 1} − (w3 + u2 + w2u1){ ˆF, ˆX1} − (w9 + u3){ ˆF, ˆX2} − (2u + w8 + w2u2) ˆF − w2{ ˆF, { ˆX1, ˆX2}} + ˆX1(w1){ ˆX2 1, ˆX2} + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' + ˆX1(w9) ˆX2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Setting the coefficients to be zero gives w1 = −u1, w2 = 0, w3 = −u2, w8 = −2u, w9 = −u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Similarly, from [C(3), ˆX2] = 0,, we have 0 =(2w4 − v1){ ˆF, ˆX3 1} + (3w5 2 − v2){ ˆF, ˆX2 1} + (w6 − v3 − u2 1){ ˆF, ˆX1} + (w7 − u1u2 − 2v) ˆF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This gives w4 = v1 2 , w5 = 2v2 3 , w6 = v3 + u2 1, w7 = u1u2 + 2v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This finishes the proof for C(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The derivation of C(5) is slightly more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We express [C(5), ˆY1] and [C(5), ˆY2] in terms of { ˆG, ˆY n 1 } and { ˆG, { ˆY1, ˆY2}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By [34, Lemma 2] and quintic commutation relations, we have [C(5), ˆY1] = − (α + ω1){ ˆG, ˆY 3 1 } − � β + ω2 − 3αω3 2 � { ˆG, ˆY 2 1 } − (δ + ω3β + ω4){ ˆG, ˆY1} − (ϵ + ω5){ ˆG, ˆY2} − ω3{ ˆG, { ˆY1, ˆY2}} + ��αϵ 2 − δ � ω3 − 2ζ − ω6 � ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Setting the coefficients of { ˆK, { ˆY1, ˆY2}} and { ˆK, ˆY2} to be zero, we obtain that ω3 = 0 and ω5 = −ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Then [C(5), ˆY1] is reduced to the form [C(5), ˆY1] =(α − ω1){ ˆG, ˆY 3 1 } + (β − ω2){ ˆG, ˆY 2 1 } + [δ − ω4)]{ ˆG, ˆY1} + (2ζ − ω6) ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' From [C(5), ˆY1] = 0 it follows that the coefficients of { ˆG, ˆY l 1} are zero for all 1 ≤ n ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Thus ω1 = −α, ω2 = −β, ω4 = −δ and ω6 = −2ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 32 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES Similarly, after some manipulations we find [C(5), ˆY2] =(3ω7 − a){ ˆG, ˆY 5 1 } + �5ω8 2 − b � { ˆG, ˆY 4 1 } − (c + 3αδ + 5ϵω7 − 2ω9){ ˆG, ˆY 3 1 } + �3α 2 �αϵ 2 − δ � − β2 + 3αδϵ 2 + 3ϵ2ω7 − e − ϵω9 + ω11 � { ˆG, ˆY1} − � βδ + ϵω8 2 − 1 2ω10 − 3αβ + d � { ˆG, ˆY 2 1 } + � β �αϵ 2 − δ � − ϵω10 2 + ϵ2ω8 + 3αβϵ 2 + (ω12 − 2z) � ˆG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It follows from [C(5), ˆY2] = 0 that ω7 = a 3, ω8 = 2b 5 , ω9 = 1 2 � c + 5ϵ 3 + 3αδ � , ω10 = 2(β(δ + 3α) + ϵb 5 − d) ω11 = �5 6 − a � ϵ2 + e + ϵc 2 + β2 − 3α 2 �αϵ 2 − δ � , ω12 = 2z + βδ + βϵ(α + δ) − bϵ2 5 − ϵd as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' we now construct realizations of these algebras in terms of the deformed oscillator algebras (1) and determine their structure functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After long computations, we obtain the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The realization ˆX1 =√u3 (N + η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆX2 = − u3 (N + η)2 − u2 √u3 (N + η) + b† + b − u u3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (18) where η is a constant parameter to be determined,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' changes the cubic algebra (16) to the deformed oscillator algebra (1) with the structure function given by Φ(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = 1 1 − 2u3 � C(3) − u2 u3 + uu2 √u3 + √u3v + (N + η)2 � −2uu1 + 2u1u2 √u3 − u2 2 + (u2 + v2)u3/2 3 − u3 � + (N + η) � 2uu1 − 2uu2 √u3 − √u3(u1u2 + 2v) + u2 2 + u3v3 � +(N + η)3 � −2u1u2 √u3 + 2u1u2 3 − 2 3v2u3/2 3 + v1u2 3 � + (N + η)4 � −2u1u2 3 + u3 3 − 1 2v1u2 3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Note that Φ is a quartic polynomial of the number operator N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The transformation ˆY1 =√ϵ(N + η),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆY2 = − α√ϵ(N + η)3 − β(N + η)2 − δ √ϵ(N + η) + b† + b − ζ ϵ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (19) where η is a constant parameter to be determined,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' maps the quintic algebra (17) to the deformed oscillator algebra with the structure function Φ(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' η) = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4ϵC(5) + (N + η)6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�aϵ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='12 − 3α2ϵ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ (N + η)5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�3α2ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2αβϵ5/2 − aϵ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='10bϵ7/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ (N + η)4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4αβ√ϵ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='8ϵ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3αδ + 5aϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2αδϵ2 + β2ϵ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4bϵ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ (N + η)3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4α(2αϵ − δ) − 3αδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2αζϵ3/2 − β2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2βδϵ3/2 − cϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4ω10ϵ5/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ (N + η)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�β(2αϵ − δ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4√ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− 3αζ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4√ϵ − βδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2√ϵ + βζϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ δ2ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 − d√ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ ω11ϵ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ (N + η) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='�δ(2αϵ − δ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4ϵ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='− βζ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2ϵ + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2δζ√ϵ − e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4ω12ϵ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='+ ζ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 + ζ(2αϵ − δ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4ϵ3/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The structure function Φ(N) is a polynomial of N of degree 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' In the next subsection, we will present new superintegrable systems in 2D Darboux spaces with cubic symmetry algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 33 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 Superintegrable systems in 2D Darboux spaces with cubic symmetry algebras In this subsection we obtain potentials in the 2D Darboux spaces which can be added to the Hamiltonians of the free superintegrable systems studied in [11] and preserve their superintegrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The free systems have only kinetic terms and possess linear and quadratic integrals of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We will determine the integrals corresponding to the superintegrable systems with potnetials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 Darboux space I The Hamiltonian of the free system in Darboux space I with separable local coordinates (x, y) studied in [11] has the form H1 = ϕ1(x)(∂2 x + ∂2 y), where ϕ1(x) = 1 αx+β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This system has linear integral X1 = ∂y and quadratic integral given by X2 = y∂x∂y − x∂2 y + 1 2∂x − 1 4αy2ϕ1(x)(∂2 x + ∂2 y), where α is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We seek new superintegrable system in Darboux space I with Hamiltonian ˆH1 = H1 + V1(x, y), where V1(x, y) is potential function, which preserves the separability of the coordinates and the superintegrability of the original system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Without loss of generality, we assume that the local separable coordinates (x, y) is an orthogonal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After some computations, we find the allowed potential V1 and the corresponding integrals ˆX1, ˆX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The results are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' ˆH1 = ϕ1(x)(∂2 x + ∂2 y) + c1ϕ1(x), ˆX1 = ∂y, ˆX2 = y∂x∂y − x∂2 y + 1 2∂x − 1 4αy2ϕ1(x)(∂2 x + ∂2 y) − 1 4c1αϕ1(x)y2 where c1 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By a direct calculation, we can show that the integrals ˆX1, ˆX2 form the cubic algebra, [ ˆX1, ˆX2] = ˆF, [ ˆX1, ˆF] = α 2 ˆH1, [ ˆX2, ˆF] = −2X3 1 + α ˆH1X1 − c1X1, (20) where explicitly ˆF = ∂x∂y − 1 2αyϕ1(x) � ∂2 x + ∂2 y � + 1 2c1αϕ1(x)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This cubic algebra is a special case of (16) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 with v1 = −2, u1 = u2 = u3 = v2 = v = 0, u = α 2 ˆH1, v3 = β ˆH1 − c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Then it follows that its Casimir operator is C(3) = ˆF 2 − X4 1 − α ˆH1 ˆX2 + (β ˆH1 − c1)X2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Since u3 = 0 it follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 that this cubic algebra does not have realization in terms of the deformed oscillator algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2 Darboux space II The Hamiltonian of the free superintegrable system in 2D Darboux space II is H2 = ϕ2(x) � ∂2 x + ∂2 y � , where ϕ2(x) = x2 a2−a1x2 , a1, a2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The system possesses the following linear and quadratic integrals of motion, X1 = ∂y, X2 = 2xy∂x∂y + (y2 − x2)∂2 y + x∂x + y∂y + a1y2H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It can be shown that we can add the potential V2(x, y) = c2 ϕ2(x), where c2 is a real constant, to the free Hamiltonian such that ˆH2 = ϕ2(x) � ∂2 x + ∂2 y � + c2 ϕ2(x) is separable and superintegrable in the 2D Darboux space II, with integrals of motion given by ˆX1 = ∂y, ˆX2 = 2xy∂x∂y + (y2 − x2 + 1)∂2 y + x∂x + y∂y + a1y2H2 + a2c2y2 a2 − a1x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 34 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES By a direct computation, we find that these integrals obey the cubic commutation relations [ ˆX1, ˆX2] = ˆF, [ ˆX1, ˆF] = 2a1 ˆH2 + 2 ˆX2 1 + 2c2, [ ˆX2, ˆF] = 4 ˆX3 1 − 2{ ˆX1, ˆX2} + (2c2 + 1 − 2a2 ˆH2)X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (21) The Casimir operator of this cubic algeba is given by C(3) = ˆF 2 − 2{X2 1, ˆX2}a + 2 ˆX4 1 + (c2 + 5 − 2a2 ˆH2)X2 1 − 4(a1 ˆH2 + c2) ˆX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='1 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4, we again find that the cubic algebra has no realization in terms of the deformed oscillator algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='3 Darboux space III In the 2D Darboux space III, the free superintegrable system Hamiltonian and its constants of motion in the separable local coordinates (u, v) are given by H3 = ϕ3(v)(∂2 u + ∂2 v), X1 = ∂u, X2 = 1 2e−v � cos u(2∂2 u + ∂v) + sin u (2∂u∂v − ∂u) � + α cos uH3, where ϕ3(v) = e−v βev−2α with α, β being real constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We seek potential of the form V3(u, v) = ϕ3(v)(f3(u) + g3(v)) such that system in Darboux space III with this potential is superintegrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' We thus expect that ˆH3 = H3 + V3(u, v) possesses linear and quadratic integrals of the form, ˆX1 = X1, ˆX2 = X2 + f3(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' After some manipulations, we find that V3(u, v) = c3 ϕ3(v) and f3(u, v) = c3 βev cos(u) 2βev−4α , where c3 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' That is, we obtain the superintegrable system in Darboux space III with Hamiltonian and integrals given by ˆH3 = ϕ3(v)(∂2 u + ∂2 v) + c3 βev − 2α, ˆX1 = ∂u, ˆX2 = 1 2 exp(−v) � cos u(2∂2 u + ∂v) + sin u(2∂u∂v − ∂u) � + α cos uH3 + c3 βev cos(u) 2βev − 4α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These integrals form the following algebra, [ ˆX1, ˆX2] = ˆF, [ ˆX1, ˆF] = − ˆX2, [ ˆX2, ˆF] = −β ˆH3 ˆX1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (22) The Casimir operator of this algebra is given by C(3) = ˆF 2−β ˆH3X2 1+ ˆX2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' It is interesting that the algebra generated by the above linear and quadratic integrals in the Darboux space III is “linear” in the generators (though with coefficient involving the Hamiltonian ˆH3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content='4 Darboux space IV In terms of separable local coordinates (u, v), the Hamiltonian of the free superintegrable system in 2D Darboux space IV is H4 = ϕ4(u) � ∂2 u + ∂2 v � , where ϕ4(u) = sin2 u β−2α cos u α, β ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The system possesses the following linear and quadratic integrals of motion, X1 = ∂v, X2 = exp(v) 2 � cos u(2∂2 v − ∂v) − sin u(2∂u∂v − ∂v) − 2αH4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' By analysis similar to previous cases, we find that the system with the Hamiltonian ˆH4 = ϕ4(u) � ∂2 u + ∂2 v � + c4 β − 2α cos u, where c4 is a constant, is superintegrable with linear and quadratic integrals given by ˆX1 = ∂v, ˆX2 = exp(v) 2 � cos u(2∂2 v − ∂v) − sin u(2∂u∂v − ∂v) − 2αH4 � + 4c4e−v β − 2α cos u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' These integrals form the cubic algebra, [X1, ˆX2] = ˆF, [X1, ˆF] = − ˆX2, [ ˆX2, ˆF] = 4X3 1 − 2β ˆH4X1 + 1 2 ˆX1 − 2αc4X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' (23) 35 ALGEBRAIC APPROACH TO SUPERINTEGRABLE SYSTEMS IN DARBOUX SPACES The Casimir operator of the algebra is C(3) = −1 2{ ˆX2, ˆF}a + β ˆH4X2 1 + X4 1 + (5 + 4c4)X2 1, which can be expressed as C(3) = ˆH2 4 + β ˆH4 + 4c4 in terms of the Hamiltonian ˆH4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Through the change of basis, X1 = (N + η), X2 = −(N + η)2 + b† + b the cubic algebra relations become those of the deformed oscillator algebra with structure function Φ(N, η) = (N + η)4 − 4(+N + η)3 + (N + η)2 − (N + η) � −2αc4 − 2βE + 1 2 � − 4c4 − E2 − βE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Here η is a constant which can be determined from the constraints on the structure function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 4 Conclusions We have presented a genuine algebraic analysis for the superintegrable systems in 2D Darboux spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The main results in this paper are following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' The first main result is the construction of the Casimir operators, deformed oscillator algebra realizations and finite-dimensional unirreps for all the 12 distinct quadratic algebras underlying the 12 superintegrable systems found in the classification of [14][15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' This allows us to give an algebraic derivation for the energy spectrum of the 12 existing classes of superintegrable systems with quadratic integrals in the 2D Darboux spaces and the determination for the structure functions of the finite-dimensional unitary irreducible representations of the deformed oscillator algebras (corresponding to the quadratic algebras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' As our results demonstrate, superintegrable systems in curved (Darboux) spaces have much richer structures than those in flat spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' For instance, the structures of energies of the systems and structure functions of the associated deformed oscillator algebras can be very complicated in the Darboux spaces, and in some cases we have to restrict the model parameter spaces in order to find explicit analytic and closed form solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Another main result of the paper is the construction of generic cubic and quintic algebras, generated by first, quadratic and cubic integrals, their Casimir operators and deformed oscillator algebra realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' As examples of applications, we obtain four classes of new superintegrable systems with non-trivial potentials and with linear and quadratic integrals in the 2D Darboux spaces, three of which have cubic algebras as their symmetry algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Acknowledgement IM and YZZ were supported by Australian 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 20200800, 21, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' [34] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Isaac and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Marquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' On realizations of polynomial algebras with three generators via deformed oscillator algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' A, 47(20):205203, 26, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} +page_content=' 38' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE2T4oBgHgl3EQfUQce/content/2301.03810v1.pdf'} diff --git a/ItE3T4oBgHgl3EQfXAqD/content/2301.04475v1.pdf b/ItE3T4oBgHgl3EQfXAqD/content/2301.04475v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b55177cdbd6907b956b56eb1fb2f6fcdd66182d7 --- /dev/null +++ b/ItE3T4oBgHgl3EQfXAqD/content/2301.04475v1.pdf @@ -0,0 +1,3 @@ +version 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a/JdAzT4oBgHgl3EQfH_vd/content/tmp_files/2301.01056v1.pdf.txt b/JdAzT4oBgHgl3EQfH_vd/content/tmp_files/2301.01056v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..383773e7da14f99a8b2f438538a6f5a501d601bd --- /dev/null +++ b/JdAzT4oBgHgl3EQfH_vd/content/tmp_files/2301.01056v1.pdf.txt @@ -0,0 +1,24525 @@ +Objectivity in continuum mechanics, an introduction +Motions, Eulerian and Lagrangian variables and functions, deformation gradient, +Lie derivatives, velocity-addition formula, Coriolis. +Gilles Leborgne, www.isima.fr/leborgne +January 4, 2023 +In classical mechanics, there are two objectivities: 1- The covariant objectivity concerns the universal +laws of physics required to be observer independent (true in any reference frame); This is a main topic in +this manuscript. 2- The isometric objectivity concerns the constitutive laws of materials once expressed +in a reference frame. +Covariant objectivity in continuum mechanics follows Maxwell’s requirements, cf. [13] page 1: “2. (...) +The formula at which we arrive must be such that a person of any nation, by substituting for the different +symbols the numerical value of the quantities as measured by his own national units, would arrive at +a true result. (...) 10. (...) The introduction of coordinate axes into geometry by Des Cartes was one +of the greatest steps in mathematical progress, for it reduced the methods of geometry to calculations +performed on numerical quantities. The position of a point is made to depend on the length of three lines +which are always drawn in determinate directions (...) But for many purposes in physical reasoning, as +distinguished from calculation, it is desirable to avoid explicitly introducing the Cartesian coordinates, +and to fix the mind at once on a point of space instead of its three coordinates, and on the magnitude +and direction of a force instead of its three components. This mode of contemplating geometrical and +physical quantities is more primitive and more natural than the other,...” +And see the (short) historical note given in the introduction of Abraham and Marsden book “Foun- +dations of Mechanics” [1], about qualitative versus quantitative theory: “Mechanics begins with a long +tradition of qualitative investigation culminating with Kepler and Galileo. Following this is the period +of quantitative theory (1687-1889) characterized by concomitant developments in mechanics, mathemat- +ics, and the philosophy of science that are epitomized by the works of Newton, Euler, Lagrange, +Laplace, Hamilton, and Jacobi. (...) For celestial mechanics (...) resolution we owe to the genius of +Poincaré, who resurrected the qualitative point of view (...) One advantage (...) is that by suppressing +unnecessary coordinates the full generality of the theory becomes evident.” +After having defined motions, Eulerian and Lagrangian variables and functions, we give the definition +of the deformation gradient as a function. We then obtain a simple understanding of the Lie derivatives +of vector fields which meet the needs of engineers. Then we get the velocity addition formula and verify +that the Lie derivatives are objective. Note that Cauchy would certainly have used the Lie derivatives if +they had existed during his lifetime: To get a stress, Cauchy had to compare two vectors, whereas one +vector is enough when using the derivatives of Lie. +We systematically start with qualitive definitions (observer independent), before quantifying with +bases and/or Euclidean dot products (observer dependent). A fairly long appendix tries to give in one +manuscript the definitions, properties and interpretations, usually scattered across several books (and +not always that easy to find). +arXiv:2301.01056v1 [physics.class-ph] 3 Jan 2023 + +2 +CONTENTS +Contents +I +Motions, Eulerian and Lagrangian descriptions, flows +11 +1 +Motions +11 +1.1 +Referential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +1.2 +Einstein’s convention (duality notation) +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +1.3 +Motion of an object +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +1.4 +Virtual and real motion +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +1.5 +Hypotheses (Newton and Einstein) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +1.6 +Configurations +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +1.7 +Definition of the Eulerian and Lagrangian variables . . . . . . . . . . . . . . . . . . . . . . +13 +1.8 +Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +1.9 +Pointed vector, tangent space, fiber, vector field, bundle . . . . . . . . . . . . . . . . . . . +13 +2 +Eulerian description (spatial description at actual time t) +14 +2.1 +The set of configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +2.2 +Eulerian variables and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +2.3 +Eulerian velocity (spatial velocity) and speed . . . . . . . . . . . . . . . . . . . . . . . . . +15 +2.4 +Spatial derivative of the Eulerian velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +2.4.1 +Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +2.4.2 +The convective derivative dEul.⃗v +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +2.4.3 +Quantification in a basis: df.⃗u is written (⃗u. ⃗ +grad)f . . . . . . . . . . . . . . . . . . +16 +2.4.4 +Representation relative to a Euclidean dot product: +⃗ +gradf . . . . . . . . . . . . . . +17 +2.4.5 +Vector valued functions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +2.5 +Streamline (current line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +2.6 +Material time derivative (dérivées particulaires) . . . . . . . . . . . . . . . . . . . . . . . . +18 +2.6.1 +Usual definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +18 +2.6.2 +Remark: About notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +2.6.3 +Definition bis: Time-space definition . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +2.6.4 +The material time derivative is a derivation . . . . . . . . . . . . . . . . . . . . . . +20 +2.6.5 +Commutativity issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +2.7 +Eulerian acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +2.8 +Time Taylor expansion of �Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +3 +Motion from an initial configuration: Lagrangian description +21 +3.1 +Initial configuration and Lagrangian “motion” . . . . . . . . . . . . . . . . . . . . . . . . . +21 +3.1.1 +Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +3.1.2 +Diffeomorphism between configurations +. . . . . . . . . . . . . . . . . . . . . . . . +22 +3.1.3 +Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +3.1.4 +Streaklines (lignes d’émission) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +3.2 +Lagrangian variables and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +3.2.1 +Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +3.2.2 +A Lagrangian function is a two point tensor . . . . . . . . . . . . . . . . . . . . . . +23 +3.3 +Lagrangian function associated with a Eulerian function . . . . . . . . . . . . . . . . . . . +24 +3.3.1 +Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +3.3.2 +Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +3.4 +Lagrangian velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +3.4.1 +Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +3.4.2 +Lagrangian velocity versus Eulerian velocity . . . . . . . . . . . . . . . . . . . . . . +24 +3.4.3 +Relation between differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +3.4.4 +Computation of d⃗v called L = +• +F.F −1 wih Lagrangian variables . . . . . . . . . . . +25 +3.5 +Lagrangian acceleration +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +3.6 +Time Taylor expansion of Φt0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +3.7 +A vector field that let itself be deformed by a motion . . . . . . . . . . . . . . . . . . . . . +26 +2 + +3 +CONTENTS +4 +Deformation gradient F := dΦ +26 +4.1 +Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +4.1.1 +Definition of the deformation gradient F . . . . . . . . . . . . . . . . . . . . . . . . +27 +4.1.2 +Push-forward (values of F) +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +4.1.3 +F is a two point tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +28 +4.1.4 +Evolution: Toward the Lie derivative (in continuum mechanics) . . . . . . . . . . . +28 +4.1.5 +Pull-back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +4.2 +Quantification with bases +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +4.3 +The unfortunate notation d⃗x = F.d ⃗X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +4.3.1 +Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +4.3.2 +Where does this unfortunate notation come from? +. . . . . . . . . . . . . . . . . . +30 +4.3.3 +Interpretation: Vector approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +4.3.4 +Interpretation: Differential approach . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +4.3.5 +The ambiguous notation +• +d⃗x = +• +F.d ⃗X . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +4.4 +Tensorial notations, warnings, remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +4.5 +Change of coordinate system at t for F +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +4.6 +Spatial Taylor expansion of Φ and F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +4.7 +Time Taylor expansion of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +4.8 +Homogeneous and isotropic material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +33 +4.9 +The inverse of the deformation gradient +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +33 +5 +Flow +34 +5.1 +Introduction: Motion versus flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +34 +5.2 +Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +34 +5.3 +Cauchy–Lipschitz theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +34 +5.4 +Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +5.5 +Composition of flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +5.5.1 +Law of composition of flows (determinism) +. . . . . . . . . . . . . . . . . . . . . . +36 +5.5.2 +Stationnary case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +37 +5.6 +Velocity on the trajectory traveled in the opposite direction . . . . . . . . . . . . . . . . . +38 +5.7 +Variation of the flow as a function of the initial time . . . . . . . . . . . . . . . . . . . . . +38 +5.7.1 +Ambiguous and non ambiguous notations . . . . . . . . . . . . . . . . . . . . . . . +38 +5.7.2 +Variation of the flow as a function of the initial time . . . . . . . . . . . . . . . . . +39 +II +Push-forward +40 +6 +Push-forward +40 +6.1 +Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +6.2 +Push-forward and pull-back of points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +6.3 +Push-forward and pull-back of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +41 +6.4 +Push-forward and pull-back of scalar functions +. . . . . . . . . . . . . . . . . . . . . . . . +41 +6.4.1 +Definitions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +41 +6.4.2 +Interpretation: Why is it useful? . . . . . . . . . . . . . . . . . . . . . . . . . . . . +42 +6.5 +Push-forward and pull-back of vector fields +. . . . . . . . . . . . . . . . . . . . . . . . . . +42 +6.5.1 +A definition by approximation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +42 +6.5.2 +The definition of the push-forward of a vector field . . . . . . . . . . . . . . . . . . +42 +6.5.3 +Pull-back of a vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +43 +6.6 +Quantification with bases +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +44 +6.6.1 +Usual result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +44 +6.6.2 +Example: Polar coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . +44 +7 +Push-forward and pull-back of differential forms +46 +7.1 +Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +46 +7.2 +Incompatibility: Riesz representation and push-forward +. . . . . . . . . . . . . . . . . . . +47 +3 + +4 +CONTENTS +8 +Push-forward and pull-back of tensors +48 +8.1 +Push-forward and pull-back of order 1 tensors . . . . . . . . . . . . . . . . . . . . . . . . . +48 +8.2 +Push-forward and pull-back of order 2 tensors . . . . . . . . . . . . . . . . . . . . . . . . . +48 +8.3 +Push-forward and pull-back of endomorphisms +. . . . . . . . . . . . . . . . . . . . . . . . +49 +8.4 +Application to derivatives of vector fields +. . . . . . . . . . . . . . . . . . . . . . . . . . . +49 +8.5 +Ψ∗(d⃗u) versus d(Ψ∗⃗u): No commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . +50 +8.6 +Application to derivative of differential forms . . . . . . . . . . . . . . . . . . . . . . . . . +50 +8.7 +Ψ∗(dα) versus d(Ψ∗α): No commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . +50 +III +Lie derivative +51 +9 +Lie derivative +51 +9.0 +Purpose and first results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +9.0.1 +Purpose? +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +9.0.2 +Basic results +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +9.1 +Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +9.1.1 +Issue (ubiquity gift)... +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +51 +9.1.2 +...Toward a solution (without ubiquity gift)... . . . . . . . . . . . . . . . . . . . . . +52 +9.1.3 +... The Lie derivative, first definition . . . . . . . . . . . . . . . . . . . . . . . . . . +52 +9.1.4 +A more general definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +52 +9.1.5 +Equivalent definition (differential geometry) . . . . . . . . . . . . . . . . . . . . . . +53 +9.2 +Lie derivative of a scalar function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +53 +9.3 +Lie derivative of a vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +54 +9.3.1 +Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +54 +9.3.2 +Interpretation: Flow resistance measurement +. . . . . . . . . . . . . . . . . . . . . +54 +9.3.3 +Autonomous Lie derivative and Lie bracket . . . . . . . . . . . . . . . . . . . . . . +55 +9.4 +Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +55 +9.4.1 +Lie Derivative of a vector field along itself . . . . . . . . . . . . . . . . . . . . . . . +55 +9.4.2 +Lie derivative along a uniform flow . . . . . . . . . . . . . . . . . . . . . . . . . . . +55 +9.4.3 +Lie derivative of a uniform vector field . . . . . . . . . . . . . . . . . . . . . . . . . +55 +9.4.4 +Uniaxial stretch of an elastic material +. . . . . . . . . . . . . . . . . . . . . . . . . +55 +9.4.5 +Simple shear of an elastic material . . . . . . . . . . . . . . . . . . . . . . . . . . . +56 +9.4.6 +Shear flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +56 +9.4.7 +Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +57 +9.4.8 +Second order Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +57 +9.5 +Lie derivative of a differential form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +57 +9.6 +Incompatibility with Riesz representation vectors . . . . . . . . . . . . . . . . . . . . . . . +59 +9.7 +Lie derivative of a tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +59 +9.7.1 +Lie derivative of a mixed tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +59 +9.7.2 +Lie derivative of a up-tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +60 +9.7.3 +Lie derivative of a down-tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +60 +IV +Velocity-addition formula +61 +10 Change of referential and velocity-addition formula +61 +10.0 Issue and result (summary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +61 +10.1 Referentials and “matrix motions” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +61 +10.1.1 Absolute and relative referentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . +61 +10.1.2 Motion of a material object Obj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +62 +10.1.3 Quantification: Absolute and relative “motion” of Obj +. . . . . . . . . . . . . . . . +62 +10.1.4 Motion of RB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +63 +10.1.5 Quantification: Drive and static “motion” of RB +. . . . . . . . . . . . . . . . . . . +63 +10.2 The translator Θt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +64 +10.2.1 Definition of Θt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +64 +10.2.2 Translation at t for the motion �Φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . +64 +10.3 dΘt +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +64 +10.3.1 Push-forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +64 +10.3.2 Θt is affine in classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . +65 +4 + +5 +CONTENTS +10.4 Translated velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +65 +10.5 Definition of Θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +66 +10.6 The “Θ-velocity” is the drive velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +66 +10.7 The velocity-addition formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +66 +10.8 Coriolis acceleration, and the acceleration-addition formula +. . . . . . . . . . . . . . . . . +67 +10.9 With an initial time +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +67 +10.10Drive and Coriolis forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +68 +10.10.1Fundamental principal: requires a Galilean referential +. . . . . . . . . . . . . . . . +68 +10.10.2Drive + Coriolis forces = the inertial force +. . . . . . . . . . . . . . . . . . . . . . +68 +10.11Summary for “Sun and Earth” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +69 +10.11.1Coriolis forces on the Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +69 +11 Objectivities +71 +11.1 “Isometric objectivity” and “Frame Invariance Principle” . . . . . . . . . . . . . . . . . . . +71 +11.2 Definition and characterization of the covariant objectivity . . . . . . . . . . . . . . . . . . +71 +11.2.1 Framework of classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . +71 +11.2.2 Covariant objectivity of a scalar function +. . . . . . . . . . . . . . . . . . . . . . . +71 +11.2.3 Covariant objectivity of a vector field +. . . . . . . . . . . . . . . . . . . . . . . . . +72 +11.2.4 Covariant objectivity of differential forms . . . . . . . . . . . . . . . . . . . . . . . +72 +11.2.5 Covariant objectivity of tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +72 +11.3 Non objectivity of the velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +73 +11.3.1 Eulerian velocity ⃗v : not covariant (and not isometric) objective . . . . . . . . . . . +73 +11.3.2 d⃗v is not objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +73 +11.3.3 d⃗v + d⃗vT is “isometric objective” +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +74 +11.3.4 Lagrangian velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +74 +11.4 The Lie derivatives are covariant objective . . . . . . . . . . . . . . . . . . . . . . . . . . . +74 +11.4.1 Scalar functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +74 +11.4.2 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +74 +11.4.3 Tensors +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +75 +11.5 Taylor expansions and ubiquity gift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +76 +11.5.1 In Rn with ubiquity +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +76 +11.5.2 General case +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +76 +V +Appendix +78 +A Classical and duality notations +78 +A.1 Contravariant vector and basis +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +78 +A.1.1 +Contravariant vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +78 +A.1.2 +Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +78 +A.1.3 +Canonical basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +78 +A.1.4 +Cartesian basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +78 +A.2 Representation of a vector relative to a basis +. . . . . . . . . . . . . . . . . . . . . . . . . +79 +A.3 Dual basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +79 +A.3.1 +Linear forms = “Covariant vectors” . . . . . . . . . . . . . . . . . . . . . . . . . . . +79 +A.3.2 +Covariant dual basis (= the functions that give the components of a vector) . . . . +80 +A.3.3 +Example: aeronautical units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +81 +A.3.4 +Matrix representation of a linear form . . . . . . . . . . . . . . . . . . . . . . . . . +81 +A.3.5 +Example: Thermodynamic +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +82 +A.4 Einstein convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +82 +A.4.1 +Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +82 +A.4.2 +Do not mistake yourself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +83 +A.5 Change of basis formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +83 +A.5.1 +Change of basis endomorphism and transition matrix +. . . . . . . . . . . . . . . . +83 +A.5.2 +Inverse of the transition matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +84 +A.5.3 +Change of dual basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +84 +A.5.4 +Change of coordinate system for vectors and linear forms +. . . . . . . . . . . . . . +84 +A.6 Bidual basis (and contravariance) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +85 +A.7 Bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +85 +5 + +6 +CONTENTS +A.7.1 +Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +85 +A.7.2 +The transposed of a bilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . +85 +A.7.3 +Symmetric and definite positive bilinear forms +. . . . . . . . . . . . . . . . . . . . +86 +A.7.4 +Inner dot product, and metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +86 +A.7.5 +Quantification: Matrice [βij] and tensorial representation +. . . . . . . . . . . . . . +86 +A.8 Linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +87 +A.8.1 +Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +87 +A.8.2 +Quantification: Matrices [Lij] = [Lij] . . . . . . . . . . . . . . . . . . . . . . . . . . +88 +A.8.3 +Trace of an endomorphism +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +89 +A.9 Transposed matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +89 +A.10 A transposed endomorphism: depends on a chosen inner dot product . . . . . . . . . . . . +90 +A.10.1 Definition (requires an inner dot product: Not objective) +. . . . . . . . . . . . . . +90 +A.10.2 Quantification with bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +90 +A.10.3 Symmetric endomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +91 +A.10.4 The general flat ♭ notation for an endomorphism: Relative to a (·, ·)g . . . . . . . . +92 +A.11 A transposed of a linear map: depends on chosen inner dot products . . . . . . . . . . . . +92 +A.11.1 Definition (subjective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +93 +A.11.2 Quantification with bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +93 +A.11.3 Deformation gradient symmetric: Absurd . . . . . . . . . . . . . . . . . . . . . . . +93 +A.11.4 Isometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +94 +A.12 The adjoint of a linear map (objective) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +94 +A.12.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +94 +A.12.2 Quantification +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +95 +A.12.3 Relation with the transposed when inner dot products are introduced +. . . . . . . +95 +A.13 Tensorial representation of a linear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . +95 +A.13.1 A tensorial representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +95 +A.13.2 Warning: Confusion between transposed and adjoint . . . . . . . . . . . . . . . . . +96 +A.14 Change of basis formulas for bilinear forms and linear maps . . . . . . . . . . . . . . . . . +96 +A.14.1 Notations for transitions matrices for bilinear forms and linear maps . . . . . . . . +96 +A.14.2 Change of coordinate system for bilinear forms ∈ L(A, B; R) +. . . . . . . . . . . . +97 +A.14.3 Change of coordinate system for bilinear forms ∈ L(A∗, B∗; R) +. . . . . . . . . . . +97 +A.14.4 Change of coordinate system for bilinear forms ∈ L(B∗, A; R) . . . . . . . . . . . . +97 +A.14.5 Change of coordinate system for tri-linear forms ∈ L(A∗, A, A; R) . . . . . . . . . . +98 +A.14.6 Change of coordinate system for linear maps ∈ L(A; B) +. . . . . . . . . . . . . . . +98 +B Euclidean Frameworks +98 +B.1 +Euclidean basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +99 +B.2 +Euclidean dot product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +99 +B.3 +Change of Euclidean basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 +B.3.1 +Two Euclidean dot products are proportional . . . . . . . . . . . . . . . . . . . . . 100 +B.3.2 +Counterexample : non existence of a Euclidean dot product . . . . . . . . . . . . . 100 +B.4 +Euclidean transposed of the deformation gradient . . . . . . . . . . . . . . . . . . . . . . . 100 +B.5 +The Euclidean transposed for endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 100 +B.6 +Unit normal vector, unit normal form +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 +B.6.1 +Framework +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 +B.6.2 +Unit normal vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 +B.6.3 +Unit normal form n♭ associated to ⃗n . . . . . . . . . . . . . . . . . . . . . . . . . . 102 +B.7 +Integration by parts (Green–Gauss–Ostrogradsky) +. . . . . . . . . . . . . . . . . . . . . . 102 +C Rate of deformation tensor and spin tensor +103 +C.1 The symmetric and antisymmetric parts of d⃗v . . . . . . . . . . . . . . . . . . . . . . . . . 103 +C.2 Quantification with a basis +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 +D Interpretation of the rate of deformation tensor +104 +6 + +7 +CONTENTS +E Rigid body motions and the spin tensor +104 +E.1 +Affine motions and rigid body motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 +E.1.1 +Affine motions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 +E.1.2 +Rigid body motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 +E.1.3 +Alternative definition of a rigid body motion: d⃗v + d⃗vT = 0 . . . . . . . . . . . . . 106 +E.2 +Representation of the spin tensor Ω: vectors, and pseudo-vectors . . . . . . . . . . . . . . 106 +E.2.1 +Reminder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 +E.2.2 +Definition of the vector product (cross product) . . . . . . . . . . . . . . . . . . . . 107 +E.2.3 +Calculation of the vector product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 +E.2.4 +Antisymmetric endomorphism represented by a vector . . . . . . . . . . . . . . . . 108 +E.2.5 +Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 +E.3 +Pseudo-cross product, and pseudo-vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 +E.3.1 +Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 +E.3.2 +Antisymmetric matrix represented by a pseudo-vector . . . . . . . . . . . . . . . . 110 +E.3.3 +Antisymmetric endomorphism and its pseudo-vectors representations . . . . . . . . 110 +E.4 +Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 +E.4.1 +Rectilinear motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 +E.4.2 +Circular motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 +E.4.3 +Motion of a planet (centripetal acceleration) +. . . . . . . . . . . . . . . . . . . . . 112 +F Riesz representation theorem +115 +F.1 +The Riesz representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 +F.2 +The Riesz representation operator +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 +F.3 +Quantification with a basis +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 +F.4 +Change of Riesz representation vector, and Euclidean case . . . . . . . . . . . . . . . . . . 117 +F.5 +A Riesz representation vector is contravariant . . . . . . . . . . . . . . . . . . . . . . . . . 118 +F.6 +What is a vector versus a (·, ·)g-vector? +. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 +F.7 +The “(·, ·)g-dual vectorial bases” of one basis (and warnings) . . . . . . . . . . . . . . . . . 119 +F.7.1 +A basis and its many associated “dual vectorial basis” +. . . . . . . . . . . . . . . . 119 +F.7.2 +Components of ⃗ejg in the basis (⃗ei) . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 +F.7.3 +Multiple admissible notations for the components of ⃗ejg . . . . . . . . . . . . . . . 121 +F.7.4 +(Huge) differences between “the (covariant) dual basis” and “a dual vectorial basis” +121 +F.7.5 +About the notation gij = shorthand notation for (g♯)ij +. . . . . . . . . . . . . . . 121 +G Cauchy–Green deformation tensor C = F T .F +122 +G.0 Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 +G.1 Transposed F T : Inner dot products required +. . . . . . . . . . . . . . . . . . . . . . . . . 122 +G.1.1 +Definition of the function F T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 +G.1.2 +Quantification with bases (matrix representation) . . . . . . . . . . . . . . . . . . . 123 +G.1.3 +Remark: F ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 +G.2 Cauchy–Green deformation tensor C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 +G.2.1 +Definition of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 +G.2.2 +Quantification +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 +G.3 Time Taylor expansion of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 +G.4 Remark: C♭ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 +G.4.1 +Definition of C♭... +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 +G.4.2 +... and remarks about C♭... and Jaumann . . . . . . . . . . . . . . . . . . . . . . . 126 +G.5 Stretch ratio and deformed angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 +G.5.1 +Stretch ratio +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 +G.5.2 +Deformed angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 +G.6 Decompositions of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 +G.6.1 +Spherical and deviatoric tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 +G.6.2 +Rigid motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 +G.6.3 +Diagonalization of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 +G.6.4 +Mohr circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 +G.7 Green–Lagrange deformation tensor E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 +G.8 Small deformations (linearization): The infinitesimal strain tensor ε +. . . . . . . . . . . . 130 +G.8.1 +Landau notations big-O and little-o +. . . . . . . . . . . . . . . . . . . . . . . . . . 130 +G.8.2 +Definition of the infinitesimal strain tensor ε +. . . . . . . . . . . . . . . . . . . . . 130 +7 + +8 +CONTENTS +H Finger tensor F.F T (left Cauchy–Green tensor) +131 +H.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 +H.2 b−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 +H.3 Time derivatives of b−1 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 +H.4 Euler–Almansi tensor a +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 +H.5 Time Taylor expansion for a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 +H.6 Almansi modified Infinitesimal strain tensor �ε . . . . . . . . . . . . . . . . . . . . . . . . . 133 +I +Polar decomposition, elasticity and objectivity +133 +I.1 +Polar decompositions of F (“isometric objectivity”) . . . . . . . . . . . . . . . . . . . . . . 133 +I.1.1 +F = R.U (right polar decomposition) . . . . . . . . . . . . . . . . . . . . . . . . . . 134 +I.1.2 +F = S.R0.U (shifted right polar decomposition for covariant objectivity) . . . . . . 134 +I.1.3 +F = V.R (left polar decomposition) . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 +I.2 +Linear elasticity: A Classical “tensorial” approach . . . . . . . . . . . . . . . . . . . . . . . 136 +I.2.1 +Classical approach (“isometric objectivity”), and an issue . . . . . . . . . . . . . . . 136 +I.2.2 +A functional (tensorial) formulation (“isometric objectivity”) . . . . . . . . . . . . . 136 +I.2.3 +Second functional formulation: With the Finger tensor . . . . . . . . . . . . . . . . 138 +I.3 +Elasticity with a covariant objective approach? . . . . . . . . . . . . . . . . . . . . . . . . 138 +J +Displacement +139 +J.1 +The displacement vector ⃗U +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 +J.2 +The differential of the displacement vector . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 +J.3 +Deformation “tensor” ε (matrix), bis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 +J.4 +Small displacement hypothesis, bis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 +J.5 +Displacement vector with differential geometry +. . . . . . . . . . . . . . . . . . . . . . . . 141 +J.5.1 +The shifter +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 +J.5.2 +The displacement vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 +K Determinants +142 +K.1 Alternating multilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 +K.2 Leibniz formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 +K.3 Determinant of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 +K.4 Determinant of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 +K.5 Volume +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 +K.6 Determinant of an endomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 +K.6.1 +Definition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 +K.6.2 +The determinant of an endomorphism is objective +. . . . . . . . . . . . . . . . . . 146 +K.7 Determinant of a linear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 +K.7.1 +Definition and first properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 +K.7.2 +Jacobian of a motion, and dilatation . . . . . . . . . . . . . . . . . . . . . . . . . . 147 +K.7.3 +Determinant of the transposed +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 +K.8 Dilatation rate +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 +K.8.1 +∂Jt0 +∂t (t, pt0) = Jt0(t, pt0) div⃗v(t, pt) +. . . . . . . . . . . . . . . . . . . . . . . . . . . 148 +K.8.2 +Leibniz formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 +K.9 ∂J/∂F = J F −T +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 +K.9.1 +Meaning of ∂ det +∂Mij ? +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 +K.9.2 +Calculation of ∂ det +∂Mij +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 +K.9.3 +∂J/∂F = J F −T usually written [ ∂J +∂Fij ] = J F −T +. . . . . . . . . . . . . . . . . . . 150 +K.9.4 +Interpretation of +∂J +∂Fij ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 +L +Transport of volumes and areas +151 +L.1 +Transformed parallelepiped +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 +L.2 +Transformed volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 +L.3 +Transformed parallelogram +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 +L.4 +Transformed surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 +L.4.1 +Deformation of a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 +L.4.2 +Euclidean dot product and unit normal vectors . . . . . . . . . . . . . . . . . . . . 152 +L.4.3 +Relations between surfaces +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 +L.5 +Piola identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 +8 + +9 +CONTENTS +L.6 +Piola transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 +M Work and power +154 +M.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 +M.1.1 Work +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 +M.1.2 And its associated power density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 +M.2 Piola–Kirchhoff tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 +M.2.1 Objective internal power for the stress: function of d⃗v . . . . . . . . . . . . . . . . 155 +M.2.2 The first Piola–Kirchhoff tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 +M.2.3 The second Piola–Kirchhoff tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 +M.3 Classical hyper-elasticity: ∂W/∂F +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 +M.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 +M.3.2 Expression with bases (quantification): The ∂W/∂Lij +. . . . . . . . . . . . . . . . 157 +M.3.3 Motions and ω-lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 +M.3.4 Application to classical hyper-elasticity: PK = ∂W/∂F . . . . . . . . . . . . . . . . 159 +M.3.5 Corollary (hyper-elasticity): SK = ∂W/∂C . . . . . . . . . . . . . . . . . . . . . . . 160 +N Conservation of mass +160 +O Balance of momentum +161 +O.1 Framework +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 +O.2 Master balance law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 +O.3 Cauchy theorem ⃗T = σ.⃗n (stress tensor σ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 +P Balance of moment of momentum +163 +Q Uniform tensors in Lr +s(E) +163 +Q.1 Tensorial product and multilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 +Q.1.1 +Tensorial product of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 +Q.1.2 +Tensorial product of linear forms: multilinear forms +. . . . . . . . . . . . . . . . . 163 +Q.2 Uniform tensors in L0 +s(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 +Q.2.1 +Definition of type +�0 +s +� +uniform tensors +. . . . . . . . . . . . . . . . . . . . . . . . . 164 +Q.2.2 +Example: Type +�0 +1 +� +uniform tensor = linear forms +. . . . . . . . . . . . . . . . . . 164 +Q.2.3 +Example: Type +�0 +2 +� +uniform tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 +Q.2.4 +Example: Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 +Q.3 Uniform tensors in Lr +s(E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 +Q.3.1 +Definition of type +�r +s +� +uniform tensors +. . . . . . . . . . . . . . . . . . . . . . . . . 165 +Q.3.2 +Example: Type +�1 +0 +� +uniform tensor: Identified with a vector . . . . . . . . . . . . . 165 +Q.3.3 +Example: Type +�1 +1 +� +uniform tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 +Q.3.4 +Example: Type +�1 +2 +� +uniform tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 +Q.4 Exterior tensorial products +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 +Q.5 Contractions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 +Q.5.1 +Contraction of a linear form with a vector . . . . . . . . . . . . . . . . . . . . . . . 166 +Q.5.2 +Contraction of a +�1 +1 +� +tensor and a vector . . . . . . . . . . . . . . . . . . . . . . . . 166 +Q.5.3 +Contractions of uniform tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 +Q.5.4 +Objective double contractions of uniform tensors . . . . . . . . . . . . . . . . . . . 168 +Q.5.5 +Non objective double contraction: Double matrix contraction . . . . . . . . . . . . 169 +Q.6 Kronecker (contraction) tensor, trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 +R Tensors in T r +s (U) +170 +R.1 Introduction, module, derivation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 +R.2 Field of functions and vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 +R.2.1 +Field of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 +R.2.2 +Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 +R.3 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 +R.4 Tensors +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 +R.5 First Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 +R.5.1 +Type +�0 +1 +� +tensor = differential forms +. . . . . . . . . . . . . . . . . . . . . . . . . . 172 +R.5.2 +Type +�1 +0 +� +tensor (identified to a vector field) . . . . . . . . . . . . . . . . . . . . . . 172 +9 + +10 +CONTENTS +R.5.3 +A metric is a +�0 +2 +� +tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 +R.6 +�1 +1 +� +tensor, identification with fields of endomorphisms . . . . . . . . . . . . . . . . . . . . 172 +R.7 Unstationary tensor +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 +S +Differential, its eventual gradients, divergences +173 +S.1 +Differential +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 +S.1.1 +Framework +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 +S.1.2 +Directional derivative and differential (observer independent) . . . . . . . . . . . . 173 +S.1.3 +Notation for the second order Differential . . . . . . . . . . . . . . . . . . . . . . . 174 +S.2 +A basis and the j-th partial derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 +S.3 +Application 1: Scalar valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 +S.3.1 +Differential of a scalar valued function (objective) . . . . . . . . . . . . . . . . . . . 175 +S.3.2 +Quantification +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 +S.3.3 +Gradients (subjective) associated with a differential through inner dot products . . 176 +S.4 +Application 2: Coordinate system basis and Christoffel symbols . . . . . . . . . . . . . . . 176 +S.4.1 +Coordinate system, and coordinate system basis +. . . . . . . . . . . . . . . . . . . 176 +S.4.2 +Parametric expression of the differential of a scalar valued function . . . . . . . . . 177 +S.4.3 +Christoffel symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 +S.5 +Application 3: Differential of a vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 +S.6 +Application 4: Differential of a differential form . . . . . . . . . . . . . . . . . . . . . . . . 180 +S.7 +Application 5: Differential of a 1 1 tensor +. . . . . . . . . . . . . . . . . . . . . . . . . . . 180 +S.8 +Divergence of a vector field: Invariant +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 +S.9 +Objective divergence for 1 1 tensors +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 +S.9.1 +Divergence of a 2 0 tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 +S.9.2 +Divergence of a 0 2 tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 +S.10 Euclidean framework and “classic divergence” of a tensor (subjective) . . . . . . . . . . . . 184 +T Natural canonical isomorphisms +184 +T.1 The adjoint of a linear map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 +T.2 An isomorphism E ≃ E∗ is never natural (never objective) . . . . . . . . . . . . . . . . . . 185 +T.3 Natural canonical isomorphism E ≃ E∗∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 +T.4 Natural canonical isomorphisms L(E; F) ≃ L(F ∗, E; R) ≃ L(E∗; F ∗) . . . . . . . . . . . . 186 +T.5 Natural canonical isomorphisms L(E; L(E; F)) ≃ L(E, E; F) ≃ L(F ∗, E, E; R) . . . . . . . 187 +U Distribution in brief: A covariant concept +188 +U.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 +U.2 Derivation of a distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 +U.3 Hilbert space H1(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 +U.3.1 +Motivation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 +U.3.2 +Definition of H1(Ω) +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 +U.3.3 +Subspace H1 +0(Ω) and its dual space H−1(Ω) . . . . . . . . . . . . . . . . . . . . . . 191 +10 + +11 +A quantity f being given then: g defined by « g equals f » is noted g := f. +Part I +Motions, Eulerian and Lagrangian +descriptions, flows +1 +Motions +The framework is classical mechanics, time being decoupled from space. R3 is the classical geometric +affine space (the space we live in), and ( ⃗R3, +, .) = { ⃗pq : p, q ∈ R3} =noted ⃗R3 is the associated vector +space of bipoint vectors equipped with its usual rules. We also consider R and R2 as subspaces of R3, i.e. +we consider Rn and ⃗Rn, n = 1, 2, 3. +1.1 +Referential +Origin: +An observer chooses an origin O ∈ Rn; Thus a point p ∈ Rn can be located by the observer +thanks to the bipoint vector −→ +Op = ⃗x ∈ ⃗Rn; Hence p = O + ⃗x, and ⃗x = −→ +Op =noted p − O. +Another observer chooses an origin � +O ∈ Rn; Thus the point p can also be located by this observer +with the bipoint vector +−→ +� +Op = �⃗x ∈ ⃗Rn; So p = O + ⃗x = � +O + �⃗x, and �⃗x = +−−→ +O � +O + ⃗x. +Cartesian coordinate system: +A Cartesian coordinate system in the affine space Rn is a set RCart = +(O, (⃗ei)i=1,...,n), where O is an origin and (⃗ei) := (⃗ei)i=1,...,n is a basis in ⃗Rn chosen by the observer. +Thus the location of a point p ∈ Rn can quantified by the observer ∃⃗x ∈ ⃗Rn s.t. +p = O + ⃗x +with +⃗x = +n +� +i=1 +xi⃗ei, +i.e. +[−→ +Op]|⃗e = [⃗x]|⃗e = +� +� +x1 +... +xn +� +� , +(1.1) +[⃗x]|⃗e = [−→ +Op]|⃗e being the column matrix containing the components xi ∈ R of −→ +Op = ⃗x in the basis (⃗ei). +Another observer with his origin Ob and his Cartesian basis (⃗bi)i=1,...,n make the Cartesian coordinate +system RCart,b = (Ob, (⃗bi)i=1,...,n), and gets for the same position p in Rn, +p = Ob + ⃗y +with +⃗y = +n +� +i=1 +�yi�⃗bi, +i.e. +[−−→ +Obp]|⃗b = [⃗y]|⃗b = +� +� +� +y1 +... +yn +� +� +� , +(1.2) +[⃗y]|⃗b = [−−→ +Obp]|⃗b being the column matrix containing the components yi ∈ R of −−→ +Obp = ⃗y in the basis (⃗bi). +And −−→ +Obp = −−→ +ObO + −→ +Op, i.e. ⃗y = +−−→ +� +OO + ⃗x, gives the relation between ⃗x and ⃗y (drawing). +Chronology: +A chronology (or temporal coordinate system) is a set Rtime = (t0, (∆t)) chosen by an +observer, where t0 ∈ R is the time origin, and (∆t) is the time unit (a basis in ⃗R). +Referentiel: +A referential R is the set +R = (Rtime, RCart) = (t0, (∆t), O, (⃗ei)i=1,...,n) = (“chronologie”,“Cartesian coordinate system”), +(1.3) +made of a chronology and a Cartesian coordinate system, chosen by an observer. +In the following, to simplify the writings, the same implicit chronology is used by all observers, and +a referential R = (Rtime, RCart) will simply be noted as the reference frame R = (O, (⃗ei)) (so := RCart). +11 + +12 +1.2. +Einstein’s convention (duality notation) +1.2 +Einstein’s convention (duality notation) +Starting point: The classical notation xi for the components of a vector ⃗x relative to a basis, cf. (1.1). +Then the duality notion is introduced: xi =noted xi (enables to see the difference between a vector and a +function when using components). So +⃗x = +n +� +i=1 +xi⃗ei +� �� � +classic not. += +n +� +i=1 +xi⃗ei +� �� � +duality not. +, +and +[⃗x]|⃗e +clas. += +� +� +x1 +... +xn +� +� dual += +� +� +x1 +... +xn +� +� . +(1.4) +The duality notation is part of the Einstein’s convention; Moreover Einstein’s convention uses the notation +�n +i=1xi⃗ei =noted xi⃗ei, i.e. the sum sign �n +i=1 can be omitted when an index (i here) is used twice, once +up and once down, details at § A.4. However this omission of the sum sign � will not be made in this +manuscript (to avoid ambiguities): The TEX-LATEX program makes it easy to print �n +i=1. +Example 1.1 The height of a child is represented on a wall by a vertical bipoint vector ⃗x starting from +the ground up to a pencil line. Question: What is the size of the child ? +Answer: It depends... +on the observer (quantitative value = subjective result). +E.g., an English +observer chooses a vertical basis vector ⃗a1 which length is one English foot (ft). So he writes ⃗x = x1⃗a1, +and for him the size of the child (size of ⃗x) is x1 in foot. E.g. x1 = 4 means the child is 4 ft tall. A +French observer chooses a vertical basis vector ⃗b1 which length is one metre (m). So he writes ⃗x = y1⃗b1, +and for him the size of the child (size of ⃗x) is y1 metre. E.g., if x1 = 4 then y1 ≃ 1.22, since 1 ft := +0.3048 m: The child is both 4 and 1.22 tall... in foot or metre. This quantification is written ⃗x = 4 ft += 1.22 m, where ft means ⃗a1 and m means ⃗b1 here. NB: The qualitative vector ⃗x is the same vector for +all observers, not the quantitative values 4 or 1.22 (depends on a choice of a unit of measurement). +With duality notation: ⃗x = x1⃗a1 = y1⃗b1, so if x1 = 4 then y1 ≃ 1.22. +This manuscript insists on covariant objectivity; Thus an English engineer (and his foot) and a French +engineer (and his metre) will be able to work together ... and be able to avoid crashes like that of the Mars +Climate Orbiter probe, see remark A.14. And they will be able to use the results of Galileo, Descartes, +Newton, Euler... +who used their own unit of length, and knew nothing about the metre defined in +1793 and adopted in 1799 in France (after 6 years of measurements), and considered by the scientific +community at the end of the ninetieth century... and couldn’t explicitly use the “Euclidean dot products” +either (which seems to have been defined mathematically by Grassmann around 1844). +1.3 +Motion of an object +Let Obj be a “real object”, or “material object”, made of particles (e.g., the Moon: Exists independently +of an observer). Let t1, t2 ∈ R, t1 < t2. +Definition 1.2 The motion of Obj in Rn is the map +�Φ : +� +� +� +� +� +[t1, t2] × Obj → Rn +(t, PObj) +� �� � +particle +→ +p = �Φ(t, PObj) +� +�� +� +its position at t in the Universe +. +(1.5) +And t is the time variable, p is the space variable, and (t, p) ∈ R × Rn is the time-space variable. And +�Φ is supposed to be C2 in time. +With an origin O (observer dependent), the motion can be described with the bi-point vector +⃗x = +−−−−−−−→ +O�Φ(t, PObj) = −→ +Op noted += +�⃗ϕ(t, PObj). +(1.6) +But then, two observers with different origins O and Ob have different description of the motion. There- +fore, in the following we won’t use �⃗ϕ. +Then (quantification) with a Cartesian basis (⃗ei) to make a +referential R, we get (1.1). +1.4 +Virtual and real motion +Definition 1.3 A virtual (or possible) motion of Obj is a function �Φ “regular enough for the calculations +to be meaningful”. Among all the virtual motions, the observed motion is called the real motion. +12 + +13 +1.5. +Hypotheses (Newton and Einstein) +1.5 +Hypotheses (Newton and Einstein) +Hypotheses of Newtonian mechanics (Galileo relativity) and general relativity (Einstein): +1- You can describe a phenomenon only at the actual time t and from the location p you are at (you +have no gift of ubiquity in time or space); +2- You don’t know the future; +3- You can use your memory, so use some past time t0 and some past position pt0; +4- You can use someone else memory (results of measurements) if you can communicate objectively. +1.6 +Configurations +Fix t ∈ [t1, t2], and define �Φt : +� +Obj → Rn +PObj �→ p = �Φt(PObj) := �Φ(t, PObj). +Definition 1.4 The “configuration at t” of Obj is the range (or image) of �Φt, i.e. is the subset of Rn +(affine space) defined by +Ωt := {p ∈ Rn : ∃PObj ∈ Obj s.t. p = �Φt(PObj)} noted += +�Φt(Obj) noted += +Im(�Φt). +(1.7) +If t is the actual time then Ωt is the actual (or current or Eulerian) configuration. +If t0 is a time in the past then Ωt0 is the past (or initial or Lagrangian) configuration. +Hypothesis: At any time t, Ωt is supposed to be a “smooth domain” in Rn, and the map �Φt is assumed +to be one-to-one (= injective): Obj does not crash onto itself. +1.7 +Definition of the Eulerian and Lagrangian variables +• If t is the actual time, then pt = �Φt(PObj) ∈ Ωt is called the Eulerian variable relative to PObj and t. +• If t0 is a time in the past, then pt0 = �Φt0(PObj) ∈ Ωt0 is called the Lagrangian variable relative to PObj +and t0. (A Lagrangian variable is a “past Eulerian variable”). (Two observers with two different origin of +time t0 and t0′ get two different Lagrangian variable while they have the same Eulerian variable.) +1.8 +Trajectories +Let �Φ be a motion of Obj, cf. (1.5), and PObj ∈ Obj (a particle in Obj = e.g. the Moon). +Definition 1.5 The (parametric) trajectory of PObj is the function +�ΦPObj : +� +[t1, t2] → Rn, +t �→ p(t) = �ΦPObj (t) := �Φ(t, PObj) +(position of PObj at t in the Universe). +(1.8) +Its geometric trajectory is the range (image) of �ΦPObj , i.e. +geometric trajectory of PObj := {q ∈ Rn : ∃t ∈ [t1, t2] s.t. q = �ΦPObj (t)} = Im(�ΦPObj ) = �ΦPObj ([t1, t2]). (1.9) +1.9 +Pointed vector, tangent space, fiber, vector field, bundle +(See e.g. Abraham–Marsden [1].) To deal with surfaces S in R3, e.g. with S = a sphere (and more +generally with manifolds in Rn), a vector cannot simply be a “bi-point vector connecting two points of S” +(would get “through the surface”). A vector is defined to be tangent to S: Consider a “regular” curve +c : s ∈] − ε, ε[→ c(s) ∈ S where S is a surface in an affine space, and the vector tangent to S at c(0) is +⃗w(c(0)) = limh→0 +c(h)−c(0) +h +(it is defined with a parametrization of c in a general manifold); Considering +all the possible curves, we get “all possible vectors on S”. +Notation: +TpS := {tangent vectors ⃗wp at S at p} = The tangent space at p ∈ S. +(1.10) +E.g., if S is a sphere in R3 and p ∈ S, then TpS is its usual tangent plane at p at S. +E.g., particular case: If S = Ω is an open set in Rn, then TpS = TpΩ = ⃗Rn is independent of p. +13 + +14 +2.1. +The set of configurations +Definition 1.6 +The fiber at p := {p} × TpS = { +(p, ⃗wp) +� �� � +pointed vector +∈ {p} × TpS}, +(1.11) +i.e., the fiber at p is the set of “pointed vectors at p”, a pointed vector being the couple (p, ⃗wp) made of +the “base point” p and the vector ⃗wp defined at p. +Drawing: A vector in ⃗Rn can be drawn anywhere in Rn; While a “pointed vectors at p” has to be +drawn at the point p in Rn. +If the context is clear, a pointed vector is simply noted �⃗w(p) =noted ⃗w(p) (lighten the writing). +Particular case: If S = Ω is an open set in Rn, then the fiber at p is TpΩ = {p} × ⃗Rn. +Definition 1.7 +The tangent bundle TS := +� +p∈S +({p} × TpS), +(1.12) +that is, is the union of the fibers. +Definition 1.8 A vector field �⃗w in S is a C∞ function (or at least C2 in the following) +�⃗w : +� +S → TS +p → �⃗w(p) = (p, ⃗w(p)). +(1.13) +If the context is clear, a vector field is simply noted �⃗w =noted ⃗w (lighten the writing). +2 +Eulerian description (spatial description at actual time t) +2.1 +The set of configurations +Let �Φ be a motion of Obj, cf. (1.5), and Ωt = �Φt(Obj) ⊂ Rn be the configuration at t, cf. (1.7). The set +of configurations is the subset C ⊂ R × Rn (the “time-space”) defined by +C := +� +t∈[t1,t2] +({t} × Ωt) +(= set in which you find particles in “time-space”) += {(t, p) ∈ R × Rn : ∃(t, PObj) ∈ [t1, t2] × Obj, p = �Φ(t, PObj)}, +(2.1) +Question: Why don’t we simply use � +t∈[t1,t2] Ωt instead of C = � +t∈[t1,t2]({t} × Ωt)? +Answer: C gives the film of the life of Obj = the succession of the photos Ωt taken at each t; And Ωt +is obtained from C thanks to the pause feature at t. Whereas � +t∈[t1,t2] Ωt ⊂ Rn is the superposition of +all the photos on the image � +t∈[t1,t2] Ωt... and we don’t distinguish the past from the present. +2.2 +Eulerian variables and functions +Definition 2.1 In short: A Eulerian function relative to Obj is a function, with m ∈ N∗, +Eul : +� +C → ⃗ +Rm (or more generally a suitable set of tensors) +(t, p) → Eul(t, p), +(2.2) +the spatial variable p being the Eulerian variable. +In details: A function Eul being given as in (2.2), the associated Eulerian function � +Eul is the function +� +Eul : +� +C → C × ⃗ +Rm (or C× some suitable set of tensors) +(t, p) → � +Eul(t, p) = ((t, p), Eul(t, p)) = (time-space position , value), +(2.3) +and is called “a field of functions”. So � +Eul(t, p) is the “pointed Eul(t, p)” at (t, p) (in time-space). +So, the range Im( � +Eul) = � +Eul(C) of an Eulerian function � +Eul is the graph of Eul. (Recall: The graph of +a function f : x ∈ A → f(x) ∈ B is the subset {(x, f(x)) ∈ A × B} ⊂ A × B: gives the “drawing of f”). +If there is no ambiguity, � +Eul =noted Eul for short. +14 + +15 +2.3. +Eulerian velocity (spatial velocity) and speed +At t, the Eulerian vector field at t is � +Eult : +� +Ωt → Ωt × ⃗Rn +p → � +Eult(p) := (p, Eult(p)) = (position , value). +Example 2.2 Eul(t, p) = θ(t, p) ∈ R = temperature of the particle PObj which is at t at p = �Φ(t, PObj); +Example 2.3 Eul(t, p) = ⃗u(t, p) ∈ ⃗Rn = force applied on the particle PObj which is at t at p. +Example 2.4 Eul(t, p) = d⃗u(t, p) ∈ L(⃗Rn : ⃗Rn) = the differential at t at p of a Eulerian function ⃗u. +Question: Why introduce � +Eul? Isn’t Eul sufficient? +Answer: The “pointed value” � +Eul(t, p) = ((t, p), Eul((t, p))) is drawn on the graph of Eul. +E.g., at t at p the velocity vector ⃗v(t, p) ∈ ⃗R3 can be drawn anywhere, while the “pointed vector” +�⃗v(t, p) = ((t, p);⃗v(t, p)) is ⃗v(t, p) drawn at t at p (and �⃗v is called the velocity field). +Moreover (2.3) emphasizes the difference between a Eulerian vector field and a Lagrangian vector +function, see (3.14). +Remark 2.5 E.g., the initial framework of Cauchy for his description of forces is Eulerian: The Cauchy +stress vector ⃗t = σ.⃗n is considered at the actual time t at a point p ∈ Ωt. (It is not Lagrangian.) +2.3 +Eulerian velocity (spatial velocity) and speed +Definition 2.6 In short: Consider a particle PObj and its (regular) trajectory �ΦPObj : t → p(t) = �ΦPObj (t), +cf. (1.8). Its Eulerian velocity at t at p(t) = �ΦPObj (t) is +⃗v(t, p(t)) := �ΦPObj +′(t) noted += +∂�Φ +∂t (t, PObj), +when +p(t) = �ΦPObj (t), +(2.4) +i.e., ⃗v(t, p(t)) is the tangent vector at t at p(t) = �ΦPObj (t) to the trajectory �ΦPObj . This defines the vector +field (in short) ⃗v : +� +C → ⃗Rn +(t, pt) → ⃗v(t, pt) +� +. +In details: cf. (2.3), the Eulerian velocity is the function �⃗v : +� +C → C × ⃗ +Rm +(t, p) → �⃗v(t, p) = ((t, p),⃗v(t, p)) +� +(pointed vector) where ⃗v(t, p) is given by (2.4). +Remark 2.7 +d�ΦPObj +dt +(t) = ⃗v(t, �ΦPObj (t)), with p(t) = �ΦPObj (t), is often written +dp +dt (t) = ⃗v(t, p(t)), +or +d⃗x +dt (t) = ⃗v(t, ⃗x(t)), +or +d⃗x +dt = ⃗v(t, ⃗x), +(2.5) +the two last notations when an origin O is chosen and ⃗x(t) = −−−→ +Op(t). +Such an equation is the pro- +totype of an ODE (ordinary differential equation) solved with the Cauchy–Lipschitz theorem, see § 5. +(A Lagrangian velocity does not produce an ODE, see (3.21).) +Definition 2.8 If an observer chooses a Euclidean dot product (·, ·)g (e.g. foot or metre built), the +associated norm being ||.||g, then the length ||⃗v(t, p)||g is the speed (or scalar velocity) of PObj (e.g. in +ft/s or in m/s). And the context must remove the ambiguities: the “velocity” is either the vector velocity +⃗v(t, p) = �ΦPObj +′(t) or the speed (the scalar velocity) ||⃗v(t, p)||g. +Exercice 2.9 Euclidean dot product (·, ·)g, ⃗x(t) = −−−→ +Op(t), ⃗T(t) = +⃗x ′(t) +||⃗x ′(t)||g , and f(t) = ||⃗x ′(t)||g (speed). +Prove : +df +dt(t) = (⃗x ′′(t), ⃗T(t))g =noted ⃗x ′′(t) • ⃗T(t) (= tangential acceleration). +Answer. +2-D and Euclidean basis: +⃗x(t) += +� x(t) +y(t) +� +gives f(t) += +(x′(t)2 + y′(t)2) +1 +2 , thus f ′(t) += +x′(t)x′′(t)+y′(t)y′′(t) +f(t) += ⃗r ′(t) • ⃗r ′′(t) +||⃗r ′(t)|| +. Idem in n-D. +2.4 +Spatial derivative of the Eulerian velocity +t ∈ [t1, t2] is fixed, Eul is a given Eulerian function, and Eult : +� +Ωt → ⃗ +Rm +p → Eult(p) := Eul(t, p) +� +is C1. +15 + +16 +2.4. +Spatial derivative of the Eulerian velocity +2.4.1 +Definition +Recall: If Ω is an open set in Rn and if f : Ω → R is differentiable at p, then its differential at p is the +linear form df(p) ∈ L(⃗Rn; R) (linear map with real values) defined by, for all ⃗u ∈ ⃗Rn (vector at p), +df(p).⃗u = lim +h→0 +f(p+h⃗u) − f(p) +h +. +(2.6) +This expression is the same for all observers (English, French...: There is no inner dot product here). +Definition 2.10 The space derivative of Eul at (t, p) is the differential dEult at p, i.e., for all t ∈ [t1, t2], +all p ∈ Ωt and all ⃗wp ∈ ⃗Rn +t (vector at p), +(dEult(p).⃗wp =) +dEul(t, p).⃗wp = lim +h→0 +Eul(t, p+h⃗wp) − Eul(t, p) +h +noted += +∂Eul +∂p (t, p).⃗wp. +(2.7) +In Ωt (the photo at t), dEul(t, p).⃗wp gives the rate of variations of Eult at p in the direction ⃗wp. +E.g., at t, the space derivative d⃗v of the Eulerian velocity field is defined by +d⃗v(t, p).⃗wp = lim +h→0 +⃗v(t, p+h⃗wp) − ⃗v(t, p) +h +(= d⃗vt(p).⃗wp). +(2.8) +Remark 2.11 In differential geometry, (2.6) is also written ⃗u(f)(p) = +d +dhf(p+h⃗u)|h=0; Don’t use this +notation if you are not at ease with differential geometry (where a vector is defined to be a derivation, +so ⃗u[f] is the derivation of f by ⃗u). +2.4.2 +The convective derivative dEul.⃗v +Definition 2.12 If ⃗v is the Eulerian velocity field, then dEul.⃗v is called the convective derivative of Eul. +2.4.3 +Quantification in a basis: df.⃗u is written (⃗u. ⃗ +grad)f +Quantification: Let f : p ∈ Rn → f(p) ∈ R be C1. Let (⃗ei) be a basis in ⃗Rn. Let (usual definition) +∂f +∂xi +(p) := df(p).⃗ei +and +[df(p)]|⃗e = ( ∂f +∂x1 (p) +... +∂f +∂xn (p) ) (line matrix). +(2.9) +(Recall: The matrix which represents a linear form is a line matrix.) And [df(p)]|⃗e is the Jacobian matrix +of f at p relative to (⃗ei). +So, with ⃗u = �n +i=1ui⃗ei a vector at p, and with the usual matrix multiplication +rule, we have +df(p).⃗u = [df(p)]|⃗e.[⃗u]|⃗e = +n +� +i=1 +∂f +∂xi +(p)ui = +n +� +i=1 +ui +∂f +∂xi +(p) noted += +(⃗u. ⃗ +grad)|ef(p), +(2.10) +where (⃗u. ⃗ +grad)|e : C1(Ω; R) → C0(Ω; R) is the differential operator defined relative to a basis (⃗ei) by +(⃗u. ⃗ +grad)|e(f) = +n +� +i=1 +ui +∂f +∂xi +. +(2.11) +If the basis (⃗ei) is unambiguously imposed, then (⃗u. ⃗ +grad)|e =noted ⃗u. ⃗ +grad +For vector valued functions ⃗f : Ω → ⃗ +Rm, the above steps apply to the components of ⃗f in a basis (⃗bi) +in ⃗ +Rm: If ⃗f = �m +i=1fi⃗bi, i.e. ⃗f(p) = �m +i=1fi(p)⃗bi, then +(⃗u. ⃗ +grad)|e(⃗f) = +m +� +i=1 +(dfi.⃗u)⃗bi = +m +� +i=1 +((⃗u. ⃗ +grad)|efi)⃗bi = +m +� +i=1 +n +� +j=1 +(uj. ∂fi +∂xj +)⃗bi. +(2.12) +16 + +17 +2.5. +Streamline (current line) +2.4.4 +Representation relative to a Euclidean dot product: +⃗ +gradf +An observer chooses a distance unit (foot, metre...) and uses the associated Euclidean dot product (·, ·)g. +Let Ω be an open set in Rn, f ∈ C1(Ω; R) (scalar valued function), and p ∈ Ω. Then the (·, ·)g-Riesz +representation vector of the differential form df(p) is called the gradient of f at p relative to (·, ·)g, and +named +⃗ +gradgf(p) ∈ ⃗Rn: It is defined by +∀⃗u ∈ ⃗Rn, +( ⃗ +gradgf(p), ⃗u)g = df(p).⃗u, +written +⃗ +gradf • ⃗u = df.⃗u, +(2.13) +the last notation iff a Euclidean dot product (·, ·)g is imposed to all observer (quite subjective: foot, +metre ?). +(The first order Taylor expansion f(p+h⃗u) = f(p) + h df(p).⃗u + o(h) can therefore, after a choice of +an Euclidean dot product, be written f(p+h⃗u) = f(p) + h +⃗ +gradgf(p) •g ⃗u + o(h).) +Quantification: Let (⃗ei) be a Cartesian basis in Rn. Then (2.13) gives [df].[⃗u] = [ ⃗ +gradf]T .[g].[⃗u], for all +⃗u ∈ ⃗Rn +t (more precisely [df]|⃗e.[⃗u]|⃗e = [ ⃗ +gradgf]T +|⃗e.[g]|⃗e.[⃗u]|⃗e), thus (since [g]|⃗e is symmetric) +[ ⃗ +gradf] = [g].[df]T +(column matrix). +(2.14) +I.e., if +⃗ +gradf = �n +i=1ai⃗ei then ai = �n +j=1gij +∂f +∂xj for all i. In particular, if (⃗ei) is a (·, ·)g-orthonormal +basis then [ ⃗ +gradf] = [df]T . +With duality notations, +⃗ +gradf = �n +i=1ai⃗ei and (2.14) gives ai = �n +j=1gij +∂f +∂xj : The Einstein convention +is not satisfied (the index j is twice bottom), which is expected since the definition of +⃗ +gradgf depends on +a subjective choice (unit of length). In comparison, df = �n +i=1 +∂f +∂xi dxi satisfies the Einstein convention (a +differential is objective). +Mind the notations: The gradient +⃗ +gradgf =noted +⃗ +gradf depends on (·, ·)g, cf. (2.13)-(2.14), while +(⃗u. ⃗ +grad)f does not (only depends on a basis), cf. (2.11) (historical notations...). +2.4.5 +Vector valued functions +For vector valued functions ⃗f : Ω → ⃗ +Rm, the above steps apply to the components fi of ⃗f relative to a +basis (⃗bi) in ⃗ +Rm... But, depending on the book you read: +1- Ambiguous: d⃗f, the differential of ⃗f, is unfortunately also sometimes called the “gradient matrix” +(although no Euclidean dot product is required). +2- Ambiguous: It could mean the differential... or the Jacobian matrix... or its transposed... because +an orthonormal basis relative to an imposed Euclidean dot product is chosen (which one?) and then +[ ⃗ +gradfi] = [dfi]T ... And calculations confuses [.] and [.]T ... +3- Non ambiguous: In the objective framework of this manuscript, we will use the differential d⃗f +(objective) to begin with; And only after an explicit choice of bases (⃗ei) for quantitative purposes, the +Jacobian matrix, which is [df]|⃗e, will be used. +Exercice 2.13 A Euclidean framework being chosen, prove: (⃗v. ⃗ +grad)⃗v = 1 +2 ⃗ +grad(||⃗v||2) + ⃗ +rot⃗v ∧ ⃗v. +Answer. Euclidean basis ( ⃗Ei), Euclidean dot product (·, ·)g =noted (·, ·), associated norm ||.||g =noted ||.||. Thus +⃗v = �n +i=1vi ⃗Ei gives ||⃗v||2 = +� +i +v2 +i , thus ∂||⃗v||2 +∂xk += +� +i +2vi ∂vi +∂xk , for any k = 1, 2, 3. And, the first component +of ⃗ +rot⃗v is ( ⃗ +rot⃗v)1 = ∂v3 +∂x2 − ∂v2 +∂x3 , idem for ( ⃗ +rot⃗v)2 and ( ⃗ +rot⃗v)3 (circular permutation). Thus (first component) +( ⃗ +rot⃗v∧⃗v)1 = ( ∂v1 +∂x3 − ∂v3 +∂x1 )v3−( ∂v2 +∂x1 − ∂v1 +∂x2 )v2, idem for ( ⃗ +rot⃗v∧⃗v)2 and ( ⃗ +rot⃗v∧⃗v)2. Thus ( 1 +2 ⃗ +grad(||⃗v||2)+ ⃗ +rot⃗v∧⃗v)1 = +v1 +∂v1 +∂x1 + v2 +∂v2 +∂x1 + v3 +∂v3 +∂x1 + ∂v1 +∂x3 v3 − ∂v3 +∂x1 v3 − ∂v2 +∂x1 v2 + ∂v1 +∂x2 v2 = v1 +∂v1 +∂x1 + v2 +∂v1 +∂x2 + v3 +∂v1 +∂x3 = (⃗v. ⃗ +grad)v1. Idem for the +other components. +2.5 +Streamline (current line) +Fix t ∈ R, and consider the photo Ωt = �Φt(Obj). Let pt ∈ Ωt, ε > 0, and consider the spatial curve in Ωt +at pt defined by: +cpt : +� +] − ε, ε[ → Ωt +s → q = cpt(s) +� +s.t. +cpt(0) = pt. +(2.15) +17 + +18 +2.6. +Material time derivative (dérivées particulaires) +So s is a curvilinear spatial coordinate (dimension of a length), and the graph of cpt is drawn in the photo +Ωt at t. +Definition 2.14 ⃗v : (t, p) → ⃗v(t, p) being the Eulerian velocity field of Obj, a streamline through a point +pt ∈ Ωt is a (parametric) spatial curve cpt solution of the differential equation +dcpt +ds (s) = ⃗vt(cpt(s)) +with +cpt(0) = pt. +(2.16) +And Im(cpt) is the geometric associated streamline (⊂ Ωt). +NB: (2.16) cannot be confused with (2.5): In (2.5) the variable is the time variable t, while in (2.16) +the variable is the space variable s. +Usual notation: If an origin O is chosen at t by an observer and ⃗x(s) := −−−−−→ +Ocpt(s) , then (2.16) is written +d⃗x +ds (s) = ⃗vt(⃗x(s)) +with +⃗x(0) = −−→ +Opt. +(2.17) +Moreover, with a Cartesian basis (⃗ei)) chosen at t by the observer, with ⃗x(s) = �n +i=1xi(s)⃗ei we get +d⃗x +ds (s) = �n +i=1 +dxi +ds (s)⃗ei, and (2.17) reads as the differential system of n equations in ⃗Rn +∀i = 1, ..., n, +dxi +ds (s) = vi(t, x1(s), ..., xn(s)) +with +xi(0) = (−−→ +Opt)i +(2.18) +(the n functions xi : s → xi(s) are the unknown). Also written +dx1 +v1 += ... = dxn +vn += ds, +(2.19) +which means: It is the differential system (2.18) of n equations and n unknowns which must be solved. +(With duality notations, dxi +ds (s) = vi(t, x1(s), ..., xn(s)) and xi(0) = (−−→ +Opt)i for all i.) +2.6 +Material time derivative (dérivées particulaires) +2.6.1 +Usual definition +Goal: To compute the variations of a Eulerian function Eul along the trajectory �ΦPObj of a particle PObj +(e.g. the temperature of a particle along its trajectory). So consider the function gPObj giving the values +of Eul relative to a PObj along its trajectory: +gPObj (t) := Eul(t, p(t)) +when +p(t) := �ΦPObj (t). +(2.20) +Definition 2.15 The Material time derivative of Eul at (t, p(t)) is gPObj +′(t) =noted DEul +Dt (t, p(t)). +So: +DEul +Dt (t, p(t)) := gPObj +′(t) +(= lim +h→0 +Eul(t+h, p(t+h)) − Eul(t, p(t)) +h +). +(2.21) +Since gPObj (t) := Eul(t, �ΦPObj (t)) we get gPObj +′(t) = ∂Eul +∂t (t, �ΦPObj (t))+dEul(t, �ΦPObj (t)).�Φ′ +PObj (t), thus, having +�Φ′ +PObj (t) = ⃗v(t, p(t)) (Eulerian velocity), DEul +Dt (t, p(t)) = ∂Eul +∂t (t, p(t)) + dEul(t, p(t)).⃗v(t, p(t)): +DEul +Dt +:= ∂Eul +∂t ++ dEul.⃗v . +(2.22) +Exercice 2.16 Prove, if Eul is C2: +D2Eul +Dt2 += ∂2Eul +∂t2 ++ 2d∂Eul +∂t .⃗v + dEul.∂⃗v +∂t + d2Eul(⃗v,⃗v) + dEul.d⃗v.⃗v +(2.23) +Answer. +D2Eul +Dt2 += D DEul +Dt +Dt += g′′ +PObj (t) = ∂( ∂Eul +∂t + dEul.⃗v) +∂t ++ d(∂Eul +∂t ++ dEul.⃗v).⃗v += ∂2Eul +∂t2 ++ ∂(dEul) +∂t +.⃗v + dEul.∂⃗v +∂t + d∂Eul +∂t .⃗v + d2Eul(⃗v,⃗v) + dEul.d⃗v.⃗v, +and Eul C2 gives +∂ +∂t ◦ d = d ◦ ∂ +∂t (Schwarz theorem), hence (2.23). +18 + +19 +2.6. +Material time derivative (dérivées particulaires) +Exercice 2.17 Prove, if Eul is C2, for any vector field ⃗w, +D(dEul.⃗w) +Dt += d∂Eul +∂t .⃗w + dEul.∂ ⃗w +∂t + d2Eul(⃗v, ⃗w) + dEul.d⃗w.⃗v. +(2.24) +Answer. D(dEul.⃗w) +Dt += ∂(dEul.⃗w) +∂t ++ d(dEul.⃗w).⃗v = ∂dEul +∂t +.⃗w + dEul.∂ ⃗w +∂t + (d(dEul).⃗v).⃗w + dEul.d⃗w.⃗v. And the +Schwarz theorem gives ∂(dEul) +∂t += d( ∂Eul +∂t ) since Eul ∈ C2. Hence (2.24). +2.6.2 +Remark: About notations +• The notation +d +dt (lowercase letters) concerns a function of one variable, e.g. +dgPObj +dt (t) := gPObj +′(t) := +limh→0 +gPObj (t+h))−gPObj (t) +h +; +• +The +notation +∂ +∂t +concerns +a +function +with +more +than +one +variable, +e.g. +∂Eul +∂t (t, p) += +limh→0 +Eul(t+h,p)−Eul(t,p) +h +; +• The notation +D +Dt (capital letters) concerns a Eulerian function differentiated along a motion, +cf. (2.21). +• Other notations, often practical but might be ambiguous if composed functions are considered: +dEul(t, p(t)) +dt +:= DEul +Dt (t, p(t)), +and +dEul(t, p(t)) +dt +|t=t0 +:= DEul +Dt (t0, p(t0)). +(2.25) +2.6.3 +Definition bis: Time-space definition +Consider a C1 time-space function f : (t, p) ∈ R × Rn → f(t, p) where t = time and p = space. +Definition 2.18 The differential of f : (t, p) ∈ Rn+1 → f(t, p) considered as a function on the Cartesian +(time×space) product R × Rn is called the “total differential”, or “total derivative”, and is written Df +(here time and space are of a different nature). +So if p+ = (t, p) ∈ R×Rn and ⃗w+ = (w0, ⃗w) ∈ ⃗R×⃗Rn (time×space) then, by definition of a differential, +Df(p+).⃗w+ := lim +h→0 +f(p+ + h⃗w+) − f(p+) +h +, +(2.26) +i.e. +Df(t, p).(w0, ⃗w) := lim +h→0 +f(t+hw0, p+h⃗w) − f(t, p) +h +. +(2.27) +Thus +Df(t, p) = ∂f +∂t (t, p) dt + df(t, p). +(2.28) +Along a trajectory �ΦPObj : t → p(t) = �ΦPObj (t) with f = Eul a Eulerian function: Consider the +time-space trajectory +�ΨPObj : +� +[t1, t2] → R × Rn +t → �ΨPObj (t) := (t, �ΦPObj (t)) +(= (t, p(t))). +(2.29) +(So Im(�ΨPObj ) = graph(�ΦPObj ).) The tangent vector to this curve at t is +�ΨPObj +′(t) = (1, �ΦPObj +′(t)) = (1,⃗v(t, p(t)) ∈ ⃗R × ⃗Rn +(2.30) +where ⃗v(t, p(t)) the Eulerian velocity at p+ = (t, p(t)). And (2.20) reads +gPObj (t) = (Eul ◦ �ΨPObj )(t) = Eul(�ΨPObj (t)), +(2.31) +thus +g′ +PObj (t) = DEul(�Ψ(t)).�ΨPObj +′(t). +(2.32) +And we recover (2.22): g′ +PObj (t) =(2.28) ∂Eul +∂t (t, p(t)).1+dEul(t, p(t)).⃗v(t, p(t)) =noted DEul +Dt (t, p(t)) : The ma- +terial time derivative is the “total derivative” DEul along the time-space trajectory �ΨPObj . +19 + +20 +2.7. +Eulerian acceleration +2.6.4 +The material time derivative is a derivation +Proposition 2.19 All the functions are Eulerian and supposed C1. +• Linearity: +D(Eul1 + λEul2) +Dt += DEul1 +Dt ++ λDEul2 +Dt +. +(2.33) +• Product rules: If Eul1, Eul2 are scalar valued functions then +D(Eul1Eul2) +Dt += DEul1 +Dt +Eul2 + Eul1 +DEul2 +Dt +. +(2.34) +And if ⃗w is a vector field and T a compatible tensor (so T.⃗w is meaningful) then +D(T.⃗w) +Dt += DT +Dt .⃗w + T.D ⃗w +Dt . +(2.35) +Proof. Let i = 1, 2, and gi defined by gi(t) := Euli(t, p(t)) where p(t) = �ΦPObj (t). +• (g1 + λg2)′ = g′ +1 + λg′ +2 gives (2.33). +• On the one hand D(T.⃗w) +Dt += ∂(T.⃗w) +∂t ++ d(T.⃗w).⃗v = ∂T +∂t .⃗w + T. ∂ ⃗w +∂t + (dT.⃗v).⃗w + T.(d⃗w.⃗v), and on the +other hand DT +Dt .⃗w + T. D ⃗w +Dt = ( ∂T +∂t + dT.⃗v).⃗w + T.( ∂ ⃗w +∂t + d⃗w.⃗v). Thus (2.34)-(2.35). +2.6.5 +Commutativity issue +The Schwarz theorem that, when Eul is C2, the derivatives ∂Eul +∂t +and dEul commute. But +Proposition 2.20 The material time derivative +D +Dt does not commute with the temporal derivation +∂ +∂t +or with the spatial derivation d: We have ∂( DEul +Dt ) +∂t +̸= D( ∂Eul +∂t ) +Dt +and d( DEul +Dt ) ̸= D(dEul) +Dt +in general (because +the variables t and p are not independent along a trajectory). In facts: +∂( DEul +Dt ) +∂t += D( ∂Eul +∂t ) +Dt ++ dEul.∂⃗v +∂t += ∂2Eul +∂t2 ++ d∂Eul +∂t .⃗v + dEul.∂⃗v +∂t +� +� +� +� +� +� +� +, +and +� +� +� +� +� +d(DEul +Dt ) = D(dEul) +Dt ++ dEul.d⃗v += ∂(dEul) +∂t ++ d2Eul.⃗v + dEul.d⃗v +(2.36) +Proof. ∂ DEul +Dt +∂t += ∂( ∂Eul +∂t + dEul.⃗v) +∂t += ∂( ∂Eul +∂t ) +∂t ++ ∂dEul +∂t +.⃗v + dEul.∂⃗v +∂t +Schwarz += +∂( ∂Eul +∂t ) +∂t ++ d∂Eul +∂t .⃗v + dEul.∂⃗v +∂t , +thus (2.36)1. +dDEul +Dt += d(∂Eul +∂t ++ dEul.⃗v) = ∂(dEul) +∂t ++ d(dEul).⃗v + dEul.d⃗v, thus (2.36)2. +So ∂( DEul +Dt ) +∂t +̸= D( ∂Eul +∂t ) +Dt +and d(DEul +Dt ) ̸= D(dEul) +Dt +in general. +Exercice 2.21 Prove (2.36) with components. +Answer. (⃗ei) is a Cartesian basis. +∂ DEul +Dt +∂t += +∂( ∂Eul +∂t +� +i +∂Eul +∂xi .vi) +∂t += ∂2Eul +∂t2 ++ � +i +∂2Eul +∂t∂xi .vi + � +i +∂Eul +∂xi . ∂vi +∂t = ∂2Eul +∂t2 ++ +� +i +∂2Eul +∂t∂xi .vi + dEul. ∂⃗v +∂t . And +D( ∂Eul +∂t ) +Dt += ∂2Eul +∂t2 ++ � +i +∂ ∂Eul +∂t +∂xi .vi = ∂2Eul +∂t2 ++ � +i +∂2Eul +∂t∂xi .vi. +And d( DEul +Dt ).⃗w = � +j +∂ DEul +Dt +∂xj wj = � +j +∂( ∂Eul +∂t +� +i +∂Eul +∂xi vi) +∂xj +wj = � +j +∂2Eul +∂t∂xj wj +� +ij +∂2Eul +∂xi∂xj viwj +� +ij +∂Eul +∂xi +∂vi +∂xj wj = +� +j +∂2Eul +∂t∂xj wj + d2Eul(⃗v, ⃗w) + dEul.d⃗v.⃗w. +And +D(dEul) +Dt +.⃗w = ( ∂(dEul) +∂t ++ d(dEul).⃗v).⃗w = +∂(dEul) +∂t +.⃗w + d2Eul(⃗v, ⃗w) = +� +i +∂2Eul +∂xi∂t wi + d2Eul(⃗v, ⃗w). Thus d( DEul +Dt ).⃗w = D(dEul) +Dt +.⃗w + dEul.d⃗v.⃗w for all ⃗w. +2.7 +Eulerian acceleration +Definition 2.22 In short: If �ΦPObj is C2, then the Eulerian acceleration of the particle PObj which is at t +at pt = �Φ(t, PObj) is +⃗γ(t, pt) := �ΦPObj +′′(t) noted += +∂2�Φ +∂t2 (t, PObj). +(2.37) +In details: as in (2.3), the Eulerian acceleration (vector) field �⃗γ is defined with (2.37) by +�⃗γ(t, pt) = ((t, pt),⃗γ(t, pt)) ∈ C × ⃗Rn +t +(pointed vector). +(2.38) +20 + +21 +2.8. +Time Taylor expansion of �Φ +Proposition 2.23 +⃗γ = D⃗v +Dt = ∂⃗v +∂t + d⃗v.⃗v . +(2.39) +And if ⃗v is C2 then +d⃗γ = ∂(d⃗v) +∂t ++ d2⃗v.⃗v + d⃗v.d⃗v = D(d⃗v) +Dt ++ d⃗v.d⃗v. +(2.40) +Proof. With g(t) = ⃗v(t, p(t)) = �ΦPObj +′(t) and (2.22) we get ⃗γ(t, p(t)) = g′(t) = +D⃗v +Dt (t, p(t)). +And ⃗v +being C2, the Schwarz theorem gives d ∂⃗v +∂t = ∂(d⃗v) +∂t . +Definition 2.24 If an observer chooses a Euclidean dot product (·, ·)g (based on a foot, a metre...), the +associated norm being ||.||g, then the length ||⃗γ(t, pt)||g is the (scalar) acceleration of PObj. +2.8 +Time Taylor expansion of �Φ +Let PObj ∈ Obj and t ∈]t1, t2[. Suppose �ΦPObj ∈ C2(]t1, t2[; Rn). Its second-order (time) Taylor expansion +of �ΦPObj is, in the vicinity of a t ∈]t1, t2[, +�ΦPObj (τ) = �ΦPObj (t) + (τ−t)�Φ′ +PObj (t) + (τ−t)2 +2 +�Φ′′ +PObj (t) + o((τ−t)2), +(2.41) +i.e. +p(τ) = p(t) + (τ−t)⃗v(t, p(t)) + (τ−t)2 +2 +⃗γ(t, p(t)) + o((τ−t)2). +(2.42) +3 +Motion from an initial configuration: Lagrangian description +Instead of working on Obj, an observer may prefer to work with an initial configuration Ωt0 = �Φ(t0, Obj) +of Obj (essential for elasticity): This is the “Lagrangian approach”. This Lagrangian approach is not +objective: Two observers may choose two different initial (times and) configurations. +3.1 +Initial configuration and Lagrangian “motion” +3.1.1 +Definition +Obj is a material object, �Φ : [t1, t2[×Obj → Rn is its motion, Ωτ = �Φτ(Obj) is its configuration at τ, +t0 ∈]t1, t2[ is an “initial time”, and Ωt0 is the initial configuration for the observer who chose t0. +Definition 3.1 The motion of Obj relative to the initial configuration Ωt0 = �Φ(t0, Obj) is the function +Φt0 : +� +[t1, t2] × Ωt0 → Rn +(t, pt0) �→ pt = Φt0(t, pt0) := �Φ(t, PObj) +when +pt0 = �Φ(t0, PObj). +(3.1) +So, pt = Φt0(t, pt0) := �Φ(t, PObj) is the position at t of the particle PObj which was at pt0 at t0. In particular +pt0 = Φt0(t0, pt0) := �Φ(t0, PObj). +Marsden and Hughes notations: +Once an initial time t0 has been chosen by an observer, then +Φt0 =noted Φ, then pt0 =noted P (capital letter for positions at t0) and pt =noted p (lowercase letter for +positions at t), so +p = Φ(t, P) ∈ Ωt. +(3.2) +(When objectivity is under concern, we need to switch back to the notations Φt0, pt0 and pt.) +NB: • Talking about the motion of a position pt0 is absurd: A position in Rn does not move. Thus +Φt0 has no existence without the definition, at first, of the motion �Φ of particles. +• The domain of definition of Φt0 depends on t0 through Ωt0: The superscript t0 recalls it. And a late +observer with initial time t0′ > t0 defines Φt0 +′ which domain of definition is [t1, t2]×Ωt0′; And Φt0 +′ ̸= Φt0 +in general because Ωt0′ ̸= Ωt0 in general. +21 + +22 +3.1. +Initial configuration and Lagrangian “motion” +• The following notation is also used: +Φt0(t, pt0) = Φ(t; t0, pt0). +(3.3) +(The couple (t0, pt0) is “the initial condition”, or t0 and pt0 are the initial conditions, see the § on flows). +• If a origin O ∈ Rn is chosen by the observer, we may also use, with (1.6), +⃗xt0 = −−→ +Opt0 = ⃗ϕ t0(t0, ⃗xt0) = ⃗X = −−→ +OP +and +⃗xt = −−→ +Opt = ⃗ϕ t0(t, ⃗xt0) = ⃗x = −→ +Op. +(3.4) +3.1.2 +Diffeomorphism between configurations +With (3.1), define +Φt0 +t : +� +Ωt0 → Ωt +pt0 → pt = Φt0 +t (pt0) := Φt0(t, pt0). +(3.5) +Hypothesis: For all t0, t ∈]t1, t2[, the map Φt0 +t +: Ωt0 → Ωt is a Ck diffeomorphism (a Ck invertible +function whose inverse is Ck), where k ∈ N∗ depends on the required regularity. +Thus (3.5) gives �Φt(PObj) = Φt0 +t (�Φt0(PObj)), true for all PObj ∈ Obj, thus Φt0 +t ◦ �Φt0 = �Φt, i.e. +Φt0 +t := �Φt ◦ (�Φt0)−1 . +(3.6) +Thus, Φt0 +t0 = I and Φt +t0 ◦ Φt0 +t = (�Φt ◦ (�Φt0)−1) ◦ (�Φt0 ◦ (�Φt)−1) = I give +Φt +t0 = (Φt0 +t )−1. +(3.7) +3.1.3 +Trajectories +Let (t0, pt0) ∈ [t1, t2] × Ωt0 (initial conditions) and with (3.1) define +Φt0 +pt0 : +� [t1, t2] → Rn +t �→ p(t) = Φt0 +pt0 (t) := �ΦPObj (t) = Φt0(t, pt0) +when +pt0 = �ΦPObj (t0). +(3.8) +Definition 3.2 Φt0 +pt0 is called the (parametric) “trajectory of pt0”, which means: Φt0 +pt0 is the trajectory +of the particle PObj that is located at pt0 = �Φ(t, PObj) at t0. And the geometric “trajectory of pt0” is +Im(Φt0 +pt0 ) = Φt0 +pt0 ([t1, t2]) = +� +t∈[t1,t2] +{Φt0 +pt0 (t)} +(= Im(�ΦPObj )). +(3.9) +NB: The terminology “trajectory of pt0” is awkward, since a position pt0 does not move: It is indeed +the trajectory �ΦPObj of a particle PObj which is at pt0 at t0 that must be understood. +3.1.4 +Streaklines (lignes d’émission) +Take a film between t0 and T (start and end). +Definition 3.3 Let Q be a fixed point in Rn (you see the point Q on each photo that make up the film). +The streakline through Q is the set +Et0,T (Q) = {p ∈ Ω : ∃τ ∈ [t0, T] : p = Φτ +T (Q) = (ΦT +τ )−1(Q)} += {p ∈ Ω : ∃u ∈ [0, T−t0] : p = ΦT −u +T +(Q) = (ΦT +T −u)−1(Q)} +(3.10) += the set at T of the positions (a line in Rn) of all the particles which were at Q at a τ ∈ [t0, T]. +Example 3.4 Smoke comes out of a chimney. Fix a camera nearby, choose a point Q at the top of +the chimney where the particles are colored, and make a film. At T stop filming. Then (at time T) +superimpose the photos in the film: The colored curve we see is the streakline. +In other words = � +τ∈[t0,T ]{Φτ +Q(T)} = � +u∈[0,T −t0]{ΦT −u +Q +(T)}. +22 + +23 +3.2. +Lagrangian variables and functions +3.2 +Lagrangian variables and functions +3.2.1 +Definition +Consider a motion �Φ, cf. (1.5). An observer chose (subjective) a t0 ∈ [t1, t2] (“in the past”); So Ωt0 = +�Φ(t0, Obj) is his initial configuration. Let m ∈ N∗. +Definition 3.5 In short: A Lagrangian function, relative to Obj, �Φ and t0, is a function +Lagt0 : +� +[t1, t2] × Ωt0 → ⃗ +Rm (or, more generally, some adequat set) +(t, pt0) → Lagt0(t, pt0), +(3.11) +and pt0 is called the Lagrangian variable relative to the (subjective) choice t0. +(To compare with (2.2): A Eulerian function does not depend on any t0.) +Example 3.6 Scalar values: Lagt0(t, pt0) = Θt0(t, pt0) = temperature at t at pt = Φt0 +t (pt0) = �Φ(t, PObj) +of the particle PObj that was at pt0 at t0. (So, continuing example 2.2, Θt0(t, pt0) = θ(t, pt).) +Example 3.7 Vectorial values: Lagt0(t, pt0) = ⃗U t0(t, pt0) = force at t at pt = Φt0 +t (pt0) = �Φ(t, PObj) +acting on the particle PObj that was at pt0 at t0. (So, continuing example 2.3, ⃗U t0(t, pt0) = ⃗u(t, pt).) +If t is fixed or if pt0 ∈ Ωt0 is fixed, then we define +Lagt0 +t : +� +Ωt0 → ⃗ +Rm (or, more generally, some adequat set) +pt0 → Lagt0 +t (pt0) := Lagt0(t, pt0), +(3.12) +Lagt0 +pt0 : +� +[t1, t2] → ⃗ +Rm (or, more generally, some adequat set) +t → Lagt0 +pt0 (t) := Lagt0(t, pt0). +(3.13) +Remark 3.8 The position pt0 is also sometimes called a “material point”, which is counter intuitive: +PObj (objective) is the material point, and pt0 is just its spatial position at t0 (subjective); And a Eulerian +variable pt is not called a “material point” at t... +By the way, the variable pt is also called the “updated Lagrangian variable”... +3.2.2 +A Lagrangian function is a two point tensor +Definition 3.9 In details: Lagt0 being defined in (3.11), a Lagrangian function is a function +� +Lag +t0 : +� [t1, t2] × Ωt0 → C × ⃗ +Rm +(t, pt0) → � +Lag +t0(t, pt0) = ((t, pt), Lagt0(t, pt0)) +when +pt = Φt0 +t (pt0). +(3.14) +I.e. � +Lag +t0(t, pt0) = ((t, Φt0 +t (pt0)), Lagt0(t, pt0)). (And ⃗ +Rm can be replaced by some set.) +Definition 3.10 (Marsden and Hughes [12].) A Lagrangian function is a “two point vector field” (or +more generally a “two point tensor”) in reference to the points pt0 ∈ Ωt0 (departure set) and pt ∈ Ωt +(arrival set) where the value Lagt0(t, pt0) is considered. +Interpretation: (3.14) tells that Lagt0(t, pt0) is not represented at (t, pt0), but at (t, pt): That is, having +graph(Lagt0) = {((t, pt0), Lagt0(t, pt0)) +and +Im(� +Lag +t0) = {((t, pt), Lagt0(t, pt0))}, +(3.15) +we have +Im(� +Lag +t0) ̸= graph(Lagt0) : +(3.16) +So a Lagrangian function does not define a tensor in the usual sense. To compare with the Eulerian +function Eul which defines a tensor (in particular Im( � +Eul) = graph(Eul)), cf. (2.3). +23 + +24 +3.3. +Lagrangian function associated with a Eulerian function +3.3 +Lagrangian function associated with a Eulerian function +3.3.1 +Definition +Let �Φ be a motion, cf. (1.5). Let Eul be a Eulerian function, cf. (2.3). Let t0 ∈ [t1, t2]. +Definition 3.11 The Lagrangian function Lagt0 associated with the Eulerian function Eul is defined by, +for all (t, PObj) ∈ [t1, t2] × Obj, +Lagt0(t, �Φ(t0, PObj)) := Eul(t, �Φ(t, PObj)), +(3.17) +i.e., for all (t, pt0) ∈ [t1, t2] × Ωt0, +Lagt0(t, pt0) := Eul(t, pt), +when +pt = �Φ(t, PObj) = Φt0 +t (pt) +(3.18) +i.e., Lagt0(t, pt0) := Eul(t, pt) when pt0 = (Φt0 +t )−1(pt) for all (t, pt) ∈ C. In other words: +Lagt0 +t := Eult ◦ Φt0 +t . +(3.19) +3.3.2 +Remarks +• If you have a Lagrangian function, then you can associate the function +Eult0 +t := Lagt0 +t ◦ (Φt0 +t )−1 +(3.20) +which thus a priori depends on t0. But, a Eulerian function is independent of any initial time t0. +• For one measurement, there is only one Eulerian function Eul, while there are as many associated +Lagrangian function Lagt0 as they are t0 (as many as observers): The Lagrangian function Lagt0 +′ of a +late observer who chooses t0′ > t0 is different from Lagt0 since the domains of definition Ωt0 and Ωt0′ are +different (in general). +3.4 +Lagrangian velocity +3.4.1 +Definition +Definition 3.12 In short: The Lagrangian velocity at t at pt = �Φ(t, PObj) of the particle PObj is the +function +⃗V t0 : +� +R × Ωt0 → ⃗Rn +(t, pt0) → ⃗V t0(t, pt0) := �ΦPObj +′(t) +when +pt0 = �Φ(t0, PObj). +(3.21) +In details: With (3.21), the Lagrangian velocity is the two point vector field given by +� +⃗V t0(t, pt0) : +� +� +� +R × Ωt0 → C × ⃗Rn +(t, pt0) → � +⃗V t0(t, pt0) := ((t, pt), ⃗V t0(t, pt0)), +when +pt = Φt0(t, pt0). +(3.22) +Thus ⃗V t0(t, pt0) = �ΦPObj +′(t) = ⃗v(t, pt) is the velocity at t at pt = �Φ(t, PObj) of the particle PObj which +was at pt0 = �Φ(pt0, PObj) at t0; And ⃗V t0(t, pt0) is not tangent to graph(⃗V t0), cf. (3.16): It is tangent to +graph(⃗v) at (t, pt). +If t is fixed, or if pt0 ∈ Ωt0 is fixed, then we define +⃗V t0 +t (pt0) := ⃗V t0(t, pt0), +or +⃗V t0 +pt0 (t) := ⃗V t0(t, pt0). +(3.23) +Remark: A usual definition is given without explicit reference to a particle; It is, instead of (3.21), +⃗V t0(t, pt0) := ∂Φt0 +∂t (t, pt0), +∀(t, pt0) ∈ R × Ωt0. +(3.24) +3.4.2 +Lagrangian velocity versus Eulerian velocity +(3.21) and (2.4) give (alternative definition), with pτ = �Φ(τ, PObj), +⃗V t0(t, pt0) = ⃗v(t, pt) +(= ∂Φt0 +∂t (t, pt0) = �ΦPObj +′(t) = velocity at t at pt of PObj). +(3.25) +In other words, +⃗V t0 +t += ⃗vt ◦ Φt0 +t . +(3.26) +24 + +25 +3.5. +Lagrangian acceleration +3.4.3 +Relation between differentials +For C2 motions (3.26) gives +d⃗V t0 +t (pt0) = d⃗vt(pt).dΦt0 +t (pt0) +when +pt = Φt0 +t (pt0). +(3.27) +I.e., with +F t0 +t += dΦt0 +t +noted += +the deformation gradient relative to t0 and t, +(3.28) +d⃗V t0 +t (pt0) = d⃗vt(pt).F t0 +t (pt0) +when +pt = Φt0 +t (pt0). +(3.29) +Abusively written (dangerous notation: At what points, relative to what times?) +d⃗V = d⃗v.F. +(3.30) +3.4.4 +Computation of d⃗v called L = +• +F.F −1 wih Lagrangian variables +The Lagrangian approach can be introduced before the Eulerian approach: ⃗V t0 being given, define +⃗vt0(t, pt) := ⃗V t0(t, pt0), +when +pt = Φt0 +t (pt0), +(3.31) +cf. (3.20). (I.e. ⃗vt0(t, pt) := ⃗V t0(t, Φt0 +t +−1(pt))). So ⃗vt0(t, Φt0 +t (pt0)) = ∂Φt0 +∂t (t, pt0), thus +d⃗vt0(t, pt).dΦt0(t, pt0) = d(∂Φt0 +∂t )(t, pt0) = ∂(dΦt0) +∂t +(t, pt0) = ∂F t0 +∂t (t, pt0), +(3.32) +when Φt0 is C2 and F t0 := dΦt0. Thus +d⃗vt0(t, pt) = ∂F t0 +∂t (t, pt0).F t0(t, pt0)−1, +written in short +d⃗v = +• +F.F −1 +(points? times?). +(3.33) +And d⃗vt0 +t +can be written Lt0 +t in classical mechanics books, so you can find +Lt0 +t (pt) := +• +F t0 +pt0 (t).F t0 +t (pt0)−1, +written in short +L = +• +F.F −1 +(at what points, what times?). +(3.34) +Here it is not obvious that Lt0 +t (pt) does not depend on t0, which is indeed the case, cf. (3.29): +Lt0 +t (pt) = d⃗vt(pt). +(3.35) +Reminder: if possible, use Eulerian quantities as long as possible1. +3.5 +Lagrangian acceleration +Let PObj ∈ Obj, t0, t ∈ R, pt0 = �ΦPObj (t0) and pt = �ΦPObj (t) (positions of PObj at t0 and t). +Definition 3.13 In short, the Lagrangian acceleration at t at pt of the particle PObj is +⃗Γ t0(t, pt0) := �ΦPObj +′′(t) +when +pt0 = �ΦPObj (t0). +(3.36) +In other words +⃗Γ t0(t, pt0) := ⃗γ(t, pt) +when +pt = Φt0(t, pt0), +(3.37) +where ⃗γ(t, pt) = �ΦPObj +′′(t) is the Eulerian acceleration at t at pt = �Φ(t, PObj), cf. (2.37). +In details, the Lagrangian acceleration is the “two point vector field” defined on R × Ωt0 by +� +⃗Γ t0(t, pt0) = ((t, pt), �ΦPObj +′′(t)), +when +pt = Φt0(t, pt0). +(3.38) +(To compare with (2.38).) In particular ⃗Γ t0(t, pt0) is not drawn on the graph of ⃗Γ t0 at (t, pt0), but on +the graph of ⃗γ at (t, pt). +1To get Eulerian results from Lagrangian computations can make the understanding of a Lie derivative quite difficult: To +introduce the “so-called” Lie derivatives in classical mechanics you can find the following steps: 1- At t consider the Cauchy +stress vector ⃗t (Eulerian), 2- then with a unit normal vector ⃗n, define the associated Cauchy stress tensor σ (satisfying +⃗t = σ.⃗n), 3- then use the virtual power and the change of variables in integrals to be back into Ωt0 to be able to work +with Lagrangian variables, 4- then introduce the first Piola–Kirchhoff (two point) tensor PK, 5- then introduce the second +Piola–Kirchhoff tensor SK (endomorphism in Ωt0), 6- then differentiate SK in Ωt0 (in the Lagrangian variables although the +initials variables are the Eulerian variables in Ωt), 7- then back in Ωt to get back to Eulerian functions (change of variables +in integrals), 8- then you get some Jaumann or Truesdell or other so called Lie derivatives type terms, the appropriate choice +among all these derivatives being quite obscure because the covariant objectivity has been forgotten en route... While, with +simple Eulerian considerations, it requires a few lines to understand the (real) Lie derivative (Eulerian concept) and its +simplicity, see § 9, and deduce second order covariant objective results. +25 + +26 +3.6. +Time Taylor expansion of Φt0 +If t is fixed, or if pt0 ∈ Ωt0 is fixed, then define +⃗Γ t0 +t (pt0) := ⃗Γ t0(t, pt0), +and +⃗Γt0 +pt0 (t) := ⃗Γ t0(t, pt0). +(3.39) +Thus +⃗Γ t0 +t += ⃗γt ◦ Φt0 +t , +and +d⃗Γ t0 +t (pt0) = d⃗γt(pt).F t0 +t (pt0), +(3.40) +when pt = Φt0 +t (pt0) and F t0 +t +:= dΦt0 +t (the deformation gradient). +Risky notation: d⃗Γ = d⃗γ.F (points? times?). +3.6 +Time Taylor expansion of Φt0 +Let pt0 ∈ Ωt0. Then, at second order, +Φt0 +pt0 (τ) = Φt0 +pt0 (t) + (τ−t)Φt0 +pt0 +′(t) + (τ−t)2 +2 +Φt0 +pt0 +′′(t) + o((τ−t)2), +(3.41) +that is, with p(τ) = �ΦPObj (τ) = Φt0 +τ (pt0), +p(τ) = p(t) + (τ−t)⃗V t0(t, pt0) + (τ−t)2 +2 +⃗Γ t0(t, pt0) + o((τ−t)2). +(3.42) +NB: There are three times involved: t0 (observer dependent), t and τ (for the Taylor expansion). To +compare with (2.41)-(2.42): p(τ) = p(t)+(τ−t)⃗v(t, p(t))+ (τ−t)2 +2 +⃗γ(t, p(t))+o((τ−t)2), independent of t0. +3.7 +A vector field that let itself be deformed by a motion +Consider a C0 Eulerian vector field ⃗w : +� +C → ⃗Rn +(t, pt) → ⃗w(t, pt) +� +. +Let t0 ∈ [t1, t2[ and let ⃗wt0 +: +� +Ωt0 → ⃗Rn +pt0 → ⃗wt0(pt0) := ⃗w(t0, pt0) +� +(vector field in Ωt0). Then define the (virtual) vector field +⃗wt0∗ : +� +C → ⃗Rn +(t, pt) → ⃗wt0∗(t, pt) := dΦt0(t, pt0).⃗wt0(pt0), +when +p(t) = Φt0(t, pt0). +(3.43) +(The push-forward = result of the deformation of ⃗wt0 by the motion, see figure 4.1.) +Proposition 3.14 For C2 motions, we have (time variation rate along a virtual trajectory) +D ⃗wt0∗ +Dt += d⃗v.⃗wt0∗, +(3.44) +i.e. L⃗v ⃗wt0∗ = ⃗0, where L⃗v⃗u := D⃗u +Dt −d⃗v.⃗u (= ∂⃗u +∂t +d⃗u.⃗v −d⃗v.⃗u) is the Lie derivative of a (unsteady) vector +field ⃗u : C → ⃗Rn along ⃗v. +Interpretation: +We will see that L⃗v ⃗w(t0, pt0) = limt→t0 +⃗w(t,p(t))− ⃗wt0∗(t,p(t)) +h +measures the “re- +sistance of ⃗w to a motion”, see § 9.3.2; +Thus the result L⃗v ⃗wt0∗(t0, pt0) += ⃗0 is “obvious” (= +limt→t0 +⃗wt0∗(t,p(t))− ⃗wt0∗(t,p(t)) +h +): If ⃗w = ⃗wt0∗ then the vector (“force”) field ⃗w does not oppose any re- +sistance to the flow. +Proof. pt0 being fixed, with dΦt0(t, pt0) =noted F(t) we have ⃗wt0∗(t, p(t)) =(3.43) F(t).⃗wt0(pt0), thus +D ⃗wt0∗ +Dt +(t, p(t)) = F ′(t).⃗wt0(pt0) = F ′(t).F(t)−1.⃗wt0∗(t, p(t)) =(3.33) d⃗v(t, p(t)).⃗wt0∗(t, p(t)), i.e. (3.44). +4 +Deformation gradient F := dΦ +Consider a motion �Φ : +� +R × Obj → Rn +(t, PObj) → pt = �Φ(t, PObj) +� +, Ωt := �Φ(t, Obj) the configuration of Obj at any t, +fix t0, t in R, and let Φt0 +t +: +� +Ωt0 → Ωt +pt0 = �Φ(t0, PObj) → pt = Φt0 +t (pt0) := �Φ(t, PObj) +� +, supposed to be a C1 +diffeomorphism. Notations for calculations (quantification), to comply with practices: +26 + +27 +4.1. +Definitions +1- Classical (unambiguous) notations as in Arnold, Germain: E.g., (⃗ai) and (⃗bi) are bases resp. in ⃗Rn +t0 +and ⃗Rn +t , ⃗wt0(pt0) = � +i wt0,i(pt0)⃗ai ∈ ⃗Rn +t0, ⃗wt,i(pt) = � +i wt,i(pt)⃗bi ∈ ⃗Rn +t ; And +2- Marsden–Hughes duality notations: Capital letters at t0, lower case letters at t, duality notation, +e.g. ( ⃗EI) and (⃗ei) are bases resp. in ⃗Rn +t0 and ⃗Rn +t , ⃗W(P) = � +I W I(P) ⃗EI ∈ ⃗Rn +t0, ⃗w(p) = � +i wi(p)⃗ei ∈ ⃗Rn +t . +4.1 +Definitions +4.1.1 +Definition of the deformation gradient F +Definition 4.1 The differential dΦt0 +t =noted F t0 +t +: +� +Ωt0 → L(⃗Rn +t0; ⃗Rn +t ) +pt0 → F t0 +t (pt0) := dΦt0 +t (pt0) +� +is called “the covari- +ant deformation gradient between t0 and t”, or simply “the deformation gradient”. And “the covariant +deformation gradient at pt0 between t0 and t”, or in short “the deformation gradient at pt0” is the linear +map F t0 +t (pt0) ∈ L(⃗Rn +t0; ⃗Rn +t ), so defined by, for all ⃗wt0(pt0) ∈ ⃗Rn +t0 (vector at pt0), +F t0 +t (pt0).⃗wt0(pt0) := lim +h→0 +Φt0 +t (pt0+h⃗wt0(pt0)) − Φt0 +t (pt0) +h +noted += +(Φt0 +t )∗(⃗wt0)(pt) noted += +⃗wt0∗(t, pt), +(4.1) +with pt = Φt0 +t (pt0). See figure 4.1. +Marsden–Hughes notations: Φ := Φt0 +t , F := dΦ, P := pt0, ⃗W(P) := ⃗wt0(pt0), p = Φ(P), thus +F(P). ⃗W(P) := lim +h→0 +Φ(P+h ⃗W(P)) − Φ(P) +h +noted += +Φ∗ ⃗W(p) noted += +⃗w∗(p). +(4.2) +Figure 4.1: ⃗w is a Eulerian vector field. At t0 define vector field ⃗wt0 in Ωt0 by ⃗wt0(pt0) := ⃗w(t0, pt0). The +(spatial) curve ct0 : s → pt0 = ct0(s) in Ωt0 is an integral curve of ⃗wt0, i.e. satisfies ct0 +′(s) = ⃗wt0(ct0(s)). +It is transformed by Φt0 +t into the (spatial) curve ct = Φt0 +t ◦ ct0 : s → pt = ct(s)=Φt0 +t (ct0(s)) in Ωt; Hence +ct′(s) = dΦt0 +t (pt0).ct0 +′(s) = dΦt0 +t (pt0).⃗wt0(pt0) =noted ⃗wt0∗(t, pt) is the tangent vector at ct at pt (the +push-forward of ⃗wt0 by Φt0 +t ). And ⃗w(t, p(t)) (actual value) is also drawn. +NB: The “deformation gradient” F t0 +t += dΦt0 +t +is not a “gradient” (its definition does not need a +Euclidean dot product); This lead to confusions when covariance-contravariance and objectivity are at +stake. It would be simpler to stick to the name “F t0 +t += the differential of Φt0 +t ”, but it is not the standard +usage, except in thermodynamics: E.g., the differential dU of the internal energy U is not called “the +gradient of U” (there is no meaningful inner dot product): It is just called “the differential of U”... +4.1.2 +Push-forward (values of F) +Definition 4.2 Let ⃗wt0 : +� +Ωt0 → ⃗Rn +t0 +pt0 → ⃗wt0(pt0) +� +be a vector field in Ωt0. Its push-forward by Φt0 +t +is the +vector field (Φt0 +t )∗(⃗wt0) in Ωt defined by +(Φt0 +t )∗ ⃗wt0(pt) = F t0 +t (pt0).⃗wt0(pt0) noted += +⃗wt0∗(t, pt) +when +pt = Φt0 +t (pt0). +(4.3) +See figure 4.1. Marsden notation: Φ∗ ⃗W(p) = F(P). ⃗W(P) =noted ⃗w∗(p) when p = Φt0 +t (P). +27 + +042 +w(p +Cto +Ct28 +4.1. +Definitions +In other words +(Φt0 +t )∗ ⃗wt0 := (F t0 +t .⃗wt0) ◦ (Φt0 +t )−1. +(4.4) +Marsden notation: Φ∗ ⃗W = (F. ⃗W) ◦ Φ−1 = ⃗w∗. +4.1.3 +F is a two point tensors +With (4.1), “the tangent map” is +� +F t0 +t +: +� Ωt0 → Ωt × L(⃗Rn +t0; ⃗Rn +t ) +pt0 → � +F t0 +t (pt0) = (pt, F t0 +t (pt0)) +when +pt = Φt0 +t (pt0). +(4.5) +Definition 4.3 (Marsden–Hughes [12].) The function � +F t0 +t +is called a two point tensor, referring to the +points pt0 ∈ Ωt0 (departure set) and pt = Φt0 +t (pt0) ∈ Ωt (arrival set where ⃗wt0∗(t, pt) = F t0 +t (pt0).⃗wt0(pt0) +is drawn). And in short � +F t0 +t +=noted F t0 +t +is said to be a two point tensor. +Remark 4.4 The name “two point tensor” is a shortcut than can create confusions and errors when +dealing with the transposed: F t0 +t +is not immediately a “tensor”: A tensor is a multilinear form, so gives +scalar results (∈ R), while F(P) := F t0 +t (P) =noted FP ∈ L(⃗Rn +t0; ⃗Rn +t ) gives vector results (in ⃗Rn +t ). However +FP can be naturally and canonically associated with the bilinear form �FP ∈ L(⃗Rn∗ +t , ⃗Rn +t0; R) defined by, +for all ⃗uP ∈ ⃗Rn +t0 and ℓp ∈ ⃗Rn∗ +t , with p = Φt0 +t (P), +�FP (ℓp, ⃗uP ) := ℓp.FP .⃗uP (∈ R), +(4.6) +see § A.13, and it is �FP which defines the so-called “two point tensor”. +But don’t forget that the transposed of a linear form (FP here) is not deduced from the transposed +of the associated bilinear form ( �FP here). So be careful with the word “transposed” and its two distinct +definitions. Indeed, the transposed of a bilinear form b(·, ·) is intrinsic to b(·, ·) (is objective), given by +bT (⃗u, ⃗w) = b(⃗w, ⃗u), while the transposed of a linear function L is not intrinsic to L (is subjective), given +by (LT .⃗u, ⃗w)g = (L.⃗w, ⃗u)h where (·, ·)g and (·, ·)h are inner dot products chosen by human beings. (details +in § A.7.2 and § A.11.1). +Remark 4.5 More generally for manifolds, the differential of Φ := Φt0 +t +at P ∈ Ωt0 is F(P) := dΦ(P) : +� +TP Ωt0 → TpΩt +⃗W(P) → ⃗w∗(p) := dΦ(P). ⃗W(P) +� +with p = Φt0 +t (P). And the tangent map is +TΦ : +� +TΩt0 → TΩt +(P, ⃗W(P)) → TΦ(P, ⃗W(P)) := (p, dΦ(P). ⃗W(P)) = (p, ⃗w∗(p)), +where +p = Φt0 +t (P), +(4.7) +called the associated two point tensor. +4.1.4 +Evolution: Toward the Lie derivative (in continuum mechanics) +Consider a Eulerian vector field ⃗w : +� +� +� +C = +� +t +({t} × Ωt) → ⃗Rn +(t, p) → ⃗w(t, p) +� +� +�, e.g. a “force field”. Then, at t0 +consider ⃗wt0 : +� +Ωt0 → ⃗Rn +t0 +pt0 → ⃗wt0(pt0) := ⃗w(t0, pt0) +� +. The push-forward of ⃗wt0 by Φt0 +t is, cf. (4.2), +⃗wt0∗(t, p(t)) = F t0 +t (pt0).⃗wt0(pt0), +where +p(t) = Φt0(t, pt0). +(4.8) +See figure 4.1. Then, without any ubiquity gift, at t at p(t) we can compare ⃗w(t, p(t)) (real value of ⃗w +at t at p(t)) with ⃗wt0∗(t, p(t)) (transported memory along the trajectory). Thus the rate +⃗w(t, p(t)) − ⃗wt0∗(t, p(t)) +t − t0 += +actual(t, p(t)) − memory(t, p(t)) +t − t0 +is meaningful at (t, p(t)) +(4.9) +(no ubiquity gift required). This rate gives, as h → 0, the Lie derivative L⃗v ⃗w (the rate of stress), and we +will see at § 9.3 that L⃗v ⃗w = D ⃗w +Dt − d⃗v.⃗w (the d⃗v term tells that a “non-uniform flow” acts on the stress). +28 + +29 +4.2. +Quantification with bases +4.1.5 +Pull-back +Formally the pull-back is the push-forward with (Φt0 +t )−1: +Definition 4.6 The pull-back (Φt0 +t )∗ ⃗wt of a vector field ⃗wt defined on Ωt is the vector field defined on Ωt0 +by, with pt0 = (Φt0 +t )−1(pt), +⃗w∗ +t (t0, pt0) = (Φt0 +t )∗ ⃗wt(pt0) := (F t0 +t )−1(pt).⃗wt(pt), +written +⃗W ∗(P) = F −1(p).⃗w(p). +(4.10) +4.2 +Quantification with bases +(Simple Cartesian framework.) (⃗ai) is a Cartesian basis in ⃗ +Rn +t0, (⃗bi) is a Cartesian basis in ⃗ +Rn +t , ot is an +origin in Rn at t, Φt0 +t =noted Φ supposed C1, ϕi : Ωt0 → R is its components in the referential (ot, (⃗bi)): +Φ(pt0) = ot + +n +� +i=1 +ϕi(pt0)⃗bi, +i.e. +−−−−−→ +otΦ(pt0) = +n +� +i=1 +ϕi(pt0)⃗bi. +(4.11) +Thus, with the classic notation dϕi(pt0).⃗aj =noted ∂ϕi +∂Xj (pt0) since (⃗ai) is a Cartesian basis, and (⃗bi) being +a Cartesian basis, +dΦ(pt0).⃗aj = +n +� +i=1 +(dϕi(pt0).⃗aj)⃗bi = +n +� +i=1 +∂ϕi +∂Xj +(pt0)⃗bi, +thus +[dΦ(pt0)][⃗a,⃗b] = [ ∂ϕi +∂Xj +(pt0)] = [F(pt0)][⃗a,⃗b], +[dΦ(pt0)][⃗a,⃗b] = [F(pt0)][⃗a,⃗b] being the Jacobian matrix of Φ at pt0 relative to the chosen bases. In short: +dΦ.⃗aj = +n +� +i=1 +∂ϕi +∂Xj +⃗bi, +thus +[dΦ][⃗a,⃗b] = [ ∂ϕi +∂Xj +] = [F][⃗a,⃗b] = [Fij], +(4.12) +Thus, if ⃗W ∈ ⃗Rn +t0 is a vector at pt0 and ⃗W = �n +j=1Wj⃗aj then, by linearity of differentials, +dΦ. ⃗W = F. ⃗W = +n +� +i=1 +FijWj⃗bi, +i.e. +[F. ⃗W]|⃗b = [F]|⃗a,⃗b.[ ⃗W]|⃗a +(4.13) +(more precisely: F t0 +t (pt0). ⃗W(pt0) = �n +i=1Fij(pt0)Wj(pt0)⃗bi). +Similarly, for the second order derivative d2Φ = dF (when Φ is C2): With ⃗U = �n +j=1Uj⃗aj and +⃗W = �n +k=1Wk⃗ak, and with (⃗ai) and (⃗bi) Cartesian bases, we get +dF(⃗U, ⃗W) = d2Φ(⃗U, ⃗W) = +n +� +i=1 +d2ϕi(⃗U, ⃗W)⃗bi = +n +� +i,j,k=1 +∂2ϕi +∂Xj∂Xk +UjWk⃗bi = +n +� +i=1 +� +[⃗U]T +|⃗a.[d2ϕi]|⃗a.[ ⃗W]|⃗a +� +⃗bi, +(4.14) +[d2ϕi(pt0)]|⃗a = [ +∂2ϕi +∂Xj∂Xk (pt0)] j=1,...,n +k=1,...,n being the Hessian matrix of ϕi at pt0 relative to the basis (⃗ai). +With Marsden duality notations: +• p = Φ(P) = ot + +n +� +i=1 +ϕi(P)⃗ei, +F i +J(P) = ∂ϕi +∂XJ (P) +(= dϕi(P). ⃗EJ), +• F(P). ⃗W = +n +� +i,J=1 +F i +J(P) W J⃗ei, +[F] = [F i +J] = [dΦ], +• dF(⃗U, ⃗W) = d2Φ(⃗U, ⃗W) = +n +� +i,J,K=1 +∂2ϕi +∂XJ∂XK U JW K⃗ei = +n +� +i=1 +� +[⃗U]T .[d2ϕi].[ ⃗W] +� +⃗ei. +. +(4.15) +Remark 4.7 J, j are dummy variables when used in a summation: E.g., df. ⃗W = �n +j=1 +∂f +∂Xj W j = +�n +J=1 +∂f +∂XJ W J = �n +α=1 +∂f +∂Xα W α = +∂f +∂X1 W 1 + +∂f +∂X2 W 2 + ... (there is no uppercase for 1, 2...). +And +Marsden–Hughes notations (capital letters for the past) are not at all compulsory, classical notations +being just as good and even preferable if you hesitate (because they are not misleading). See § A. +29 + +30 +4.3. +The unfortunate notation d⃗x = F.d ⃗X +4.3 +The unfortunate notation d⃗x = F.d ⃗X +4.3.1 +Issue +(4.3), i.e. ⃗w∗(p) := F(P). ⃗W(P), is sometimes written +d⃗x = F.d ⃗X : “a very unfortunate and misleading notation” +(4.16) +which amounts to “confuse a length and a speed”... +And you also the phrase “(4.16) is still true if +||d ⃗X|| = 1”... while d ⃗X is supposed to be small... +4.3.2 +Where does this unfortunate notation come from? +The notation (4.16) comes from the first order Taylor expansion Φ(Q) = Φ(P) + dΦt0 +t (P).(Q−P) + +o(||Q−P||), where P, Q ∈ Ωt0, i.e., with p = Φt0 +t (P) and q = Φt0 +t (Q) and h = ||Q−P||, +q − p = F(P).(Q−P) + o(h), +written +δ⃗x = F.δ ⃗X + o(δ ⃗X), +(4.17) +or −→ +pq = F(P).−−→ +PQ + o(h). So as Q → P we get 0 = 0... Quite useless, isn’t it? +While +q − p +h += F(P).Q − P +h ++ o(1) +is useful: +(4.18) +As Q → P we get ⃗w∗ = F(P). ⃗W which relates tangent vectors, see figure 4.1 Details: +4.3.3 +Interpretation: Vector approach +Consider a spatial curve ct0 : +� +[s1, s2] → Ωt0 +s → P := ct0(s) +� +in Ωt0, cf. figure 4.1. It is deformed by Φt0 +t +to +become the spatial curve defined by ct := Φt0 +t ◦ ct0 : +� +[s1, s2] → Ωt +s → p := ct(s) = Φt0 +t (ct0(s)). +in Ωt. Hence, +relation between tangent vectors: +dct +ds (s) = dΦt0 +t (ct0(s)).dct0 +ds (s), , +i.e. +⃗w∗(p) = F(P). ⃗W(P) +written +d⃗x +ds = F.d ⃗X +ds , +(4.19) +But you can’t simplify by ds to get d⃗x = F.d ⃗X: It is absurd to confuse “a slope d ⃗ +X +ds (s)” and “a length δ ⃗X”. +NB: || dct +ds (s)|| = || d ⃗ +X +ds (s)|| = 1 is meaningful in (4.19): It means that the parametrization of the +curve ct0 in Ωt0 uses a spatial parameter s such that ||ct0 +′(s)|| = 1 for all s, i.e. s.t. || ⃗WP || = 1 in +figure 4.1. You cannot simplify by ds: ||d ⃗X|| = 1 is absurd together with d ⃗X “small”. +4.3.4 +Interpretation: Differential approach +(4.16) is a relation between differentials... if you adopt the correct notations; Let us do it: With (4.11), +⃗x = −→ +otp = −−−−−−→ +otΦt0 +t (P) = +n +� +i=1 +ϕi(P)⃗bi +noted += +n +� +i=1 +xi(P)⃗bi, +where +ϕi +noted += +xi +(function of P). +(4.20) +Thus, with (πai) = (dXi) the (covariant) dual basis of (⃗ai) we get the system of n equations (functions): +dΦ = F, +i.e. +� +� +� +� +� +� +� +� +� +dϕ1(P) = �n +j=1 +∂ϕ1 +∂Xj (P) dXj +... +dϕn(P) = �n +j=1 +∂ϕn +∂Xj (P) dXj +� +� +� +� +� +� +� +� +� +, +which is noted +d⃗x = F.d ⃗X, +(4.21) +this last notation being often misunderstood2: It is nothing more than dΦ = F (coordinate free notation). +2Spivak [17] chapter 4: Classical differential geometers (and classical analysts) did not hesitate to talk about “infinitely +small” changes dxi of the coordinates xi, just as Leibnitz had. No one wanted to admit that this was nonsense, because +true results were obtained when these infinitely small quantities were divided into each other (provided one did it in the +right way). Eventually it was realized that the closest one can come to describing an infinitely small change is to describe +a direction in which this change is supposed to occur, i.e., a tangent vector. Since df is supposed to be the infinitesimal +change of f under an infinitesimal change of the point, df must be a function of this change, which means that df should +be a function on tangent vectors. The dXi themselves then metamorphosed into functions, and it became clear that they +must be distinguished from the tangent vectors ∂/∂Xi. Once this realization came, it was only a matter of making new +definitions, which preserved the old notation, and waiting for everybody to catch up. +30 + +31 +4.4. +Tensorial notations, warnings, remarks +4.3.5 +The ambiguous notation +• +d⃗x = +• +F.d ⃗X +The bad notation d⃗x = F.d ⃗X gives the unfortunate and misunderstood notations +• +d⃗x = +• +F.d ⃗X, and then +• +d⃗x = L.d⃗x +where +L = +• +F.F −1. +(4.22) +Question: What is the meaning (and legitimate notation) of (4.22)? +Answer: +• +d⃗x = L.d⃗x means +D ⃗wt0∗ +Dt += d⃗v.⃗wt0∗ += evolution rate of tangent vectors along a trajectory +(4.23) +see figure 4.1. Indeed, ⃗wt0∗(t, p(t)) =(4.8) F t0(t, pt0).⃗wt0(pt0) gives +D ⃗wt0∗ +Dt +(t, p(t)) = ∂F t0 +∂t (t, pt0).⃗wt0(pt0) = ∂F t0 +∂t (t, pt0).(F t0 +t (pt0)−1.⃗wt0∗(t, p(t))), +(4.24) +i.e. D ⃗wt0∗ +Dt +(t, p(t)) = d⃗v(t, p(t)).⃗wt0∗(t, p(t)), i.e. (4.23). In particular D ⃗wt0∗ +Dt +(t0, pt0) = d⃗v(t0, pt0).⃗wt0(pt0) +is the evolution rate of tangent vectors at t0 at pt0. +4.4 +Tensorial notations, warnings, remarks +As already noted, cf. (4.6), the linear map F := dΦt0 +t (pt0) ∈ L(⃗Rn +t0; ⃗Rn +t ) is naturally canonically associated +with the bipoint tensor �F ∈ L(⃗Rn∗ +t , ⃗Rn +t0; R) defined by, for all (ℓ, ⃗W) ∈ ⃗Rn∗ +t +× ⃗Rn +t0, +�F(ℓ, ⃗W) := ℓ.F. ⃗W, +(4.25) +Quantification of �F: basis (⃗ai) with dual basis (πai) in ⃗Rn +t0, basis (⃗bi) in ⃗Rn +t : +if +F.⃗aj = +n +� +i=1 +∂ϕi +∂Xj +⃗bj +then +�F = +n +� +i,j=1 +∂ϕi +∂Xj +⃗bi ⊗ πaj = +n +� +i=1 +⃗bi ⊗ dϕi. +(4.26) +And similarly +d �F = +n +� +i,j,k=1 +∂2ϕi +∂Xj∂Xk +⃗bi ⊗ (πaj ⊗ πak) = +n +� +i=1 +⃗bi ⊗ d2ϕi. +(4.27) +Warning: The tensorial notation can be misleading, in particular if you use the transposed, see re- +mark 4.4. So, you should always use the standard notation for the linear form F ∈ L(⃗Rn +t0; ⃗Rn +t ) to begin +with, i.e. use F.⃗aj = �n +j=1Fij⃗bi or F. ⃗EJ = �n +i,j=1F i +J⃗ei (Marsden notations). And only use the tensorial +notations for calculations purposes at the end (after application of the proper definitions). +Remark 4.8 In some manuscripts you find the notation F = dΦ =noted Φ ⊗ ∇X. It does not help to +understand what F is (it is the differential dΦ), and should not be used as far as objectivity is concerned: +• A differentiation is not a tensorial operation, see example R.1, so why use the tensor product +notation Φ ⊗ ∇X, when the standard notation dΦ ≃ �F = �n +i=1⃗ei ⊗ dϕi is legitimate, explicit, objective +and easy to manipulate? +• And it could be misinterpreted, since, in mechanics, ∇f is often understood to be the vector � +i +∂f +∂xi⃗ei +(contravariant) which needs a Euclidean dot product to be defined (which one?), while the differential df +is covariant (a differential is unmissable in thermodynamics because you can’t use gradients). +• It gives the confusing notation Φ ⊗ ∇X ⊗ ∇X, instead of the legitimate d2Φ = �n +i=1⃗bi ⊗ d2ϕi which +is explicit, objective and easy to manipulate: d2Φ(⃗U, ⃗W) = �n +i=1d2ϕi(⃗U, ⃗W)⃗bi. +Exercice 4.9 Use Marsden duality notations for (4.26)-(4.27). +Answer. Cartesian bases, with (dXi) the (covariant) dual basis of ( ⃗Ei): with F i +J = +∂ϕi +∂XJ , we get dΦ =noted �F = +�n +i=1⃗ei ⊗ dϕi = �n +i,J=1F i +J ⃗ei ⊗ dXJ, and d2Φ = �n +i=1⃗ei ⊗ d2ϕi = �n +i,J,K=1 +∂2ϕi +∂XJ ∂XK ⃗ei ⊗ (dXJ ⊗ dXK). +31 + +32 +4.5. +Change of coordinate system at t for F +4.5 +Change of coordinate system at t for F +Let pt0 ∈ Ωt0, pt = Φt0 +t (pt0) ∈ Ωt, ⃗W(pt0) ∈ ⃗Rn +t0, ⃗w(pt) = F t0 +t (pt0). ⃗W(pt0) ∈ ⃗Rn +t (its push-forward), +written ⃗w = F. ⃗W for short. The observer at t0 used a basis (⃗ai) in ⃗Rn +t0. At t, in ⃗Rn +t , a first observer +chooses a Cartesian basis (⃗bold,i), and a second observer chooses a Cartesian basis (⃗bnew,i). Let P = [Pij] +be the transition matrix from (⃗bold,i) to (⃗bnew,i), i.e. ⃗bnew,j = �n +i=1Pij⃗bold,i for all j. The change of basis +formula for vectors from (⃗bold,i) to (⃗bnew,i) (in ⃗Rn +t ) gives +[⃗w]|⃗bnew = P −1.[⃗w]|⃗bold, +thus +[F. ⃗W]|⃗bnew = P −1.[F. ⃗W]|⃗bold. +(4.28) +Thus +[F]|⃗a,⃗bnew.[ ⃗W]|⃗a = P −1.[F]|⃗a,⃗bold.[ ⃗W]|⃗a, +(4.29) +true for all ⃗W, thus +[F]|⃗a,⃗bnew = P −1.[F]|⃗a,⃗bold . +(4.30) +NB: (4.30) is not the change of basis formula [L]|new = P −1.[L]|old.P for endomorphisms, which would +be nonsense since F := F t0 +t (pt0) : ⃗Rn +t0 → ⃗Rn +t is not an endomorphism; (4.30) is just the usual change of +basis formula for vectors ⃗w in ⃗Rn +t , cf. (4.28). +4.6 +Spatial Taylor expansion of Φ and F +Φt0 +t =noted Φ is supposed to be C2 for all t0, t. Let P ∈ Ωt0, dΦ = F, and ⃗W ∈ ⃗Rn +t0 vector at P. Then, +in Ωt, +Φ(P+h ⃗W) = Φ(P) + h F(P). ⃗W + h2 +2 dF(P)( ⃗W, ⃗W) + o(h). +(4.31) +4.7 +Time Taylor expansion of F +The motion �Φ is supposed to be C3. +t0 ∈ R, Φt0 be the associated motion, p(t) = �Φ(t, PObj) and +pt0 = �Φ(t0, PObj), with ⃗v(t, pt) = ∂�Φ +∂t (t, PObj) the Eulerian velocity and ⃗V t0(t, pt0) := ∂Φt0 +∂t (t, pt0) = ⃗v(t, pt) +the Lagrangian velocity, and F t0 +pt0 (t) = F t0(t, pt0) = dΦt0(t, pt0) =noted F(t). We have +∂F t0 +∂t (t, pt0) = ∂(dΦt0) +∂t +(t, pt0) = d(∂Φt0 +∂t )(t, pt0) += d⃗V t0(t, pt0) = d⃗v(t, p(t)).F(t), +in short +• +F = d⃗V = d⃗v.F . +(4.32) +Then ∂2Φt0 +∂t2 (t, pt0) = ⃗At0(t, pt0) = ⃗γ(t, p(t)) (Lagrangian and Eulerian accelerations), hence +∂2F t0 +∂t2 (t, pt0) = d ⃗At0(t, pt0) = d⃗γ(t, pt).F(t), +in short +•• +F = d ⃗A = d⃗γ.F. +(4.33) +Thus, the second order time Taylor expansion of F t0 +pt0 =noted F is, in the vicinity of t, +F(t+h) = F(t) + h d⃗V (t) + h2 +2 d ⃗A(t) + o(h2) += +� +I + h d⃗v + h2 +2 d⃗γ +� +(t, p(t)).F(t) + o(h2) +when +p(t) = Φt0(t, pt0). +(4.34) +NB: They are three times are involved: t and t+h as usual, and t0 through F := F t0 +pt0 , ⃗V := ⃗V t0 +pt0 and +⃗A := ⃗At0 +pt0 (observer dependent), as for (3.41). +In particular F(pt0) := F t0 +t0 (pt0) = I gives, in the vicinity of t0, +F(t0+h) = +� +I + h d⃗v + h2 +2 d⃗γ +� +(t0, pt0) + o(h2). +(4.35) +Remark 4.10 γ = ∂⃗v +∂t + d⃗v.⃗v is not linear in ⃗v. Idem, +d⃗γ = d(D⃗v +Dt ) = d(∂⃗v +∂t + d⃗v.⃗v) = d∂⃗v +∂t + d2⃗v.⃗v + d⃗v.d⃗v +(= D(d⃗v) +Dt ++ d⃗v.d⃗v) +(4.36) +is non linear in ⃗v, and gives F t0 +pt0 +′′(t) = (d ∂⃗v +∂t + d2⃗v.⃗v + d⃗v.d⃗v)(t, pt).F t0 +pt0 (t), non linear in ⃗v. +32 + +33 +4.8. +Homogeneous and isotropic material +Exercice 4.11 Directly check that F ′ = d⃗v.F gives F ′′ = d⃗γ.F. +Answer. F ′(t) = d⃗v(t, p(t)).F(t) gives F ′′(t) = +D(d⃗v) +Dt (t, p(t)).F(t) + d⃗v(t, p(t)).F ′(t) with +D(d⃗v) +Dt += d⃗γ − d⃗v.⃗v, +cf. (4.36), thus F ′′(t) = (d⃗γ − d⃗v.d⃗v)(t, p(t)).F(t) + d⃗v(t, p(t)).d⃗v(t, p(t)).F(t) = d⃗γ(t, p(t)).F(t). +4.8 +Homogeneous and isotropic material +Let P ∈ Ωt0, let F t0 +t (P) := dΦt0 +t (P); Suppose that the “Cauchy stress vector” ⃗ft(pt) à t at pt = Φt0 +t (P) +only depends on P, i.e. there exists a function ⃗ +fun such that +⃗ft(pt) = ⃗ +fun(P, F t0 +t (P)). +(4.37) +Definition 4.12 A material is homogeneous iff ⃗ +fun doesn’t depend on the first variable P, i.e., iff, for +all P ∈ Ωt0, +⃗ +fun(P, F t0 +t (P)) = ⃗ +fun(F t0 +t (P)). +(4.38) +(Same mechanical property at any point.) +Definition 4.13 (Isotropy.) Consider a Euclidean dot product, the same at all time. A material is +isotropic at P ∈ Ωt0 iff ⃗ +fun is independent of the direction you consider, i.e., iff, for any rotation Rt0(P) +in ⃗Rn +t0, +⃗ +fun(P, F t0 +t (P) = ⃗ +fun(P, F t0 +t (P).Rt0(P)). +(4.39) +(Mechanical property unchanged when rotating the material first.) +Definition 4.14 A material is isotropic homogeneous iff it is isotropic and homogeneous. +4.9 +The inverse of the deformation gradient +((Φt0 +t )−1 ◦ Φt0 +t )(P) = P gives, with p = Φt0 +t (P), +d(Φt0 +t )−1(p).dΦt0 +t (P) = It0, +thus +d(Φt0 +t )−1(p) = dΦt0 +t (P)−1 = F t0 +t (P)−1, +(4.40) +where F t0 +t += dΦt0 +t is the deformation gradient. We have thus define the two point tensor +Ht0 +t := (F t0 +t )−1 : +� +� +� +Ωt → L(⃗Rn +t ; ⃗Rn +t0) +p → Ht0 +t (p) = (F t0 +t )−1(p) := (F t0 +t (P))−1 +when +p = Φt0 +t (P). +(4.41) +So +Ht0 +t (p).⃗w(p) = (F t0 +t )−1(p).⃗w(p) := F t0 +t (P)−1.⃗w(p) ∈ ⃗Rn +t0, +in short +H.⃗w = F −1.⃗w, +(4.42) +for all ⃗w(p) ∈ ⃗Rn +t vector at p. This defines, with pt = Φt0(t, P), +Ht0 : +� +� +� +C = +� +t +({t} × Ωt) → L(⃗Rn +t ; ⃗Rn +t0) +(t, pt) → Ht0(t, pt) := Ht0 +t (pt) = (F t0(t, P))−1. +(4.43) +NB: Ht0 looks like a Eulerian map, but isn’t: Ht0 depends on a initial time t0 and is a two point tensor +(starts in ⃗Rn +t , arrives in ⃗Rn +t0). We will however use the material time derivative +D +Dt notation in this case, +that is, we define, along a trajectory t → p(t) = Φt0(t, P), +DHt0 +Dt (t, p(t)) := ∂Ht0 +∂t (t, p(t)) + dHt0(t, p(t)).⃗v(t, p(t)), +i.e. +DHt0 +Dt += ∂Ht0 +∂t ++ dHt0.⃗v, +(4.44) +which is the time derivative g′(t) of the function g : t → g(t) = Ht0(t, Φt0(t, P)) (i.e. g(t) = Ht0(t, p(t))). +Hence, with p(t) = Φt0(t, P) and Ht0(t, p(t)).F t0(t, P) = It0, written H.F = I, we get +DH +Dt .F + H.∂F +∂t = 0, +thus +DH +Dt = −H.d⃗v, +(4.45) +since ∂F +∂t (t, P).F −1(t, p(t)) = d⃗v(t, p(t)) cf. (4.32). +33 + +34 +5.1. +Introduction: Motion versus flow +Exercice 4.15 With ⃗wt0∗(t, p(t)) = F t0(t, P). ⃗W(P), i.e. Ht0(t, p(t)).⃗wt0∗(t, p(t)) = ⃗W(P), when p(t) = +Φt0(t, P), prove (4.45). +Answer. +D ⃗wt0∗ +Dt +(t, p(t)) = d⃗v(t, p(t)).⃗wt0∗(t, p(t)), cf. (4.23); And (Ht0.⃗wt0∗)(t, p(t)) = ⃗W(P) gives DHt0 +Dt .⃗wt0∗ + +Ht0. +D ⃗wt0∗ +Dt += 0; Thus DHt0 +Dt .⃗wt0∗ + Ht0.d⃗v.⃗wt0∗ = 0, thus DH +Dt = −H.d⃗v. +Exercice 4.16 Prove: Ht0 +t += Ht0 +t1 ◦ Ht1 +t +and DHt0 +Dt (t, p(t)) = Ht0 +t1 (pt1). DHt1 +Dt (t, p(t)) for all t0, t1 with +pt1 = Φt0 +t1(pt0). +Answer. We have Φt0 +t (pt0) = Φt1 +t (Φt0 +t1(pt0)), cf. (5.18), hence F t0 +t (pt0) = F t1 +t (pt1).F t0 +t1 (pt0), thus F t0 +t (pt0)−1 = +F t0 +t1 (pt0)−1.F t1 +t (pt1)−1, i.e. Ht0 +t (pt) = Ht0 +t1 (pt1).Ht1 +t (p(t)), thus, Ht0(t, p(t)) = Ht0 +t1 (pt1).Ht1(t, p(t)), thus +DHt0 +Dt (t, p(t)) = Ht0 +t1 (pt1). DHt1 +Dt (t, p(t)). +5 +Flow +5.1 +Introduction: Motion versus flow +• Motion: A motion �Φ : (t, PObj) → pt = �Φ(t, PObj) locates at t a particle PObj in the affine space Rn, +cf. (1.5); From which the Eulerian velocity field ⃗v is deduced: ⃗v(t, pt) := +d�ΦPObj +dt +(t, PObj), cf. (2.4). +• Flow: A flow starts with a Eulerian velocity field ⃗v, from which we deduce a motion by solving the +ODE (ordinary differential equation) dΦ +dt (t) = ⃗v(t, Φ(t)). +5.2 +Definition +Let ⃗v : +� +R × Rn → ⃗Rn +(t, p) → ⃗v(t, p) +� +be a unstationary vector field (e.g., a Eulerian velocity field which definition +domain is C = � +t∈[t1,t2]({t} × Ωt)). We look for maps Φ : +� +R → Rn +t → p = Φ(t) +� +which are locally (i.e. in +the vicinity of some t0) solutions of the ODE (ordinary differential equation) +dΦ +dt (t) = ⃗v(t, Φ(t)), +also written +dp +dt (t) = ⃗v(t, p(t)), +or +d⃗x +dt (t) = ⃗v(t, ⃗x(t)) +(5.1) +where ⃗x(t) = −−−→ +Op(t) after a choice of an origin. Also written dp +dt = ⃗v(t, p) or d⃗x +dt = ⃗v(t, ⃗x). +Definition 5.1 A solution Φ of (5.1) is a flow of ⃗v; Also called an integral curve of ⃗v since (5.1) also +reads Φ(t) = +� t +τ=t1 ⃗v(τ, Φ(τ)) dτ + Φ(t1). +Remark 5.2 Improper notation for (5.1): +dp +dt (t) noted += +dp(t) +dt +(= ⃗v(t, p(t))). +(5.2) +Question: If the notation dp(t) +dt +is used, then what is the meaning of dp(f(t)) +dt +? +Answer: It means, either dp +dt (f(t)), or d(p◦f) +dt +(t) = dp +dt (f(t)) df +dt(t): Ambiguous. So it is better to use +dp +dt (t), and to avoid dp(t) +dt , unless the context is clear (no composite functions). +5.3 +Cauchy–Lipschitz theorem +Let (t0, pt0) be in the definition domain of ⃗v. We look for Φ solution of “the ODE with initial condition +(t0, pt0)”, in some vicinity of t0, i.e. such that +dΦ +dt (t) = ⃗v(t, Φ(t)) +and +Φ(t0) = pt0. +(5.3) +(The couple (t0, pt0) is the initial condition, and the values t0 and pt0 are the initial conditions.) +Definition 5.3 Let t1, t2 ∈ R, t1 < t2. Let Ω be an open set in Rn and Ω its closure supposed to be a +regular domain. Let ||.|| be a norm in ⃗Rn. A continuous map ⃗v : [t1, t2] × Ω → ⃗Rn is Lipschitzian iff it is +“space Lipschitzian, uniformly in time”, that is, iff +∃k > 0, ∀t ∈ [t1, t2], ∀p, q ∈ Ω, ||⃗v(t, q) − ⃗v(t, p)|| ≤ k||q − p||. +(5.4) +So, ||⃗vt(q)−⃗vt(p)|| +||q−p|| +≤ k, for all t and all p ̸= q (the variations of ⃗v are bounded in space, uniformly in time). +34 + +35 +5.3. +Cauchy–Lipschitz theorem +Theorem 5.4 (and definifion) (Cauchy–Lipschitz). +If ⃗v : [t1, t2] × Ω → ⃗Rn is Lipschitzian and +(t0, pt0) ∈]t1, t2[×Ω, then there exists ε = εt0,pt0 > 0 s.t. (5.3) has a unique solution Φ :]t0−ε, t0+ε[→ Rn, +noted Φt0 +pt0 : +dΦt0 +pt0 +dt +(t) = ⃗v(t, Φt0 +pt0 (t)) +and +Φt0 +pt0 (t0) = pt0. +(5.5) +Moreover, if ⃗v is Ck then Φ is Ck+1. +Proof. See e.g. Arnold [2], or any ODE course. +In particular ||⃗v||∞ := +sup +t∈]t0−ε,t0+ε[, p∈Ω +||⃗v(t, p)||Rn +(maximum speed) exists since ⃗v ∈ C0 on the compact [t1, t2]×Ω), see definition 5.3, hence we can choose +ε = min(t0−t1, t2−t0, d(pt0,∂Ω) +||⃗v||∞ +) (the time needed to reach the border ∂Ω from pt0). +We have thus defined the function, also called “a flow”, +Φ : +� ]t1, t2[×]t1, t2[× Ωt0 → Ω +(t, t0, pt0) → p = Φ(t, t0, pt0) := Φt0 +pt0 (t) noted += +Φ(t; t0, pt0). +(5.6) +And (5.5) reads +∂Φ +∂t (t; t0, pt0) = ⃗v(t, Φ(t; t0, pt0)), +with +Φ(t0; t0, pt0) = pt0. +(5.7) +We have thus defined the function, also called “a flow”, +Φt0 : +� +[t0−ε, t0+ε] × Ωt0 → Rn +(t, pt0) → p = Φt0(t, pt0) := Φt0 +pt0 (t) : +(5.8) +And (5.5) reads +∂Φt0 +∂t (t, pt0) = ⃗v(t, Φt0(t, pt0)), +and +Φt0(t0, pt0) = pt0. +(5.9) +Other definition and notation (can be ambiguous): Φt;t0 = Φt0 +t : Ωt0 → Rn, and (5.7) is written +dΦt;t0(pt0) +dt += ⃗v(t, Φt;t0(pt0)), +and +Φt0;t0(pt0) = pt0. +(5.10) +Theorem 5.5 Let ⃗v be Lipschitzian, let t0 ∈]t1, t2[, and let Ωt0 be an open set s.t. Ωt0 ⊂⊂ Ω (i.e. there +exists a compact set K ∈ Rn s.t. Ωt0 ⊂ K ⊂ Ω). Then there exists ε > 0 s.t. a flow Φt0 exists on +]t0−ε, t0+ε[×Ωt0. +Proof. Let d = d(K, Rn−Ω) (la distance of K to the border of Ω. +Let ||⃗v||∞ := +sup +t∈[t1,t2],p∈Ω +||⃗v(t, p)||Rn (exists since ⃗v ∈ C0 on the compact [t1, t2] × Ω). +Let ε = min(t0−t1, t2−t0, +d +||⃗v||∞ ) (less that the minimum time to reach the border from K at maximum +speed ||v||∞). +Let pt0 ∈ K and t ∈]t0−ε, t0+ε[. Then Φt0 +pt0 exists, cf.theorem 5.4, and ||Φt0 +pt0 (t) − Φt0 +pt0 (t0)||Rn ≤ +[t−t0| supτ∈]t0−ε,t0+ε[(||(Φt0 +pt0 )′(τ)||Rn) (mean value theorem since, ⃗v being C0, Φ is C1). Thus ||Φt0 +pt0 (t)− +Φt0 +pt0 (t0)||Rn ≤ [t − t0| ||v||∞, thus Φt0 +pt0 (t) ∈ Ω. Thus Φt0 +pt0 exists on ]t0−ε, t0+ε[, for all pt0 ∈ K. +Remark 5.6 The definition of a flow starts with a Eulerian velocity (independent of any initial time), +and then, due to the introduction of initial conditions, leads to the Lagrangian functions Φt0, cf. (5.8). +Once again, Lagrangian functions are the result of Eulerian functions. +35 + +36 +5.4. +Examples +5.4 +Examples +Example 1 +R2 with an origin O, a Euclidean basis (⃗e1,⃗e2) and Ω = [0, 2]×[0, 1] (observation window). +Let p ∈ R2, −→ +Op =noted ⃗x = x⃗e1 + y⃗e2 =noted (x, y). Let t1 = −1, t2 = 1, t0 ∈]t1, t2[, a, b ∈ R, a ̸= 0, and +⃗v(t, p) = +� +v1(t, x, y) = ay, +v2(t, x, y) = b sin(t−t0). +(5.11) +(b = 0 corresponds to the stationary case = shear flow.) ⃗x(t0) = +� +x0 +y0 +� +, ⃗x(t) = +� +x(t) +y(t) +� += −−−−−−→ +OΦt0 +pt0 (t) and +(5.9) give +� +� +� +� +� +dx +dt (t) = v1(t, x(t), y(t)) = ay(t), +dy +dt (t) = v2(t, x(t), y(t)) = b sin(t−t0), +with +� +x(t0) = x0, +y(t0) = y0. +(5.12) +Thus +⃗x(t) = −−−→ +Op(t) = −−−−−−→ +OΦt0 +pt0 (t) = +� +x(t) = x0 + a(y0 + b)(t−t0) − ab sin(t−t0) +y(t) = y0 + b − b cos(t−t0) +� +. +(5.13) +Example 2 +Similar framework. Let ω > 0 and consider (spin vector field) +⃗v(t, x, y) = +� +−ωy +ωx +� += ω +� +0 +−1 +1 +0 +� � +x +y +� +noted += +⃗v(x, y). +(5.14) +With −−→ +Opt0 = ⃗xt0 = +� +xt0 +yt0 +� +, rt0 = +� +x2 +t0 + y2 +t0, and θ0 s.t. ⃗xt0 = +� +xt0 = rt0 cos(ωt0) +yt0 = rt0 sin(ωt0) +� +, the solution Φt0 +pt0 +of (5.9) is +⃗x(t) = −−−→ +Op(t) = −−−−−−→ +OΦt0 +pt0 (t) = +� +x(t) = rt0 cos(ωt) +y(t) = rt0 sin(ωt) +� +. +(5.15) +Indeed, +� ∂x +∂t (t, ⃗x0) +∂y +∂t (t, ⃗x0) +� += +� +v1(t, x(t, ⃗x0), y(t, ⃗x0)) +v2(t, x(t, ⃗x0), y(t, ⃗x0)) +� += +� +−ωy(t, ⃗x0) +ωx(t, ⃗x0) +� +, thus +∂x +∂t (t, ⃗x0) = −ωy(t, ⃗x0) +and +∂y +∂t (t, ⃗x0) = ωx(t, ⃗x0), thus +∂2y +∂t2 (t, ⃗x0) = −ω2y(t, ⃗x0), hence y; Idem for x. +Here d⃗v(t, x, y) = +ω +� +0 +−1 +1 +0 +� += ω +� +cos π +2 +− sin π +2 +sin π +2 +cos π +2 +� +is the π/2-rotation composed with the homothety with ratio ω. +5.5 +Composition of flows +Let ⃗v be a vector field on R × Ω and Φt0 +pt0 solution of (5.5). We use the notations +pt = Φt0 +t (pt0) = Φt;t0(pt0) := Φt0 +pt0 (t) = Φt0(t, pt0) = Φ(t; t0, pt0) = Φt0,pt0 (t). +(5.16) +5.5.1 +Law of composition of flows (determinism) +Proposition 5.7 For all t0, t1, t2 ∈ R, we have (determinism) +Φt1 +t2 ◦ Φt0 +t1 = Φt0 +t2, +i.e. +Φt2;t1 ◦ Φt1;t0 = Φt2;t0. +(5.17) +(“The composition of the photos gives the film”). So, +pt2 = Φt1 +t2(pt1) = Φt0 +t2(pt0) +when +pt1 = Φt0 +t1(pt0), +(5.18) +i.e., +pt2 = Φt2;t1(pt1) = Φt2;t0(pt0) +when +pt1 = Φt1;t0(pt0). +(5.19) +Thus +dΦt1 +t2(pt1).dΦt0 +t1(pt0) = dΦt0 +t2(pt0), +i.e. +dΦt2;t1(pt1).dΦt1;t0(pt0) = dΦt2;t0(pt0). +(5.20) +Summary with commutative diagrams: +pt1 +Φt1 +t2 +� +pt0 +Φt0 +t1 +� +Φt0 +t2 +� pt2 +i.e. +pt1 +Φt2;t1 +� +pt0 +Φt1;t0 +� +Φt2;t0 +� pt2 +36 + +37 +5.5. +Composition of flows +Proof. Let pt1 = Φt0 +pt0 (t1). (5.9) gives +� +� +� +� +� +� +� +dΦt0 +pt0 +dt +(t) = ⃗v(t, Φt0 +pt0 (t)), +dΦt1 +pt1 +dt +(t) = ⃗v(t, Φt1 +pt1 (t)), +� +� +� +� +� +� +� +with +pt1 = Φt0 +pt0 (t1) = Φt1 +pt1 (t1). +Thus Φt0 +pt0 and Φt1 +pt1 satisfy the same ODE with the same value at t1; Thus they are equal (uniqueness +thanks to Cauchy–Lipschitz theorem), thus Φt1 +pt1 (t) = Φt0 +pt0 (t) when pt1 = Φt0 +t1(pt0), that is, Φt1 +t (pt1) = +Φt0 +t (pt0) when pt1 = Φt0 +t1(pt0), which is (5.17) for any t = t2. Thus (5.20). +Corollary 5.8 A flow is compatible with the motion �Φ of an object Obj: (3.6) gives Φt1 +t2 ◦ Φt0 +t1 = (�Φt2 ◦ +(�Φt1)−1) ◦ (�Φt1 ◦ (�Φt0)−1) = �Φt2 ◦ (�Φt0)−1 = Φt0 +t2, that is (5.17). +5.5.2 +Stationnary case +Definition 5.9 ⃗v is a stationary vector field iff ∂⃗v +∂t = 0; And then ⃗v(t, p) =noted ⃗v(p). And the associated +flow Φt0 which satisfies +∂Φt0 +∂t (t, pt0) = ⃗v(pt) +when +pt = Φt0(t, pt0), +(5.21) +is said to be stationary. +Proposition 5.10 If ⃗v is a stationary vector field then, for all t0, t1, h, when meaningful (h small enough +and t1 close enough to t0), +Φt1 +t1+h = Φt0 +t0+h, +i.e. +Φt1+h;t1 = Φt0+h;t0, +(5.22) +i.e. Φt1 +t1+h(q) = Φt0 +t0+h(q), i.e. Φ(t1+h; t1, q) = Φ(t0+h; t0, q) for all q ∈ Ωt0 (see theorem 5.5). In other +words, +Φt0+h +t1+h = Φt0 +t1, +i.e. +Φt1+h;t0+h = Φt1;t0, +(5.23) +i.e. Φt0+h +t1+h(q) = Φt0 +t1(q), i.e. Φ(t1+h; t0+h, q) = Φ(t1; t0, q) for all q ∈ Ωt0. +Proof. Let q ∈ Ωt0, α(h) = Φt0 +t0+h(q) = Φt0 +q (t0+h) and β(h) = Φt1 +t1+h(q) = Φt1 +q (t1+h). +Thus α′(h) = +dΦt0 +q +dt (t0+h) = ⃗v(t0+h, Φt0 +q (t0+h)) = ⃗v(Φt0 +q (t0+h)) = ⃗v(α(h)) (stationary flow), and +β′(h) = dΦt1q +dt (t1+h) = ⃗v(t1+h, Φt1 +q (t1+h)) = ⃗v(Φt1 +q (t1+h)) = ⃗v(β(h)) (stationary flow). +Thus α and β satisfy the same ODE with the same initial condition α(0) = β(0) = q. Thus α = β. +Hence (5.22). Thus, with h = t1−t0, i.e. with t1 = t0+h and t0+h = t1, we get (5.23). +Corollary 5.11 If ⃗v is a stationary vector field, cf. (5.21), then +dΦt0 +t (pt0).⃗v(pt0) = ⃗v(pt) +when +pt = Φt0 +t (pt0), +(5.24) +that is, if ⃗v is stationary, then ⃗v is transported (push-forwarded by Φt0 +t ) along itself. +Proof. (5.18), t2 = t1+s and t1 = t0+s give Φt0+s +t1+s(Φt0 +t0+s(pt0)) = Φt0 +t1+s(pt0), and ⃗v is stationary, thus +Φt0 +t1(Φt0 +t0+s(pt0)) = Φt0 +t1+s(pt0), i.e. Φ(t1; t0, Φt0,pt0 (t0+s)) = Φt0,pt0 (t1+s), thus (s derivative) +dΦ(t1; t0, Φ(t0+s; t0, pt0)).Φt0,pt0 +′(t0+s) = Φt0,pt0 +′(t1+s), +thus dΦt0 +t1(Φ(t0+s; t0, pt0)).⃗v(t0+s, Φt0,pt0 (t0+s)) = ⃗v(t1+s, Φt0,pt0 (t1+s)). Thus with s = 0, and ⃗v being +stationary, dΦt0 +t1(Φ(t0; t0, pt0)).⃗v(Φt0,pt0 (t0)) = ⃗v(Φt0,pt0 (t1)), thus (5.24). +37 + +38 +5.6. +Velocity on the trajectory traveled in the opposite direction +5.6 +Velocity on the trajectory traveled in the opposite direction +Let t0, t1 ∈ R, t1 > t0, and pt0 ∈ Rn. Consider the trajectory Φt0 +pt0 : +� +[t0, t1] → Rn +t → p(t) = Φt0 +pt0 (t) +� +. So pt0 +is the beginning of the trajectory, pt1 = Φt0 +t1(pt0) at the end, ⃗v(t, p(t)) = +dΦt0 +pt0 +dt +(t) being the velocity. +Define the trajectory traveled in the opposite direction, i.e. define +Ψt1 +pt1 : +� +[t0, t1] → Rn +u → q(u) = Ψt1 +pt1 (u) := Φt0 +pt0 (t0+t1−u) = Φt0 +pt0 (t) = p(t) +when +t = t0+t1−u. +(5.25) +In particular q(t0) = Ψt1 +pt1 (t0) = Φt0 +pt0 (t1) = p(t1) and q(t1) = Ψt1 +pt1 (t1) = Φt0 +pt0 (t0) = p(t0). +Proposition 5.12 The velocity on the trajectory traveled in the opposite direction is the opposite of +the velocity on the initial trajectory: +dΨt1 +pt1 +du +(u) = q′(u) = −p′(t) = −⃗v(t, p(t)) +when +t = t0+t1−u, +(5.26) +Proof. Ψt1 +pt1 (u) = Φt0 +pt0 (t0+t1−u) gives +dΨt1 +pt1 +du (u) = − +dΦt0 +pt0 +dt +(t0+t1−u) = −⃗v(t0+t1−u, Φt0 +pt0 (t0+t1−u)) = +−⃗v(t, Φt0 +pt0 (t)) when t = t0+t1−u. +5.7 +Variation of the flow as a function of the initial time +5.7.1 +Ambiguous and non ambiguous notations +Let Φ : (t, u, p) ∈ R × R × Rn → Φ(t, u, p) ∈ Rn be a C1 function. The partial derivatives are +∂1Φ(t, u, p) := lim +h→0 +Φ(t+h, u, p) − Φ(t, u, p) +h +, +(5.27) +∂2Φ(t, u, p) := lim +h→0 +Φ(t, u+h, p) − Φ(t, u, p) +h +, +(5.28) +and ∂3Φ(t, u, p), defined for all ⃗w ∈ ⃗Rn (a vector at p) by, +∂3Φ(t, u, p).⃗w := lim +h→0 +Φ(t, u, p+h⃗w) − Φ(t, u, p) +h +noted += +dΦ(t, u, p).⃗w, +(5.29) +When the name of the first variable is systematically noted t, then +∂1Φ(t, u, p) noted += +∂Φ +∂t (t, u, p) noted += +∂Φ(t, u, p) +∂t +. +(5.30) +NB: This notation can be ambiguous: What is the meaning of ∂Φ +∂t (t; t, p)? In ambiguous situations, use +the notation ∂1Φ, or (if no composed functions inside) use ∂Φ(t,u,p) +∂t +|u=t (so t is the derivation variable, +and after the calculation you take u = t). +When the name of the second variable is systematically noted u, then +∂2Φ(t, u, p) noted += +∂Φ +∂u (t, u, p) noted += +∂Φ(t, u, p) +∂u +. +(5.31) +NB: Idem this notation can be ambiguous: What is the meaning of ∂Φ +∂u (u; u, p)? In ambiguous situations, +use the notation ∂2Φ, or use ∂Φ(t,u,p) +∂u +|t=u. +When the name of the third variable is systematically a space variable noted p, then +∂3Φ(t, u, p) noted += +dΦ(t, u, p) noted += +∂Φ +∂p (t, u, p) noted += +∂Φ(t, u, p) +∂p +. +(5.32) +38 + +39 +5.7. +Variation of the flow as a function of the initial time +5.7.2 +Variation of the flow as a function of the initial time +The law of composition of the flows gives (5.19) gives Φ(t; u, Φ(u; t0, p0)) = Φ(t; t0, p0). Thus the derivative +in u gives +∂2Φ(t; u, Φ(u; t0, p0)) + dΦ(t; u, Φ(u; t0, p0)).∂1Φ(u; t0, p0) = 0, +i.e. +∂2Φ(t; u, p(u)) = −dΦ(t; u, p(u)).⃗v(u, p(u)) +when +p(u) = Φ(u; t0, p0). +(5.33) +In particular u = t0 gives, for all (t, t0, p0) ∈ R2 × Ωt0, +(∂Φ(t; t0, p0) +∂t0 +=) +∂2Φ(t; t0, p0) = −dΦ(t; t0, p0).⃗v(t0, p0). +(5.34) +In particular +(dΦ(t; t0, p0) +dt0 +|t=t0 +=) +∂2Φ(t0; t0, p0) = −⃗v(t0, p0). +(5.35) +39 + +40 +Part II +Push-forward +6 +Push-forward +The general tool to describe “transport” is “push-forward by a motion” (the “take with you” operator), +cf. § 4.1 and figure 4.1. The push-forward also gives the tool needed to understand the velocity addition +formula: In that case, the push-forward is the translator between observers. The push-forward can also +be used to write coordinate systems. As usual, we start with qualitative results (observer independent +results); Then, quantitative results are deduced. +6.1 +Definition +E and F are affine spaces, E and F are the associated vector spaces equipped with norms ||.||E and ||.||F +with dim E = dim F = n, UE and UF are open sets in the affine space E and F, or possibly the vector +spaces E and F, and +Ψ : +� +UE → UF +pE → pF = Ψ(pE) +is a diffeomorphism +(6.1) +(a C1 invertible map which inverse is C1), called the push-forward, and Ψ−1 is the pull-back (push-forward +with Ψ−1). +Figure 6.1: cE : s → pE = cE(s) is a curve in UE. Push-forwarded by Ψ it becomes the curve cE∗ := Ψ ◦ cE +in UF. The tangent vector at pE = cE(s) is ⃗wE(pE) = cE ′(s), and the tangent vector at pF = cF(s) = +Ψ(cE(s)) is ⃗wE∗(pF) = cF ′(s) = dΨ(pE).⃗wE(pE). Other illustation: See figure 4.1. +Example: Ψ = Φt0 +t : Ωt0 → Ωt, the motion that transforms Ωt0 into Ωt, cf. (3.5). +Example: Ψ : UE → UF a coordinate system, see example 6.11. +Example: Ψ = Θt : RB → RA, a change of referential at t (change of observer), see § 10. +6.2 +Push-forward and pull-back of points +Definition 6.1 If pE ∈ UE (a point in UE) then its push-forward by Ψ is the point +pF = Ψ∗pE := Ψ(pE) = pE∗ ∈ UF, +(6.2) +see figure 6.1, the last notation if Ψ is implicit. And if pF ∈ UF then its pull-back by Ψ is the point +pE = Ψ∗pF := Ψ−1(pF) = pF +∗ ∈ UE. +(6.3) +We immediately have Ψ∗ ◦ Ψ∗ = I. +The notations ∗ for push-forward and ∗ for pull-back have been proposed by Spivak; Also see Abraham +and Marsden [1] (second edition) who adopt this notation. +40 + +Us +亚 +we(pe +p =C(s +P = 亚(p) +Im(c* +Im(C41 +6.3. +Push-forward and pull-back of curves +6.3 +Push-forward and pull-back of curves +We push-forward (and pull-back) the points on a curve: +Definition 6.2 Let cE : +� +] − ε, ε[ → UE +s → pE = cE(s) +� +be a curve in UE. Its push-forward by Ψ is the curve +Ψ∗cE := Ψ ◦ cE : +� ] − ε, ε[ → UF +s → pF = Ψ∗cE(s) := Ψ(cE(s)) noted += +cE∗(s) +(= Ψ(pE)), +(6.4) +see figure 6.1. (Ψ∗cE =noted cE∗ when Ψ is implicit.) This defines +Ψ∗ : +� F(] − ε, ε[; UE) → F(] − ε, ε[; UF) +cE → Ψ∗(cE) := Ψ ◦ cE +noted += +Ψ∗cE = cE∗. +(6.5) +Definition 6.3 Let cF : +� +] − ε, ε[ → UF +s → pF = cF(s) +� +is a curve in UF. Its pull-back by Ψ is +Ψ∗cF := Ψ−1 ◦ cE +� ] − ε, ε[ → UE +s → pE = Ψ∗cF(s) := Ψ−1(cF(s)) noted += +cF +∗(s) +(= Ψ−1(pF)). +(6.6) +We have thus defined +Ψ∗ : +� F(C1(] − ε, ε[; UF) → F(C1(] − ε, ε[; UE) +cF → Ψ∗(cF) := Ψ−1 ◦ cF +noted += +Ψ∗cF = cF +∗. +(6.7) +6.4 +Push-forward and pull-back of scalar functions +6.4.1 +Definitions +Definition 6.4 Let fE : +� +UE → R +pE → fE(pE) +� +(scalar valued function). Its push-forward by Ψ is the (scalar +valued) function +Ψ∗fE := fE ◦ Ψ−1 : +� UF → R +pF → Ψ∗fE(pF) := fE(pE) noted += +fE∗(pF) +when +pE = Ψ−1(pF), +(6.8) +(noted fE∗ when Ψ is implicit), i.e. Ψ∗fE(Ψ∗pE) := fE(pE), or fE∗(pE∗) := fE(pE) when pE∗ = Ψ(pE). We +have thus defined +Ψ∗ : +� F(UE; R) → F(UF; R) +fE → fF := Ψ∗(fE) = fE ◦ Ψ−1 noted += +Ψ∗fE, +(6.9) +the notation Ψ∗(fE) = Ψ∗fE since Ψ∗ is linear: ((fE + λgE) ◦ Ψ−1)(pF) = (fE + λgE)(pE) = fE(pE) + +λgE(pE) = (fE ◦ Ψ−1)(pF) + λ(gE ◦ Ψ−1)(pF) gives Ψ∗(fE + λgE) = Ψ∗(fE) + λΨ∗(gE). +Definition 6.5 Let fF : +� +UF → R +pF → fF(pF) +� +. Its pull-back by Ψ is the push-forward by Ψ−1, i.e. is +Ψ∗fF := fF ◦ Ψ : +� UE → R +pE → Ψ∗fF(pE) := fF(pF) noted += +fF +∗(pE) +when +pF = Ψ(pE), +(6.10) +i.e. Ψ∗fF(Ψ∗pF) := fF(pF), i.e. fF ∗(pF ∗) := fF(pF) when pF = Ψ∗(pF). We have thus defined +Ψ∗ : +� F(UF; R) → F(UE; R) +fF → Ψ∗(fF) = fF +∗ := fF ◦ Ψ noted += +Ψ∗fF. +(6.11) +We immediately have Ψ∗ ◦ Ψ∗ = I +and +Ψ∗ ◦ Ψ∗ = I (the first I is the identity in F(UE; R), the +second I is the identity in F(UF; R)). +NB: We used the same notations Ψ∗ and Ψ∗ than for the push-forward and pull-backs of points: The +context removes ambiguities. +41 + +42 +6.5. +Push-forward and pull-back of vector fields +6.4.2 +Interpretation: Why is it useful? +E.g.: Let �Φ : R × Obj → Rn be a motion of an object Obj. An observer records the temperature θ at +all t ∈ [t0, T] and all p = �Φ(t, Obj): He gets θ : +� +� +� +C = +� +t +({t} × Ωt) → R +(t, p) → θ(t, p) +� +� +� a Eulerian scalar valued +function, cf. (2.2). Then he chooses an initial time t0 and considers the associated motion Φt0, cf. (3.1), +and considers θt0 : +� +Ωt0 → R +pt0 → θt0(pt0) := θ(t0, pt0) +� +(snapshot of the temperatures at t0 in Ωt0). The +push-forward of θt0 by Φt0 +t is (Φt0 +t )∗θt0 := θt0 ◦ (Φt0 +t )−1 defines the “memory function” +(Φt0 +t )∗θt0 : +� +Ωt → R +pt → (Φt0 +t )∗θt0(pt) := θt0(pt0) +when +pt = Φt0 +t (pt0), +(6.12) +And he writes (Φt0 +t )∗θt0(pt) =noted θt0∗(t, pt), so the memory transported is at t at pt (along a trajectory) +by +θt0∗(t, p(t)) = θt0(pt0). +(6.13) +Question: Why do we introduce θt0∗ since we have θt0? +Answer: An observer does not have the gift of temporal and/or spatial ubiquity; He has to do with +values at the actual time t and position pt where he is (Newton and Einstein’s point of view). So, when +he was at t0 at pt0 the observer wrote the value θt0(pt0) on a piece of paper (for memory), puts the piece +of paper is his pocket, then once at t at p(t) = Φt0(t, pt0), he takes the paper out of his pocket, and +renames the value he reads as θt0∗(t, pt) because he is now at t at pt. And, now at t at pt, he can compare +the past and present value. In particular the rate +θ(t, p(t)) − θt0∗(t, p(t)) +t − t0 += +actual(t, p(t)) − memory∗(t, p(t)) +t − t0 +(6.14) +is physically meaningful for one observer at t at pt (no ubiquity gift required). For scalar value functions, +we get the usual rate θ(t,p(t))−θ(t0,p(t0)) +t−t0 +, but it isn’t that simple for vector valued functions. +And the limit t → t0 in (6.14) defines the Lie derivative for scalar valued functions. +6.5 +Push-forward and pull-back of vector fields +This is one of the most important concept for mechanical engineers. +6.5.1 +A definition by approximation +Elementary introduction. Let pE and qE be points in UE, and let pF = pE∗ = Ψ(pE) and qF = qE∗ = Ψ(qE) +in UF be the push-forwards by Ψ cf. (6.1). The first order Taylor expansion gives +(Ψ(qE) − Ψ(pE) =) +qF − pF = dΨ(pE).(qE − pE) + o(||qE − pE||E), +(6.15) +thus, +−−→ +pFqF +||−−→ +pEqE||E += dΨ(pE). +−−→ +pEqE +||−−→ +pEqE||E ++ o(1). +(6.16) +And the definition of the push-forward is obtained by “neglecting” the o(1) (limit as qE → pE): +Definition 6.6 If ⃗wE(pE) ∈ E is a vector at pE ∈ U then its push-forward by Ψ is the vector +⃗wF(pF) =noted ⃗wE∗(pF) =noted Ψ∗ ⃗wE(pF) ∈ F defined at pF = pE∗ = Ψ(pE) ∈ UF by +⃗wF(pF) = ⃗wE∗(pF) := dΨ(pE).⃗wE(pE) noted += +Ψ∗ ⃗wE(pF). +(6.17) +6.5.2 +The definition of the push-forward of a vector field +To fully grasp the definition, and to avoid making interpretation errors as in § 4.3 (the unfortunate +notation d⃗x = F.d ⃗X), we use the following definition of “a vector”: It is a “tangent vector to a curve” +(needed for surfaces and manifolds). Details: +42 + +43 +6.5. +Push-forward and pull-back of vector fields +• Let cE : +� +] − ε, ε[ → UE +s → pE = cE(s) +� +be a C1 curve in UE. Its tangent vector at pE = cE(s) is +⃗wE(pE) := cE +′(s) +(= lim +h→0 +cE(s + h) − cE(s) +h +), +(6.18) +see figure 6.1. This defines the function ⃗wE : +� +Im(cE) → E +pE → ⃗wE(pE) +� +called a vector field along Im(cE)⊂UE. +• The push-forward of cE by Ψ being the image curve cE∗ = Ψ ◦ cE (the curve transformed by Ψ) +cf. (6.4), its tangent vector at pF = cE∗(s) is +⃗wE∗(pF) := cE∗ +′(s) +thus += dΨ(pE).cE +′(s) = dΨ(pE).⃗wE(pE). +(6.19) +Thus we have defined the vector field ⃗wE∗ along Im(cE∗) called the push-forward of ⃗wE by Ψ. +With all the integral curves of a vector field defined in UE, we get: +Definition 6.7 The push-forward by Ψ of a C0 vector field ⃗wE : +� +UE → E +pE → ⃗wE(pE) +� +is the vector field +Ψ∗ ⃗wE = ⃗wE∗ : +� +� +� +UF → F +pF → Ψ∗ ⃗wE(pF) := dΨ(pE).⃗wE(pE) noted += +⃗wE∗(pF) +when +pF = Ψ(pE), +(6.20) +see figure 6.1. (Ψ∗ ⃗wE =noted ⃗wE∗ if Ψ is implicit). In other words, +Ψ∗ ⃗wE := (dΨ.⃗wE) ◦ Ψ−1. +(6.21) +This defines the map Ψ∗ : +� +C∞(UE; E) → C∞(UF; F) +⃗wE → Ψ∗(⃗wE) := Ψ∗ ⃗wE = ⃗wE∗ +� +. (We use the same notation Ψ∗ as +in definition 6.4 for scalar valued functions: The context removes ambiguity.) +Remark 6.8 Unlike scalar functions, cf. § 6.4.2: At t0 at pt0 you cannot just draw a vector ⃗wt0(pt0) +on a piece of paper, put the paper in your pocket, then let yourself be carried by the flow Ψ = Φt0 +t +(push-forward), then, once arrived at t at pt, take the paper out of your pocket and read it to get the +push-forward: The direction and length of the vector ⃗wt0∗(t, pt) are modified by the flow (a vector is not +just a collection of scalar components). +Exercice 6.9 Prove: +⃗cE +′′(s) = d⃗wE(pE).⃗wE(pE), +(6.22) +and +d⃗wE∗(pF).dΨ(pE) = dΨ(pE).d⃗wE(pE) + d2Ψ(pE).⃗wE(pE), +(6.23) +and +cE∗ +′′(s) = d⃗wE∗(pF).⃗wE∗(pF) +(= dΨ(pE). ⃗cE +′′(s) + d2Ψ(pE). ⃗cE +′(s). ⃗cE +′(s)). +(6.24) +Answer. ⃗cE +′(s) = ⃗wE(cE(s)) gives ⃗cE +′′(s) = d⃗wE(cE(s)). ⃗cE +′(s), hence (6.22). +⃗wE∗(Ψ(pE)) = dΦ(pE).⃗wE(pE) by definition of ⃗wE∗, hence (6.23). +cF(s) = Ψ(cE(s)) gives ⃗cF +′(s) = dΨ(cE(s)). ⃗cE +′(s) = dΨ(cE(s)).⃗wE(cE(s)) = ⃗wE∗(cF(s)). +Thus ⃗cF +′′(s) = +(d2Ψ(cE(s)). ⃗cE +′(s)). ⃗cE +′(s) + dΨ(cE(s)). ⃗cE +′′(s) = d⃗wE∗(cF(s)). ⃗cF +′(s), hence (6.24). +6.5.3 +Pull-back of a vector field +Definition 6.10 If ⃗wF : +� +UF → F +pF → ⃗wF(pF) +� +is a vector field on UF, then its pull-back by Ψ is the +push-forward by Ψ−1, i.e. is the vector field on UE defined by +Ψ∗ ⃗wF : +� +� +� +UE → E +pE → Ψ∗ ⃗wF(pE) := dΨ−1(pF).⃗wF(pF) noted += +⃗wF +∗(pE), +when +pF = Ψ(pE). +(6.25) +In other words, +Ψ∗ ⃗wF := (dΨ−1.⃗wF) ◦ Ψ noted += +⃗wF +∗. +(6.26) +43 + +44 +6.6. +Quantification with bases +And we get +Ψ∗ ◦ Ψ∗ = I +and +Ψ∗ ◦ Ψ∗ = I. +(6.27) +Indeed, Ψ∗(Ψ∗ ⃗wE)(pE) = dΨ−1(pF).Ψ∗ ⃗wE(pF) = dΨ−1(pF).dΨ(pE).⃗wE(pE) = ⃗wE(pE), for all pE. Idem for +the second equality. +6.6 +Quantification with bases +6.6.1 +Usual result +(⃗ai) is a Cartesian basis in E, OF and (⃗bi) are an origin in F and a Cartesian basis in F, pE ∈ UE, +pF = Ψ(pE) = OF + +n +� +i=1 +ψi(pE)⃗bi, +i.e. +[−−−→ +OFpF]|⃗b = +� +� +� +ψ1(pE) +... +ψn(pE) +� +� +� . +(6.28) +Then, if ⃗wE is a vector field in UE and ⃗wE += � +i wj⃗ai, we get Ψ∗ ⃗wE(pF) = dΨ(pE).⃗wE(pE) = +�n +i=1(dψi(pE).⃗wE(pE))⃗bi = �n +i,j=1wj(pE)(dψi(pE).⃗aj)⃗bi = �n +i,j=1 +∂ψi +∂xj (pE)wj(pE)⃗bi, so +[Ψ∗ ⃗wE(pF)]|⃗b = [dΨ(pE)]|⃗a,⃗b.[⃗wE(pE)]|⃗a, +(6.29) +where [dΨ(pE)]|⃗a,⃗b = [dψi(pE).⃗aj] =noted [ ∂ψi +∂xj (pE)] is the Jacobian matrix. +6.6.2 +Example: Polar coordinate system +Example 6.11 Change of coordinate system interpreted as a push-forward: Paradigmatic example of +the polar coordinate system (model generalized for the parametrization of any manifold). +Parametric Cartesian vector space R × R =noted ⃗R2 +p = {⃗q = (r, θ)}, with its canonical basis (⃗a1,⃗a2), +and ⃗q = r⃗a1 + θ⃗a2 =noted (r, θ), so [⃗q]|⃗a = +� +r +θ +� +. Geometric affine space R2 (of positions), p ∈ R2, +associated vector space ⃗R2, O ∈ R2 (origin), ⃗x = −→ +Op, and a Euclidean basis (⃗b1,⃗b2) in ⃗R2. The “polar +coordinate system” is the associated map Ψ : +� ⃗R∗ ++ × R ⊂ ⃗R2 +p → ⃗R2 +⃗q = (r, θ) → ⃗x = Ψ(⃗q) = Ψ(r, θ), +� +defined by +⃗x = Ψ(⃗q) := r cos θ⃗b1 + r sin θ⃗b2, +i.e. +[⃗x]|⃗b = +� +x = r cos θ +y = r sin θ +� +. +(6.30) +The i-th coordinate line at ⃗q in ⃗R2 +p (parametric space) is the straight line ⃗c⃗q,i : +� +R → ⃗R2 +p +s → ⃗c⃗q,i(s) = ⃗q + s⃗ai +� +, +and its tangent vector at ⃗c⃗q,i(s) is ⃗c⃗q,i′(s) = ⃗ai for all s. This line is transformed by Ψ into the curve +Ψ∗(cq,i) = Ψ ◦ ⃗c⃗q,i =noted c⃗x,i : +� +R → R2 +s → c⃗x,i(s) = Ψ(⃗q + s⃗ai) +� +(in particular c⃗x,i(0) = ⃗x). So +[−−−−−→ +Oc⃗x,1(s)]|⃗b = +� +(r+s) cos θ +(r+s) sin θ +� +(straight line), +and +[−−−−−→ +Oc⃗x,2(s)]|⃗b = +� +r cos(θ+s) +r sin(θ+s) +� +(circle). +(6.31) +And the tangent vector at c⃗x,i(s) is c⃗x,i′(s) =noted ⃗ai∗(⃗x) (push-forward by Ψ), so +⃗a1∗(⃗x) := Ψ∗⃗a1(⃗x) = dΨ(⃗q).⃗a1 = lim +h→0 +Ψ(⃗q+h⃗a1) − Ψ(⃗q) +h += lim +h→0 +Ψ(r+h, θ) − Ψ(r, θ) +h += ∂Ψ +∂r (⃗q), +⃗a2∗(⃗x) := Ψ∗⃗a2(⃗x) = dΨ(⃗q).⃗a2 = lim +h→0 +Ψ(⃗q+h⃗a2) − Ψ(⃗q) +h += lim +h→0 +Ψ(r, θ+h) − Ψ(r, θ) +h += ∂Ψ +∂θ (⃗q), +(6.32) +Thus +⃗a1∗(⃗x) = cos θ⃗b1 + sin θ⃗b2 +and +⃗a2∗(⃗x) = −r sin θ⃗b1 + r cos θ⃗b2, +(6.33) +i.e. +[⃗a1∗(⃗x)]|⃗b = +� +cos θ +sin θ +� +and +[⃗a2∗(⃗x)]|⃗b = +� +−r sin θ +r cos θ +� +. +(6.34) +The basis (⃗a1∗(⃗x),⃗a2∗(⃗x)) is called the basis of the polar coordinate system at ⃗x (it is orthogonal +but not orthonormal since ||⃗a2∗(⃗x)|| = r ̸= 1 in general); And [dΨ(⃗q)]|⃗a,⃗b = +� [ ∂Ψ +∂r (⃗q)]|⃗b +[ ∂Ψ +∂θ (⃗q)]|⃗b +� += +� [⃗a1∗(⃗x)]|⃗b +[⃗a2∗(⃗x)]|⃗b +� += +� +cos θ +−r sin θ +sin θ +r cos θ +� += [ ∂Ψi +∂qj (⃗q)] is the Jacobian matrix of Ψ at ⃗q. +44 + +45 +6.6. +Quantification with bases +And the dual basis of the polar system basis (⃗a1∗(⃗x),⃗a2∗(⃗x)) is called (dq1(⃗x), dq2(⃗x)) (defined by +dqi(⃗x).⃗aj∗(⃗x) = δij), so +dq1(⃗x) = cos θ dx1 + sin θ dx2 +and +dq2(⃗x) = −1 +r sin θ dx1 + 1 +r cos θ dx2, +(6.35) +i.e. [dq1(⃗x)]|⃗b = ( cos θ +sin θ ) and [dq2(⃗x)]|⃗b = − 1 +r ( sin θ +cos θ ) (row matrices) when ⃗x = Ψ(⃗q). +Remark 6.12 The components γk +ij(⃗x) of the vector d⃗aj∗(⃗x).⃗ai∗(⃗x) ∈ ⃗R2 in the basis (⃗ai∗(⃗x)) are the +Christoffel symbols of the polar coordinate system (with duality notations as it is usually presented): +d⃗aj∗(⃗x).⃗ai∗(⃗x) = +n +� +k=1 +γk +ij(⃗x)⃗ak∗(⃗x). +(6.36) +At ⃗x = Ψ(⃗q), with ⃗aj∗(⃗x) = dΨ(⃗q).⃗aj, i.e. (⃗aj∗ ◦ Ψ)(⃗q) = ∂Ψ +∂qj , we get +d⃗aj∗(⃗x).⃗ai∗(⃗x) = +∂2Ψ +∂qi∂qj (⃗q) = d⃗ai∗(⃗x).⃗aj∗(⃗x), +so +γk +ij = γk +ji +(6.37) +for all i, j (symmetry of the bottom indices as soon as Ψ is C2). +Here for the polar coordinates, +∂Ψ +∂r (⃗q) = cos θ⃗b1 + sin θ⃗b2 gives +∂2Ψ +∂r2 (⃗q) = ⃗0, thus γ1 +11 = γ2 +11 = 0, +and +∂2Ψ +∂θ∂r(⃗q) = − sin θ⃗b1 + cos θ⃗b2 = 1 +r⃗a2∗(⃗x), thus γ1 +12 = 0 = γ1 +21 and γ2 +12 = 1 +r = γ2 +21. And ∂Ψ +∂θ (⃗q) = +−r sin θ⃗b1 + r cos θ⃗b2 gives ∂2Ψ +∂θ2 (⃗q) = −r cos θ⃗b1 − r sin θ⃗b2 = −r⃗a1∗(⃗x), thus γ1 +22 = −r and γ2 +22 = 0. +Remark 6.13 The (widely used) normalized polar coordinate basis (⃗n1(⃗x),⃗n2(⃗x)) = (⃗a1∗(⃗x), 1 +r⃗a2∗(⃗x)) +is not holonomic, i.e. is not the basis of a coordinate system (and its use makes higher deriva- +tion formulas complicated). +Indeed ⃗n2(⃗x) = +1 +r⃗a2∗(⃗x) gives d⃗n2(⃗x).⃗n1(⃗x) = (d( 1 +r)(⃗x).⃗n1(⃗x))⃗a2∗(⃗x) + +1 +rd⃗a2∗(⃗x).⃗n1(⃗x), and ⃗n1(⃗x) += ⃗a1∗(⃗x) gives d⃗n1(⃗x).⃗n2(⃗x) += +d⃗a1∗(⃗x).( 1 +r⃗a2∗), thus d⃗n2(⃗x).⃗n1(⃗x) − +d⃗n1(⃗x).⃗n2(⃗x) += +(d( 1 +r)(⃗x).⃗n1(⃗x))⃗a2∗(⃗x) +̸= +⃗0, +since +1 +r += +(x2 + y2)− 1 +2 +gives d( 1 +r)(⃗x).⃗n1(⃗x) += +( −x(x2 + y2)− 3 +2 +−y(x2 + y2)− 3 +2 ) . +� +cos θ +sin θ +� += +1 +r3 (−r cos2 θ − r sin2 θ) = −1 +r2 ̸= 0. +Remark 6.14 (Pay attention to the notations.) Let f : ⃗q ∈ ⃗R2 +p → f(⃗q) ∈ R be C2. Call g its push- +forward by Ψ, i.e. g : ⃗x ∈ R2 → g(⃗x) = f(⃗q) ∈ R when ⃗x = Ψ(⃗q). So f(⃗q) = (g ◦ Ψ)(⃗q)and +df(⃗q).⃗aj = dg(Ψ(⃗q)).dΨ(⃗q).⃗aj = dg(⃗x).⃗aj∗(⃗x). +(6.38) +With df(⃗q).⃗aj =noted +∂f +∂qj (⃗q) and dg(⃗x).⃗bj =noted +∂g +∂xj (⃗x) and ⃗aj∗(⃗x) = dΨ(⃗q).⃗aj = � +i +∂Ψi +∂qj (⃗q)⃗aj, we get +∂f +∂qj (⃗q) = +� +i +∂g +∂xi (⃗x)∂Ψi +∂qj (⃗q) noted += +∂g +∂qj (⃗x) ... (!!) +(6.39) +Mind this notation!! g is a function of ⃗x, not of ⃗q, so ∂g +∂qi (⃗x) means += +∂f +∂qi (⃗q), i.e. ∂g +∂qi (⃗x) means += +∂(g ◦ Ψ) +∂qi +(⃗q)... +which is [df(⃗q)] = [dg(⃗x)].[dΨ(⃗q)... +Then (with f and Ψ C2) +∂ ∂g +∂qi +∂qj (⃗x) means += +∂ ∂(g◦Ψ) +∂qi +∂qj +(⃗q) = d(dg.⃗ai∗)(⃗x).dΨ(⃗q).⃗aj = d(dg.⃗ai∗)(⃗x).⃗aj∗(⃗x) += d((dg(⃗x).⃗aj∗(⃗x)).⃗ai∗(⃗x) + dg(⃗x).(d⃗ai∗(⃗x).⃗aj(⃗x)) noted += +∂2g +∂qi∂qj (⃗x). +(6.40) +So +∂2g +∂qi∂qj (⃗x) means += +d2g(⃗x)(⃗ai∗(⃗x),⃗aj∗(⃗x)) + +n +� +k=1 +∂g +∂xk (⃗x)γk +ij(⃗x)⃗ak(⃗x), +(6.41) +and +∂2g +∂qi∂qj (⃗x) is not reduced to d2g(⃗x)(⃗ai∗(⃗x),⃗aj∗(⃗x)) (the Christoffel symbols have appeared): First +order derivatives +∂g +∂xk are still alive. (Contrary to +∂2g +∂xi∂xj (⃗x) = d2g(⃗x)(⃗bi,⃗bj) with a Cartesian basis (⃗bi).) +NB: The independent variables r and θ don’t have the same dimension (a length and an angle): There +is no physical meaningful inner dot product in the parameter space ⃗R2 +p = R×R = {(r, θ)}, but this space +is very useful... (As in thermodynamics: No meaningful inner dot product in the (T, P) space.) +45 + +46 +7.1. +Definition +7 +Push-forward and pull-back of differential forms +7.1 +Definition +Setting of § 6.1. Consider a differential form αE : +� +UE → E∗ = L(E; R) +pE → αE(pE) +� +on UE (a field of linear forms), +and a vector field ⃗wE : +� +UE → E +pE → ⃗wE(pE) +� +. Hence +fE = αE.⃗wE : +� +UE → R +pE → fE(pE) = (α.⃗wE)(pE) = αE(pE).⃗wE(pE) +is a scalar valued function (value of ⃗wE given by αE). And (6.8) gives (push-forward fE = αE.⃗wE by Ψ) +Ψ∗(αE.⃗wE)(pF) = (αE.⃗wE)(pE) = αE(pE).⃗wE(pE) +when +pF = Ψ(pE). +(7.1) +With ⃗wE∗(pF) = dΨ(pE).⃗wE(pE) cf. (6.20) (push-forward of ⃗wE), we get +Ψ∗(αE.⃗wE)(pF) = αE(pE).dΨ(pE)−1 +� +�� +� +=noted αE∗(pF) +.⃗wF(pF) +when +pF = Ψ(pE) : +(7.2) +Definition 7.1 The push-forward of a differential form αE ∈ Ω1(UE) is the differential form ∈ Ω1(UF) +given by +Ψ∗αE : +� +� +� +UF → F ∗ = L(F; R) +pF → Ψ∗αE(pF) := αE(pE).dΨ(pE)−1 +noted += +αE∗(pF) +when +pF = Ψ(pE), +(7.3) +the last notation when Ψ is implicit. In other words, Ψ∗αE(pF) = αE(Ψ−1(pF)).dΨ−1(pF), i.e. +Ψ∗αE := (αE ◦ Ψ−1).dΨ−1. +(7.4) +(Once again, we used the same notation Ψ∗ than for the push-forward of vector fields and functions: The +context removes any ambiguities.) +Remark 7.2 We cannot always see a vector field (e.g. we can’t see an internal force field): To know it we +need to measure it with a well defined tool, the tool being here a differential form; And the definition 7.1 +is a compatbility definition so that we can recover the push-forward of the vector field. +Definition 7.3 The pull-forward of a a differential form αF ∈ Ω1(UF) is the differential form +Ψ∗αF : +� UE → L(E; R) +pE → Ψ∗αF(pE) := αF(pF).dΨ(pE) noted += +αF +∗(pE) +when +pF = Ψ(pE), +(7.5) +In other words, +Ψ∗αF := (αF ◦ Ψ).dΨ. +(7.6) +(For an alternative definition, see remark 7.5.) +Proposition 7.4 For all αE ∈ Ω1(UE) and αF ∈ Ω1(UF) (differential forms), and ⃗wE ∈ Γ(UE) and +⃗wF ∈ Γ(UF) (vector fields), we have (objectivity result) +(Ψ∗αE)(pF).⃗wF(pF) = αE(pE).(Ψ∗ ⃗wF)(pE) +when +pF = Ψ(pE), +(7.7) +i.e. αE∗(pF).⃗wF(pF) = αE(pE).⃗wF ∗(pE). +In particular with αE = df (exact differential form) where +f ∈ C1(UE; R), +d(Ψ∗f) = Ψ∗(df). +(7.8) +(This commutativity result is very particular to the case α = df: In general d(Ψ∗T) ̸= Ψ∗(dT) for a +tensor of order ≥ 2, see e.g. (8.19)). +46 + +47 +7.2. +Incompatibility: Riesz representation and push-forward +Proof. αE∗(pF).⃗wF(pF) = (αE(pE).dΨ−1(pF)).⃗wF(pF) = αE(pE).(dΨ−1(pF).⃗wF(pF)) = αE(pE).⃗w∗ +F(pE), +for all pF = Ψ(pE) ∈ UF. +And Ψ∗f(pF) := f(pE) = f(Ψ−1(pF)), thus d(Ψ∗f)(pF) = df(pE).dΨ−1(pF) = Ψ∗(df)(pF). +And we have +Ψ∗ ◦ Ψ∗ = I +and +Ψ∗ ◦ Ψ∗ = I. +(7.9) +Indeed Ψ∗(Ψ∗αE)(pE) = Ψ∗αE(pF).dΨ(pE) = αE(pE).dΨ−1(pF).dΨ(pE) = αE(pE). Idem for Ψ∗ ◦ Ψ∗ = I. +Remark 7.5 The pull-back αF ∗ can also be defined thanks to the natural canonical isomorphism +� +L(E; F) → L(F ∗; E∗) +L → L∗ +� +given by L∗(ℓF ).⃗uE = ℓF .(L.⃗uE) for all (⃗uE, ℓF ) ∈ E×F ∗, and L∗(ℓF ) = ℓF .L +is called the pull-back of ℓF by L. +In particular with ℓF += αF(pF) and L = dΨ(pE) we get +dΨ(pE)∗(αF(pF)) = αF(pF).dΨ(pE), i.e. (7.5). +7.2 +Incompatibility: Riesz representation and push-forward +A push-forward is independent of any inner dot product: It is objective. +But here we introduce inner dot products (·, ·)g in E and (·, ·)h in F, e.g. Euclidean dot products +in ⃗Rn +t0 and ⃗Rn +t (observer dependent therefore subjective), because some mechanical engineers can’t begin +with their beloved Euclidean dot products. +Let αE ∈ Ω1(UE) and call βF := Ψ∗αE its push-forward by Ψ, i.e. +βF(pF) := αE(pE).dΨ(pE)−1 +when +pF = Ψ(pE). +(7.10) +Then call ⃗ag(pE) ∈ E and ⃗bh(pF) ∈ F the (·, ·)g and (·, ·)h-Riesz representation vectors of αE and βF, so, +for all ⃗uE ∈ Γ(UE) and all ⃗wF ∈ Γ(UF), in short, +αE.⃗uE = (⃗ag, ⃗uE)g, +and +βF.⃗wF = (⃗bh, ⃗wF)h, +(7.11) +which means αE(pE).⃗uE(pE) = (⃗ag(pE), ⃗uE(pE))g and βF(pF).⃗wF(pF) = (⃗bh(pF), ⃗wF(pF))h, for all pE ∈ UE +and pF ∈ UF. This defines the vector fields ⃗ag ∈ Γ(UE) and ⃗bh ∈ Γ(UF). +Proposition 7.6 ⃗bh ̸= Ψ∗⃗ag in general (although βF = Ψ∗αE), because +⃗bh(pF) = dΨ(pE)−T .⃗ag(pE) +̸= dΨ(pE).⃗ag(pE) in general +(7.12) +(unless dΨ(pE)−T = dΨ(pE), i.e. dΨ(pE)T .dΨ(pE)−1 = I, as a rigid body motion). +So the Riesz representation vector of the push-forwarded linear form is not the push-forwarded rep- +resentation vector of the linear form push-forwarded. +This is not a surprise: A push-forward is independent of any inner dot product, while a Riesz repre- +sentation vector depends on a chosen inner dot product (E.g. Euclidean foot? metre?). +So, as long as possible (not before you need to quantify), you should avoid using a Riesz representation +vector, i.e. you should use the original (the qualitative differential form) as long as possible, and delay +the use of a representative (quantification with which dot product?) as late as possible. +Proof. Recall: The transposed relative to (·, ·)g and (·, ·)h of the linear map dΨ(pE) ∈ L(E; F) is the +linear map dΨ(pE)T +gh =noted dΨ(pE)T ∈ L(F; E) defined by, for all ⃗uE ∈ E and ⃗wF ∈ F vectors at pE +and pF, cf. (A.68), +(dΨ(pE)T .⃗wF, ⃗uE)g = (⃗wF, dΨ(pE).⃗uE)h. +(7.13) +(7.11) gives, with pF = Ψ(pE), +(⃗ag(pE), ⃗uE)g = αE(pE).⃗uE = +� +βF(pF).dΨ(pE) +� +.⃗uE = βF(pF). +� +dΨ(pE).⃗uE +� += (⃗bh(pF), dΨ(pE).⃗uE)h = (dΨ(pE)T .⃗bh(pF), ⃗uE)g, +(7.14) +true for all ⃗uE, thus ⃗ag(pE) = dΨ(pE)T .⃗bh(pF), thus (7.12). +47 + +48 +8.1. +Push-forward and pull-back of order 1 tensors +8 +Push-forward and pull-back of tensors +To lighten the presentation, we only deal with order 1 and 2 tensors. Similar approach for any tensor. +8.1 +Push-forward and pull-back of order 1 tensors +Proposition 8.1 If T is either a vector field or a differential form, then its push-forward satisfies, for +all ξ vector field or differential form (when required) in UF, +in short: +(Ψ∗T)(ξ) = T(Ψ∗ξ), +written +Ψ∗T(.) = T(Ψ∗.), +(8.1) +i.e. (Ψ∗T)(pF).ξ(pF) = T(pE).Ψ∗ξ(pE) when pF = Ψ(pE). Similarly: +in short: +(Ψ∗T)(ξ) = T(Ψ∗ξ), +written +Ψ∗T(.) = T(Ψ∗.), +(8.2) +i.e. (Ψ∗T)(pE).ξ(pE) = T(pF).Ψ∗ξ(pF) when pF = Ψ(pE). +Proof. • Case T = αE ∈ Ω1(UE) (differential form = a +�0 +1 +� +tensor), then here ξ = ⃗wF ∈ Γ(UF) +and we have to check: +(Ψ∗αE)(pF).⃗wF(pF) = αE(pE).Ψ∗ ⃗wF(pE), i.e. (αE(pE).dΨ−1(pE)).⃗wF(pF) = +αE(pE).(dΨ−1(pE).⃗wF(pF)): True. +• Case T = ⃗wE ∈ Γ(UE) (vector field ≃ a +�1 +0 +� +tensor), then here ξ = αF ∈ Ω1(UF) we have to check: +(Ψ∗ ⃗wE)(pF).αF(pF) = ⃗wE(pE).Ψ∗(αF)(pE), where we implicitly use to the natural canonical isomorphism +J : +� E → E∗∗ +⃗w → w noted += +⃗w +� +defined by w(ℓ) = ℓ.⃗w for all ℓ ∈ E∗. So we have to check: αF(pF).(Ψ∗ ⃗wE)(pF) = +Ψ∗(αF)(pE).⃗wE(pE), i.e. αF(pF).(dΨ(pE).⃗wE(pE)) = (αF(pF).dΨ(pE)−1).⃗wE)(pE) : True. +For (8.2), use Ψ−1 instead of Ψ. +8.2 +Push-forward and pull-back of order 2 tensors +Definition 8.2 Let T be an order 2 tensor in UE. Its push-forward by Ψ is the order 2 tensor Ψ∗T in UF +defined by, for all ξ1, ξ2 vector field or differential form (when required) in UF, +in short: +Ψ∗T(ξ1, ξ2) := T(Ψ∗ξ1, Ψ∗ξ2) +written +Ψ∗T(·, ·) := T(Ψ∗·, Ψ∗·), +(8.3) +i.e. Ψ∗T(pF)(ξ1(pF), ξ2(pF)) := T(pE)(Ψ∗ξ1(pE), Ψ∗ξ2(pE)) when pF = Ψ(pE). +Let T be an order 2 tensor in UF. Its pull-back by Ψ is the order 2 tensor Ψ∗T in UE defined by, for +all ξ1, ξ2 vector field or differential form (when required) in UE, +in short: +Ψ∗T(ξ1, ξ2) := T(Ψ∗ξ1, Ψ∗ξ2) +written +Ψ∗T(·, ·) := T(Ψ∗·, Ψ∗·), +(8.4) +i.e., Ψ∗T(pE)(ξ1(pE), ξ2(pE)) := T(pF)(Ψ∗ξ1(pF), Ψ∗ξ2(pF)) when pF = Ψ(pE). +Example 8.3 If T ∈ T 0 +2 (UE) (e.g., a metric) then, for all vector fields ⃗w1, ⃗w2 in UF, +T∗(⃗w1, ⃗w2) +(8.3) += T(⃗w1 +∗, ⃗w2 +∗) = T(dΨ−1.⃗w1, dΨ−1.⃗w2), +(8.5) +i.e., T∗(pF)(⃗w1(pF), ⃗w2(pF)) = T(pE)(dΨ−1(pF).⃗w1(pF), dΨ−1(pF).⃗w2(pF)) when pF = Ψ(pE). +Expression with bases (⃗ai) in E and (⃗bi) in F: In short we have (T∗)ij = T∗(⃗bi,⃗bj) = T(⃗bi∗,⃗bj∗) = +[⃗b∗ +i ]T +|⃗a.[T]|⃗a.[⃗b∗ +j]|⃗a = ([⃗bi]T +|⃗b.[dΨ]−T +|⃗a,⃗b).[T]|⃗a.([dΨ]−1 +|⃗a,⃗b.[⃗bj]|⃗b) = ([dΨ]−T +|⃗a,⃗b.[T]|⃗a.[dΨ]−1 +|⃗a,⃗b)ij, thus +[T∗]|⃗b = [dΨ]−T +|⃗a,⃗b.[T]|⃗a.[dΨ]−1 +|⃗a,⃗b, +(8.6) +which means [(Ψ∗T)(pF)]|⃗b = ([dΨ(pE)]|⃗a,⃗b)−T .[T(pE)]|⃗a.([dΨ(pE)]|⃗a,⃗b)−1 when pF = Ψ(pE). +Particular case of an elementary tensor T = α1 ⊗ α2 ∈ T 0 +2 (UE), where α1, α2 ∈ Ω1(UE), so T(⃗u1, ⃗u2) = +(α1 ⊗ α2)(⃗u1, ⃗u2) = (α1.⃗u1)(α2.⃗u2): For all ⃗w1, ⃗w2 ∈ Γ(UF), +(α1 ⊗ α2)∗(⃗w1, ⃗w2) +(8.3) += (α1 ⊗ α2)(⃗w∗ +1, ⃗w∗ +2) = (α1.⃗w∗ +1)(α2.⃗w∗ +2) +(7.7) += (α1∗.⃗w1)(α2∗.⃗w2), +(8.7) +thus +(α1 ⊗ α2)∗ = α1∗ ⊗ α2∗. +(8.8) +(And any tensor is a finite sum of elementary tensors.) +48 + +49 +8.3. +Push-forward and pull-back of endomorphisms +And for the pull-back: For all vector fields ⃗u1, ⃗u2 in UE, +T ∗(⃗u1, ⃗u2) +(8.3) += T(⃗u1∗, ⃗u2∗) = T(dΨ.⃗u1, dΨ.⃗u2). +(8.9) +Example 8.4 If T ∈ T 1 +1 (UE) then for all vector fields ⃗w ∈ Γ(UF) and differential forms β ∈ Ω1(UF), +T∗(β, ⃗w) = T(β∗, ⃗w∗) = T(β.dΨ, dΨ−1.⃗w), +(8.10) +i.e., T∗(pF)(β(pF), ⃗w(pF)) = T(pE)(β(pF).dΨ(pE), dΨ−1(pF).⃗w(pF)) when pF = Ψ(pE). +For the elementary tensor T = ⃗u ⊗ α ∈ T 1 +1 (UE), made of the vector field ⃗u ∈ Γ(UE) and of the +differential form α ∈ Ω1(UE): For all β, ⃗w ∈ Ω1(UF) × Γ(UF), in short, +(⃗u ⊗ α)∗(β, ⃗w) +(8.3) += (⃗u ⊗ α)(β∗, ⃗w∗) = (⃗u.β∗)(α.⃗w∗) +(7.7) += (⃗u∗.β)(α∗.⃗w) = (⃗u∗ ⊗ α∗)(β, ⃗w), +(8.11) +thus +(⃗u ⊗ α)∗ = ⃗u∗ ⊗ α∗. +(8.12) +Expression with bases (⃗ai) in E and (⃗bi) in F: +In short we have (T∗)ij += +T∗(bi,⃗bj) += +T(Ψ∗(bi), Ψ∗(⃗bj)) = [Ψ∗(bi)].[T].[Ψ∗(⃗bj)] = [bi].[dΨ].[T].[dΨ−1].[⃗bj] = ([dΨ].[T].[dΨ−1])ij, thus +[T∗]|⃗b = [dΨ]|⃗a,⃗b.[T]|⃗a.[dΨ]−1 +|⃗a,⃗b, +(8.13) +which means [(Ψ∗T)(pF)]|⃗b = [dΨ(pE)]|⃗a,⃗b.[T(pE)]|⃗a.[dΨ(pE)]−1 +|⃗a,⃗b when pF = Ψ(pE). +8.3 +Push-forward and pull-back of endomorphisms +We have the natural canonical isomorphism +J2 : +� +L(E; E) → L(E∗, E; R) +L → TL = J2(L) +where +TL(α, ⃗u) := α.L.⃗u, +∀(α, ⃗u) ∈ E∗ × E. +(8.14) +Thus Ψ∗TL(m, ⃗w) = TL(Ψ∗m, Ψ∗ ⃗w) = (Ψ∗m).L.(Ψ∗ ⃗w) = m.dΨ.L.dΨ−1.⃗w, thus: +Definition 8.5 The push-forward by Ψ of a field of endomorphisms L on UE is the field of endomorphisms +Ψ∗L = L∗ on UF defined by +in short: +Ψ∗L = L∗ = dΨ.L.dΨ−1 , +(8.15) +i.e., L∗(pF) = dΨ(pE).L(pE).dΨ−1(pF) when pF = Ψ(pE). +Thus with bases we get [L∗]|⃗b = [dΨ]|⃗a,⃗b.[L]|⃗a.[dΨ]−1 +|⃗a,⃗b, “as in (8.13)”. +Example 8.6 Elementary field of endomorphisms L = (J2)−1(⃗u ⊗ α), where ⃗u ∈ Γ(E) and α ∈ Ω1(E): +So TL = ⃗u ⊗ α and L.⃗u2 = (α.⃗u2)⃗u for all ⃗u2 ∈ Γ(UE)). Thus L∗.⃗w2 = dΨ.L.dΨ−1.⃗w2 = dΨ.L.⃗w2∗ = +(α.⃗w2∗)dΨ.⃗u = (α∗.⃗w2)⃗u∗ for all ⃗w2 ∈ Γ(E), thus (TL)∗ = ⃗u∗ ⊗ α∗. +Definition 8.7 Let L be a field of endomorphisms on UF. Its pull-back by Ψ is the field of endomorphisms +Ψ∗L = L∗ on UE defined by +in short: +Ψ∗L = L∗ = dΨ−1.L.dΨ , +(8.16) +i.e., L∗(pE) = dΨ−1(pF).L(pF).dΨ(pE) when pF = Ψ(pE). +8.4 +Application to derivatives of vector fields +⃗u ∈ Γ(UE) is a C1 vector field in UE), pE ∈ UE, so d⃗u : UE → L(E; E) (given by d⃗u(pE).⃗w(pE) = +limh→0 +⃗u(pE+h⃗w(pE))−⃗u(pE) +h +for all ⃗w ∈ Γ(UE)). Thus its push-forward: +((d⃗u)∗ =) +Ψ∗(d⃗u) = dΨ.d⃗u.dΨ−1 +(8.17) +i.e. (d⃗u)∗(pF) = dΨ(pE).d⃗u(pE).dΨ(pE)−1 when pF = Ψ(pE). +49 + +50 +8.5. +Ψ∗(d⃗u) versus d(Ψ∗⃗u): No commutativity +8.5 +Ψ∗(d⃗u) versus d(Ψ∗⃗u): No commutativity +Here Ψ is C2, ⃗u ∈ Γ(UE), pE ∈ UE, pF = Ψ(pE), so Ψ∗⃗u(pF) = dΨ(pE).⃗u(pE) = (dΨ(Ψ−1(pF)).(⃗u(Ψ−1(pF)), +and, for all ⃗w ∈ Γ(UF), +d(Ψ∗⃗u)(pF).⃗w(pF) = (d2Ψ(pE).(dΨ−1(pF).⃗w(pF))).⃗u(pE) + dΨ(pE).d⃗u(pE).dΨ−1(pF).⃗w(pF), +(8.18) +with Ψ∗(d⃗u)(pF) = dΨ(pE).d⃗u(pE).dΨ−1(pF), thus, in short, +d(Ψ∗⃗u).⃗w = Ψ∗(d⃗u).⃗w + d2Ψ(Ψ∗ ⃗w, ⃗u) ̸= Ψ∗(d⃗u) +in general. +(8.19) +So the differentiation d and the push-forward ∗ do not commute (d(Ψ∗⃗u) = Ψ∗(d⃗u) iff Ψ is affine). +8.6 +Application to derivative of differential forms +Let α ∈ Ω1(UE) (a differential form on UE). Its derivative dα : UE → L(E; E∗) is given by dα(pE).⃗u(pE) = +limh→0 +α(pE+h⃗u(pE))−α(pE) +h +∈ E∗, for all ⃗u ∈ Γ(UE), i.e., for all ⃗u1, ⃗u2 ∈ Γ(UE), +(dα(pE).⃗u1(pE)).⃗u2(pE) = lim +h→0 +α(pE + h⃗u1(pE)).⃗u2(pE) − (α(pE).⃗u1(pE)).⃗u2(pE) +h +∈ R. +(8.20) +With the natural canonical isomorphism L(E; E∗) ≃ L(E, E; R), cf. (T.16) with E∗∗ ≃ E, we can write +dα(pE)(⃗u1(pE)).⃗u2(pE) = dα(pE)(⃗u1(pE), ⃗u2(pE)), i.e. +dα(⃗u1).⃗u2 = dα(⃗u1, ⃗u2). +(8.21) +Thus the push-forward Ψ∗(dα) =noted (dα)∗ of dα, is given by, for all ⃗w1, ⃗w2 ∈ Γ(UF), in short, +(dα)∗(⃗w1, ⃗w2) = dα(⃗w∗ +1, ⃗w∗ +2), +(8.22) +i.e., with pF = Ψ(pE), (dα)∗(pF).⃗w1(pF)).⃗w2(pF) = (dα(pE).dΨ−1(pF).⃗w1(pF)).dΨ−1(pF).⃗w2(pF). +In particular, (d2f)∗(⃗w1, ⃗w2) = d2f(dΨ−1.⃗w1, dΨ−1.⃗w2) (= d2f(⃗w∗ +1, ⃗w∗ +2)). +8.7 +Ψ∗(dα) versus d(Ψ∗α): No commutativity +Here Ψ is C2, ⃗u ∈ Γ(UE), pE ∈ UE and pF = Ψ(pE). +We have Ψ∗α(pF) = α(pE).dΨ−1(pF) = +α(Ψ−1(pF)).dΨ−1(pF), thus, for all ⃗w1 ∈ Γ(UF), +d(ψ∗α)(pF).⃗w1(pF) = (dα(pE).dΨ−1(pF).⃗w1(pF)).dΨ−1(pF) + α(pE).d2Ψ−1(pF).⃗w1(pF) ∈ F ∗, +(8.23) +thus, for all ⃗w1, ⃗w2 ∈ Γ(UF), in short +d(ψ∗α)(⃗w1, ⃗w2) = dα(dΨ−1.⃗w1, dΨ−1.⃗w2) + α.d2Ψ−1(⃗w1, ⃗w2) ̸= dα(⃗w∗ +1, ⃗w∗ +2) +in general. +(8.24) +So the differentiation d and the push-forward ∗ do not commute (d(Ψ∗α) = Ψ∗(dα) iff Ψ is affine). +50 + +51 +Part III +Lie derivative +9 +Lie derivative +9.0 +Purpose and first results +9.0.1 +Purpose? +Cauchy’s approach may be insufficient, e.g.: +1. - Cauchy’s approach aims to compare two vectors deformed by a motion, thanks to a Euclidean dot +product and the deformation gradient F, with the deformation tensor C defined by (C. ⃗W1) • ⃗W2 := +(F. ⃗W1) • (F. ⃗W2). It is a quantitative approach (needs a chosen Euclidean dot product: foot? metre?). +- Cauchy’s approach is a first order method (dedicated to linear material): Only the first order Taylor +expansion of the motion is used: Only dΦ = F is used (the “slope”), not d2Φ = dF (the “curvature”) +or higher derivatives. +2. - Lie’s approach aims to build qualitative “covariant objective constitutive laws” (some will be discred- +ited afterward, because of invariance or thermodynamical requirements). +- Lie’s approach “naturally” applies to non-linear materials thanks to second order Lie derivatives which +uses the second order Taylor expansion of the motion. +- In a non planar surface S, you need the Lie derivative if you want to derive along a trajectory. +- In a Galilean Euclidean framework (quantification), the first order Lie derivatives approach give the +same results than Cauchy’s approach. +(Cauchy died in 1857, and Lie was born in 1842: Unfortunately Cauchy could not use the Lie derivative.) +9.0.2 +Basic results +The Eulerian velocity of the motion is ⃗v. With the material derivative is DEul +Dt := ∂Eul +∂t + dEul.⃗v: +1. The Lie derivative L⃗vf of a Eulerian scalar valued function f is the material derivative +L⃗vf = Df +Dt . +(9.1) +2. The Lie derivative L⃗v ⃗w of a (Eulerian) vector field ⃗w is more than just the material derivative D ⃗w +Dt : +L⃗v ⃗w = D ⃗w +Dt − d⃗v.⃗w. +(9.2) +L⃗v ⃗w gives the rate of stress on ⃗w due to a flow, and in particular the −d⃗v.⃗w term in L⃗v ⃗w tells that +the spatial variations of ⃗v (variations of the flow) act on the evolution of the stress (anticipated). +3. (9.1)-(9.2) enable to define the Lie derivatives of tensors of any order. +9.1 +Definition +9.1.1 +Issue (ubiquity gift)... +�Φ is supposed to be regular. ⃗v(t, p(t)) = ∂�Φ +∂t (t, PObj) is the Eulerian velocity at t at p(t) = �Φ(t, PObj). +Recall: If Eul is a Eulerian function then its material time derivative is +DEul +Dt (t, p(t)) = lim +h→0 +Eul(t+h, p(t+h)) − Eul(t, p(t)) +h +. +(9.3) +Issue: The rate Eul(t+h,p(t+h))−Eul(t,p(t)) +h +raises questions: +1- The difference Eul(t+h, p(t+h)) − Eul(t, p(t)) requires the time and space ubiquity gift to be cal- +culated by an observer, since it mixes two distinct times, t and t+h, and two distinct locations, p(t) +and p(t+h). +2- The difference Eul(t+h, p(t+h)) − Eul(t, p(t)) can be impossible: E.g. if Eul = ⃗w is a vector field +in a “non planar surface considered on its own” (manifold) then Eul(t+h, p(t+h)) and Eul(t, p(t)) don’t +belong to the same (tangent) vector space (so the difference ⃗w(t+h, p(t+h)) − ⃗w(t, p(t)) is meaningless). +51 + +52 +9.1. +Definition +9.1.2 +...Toward a solution (without ubiquity gift)... +To compare Eul(t+h, p(t+h)) and Eul(t, p(t)) (to get the evolution of Eul along a trajectory), you need +the duration h to get from t to t+h and to move from p(t) to p(t+h). So, you must: +• take the value Eul(t, pt)) with you (for memory), +• move along the considered trajectory, and doing so, the value Eul(t, pt) has possibly changed to, +with τ = t+h, +((Φt +τ)∗Eult)(pτ) noted += +Eult∗(τ, pτ) +(push-forward); +(9.4) +• And now, at (τ, pτ) where you are, you can compare the actual value Eul(τ, pτ) with the value +Eult∗(τ, pτ) you arrived with (the transported memory), thus the difference +Eul(τ, pτ) − Eult∗(τ, pτ) +(9.5) +is meaningful for a human being since it is computed at a unique time τ and at a unique point pτ (no +gift of ubiquity required). +Figure 9.1: To compute (9.5) with Eul = ⃗w a (Eulerian) vector field: At t define the vector field ⃗wt in Ωt +by ⃗wt(pt) := ⃗w(t, pt). The (spatial) curve ct : s → pt = ct(s) in Ωt is an integral curve of ⃗wt, i.e. satisfies +ct′(s) = ⃗wt(ct(s)). ct is transformed by Φt +τ into the (spatial) curve cτ = Φt +τ ◦ct : s → pτ = cτ(s)=Φt +τ(ct(s)) +in Ωτ; Hence cτ ′(s) = dΦt +τ(pt).c′(s) = dΦt +τ(pt).⃗wt(pt) =noted ⃗wt∗(τ, pτ) is the tangent vector at cτ at pτ +(push-forward). Thus the difference ⃗w(τ, pτ)− ⃗wt∗(τ, pτ) can be computed by a human being, i.e. without +ubiquity gift. +9.1.3 +... The Lie derivative, first definition +Definition 9.1 The Lie derivative L⃗vEul along ⃗v of an Eulerian function Eul is the Eulerian function +L⃗vEul defined by, at t at pt = �Φ(t, PObj), +L⃗vEul(t, pt) := lim +h→0 +Eul(t+h, p(t+h)) − (Φt +t+h)∗Eult(p(t+h)) +h +. +(9.6) +Interpretation: L⃗vEul measures the rate of change of Eul along a trajectory: +• Eul(t+h, p(t+h)) is the value of Eul at t+h at p(t+h), see figure 9.1. +• Eult∗(t+h, p(t+h)) = ((Φt +t+h)∗Eult)(t+h, p(t+h)) is exclusively strain related: It is the memory +transported along a flow, i.e. the value Eul(t, pt) distorted by the flow. +So, with g defined by g(τ) = ((Φt +τ)∗Eult)(pτ) (in particular g(t) = Eult(pt)): +L⃗vEul(t, pt) := g′(t) = lim +τ→t +g(τ) − g(t) +τ − t +also written += +d((Φt +t+h)∗Eult)(p(t+h)) +dt +|τ=t. +(9.7) +9.1.4 +A more general definition +The rate in (9.6) has to be slightly modified to be adequate in all situations: +Eul(t+h, p(t+h)) − +Eul∗(t+h, p(t+h)) is computed at (t+h, p(t+h)) which moves as h → 0, and on a “non-planar mani- +fold” this is problematic. The “natural” definition is to arrive with the memory: +52 + +ex(E, Pe) +2 +2 +pz= 中(p)= Ex (b) +Pt=C (p)53 +9.2. +Lie derivative of a scalar function +Definition 9.2 The Lie derivative L⃗vEul along ⃗v of an Eulerian function Eul is the Eulerian function +L⃗vEul defined by, at t at pt = �ΦPObj (t), +L⃗vEul(t, pt) := lim +h→0 +Eul(t, pt) − (Φt−h +t +)∗Eult−h(pt) +h +. +(9.8) +I.e. with �g defined by �g(τ) = ((Φτ +t )∗Eulτ)(pt) (in particular �g(t) = Eul(t, pt)): +L⃗vEul(t, pt) := �g′(t) = lim +τ→t +�g(t) − �g(τ) +t − τ += lim +τ→t +�g(τ) − �g(t) +τ − t +also written += +d((Φτ +t )∗Eulτ)(pt) +dτ +|τ=t. +(9.9) +Here the observer must: +• At t−h at p(t−h) = �ΦPObj (t−h), take the value Eul(t−h, p(t−h)), +• travel along the trajectory �ΦPObj , +• once at t at pt = �ΦPObj (t), this value has become ((Φt +t−h)∗Eult−h)(pt) (transported memory), +• and then the comparison with Eul(t, pt) can be done in Ωt (no ubiquity gift required). +Exercice 9.3 Prove: (9.6) and (9.8) are equivalent. +Answer. +With (Φt +t+h)∗.(Φt +t+h)∗ += +I, +(9.6) gives L⃗vEul(t, pt) += +limh→0 +(Φt +t+h)∗Eul(t,pt)−Eult(t,pt) +h += +limh→0 +(Φt +t−h)∗Eult−h)(pt)−Eult(pt) +−h += limh→0 +Eult(pt)−((Φt +t−h)∗Eult−h)(pt) +h +, and (Φt +t−h)∗ = (Φt−h +t +)∗. +9.1.5 +Equivalent definition (differential geometry) +Definition 9.4 The Lie derivative of a Eulerian function Eul along a flow of Eulerian velocity ⃗v is the +Eulerian function L⃗vEul defined at (t, pt) by +L⃗vEul(t, pt) := lim +h→0 +((Φt +t+h)∗Eult+h)(pt) − Eul(t, pt) +h +, +rate in Ωt. +(9.10) +In other words, if ˆg is defined by ˆg(τ) = ((Φt +τ)∗Eulτ)(pt) (in particular ˆg(t) = Eul(t, pt)), then +L⃗vEul(t, pt) := ˆg′(t) = lim +τ→t +ˆg(τ) − ˆg(t) +τ − t +also written += +d((Φt +τ)∗Eulτ)(pt) +dτ +|τ=t. +(9.11) +Exercice 9.5 Prove: (9.8) and (9.10) are equivalent. +Answer. (9.10) also reads L⃗vEul(t, pt) = limh→0 +((Φt +t−h)∗Eult−h)(pt)−Eult(pt) +−h +, and (Φt +t−h)∗.(Φt−h +t +)∗ = I. +Remark 9.6 More precise definition, as in (2.3): E.g. with (9.10), the Lie derivative �L⃗vEul of a Eulerian +function � +Eul along a flow of Eulerian velocity ⃗v is the Eulerian function defined by, at t at pt = �Φ(t, PObj), +�L⃗vEul(t, pt) := ((t, pt), L⃗vEul(t, pt) +(pointed function at (t, pt)), +(9.12) +And, to lighten the notation, �L⃗vEul(t, pt) =noted L⃗vEul(t, pt) (second component of �L⃗vEul(t, pt)). +9.2 +Lie derivative of a scalar function +Let f be a C1 Eulerian scalar valued function. With (Φt−h +t +)∗ft−h(pt) = ft−h(p(t−h)), cf. (6.10), we get +L⃗vf(t, pt) +(9.8) += +lim +h→0 +f(t, pt) − f(t−h, p(t−h)) +h +, +i.e. +L⃗vf = Df +Dt += ∂f +∂t + df.⃗v. +(9.13) +So, for scalar functions, the Lie derivative is the material derivative. +Interpretation: L⃗vf measures the rate of change of f along a trajectory. +Proposition 9.7 L⃗vf = 0 iff f is constant along any trajectory (the real value is the memory value): +L⃗vf = 0 +⇐⇒ +∀t, τ ∈ [t0, T], (Φt +τ ∗)ft(pτ) = f(t, p(t)) when pτ = Φt +τ(pt), +(9.14) +i.e. iff f(t, p(t)) = f(t0, pt0) when p(t) = Φt0(t, pt0), i.e. iff f let itself be carried by the flow (unchanged). +53 + +54 +9.3. +Lie derivative of a vector field +Proof. Let p(t) = �Φ(t, PObj) = pt for all t, so p(τ) = �Φ(τ, PObj) = pτ = Φt +t+h(pt) = Φt(τ, pt). +⇐: If fτ = (Φt +t+h)∗ft, then fτ(pτ) = ft(pt), thus limτ→t +f(τ,p(τ))−f(t,p(t)) +τ−t += 0, that is, Df +Dt = 0. +⇒: If Df +Dt = 0 then f(t, p(t)) is a constant function on the trajectory t → �Φ(t, PObj), for any parti- +cle PObj, so f(τ, p(τ)) = f(t, pt) when p(τ) = Φt +t+h(pt), that is, f(τ, pτ) = (Φt +t+h)∗ft(pτ). +Exercice 9.8 Prove: L⃗v(L⃗vf) = D2f +Dt2 = ∂2f +∂t2 + 2d( ∂f +∂t ).⃗v + d2f(⃗v,⃗v) + df.( ∂⃗v +∂t + d⃗v). +Answer. See (2.23). +9.3 +Lie derivative of a vector field +9.3.1 +Formula +Let ⃗w be a C1 (Eulerian) vector field (interpreted as an “internal force field” in the following). +Proposition 9.9 +L⃗v ⃗w = D ⃗w +Dt − d⃗v.⃗w = ∂ ⃗w +∂t + d⃗w.⃗v − d⃗v.⃗w. +(9.15) +So the Lie derivative is not reduced to the material derivative D ⃗w +Dt (unless d⃗v = 0, i.e. unless ⃗v is uniform): +The spatial variations d⃗v of ⃗v influences the rate of stress: ⃗v tries to bend ⃗w (which is expected). +Proof. Let ⃗g : τ → ⃗g(τ) = (Φt∗ +τ ⃗w)(t, p(t)) = dΦt +τ(pt)−1.⃗w(τ, p(τ)) when p(τ) = Φt(τ, pt), so (9.10) reads +L⃗v ⃗w(t, pt) = ⃗g ′(t). With ⃗z(τ) := ⃗w(τ, p(τ)) = dΦt(τ, pt).⃗g(τ), +⃗z ′(τ) = D ⃗w +Dτ (τ, p(τ)) = ∂(dΦt) +∂τ +(τ, pt).⃗g(τ) + dΦt(τ, pt).⃗g ′(τ) += (d⃗v(τ, p(τ)).F t(τ, pt)).(F t(τ, pt)−1.⃗w(τ, p(τ))) + F t +τ(pt).⃗g ′(τ) += d⃗v(τ, p(τ)).⃗w(τ, p(τ)) + F t +τ(pt).⃗g ′(τ). +(9.16) +Thus D ⃗w +Dt (t, pt) = d⃗v(t, pt).⃗w(t, pt) + I.⃗g ′(t), thus ⃗g ′(t) = D ⃗w +Dt (t, pt) − d⃗v(t, pt).⃗w(t, pt). +Quantification: Basis (⃗ei), ⃗v = � +i vi⃗ei, ⃗w = � +i wi⃗ei, d⃗v.⃗ej = � +ij vi|j⃗ei, d⃗w.⃗ej = � +ij wi|j⃗ei; Then +L⃗v ⃗w = +n +� +i=1 +∂wi +∂t ⃗ei + +n +� +i,j=1 +wi|jvj⃗ei − +n +� +i,j=1 +vi|jwj⃗ei. +(9.17) +So (column matrix), with [·] := [·]|⃗e, +[L⃗v ⃗w] = [D ⃗w +Dt ] − [d⃗v].[⃗w] +(= [∂ ⃗w +∂t ] + [d⃗w.⃗v] − [d⃗v].[⃗w]). +(9.18) +(And [d⃗w.⃗v] = [d⃗w].[⃗v].) Duality notations: L⃗v ⃗w = � +i +∂wi +∂t ⃗ei + � +ij wi +|jvj⃗ei − � +ij vi +|jwj⃗ei. +9.3.2 +Interpretation: Flow resistance measurement +Proposition 9.10 Φt0 is supposed to be a C2 motion and a C1 diffeomorphism in space, and ⃗w is a +vector field. +L⃗v ⃗w = 0 +⇐⇒ +∀t ∈ [t0, T], +⃗wt = (Φt0 +t )∗ ⃗wt0. +(9.19) +i.e., D ⃗w +Dt = d⃗v.⃗w ⇔ the actual vector ⃗w(t, p(t)) is equal to F t0 +t (pt0).⃗wt0(pt0) = ⃗wt0∗(t, p(t)) the deformed +vector by the flow, see figure 9.1. So: The Lie derivative L⃗v ⃗w vanishes iff ⃗w does not resist the flow (let +itself be deformed by the flow), i.e. iff ⃗w(t, pt) = ⃗wt0∗(t, pt). +Proof. We have L⃗v ⃗w = D ⃗w +Dt − d⃗v.⃗w and ∂F t0 +∂t (t, pt0) = d⃗v(t, p(t)).F t0 +t (pt0), cf. (3.33). +⇐ (derivation): Suppose ⃗w(t, p(t)) = F t0(t, pt0).⃗w(t0, pt0) when p(t) = Φt0 +t (pt0). Then D ⃗w +Dt (t, p(t)) = +∂F t0 +∂t (t, pt0).⃗w(t0, pt0) = (d⃗v(t, p(t)).F t0 +t (pt0)).(F t0 +t (pt0)−1.⃗w(t, p(t))) = d⃗v(t, p(t)).⃗w(t, p(t)), thus +D ⃗w +Dt − +d⃗v.⃗w = 0. (See proposition 3.14.) +⇒ (integration): +Suppose +D ⃗w +Dt += +d⃗v.⃗w. +Let +⃗f(t) += +(F t0 +t (pt0))−1.⃗w(t, p(t)) (= pull-back +(Φt0 +t )∗ ⃗w(t0, pt0)) when p(t) += +Φt0(t, pt0); +So +⃗w(t, p(t)) += +F t0(t, pt0).⃗f(t) and +D ⃗w +Dt (t, p(t)) += +∂F t0 +∂t (t, pt0).⃗f(t) + F t0 +t (pt0).⃗f ′(t) = d⃗v(t, p(t)).F t0 +t (pt0).⃗f(t) + F t0 +t (pt0).⃗f ′(t) = d⃗v(t, p(t)).⃗w(t, p(t)) + +F t0 +t (pt0).⃗f ′(t) =hyp. D ⃗w +Dt (t, p(t))+F t0 +t (pt0).⃗f ′(t) for all t; Thus F t0 +t (pt0).⃗f ′(t) = ⃗0, thus ⃗f ′(t) = ⃗0 (because +Φt0 +t is a diffeomorphism), thus ⃗f(t) = ⃗f(t0), i.e. ⃗wt = (Φt0 +t )∗ ⃗wt0, for all t. +54 + +55 +9.4. +Examples +9.3.3 +Autonomous Lie derivative and Lie bracket +The Lie bracket of two vector fields ⃗v and ⃗w is +[⃗v, ⃗w] := d⃗w.⃗v − d⃗v.⃗w noted += +L0 +⃗v ⃗w. +(9.20) +And L0 +⃗v ⃗w = [⃗v, ⃗w] is called the autonomous Lie derivative of ⃗w along ⃗v. Thus +L⃗v ⃗w = ∂ ⃗w +∂t + [⃗v, ⃗w] = ∂ ⃗w +∂t + L0 +⃗v ⃗w. +(9.21) +NB: L0 +⃗v ⃗w is used when ⃗v et ⃗w are stationary vector fields, thus does not concern objectivity: A stationary +vector field in a referential is not necessary stationary in another (moving) referential. +9.4 +Examples +9.4.1 +Lie Derivative of a vector field along itself +(9.15) with ⃗w = ⃗v gives L⃗v⃗v = ∂⃗v +∂t . In particular, if ⃗v is a stationary vector field then L⃗v⃗v = ⃗0 (= [⃗v,⃗v]). +9.4.2 +Lie derivative along a uniform flow +Here d⃗v = 0, thus +L⃗v ⃗w = D ⃗w +Dt = ∂ ⃗w +∂t + d⃗w.⃗v +(when d⃗v = 0). +(9.22) +Here the flow is rectilinear (d⃗v = 0): there is no curvature (of the flow) to influence the stress on ⃗w. +Moreover, if ⃗w is stationary, that is ∂ ⃗w +∂t = 0, then L⃗v ⃗w = d⃗w.⃗v = the directional derivative ∂ ⃗w +∂⃗v of the +vector field ⃗w in the direction ⃗v. +9.4.3 +Lie derivative of a uniform vector field +Here d⃗w(t, p) = 0, thus +L⃗v ⃗w = ∂ ⃗w +∂t − d⃗v.⃗w +(when d⃗w = 0), +(9.23) +thus the stress on ⃗w is due to the space variations of ⃗v. Moreover, is ⃗w is stationary then L⃗v ⃗w = −d⃗v.⃗w. +9.4.4 +Uniaxial stretch of an elastic material +• Strain. With [−−→ +OP]|⃗e = [ ⃗X]|⃗e = +� +X +Y +� +, with ξ > 0, t ≥ t0, p(t) = Φt0(t, P) and [⃗x]|⃗e = [−−−→ +Op(t)]|⃗e: +[⃗x]|⃗e = +� +x +y +� += +� +X +Y +� ++ ξ(t−t0) +� +X +0 +� += +� +X(1 + ξ(t−t0)) +Y +� +. +(9.24) +• Eulerian velocity ⃗v(t, p) = +� +ξX +0 +� += +� +ξ +1+ξ(t−t0)x +0 +� +, d⃗v(t, p) = +� +ξ +1+ξ(t−t0) +0 +0 +0 +� +(independent of p). +• Deformation gradient (independent of P), with κt = ξ(t−t0): +Ft = dΦt0 +t (P) = +� +1 + κt +0 +0 +1 +� += I + κt +� +1 +0 +0 +0 +� +. +(9.25) +Infinitesimal strain tensor, with F T +t = Ft here: +εt0 +t (P) = Ft − I = κt +� +1 +0 +0 +0 +� += εt. +(9.26) +• Stress. Constitutive law = Linear isotropic elasticity: +σt(pt) = λTr(εt)I + 2µεt = κt +� +λ+2µ +0 +0 +λ +� += σt. +(9.27) +Cauchy stress vector ⃗T on a surface at p with normal ⃗nt(p) = +� +n1 +n2 +� += ⃗n: +⃗Tt(pt) = σt.⃗n = κt +� +(λ+2µ)n1 +λn2 +� += ξ(t−t0) +� +(λ+2µ)n1 +λn2 +� += ⃗Tt. +(9.28) +• Push-forwards: ⃗Tt0(pt0) = 0, thus F t0 +t0+h(pt0).⃗Tt0(pt0) = ⃗0. +55 + +56 +9.4. +Examples +• Lie derivative: +L⃗v ⃗T(t0, pt0) = lim +t→t0 +⃗Tt(pt) − F t0 +t (pt0).⃗Tt0(pt0) +t − t0 += ξ +� +(λ+2µ)n1 +λn2 +� +(rate of stress at (t0, pt0)). +(9.29) +• Generic computation with L⃗v ⃗T += +∂ ⃗T +∂t + d⃗T.⃗v − d⃗v.⃗T: +(9.28) gives +∂ ⃗T +∂t += ξ +� +(λ+2µ) n1 +λ n2 +� +and +d⃗T = 0 and d⃗vt.⃗Tt = +� +ξ +1+ξ(t−t0) +0 +0 +0 +� +.ξ(t−t0) +� +(λ+2µ) n1 +λ n2 +� += +ξ2(t−t0) +1+ξ(t−t0) +� +(λ+2µ) n1 +0 +� +. +In particu- +lar, d⃗v(t0, pt0).⃗T(t0, pt0) = ⃗0. Thus L⃗v ⃗T(t0, pt0) = ξ +� +(λ+2µ) n1 +λ n2 +� += rate of stress at the initial (t0, pt0). +9.4.5 +Simple shear of an elastic material +Euclidean basis (⃗e1,⃗e2) in R2, the same basis at any time. Initial configuration Ωt0 = [0, L1] ⊗ [0, L2]. +Initial position [−−→ +OP]⃗e = [−−→ +Opt0]⃗e = [ ⃗X]⃗e = +� +X +Y +� +. Let ξ ∈ R∗, pt = Φt0 +t (pt0), [⃗x]|⃗e = [−−−→ +Op(t)]|⃗e, and +[⃗x]⃗e = +� +x = ϕ1(t, X, Y ) = X +y = ϕ2(t, X, Y ) +� += +� +X + ξ(t−t0)Y +Y +� +. +(9.30) +• Eulerian velocity ⃗vt(pt) = +� +ξY +0 +� += +� +ξy +0 +� +, thus d⃗vt(pt) = +� +0 +ξ +0 +0 +� +. +• Strain. With κt = ξ(t−t0), deformation gradient (independent of P): +dΦt0 +t (P) = +� +1 +κt +0 +1 +� += F t0 +t , +thus +F t0 +t +− I = κt +� +0 +1 +0 +0 +� +. +(9.31) +• Infinitesimal strain tensor: +εt0 +t (P) = F t0 +t (P)−I + (F t0 +t (P)−I)T +2 += κt +2 +� +0 +1 +1 +0 +� += εt. +(9.32) +• Stress. Constitutive law, usual linear isotropic elasticity (requires a Euclidean dot product): +σ(t, pt) = λTr(εt)I + 2µεt = µκt +� +0 +1 +1 +0 +� += σt. +(9.33) +Cauchy stress vector ⃗T(t, pt) (at t at pt) on a surface at p with normal ⃗nt(p) = +� +n1 +n2 +� += ⃗n: +⃗Tt = σt.⃗n = µκt +� +n2 +n1 +� += µξ(t−t0) +� +n2 +n1 +� += ⃗T(t) +(stress independent of pt). +(9.34) +• Lie derivative, with ⃗Tt0 = ⃗0: +L⃗v ⃗T(t0, pt0) = lim +t→t0 +⃗Tt(pt) − F t0 +t (pt0).⃗Tt0(pt0) +t − t0 += µξ +� +n2 +n1 +� +(rate of stress at (t0, pt0)). +(9.35) +• Generic computation: L⃗v ⃗T = ∂ ⃗T +∂t + d⃗T.⃗v − d⃗v.⃗T. (9.34) gives ∂ ⃗T +∂t (t, p) = µξ +� +n2 +n1 +� +and d⃗T = 0. With +d⃗vt0.⃗Tt0 = ⃗0. Thus L⃗v ⃗T(t0, pt0) = µξ +� +n2 +n1 +� +. +9.4.6 +Shear flow +Stationary shear field, see (5.11) with α = 0 and t0 = 0: +⃗v(x, y) = +� +v1(x, y) = λy, +v2(x, y) = 0, +d⃗v(x, y) = +� +0 +λ +0 +0 +� +. +(9.36) +Let ⃗w(t, p) = +� +0 +b +� += ⃗w(t0, pt0) (constant in time and uniform in space). Then L⃗v ⃗w = −d⃗v.⃗w = +� +−λb +0 +� +measures “the resistance to deformation due to the flow”. See figure 9.2, the virtual vector ⃗w∗(t, p) = +dΦ(t0, pt0).⃗w(t0, pt0) being the vector that would have let itself be carried by the flow (the push-forward). +56 + +57 +9.5. +Lie derivative of a differential form +Figure 9.2: Shear flow, cf. (9.36), with ⃗w constant and uniform. L⃗v ⃗w measures the resistance to the +deformation. +9.4.7 +Spin +Rotating flow: Continuing (5.14): +⃗v(x, y) = ω +� +0 +−1 +1 +0 +� � +x +y +� +, +d⃗v(x, y) = ω +� +0 +−1 +1 +0 +� += ω Rot(π/2). +(9.37) +In particular d2⃗v = 0. With ⃗w = ⃗w0 constant and uniform we get +L⃗v ⃗w0 = −d⃗v(p).⃗w0 = −ω Rot(π/2).⃗w0 +(⊥ +� +a +b +� += ⃗w0). +(9.38) +gives “the force at which ⃗w refuses to turn with the flow”. +9.4.8 +Second order Lie derivative +Exercice 9.11 Let ⃗v, ⃗w be C2. Prove: +L⃗v(L⃗v ⃗w) = D2 ⃗w +Dt2 − 2d⃗v.D ⃗w +Dt − D(d⃗v) +Dt +.⃗w + d⃗v.d⃗v.⃗w, += ∂2 ⃗w +∂t2 + 2d∂ ⃗w +∂t .⃗v − 2d⃗v.∂ ⃗w +∂t + d⃗w.∂⃗v +∂t − d∂⃗v +∂t .⃗w ++ (d2 ⃗w.⃗v).⃗v + d⃗w.d⃗v.⃗v − 2d⃗v.d⃗w.⃗v − (d2⃗v.⃗v).⃗w + d⃗v.d⃗v.⃗w. +(9.39) +Answer. +L⃗v(L⃗v ⃗w) = D(L⃗v ⃗w) +Dt +− d⃗v.(L⃗v ⃗w) = D( D ⃗w +Dt − d⃗v.⃗w) +Dt +− d⃗v.(D ⃗w +Dt − d⃗v.⃗w) += D2 ⃗w +Dt2 − D(d⃗v) +Dt +.⃗w − d⃗v.D ⃗w +Dt − d⃗v.D ⃗w +Dt + d⃗v.d⃗v.⃗w, +thus (9.39)1, thus (9.39)2. +9.5 +Lie derivative of a differential form +When the Lie derivative of a vector field ⃗w cannot be obtained by direct measurements, you need to use +a “measuring device” (Germain: To know the weight of a suitcase you have to lift it: You use work). +Here we consider a measuring device which is a differential form α. So, if ⃗w is a vector field then +f = α.⃗v is a scalar function, and (9.13) gives L⃗v(α.⃗w) = D(α.⃗w) +Dt += Dα +Dt .⃗w + α. D ⃗w +Dt , thus +L⃗v(α.⃗w) = Dα +Dt .⃗w + α.d⃗v.⃗w +� +�� +� +→(L⃗vα).⃗w ++ α.D ⃗w +Dt − α.d⃗v.⃗w +� +�� +� +=α.L⃗v ⃗w +: +(9.40) +Definition 9.12 Let α be a differential form. The Lie derivative of α along ⃗v is the differential form +L⃗vα := Dα +Dt + α.d⃗v = ∂α +∂t + dα.⃗v + α.d⃗v. +(9.41) +(An equivalent definition is given at (9.47).) I.e., for all vector field ⃗w, +L⃗vα.⃗w := Dα +Dt .⃗w + α.d⃗v.⃗w +(= ∂α +∂t .⃗w + (dα.⃗v).⃗w + α.d⃗v.⃗w). +(9.42) +57 + +(B ) A +q=c +(t +8.% +w(t) +w(t,/p) +w.(t,p) +od +T +V(B) +(t)58 +9.5. +Lie derivative of a differential form +The definition of L⃗vα, cf. (9.41), immediately gives the “derivation property” +L⃗v(α.⃗w) = (L⃗vα).⃗w + α.(L⃗v ⃗w) +(i.e. L⃗v is a derivation). +(9.43) +Quantification: Relative to a basis (⃗ei) and with [·] := [·]|⃗e, +[L⃗vα] = [Dα +Dt ] + [α].[d⃗v] +(row matrix) = [∂α +∂t ] + [dα.⃗v] + [α].[d⃗v]. +(9.44) +Thus +[L⃗vα.⃗w] = [L⃗vα].[⃗w] = [∂α +∂t ].[⃗w] + [dα.⃗v].[⃗w] + [α].[d⃗v].[⃗w]. +(9.45) +Exercice 9.13 Prove (9.44) with components. +And prove [dα.⃗v] = [⃗v]T .[dα]T (row matrix), thus +[dα.⃗v].[⃗w] = [⃗v]T .[dα]T .[⃗w] = [⃗w]T .[dα].[⃗v]. +Answer. +Basis (⃗ei), dual basis (πei), thus (9.41) gives [L⃗vα] = [ Dα +Dt ] + [α.d⃗v]. +Let α = � +i αiπei, +⃗v = � +i vi⃗ei, d⃗v = � +ij vi|j⃗ei ⊗ πej (tensorial writing convenient for calculations), i.e. [d⃗v]|⃗e = [vi|j], thus +α.d⃗v = � +ij αivi|jπej, thus [α.d⃗v]|πe = [α]|πe.[d⃗v]|⃗e (row matrix). And dα = � +ij αi|jπei ⊗ πej, i.e. [dα]|πe = [αi|j], +gives dα.⃗v = � +ij αi|jvjπei = � +ij viαj|iπej, and [dα.⃗v]|πe is a row matrix (dα.⃗v is a differential form), thus +[dα.⃗v]|πe = [⃗v]T +|⃗e.[dα]T +|πe. (Or compute (dα.⃗v).⃗w = � +ij αi|jvjwi = [⃗w]T +|⃗e.[dα]|⃗e.[⃗v]|⃗e = [⃗v]T +|⃗e.[dα]T +|πe.[⃗w]|⃗e.) +Exercice 9.14 Let α be a differential form, and let αt(p) := α(t, p). Prove, when Φt0 +t is a diffeomorphism, +L⃗vα = 0 +⇐⇒ +∀t ∈ [t0, T], αt = (Φt0 +t )∗αt0. +(9.46) +I.e.: +Dα +Dt = −α.d⃗v ⇐⇒ αt(pt) = αt0(pt0).F t0 +t (pt0)−1 for all t, when pt = Φt0 +t (pt0). +Answer. ⇐: If αt(p(t)) = αt0(pt0).F t0 +t (pt0)−1, then α(t, p(t)).F t0(t, pt0) = αt0(pt0), thus Dα +Dt (t, pt).F t0 +t (pt0) + +αt(pt). ∂F t0 +∂t (t, pt0) = 0, thus +Dα +Dt (t, p(t)).F t0 +t (pt0) + αt(pt).d⃗v(t, pt).F t0 +t (pt0) = 0, thus L⃗vα = 0, since Φt0 +t +is a +diffeomorphism. +⇒: If β(t) := (Φt0 +t )∗αt0(pt0) = αt(p(t)).F t0 +t (pt0) (pull-back at (t0, pt0)), then β(t) = α(t, p(t)).F t0(t, pt0), thus +β′(t) = Dα +Dt (t, pt).F t0 +t (pt0) + α(t, pt).d⃗v(t, pt).F t0 +t (pt0) = 0 (hypothesis L⃗vα = 0), thus β(t) = β(t0) = αt0(pt0). +Remark 9.15 A definition equivalent to (9.41) is, cf. (9.10), +L⃗vα(t, pt) := lim +τ→t +(Φt +τ)∗ατ(pt) − αt(pt) +τ − t +(= lim +τ→t +ατ(pτ).dΦt +τ(pt) − αt(pt) +τ − t +) +noted += +D(Φt∗ +τ ατ(pt)) +Dτ +|τ=t +noted += +D(α∗ +τ(pt)) +Dτ +|τ=t +(= D(ατ(pτ).dΦt +τ(pt)) +Dτ +|τ=t). +(9.47) +Indeed, if β(τ) = (Φt +τ)∗ατ(pt) = ατ(pτ).dΦt +τ(pt), then β′(τ) and then τ = t give (9.41). +Exercice 9.16 ⃗v and α being C2, prove: +L⃗v(L⃗vα) = ∂2α +∂t2 + 2d∂α +∂t .⃗v + 2∂α +∂t .d⃗v + dα.∂⃗v +∂t + α.∂d⃗v +∂t ++ (d2α.⃗v).⃗v + dα.(d⃗v.⃗v) + 2(dα.⃗v).d⃗v + α.(d2⃗v.⃗v) + (α.d⃗v).d⃗v. +(9.48) +Answer. (9.41) gives +L⃗v(L⃗vα) = L⃗v(∂α +∂t ) + L⃗v(dα.⃗v) + L⃗v(α.d⃗v) += ∂2α +∂t2 + d∂α +∂t .⃗v + ∂α +∂t .d⃗v + ∂(dα.⃗v) +∂t ++ d(dα.⃗v).⃗v + (dα.⃗v).d⃗v + ∂(α.d⃗v) +∂t ++ d(α.d⃗v).⃗v + (α.d⃗v).d⃗v += ∂2α +∂t2 + d∂α +∂t .⃗v + ∂α +∂t .d⃗v + ∂dα +∂t .⃗v + dα.∂⃗v +∂t + (d2α.⃗v).⃗v + dα.(d⃗v.⃗v) + (dα.⃗v).d⃗v ++ ∂α +∂t .d⃗v + α.∂d⃗v +∂t + (dα.⃗v).d⃗v + α.d2⃗v.⃗v + (α.d⃗v).d⃗v += ∂2α +∂t2 + 2d∂α +∂t .⃗v + 2∂α +∂t .d⃗v + dα.∂⃗v +∂t + (d2α.⃗v).⃗v + dα.(d⃗v.⃗v) + 2(dα.⃗v).d⃗v + α.∂d⃗v +∂t ++ α.(d2⃗v.⃗v) + (α.d⃗v).d⃗v. +58 + +59 +9.6. +Incompatibility with Riesz representation vectors +9.6 +Incompatibility with Riesz representation vectors +The Lie derivative has nothing to do with any inner dot product (the Lie derivative does not compare +two vectors, contrary to a Cauchy type approach). +Here we introduce a Euclidean dot product (·, ·)g and show that the Lie derivative of a linear form α +is not trivially deduced from the Lie derivative of a Riesz representation vector of α (which one?). (Same +issue as at § 7.2.) +Let α be a Eulerian differential form; Then let ⃗ag(t, p) ∈ ⃗Rn be the (·, ·)g-Riesz representation vector +of the linear form α(t, p) ∈ Rn∗: So, for all Eulerian vector field ⃗w, +α.⃗w = (⃗ag, ⃗w)g +(= ⃗ag •g ⃗w), +(9.49) +which means α(t, p).⃗w(t, p) = (⃗ag(t, p), ⃗w(t, p))g at all admissible (t, p). This defines the Eulerian vector +field ⃗ag (not intrinsic to α: ⃗ag depends on the choice of (·, ·)g, cf. (F.12)). +Proposition 9.17 For all ⃗v, ⃗w ∈ ⃗Rn, +∂α +∂t .⃗w = (∂⃗ag +∂t , ⃗w)g, +(dα.⃗v).⃗w = (d⃗ag.⃗v, ⃗w)g, +Dα +Dt .⃗w = (D⃗ag +Dt , ⃗w)g. +(9.50) +Thus +L⃗vα.⃗w = (L⃗v⃗ag, ⃗w)g + (⃗ag, (d⃗v+d⃗vT ).⃗w)g, +and +L⃗vα.⃗w ̸= (L⃗v⃗ag, ⃗w)g +in general. +(9.51) +So L⃗v⃗ag is not the Riesz representation vector of L⃗vα (but for solid body motions). (Expected: A Lie +derivative is covariant objective, see § 11.4, and the use of an inner dot product ruins this objectivity.) +Proof. A Euclidean dot product g(·, ·) is bilinear constant and uniform, thus: +α.⃗w = (⃗ag, ⃗w)g gives ∂α +∂t .⃗w + α. ∂ ⃗w +∂t = ( ∂⃗ag +∂t , ⃗w)g + (⃗ag, ∂ ⃗w +∂t )g, with α. ∂ ⃗w +∂t = (⃗ag, ∂ ⃗w +∂t )g, thus we are left +with ∂α +∂t .⃗w = ( ∂⃗ag +∂t , ⃗w)g, for all ⃗w. +α.⃗w = (⃗ag, ⃗w)g gives d(α.⃗w).⃗v = d(⃗ag, ⃗w)g.⃗v for all ⃗v, ⃗w, thus (dα.⃗v).⃗w + α.(d⃗w.⃗v) = (d⃗ag.⃗v, ⃗w)g + +(⃗ag, d⃗w.⃗v)g, with α.(d⃗w.⃗v) = (⃗ag, d⃗w.⃗v)g, thus we are left with (dα.⃗v).⃗w = (d⃗ag.⃗v, ⃗w)g. +Thus Dα +Dt .⃗w = ( D⃗ag +Dt , ⃗w)g. +Thus (L⃗vα).⃗w = Dα +Dt .⃗w + α.d⃗v.⃗w = ( D⃗ag +Dt , ⃗w)g + (⃗ag, d⃗v.⃗w)g = (L⃗v⃗ag + d⃗v.⃗ag, ⃗w)g + (d⃗vT +g .⃗ag, ⃗w)g. +Remark 9.18 Chorus: a “differential form” (measuring instrument, covariant) should not be confused +with a “vector field” (object to be measured, contravariant); Thus, the use of a dot product (which one?) +and the Riesz representation theorem should be restricted for computational purposes, after an objective +equation has been established. See also remark F.13. +9.7 +Lie derivative of a tensor +The Lie derivative of any tensor of order ≥ 2 is defined thanks to +L⃗v(T ⊗ S) = (L⃗vT) ⊗ S + T ⊗ (L⃗vS) +(derivation formula). +(9.52) +(Or direct definition: L⃗vT(t0, pt0) = D((Φt0 +t )∗Tt)(pt0) +Dt +|t=t0). +9.7.1 +Lie derivative of a mixed tensor +Let Tm ∈ T 1 +1 (Ω), and Tm is called a mixed tensor; Its Lie derivative, called the Jaumann derivative, is +given by +L⃗vTm = DTm +Dt +− d⃗v.Tm + Tm.d⃗v = ∂Tm +∂t ++ dTm.⃗v − d⃗v.Tm + Tm.d⃗v. +(9.53) +Can be checked with an elementary tensor T = ⃗w ⊗α: we have d(⃗w ⊗α).⃗v = (d⃗w.⃗v)⊗α+ ⃗w ⊗(dα.⃗v) and +(d⃗v.⃗w)⊗α = d⃗v.(⃗w⊗α), and ⃗w⊗(α.d⃗v) = (⃗w⊗α).d⃗v , thus (9.52) gives L⃗v(⃗w⊗α) = (L⃗v ⃗w)⊗α+ ⃗w⊗(L⃗vα) += ∂ ⃗w +∂t ⊗ α + (d⃗w.⃗v) ⊗ α − (d⃗v.⃗w) ⊗ α + ⃗w ⊗ ∂α +∂t + ⃗w ⊗ (dα.⃗v) + ⃗w ⊗ (α.d⃗v) += ∂ ⃗w⊗α +∂t ++ d(⃗w ⊗ α).⃗v − d⃗v.(⃗w ⊗ α) + (⃗w ⊗ α).d⃗v. +59 + +60 +9.7. +Lie derivative of a tensor +Quantification. Relative to a basis (⃗ei): +[L⃗vTm] = [DTm +Dt ] − [d⃗v].[Tm] + [Tm].[d⃗v] +(9.54) +(the signs ∓ are mixed). “Mixed” also refers to positions of indices (up and down with duality notations): +Tm = �n +i,j=1T ij⃗ei ⊗ ej with the dual basis (ei), i.e. [Tm]|⃗e = [T ij]. +Exercice 9.19 With components, prove (9.54). +Answer. +∂Tm +∂t += � +ij +∂T ij +∂t ⃗ei ⊗ ej, dTm = � +ijk T i +j|k⃗ei ⊗ ej ⊗ ek, ⃗v = � +i vi⃗ei, d⃗v = � +ij vi +|j⃗ei ⊗ ej, thus +dTm.⃗v = � +ijk T i +j|kvk⃗ei ⊗ ej, d⃗v.Tm = � +ijk vi +|kT k +j⃗ei ⊗ ej, Tm.d⃗v = � +ijk T i +kvk +|j⃗ei ⊗ ej. +9.7.2 +Lie derivative of a up-tensor +Recall: If L ∈ L(E; F) (a linear map) then its adjoint L∗ ∈ L(F ∗; E∗) is defined by, cf. § A.12, +∀m ∈ F ∗, +L∗.m := m.L , +i.e., +∀m, ⃗u ∈ (F ∗ × E), +(L∗.m).⃗u = m.L.⃗u. +(9.55) +(There is no inner dot product involved here.) +In particular, d⃗v∗.m := m.d⃗v for all m ∈ ⃗Rn∗ +t , i.e. +(d⃗v∗.m).⃗u = (m.d⃗v).⃗u = m.(d⃗v.⃗u) for all m ∈ ⃗Rn∗ +t +and all ⃗u ∈ ⃗Rn +t . +Let Tu ∈ T 2 +0 (Ω), and Tu is called a up tensor; Its Lie derivative is called the upper-convected (Maxwell) +derivative or the Oldroyd derivative and is given by +L⃗vTu = DTu +Dt − d⃗v.Tu − Tu.d⃗v∗ = ∂Tu +∂t + dTu.⃗v − d⃗v.Tu − Tu.d⃗v∗. +(9.56) +Can be checked with an elementary tensor T = ⃗u ⊗ ⃗w and L⃗v(⃗u ⊗ ⃗w) = (L⃗v⃗u) ⊗ ⃗w + ⃗u ⊗ (L⃗v ⃗w). +Quantification. Relative to a basis (⃗ei): +[L⃗vTu] = [DTu +Dt ] − [d⃗v].[Tu] − [Tu].[d⃗v]T . +(9.57) +“up” also refers to positions of indices (with duality notations): Tu = �n +i,j=1T ij⃗ei ⊗ ⃗ej with the dual +basis (ei), i.e. [Tu]|⃗e = [T ij]. +Exercice 9.20 With components, prove (9.56). +Answer. +∂Tu +∂t = � +ij +∂T ij +∂t ⃗ei⊗⃗ej, dTu = � +ijk T ij +|k⃗ei⊗⃗ej⊗ek, ⃗v = � +i vi⃗ei, d⃗v = � +ij vi +|j⃗ei⊗ej, d⃗v∗ = � +ij vj +|iei⊗⃗ej, +thus dTu.⃗v = � +ijk T ij +|k vk⃗ei ⊗ ej, d⃗v.Tu = � +ijk vi +|kT kj⃗ei ⊗ ⃗ej, Tu.d⃗v∗ = � +ijk T ikvj +|kei ⊗ ⃗ej. +9.7.3 +Lie derivative of a down-tensor +Let Td ∈ T 0 +2 (Ω), and Td is called a down tensor; The Lie derivative is called the lower-convected Maxwell +derivative and is given by +L⃗vTd = DTd +Dt + Td.d⃗v + d⃗v∗.Td = ∂Td +∂t + dTd.⃗v + Td.d⃗v + d⃗v∗.Td. +(9.58) +Can be checked with an elementary tensor T = ℓ ⊗ m and L⃗v(ℓ ⊗ m) = (L⃗vℓ) ⊗ m + ℓ ⊗ (L⃗vm). +Quantification. Relative to a basis (⃗ei): +[L⃗vTd] = [DTd +Dt ] + [Td].[d⃗v] + [d⃗v]T .[Td]. +(9.59) +“down” also refers to positions of indices (with duality notations): Td = �n +i,j=1Tijei ⊗ ej with the dual +basis (ei), i.e. [Td]|⃗e = [Tij]. +Exercice 9.21 With components, prove (9.59). +Answer. +∂Td +∂t = � +ij +∂Tij +∂t ei⊗ej, dTd = � +ijk Tij|kei⊗ej⊗ek, ⃗v = � +i vi⃗ei, d⃗v = � +ij vi +|j⃗ei⊗ej, d⃗v∗ = � +ij vj +|iei⊗⃗ej, +thus dTd.⃗v = � +ijk Tij|kvkei ⊗ ej, Td.d⃗v = � +ijk Tikvk +|jei ⊗ ⃗ej, d⃗v∗.Td = � +ijk vk +|iTkjei ⊗ ⃗ej. +Example 9.22 Let g = (·, ·)g ∈ T 0 +2 (Ω) be a constant and uniform metric (a unique inner dot product +for all t, p, e.g., a Euclidean dot product at all t). Then Dg +Dt = 0, thus L⃗vg = 0 + g.d⃗v + d⃗v∗.g, thus +[L⃗vg] = [g].[d⃗v] + [d⃗v]T .[g]. +60 + +61 +Part IV +Velocity-addition formula +10 +Change of referential and velocity-addition formula +10.0 +Issue and result (summary) +The velocity-addition formula is (in classical mechanics) +⃗vA = ⃗vB + ⃗vD, +(10.1) +where ⃗vA, ⃗vB and ⃗vD are the absolute, relative and drive velocity, ⃗vA and ⃗vD being velocities described by +an observer A with his referential RA = (OA, ( ⃗Ai)) and ⃗vB being a velocity described by an observer B +with his referential RB = (OB, ( ⃗Bi)). But (10.1) is problematic (inconsistent): +• The velocities ⃗vA and ⃗vD are quantified in RA, e.g. expressed in foot/s by the absolute observer, +• The velocity ⃗vB is a quantified in RB, e.g. expressed in metre/s by the relative observer, +Thus (10.1) with ⃗vB + ⃗vD tells that you add metre/s and foot/s... absurd. So: +Question: What are we missing (and what does (10.1) really mean)? +Answer: We miss a functional link: The translator between A and B. Summary: +Call �Φ the motion of a observed object Obj; �Φ is quantified by A in his referential RA = (OA, ( ⃗Ai)) +as the “motion” ⃗ϕA = [ +−−→ +OA�Φ]| ⃗A, and is quantified by B in his referential RB = (OB, ( ⃗Bi)) as the “motion” +⃗ϕB = [ +−−→ +OB �Φ]| ⃗B. At t, the translator Θ connects these numerical values: ⃗ϕA(t, PObj) = Θ(t, ⃗ϕB(t, PObj)). +Thus ∂ ⃗ϕA +∂t (t, PObj) = ∂Θ +∂t (t, ⃗xBt) + dΘ(t, ⃗xBt). ∂ ⃗ϕB +∂t (t, PObj), i.e. ⃗vA(t, ⃗xAt) = ∂Θ +∂t (t, ⃗xBt) + dΘ(t, ⃗xBt).⃗vB(t, ⃗xBt) +where ⃗xAt = ⃗ϕA(t, PObj) and ⃗xBt = ⃗ϕB(t, PObj). Then call +dΘt(⃗xBt).⃗vBt(⃗xBt) = ⃗vBt∗(⃗xAt) = “the translated relative velocity at t from B to A”, +(10.2) +thus, with ∂Θ +∂t (t, ⃗xBt) = ⃗vD(t, ⃗xAt) the drive velocity, which gives ⃗vA(t, ⃗xAt) = ⃗vB∗(t, ⃗xAt) + ⃗vD(t, ⃗xAt): so +⃗vA = ⃗vB∗ + ⃗vD += +the velocity addition formula in RA, +(10.3) +i.e.: (Absolute velocity) = (Translated relative velocity) + (Drive velocity). +In other words, with ⃗v the velocity of Obj and with ⃗vRB the velocity of RB in RA: For all pt = �Φ(t, PObj), +[⃗vt(pt)]| ⃗A = dΘt.[⃗vt(pt)] ⃗B + [⃗vRBt(pt)]| ⃗A, +(10.4) +relation between the numerical values of the velocities stored by A and B. +Example 10.1 Translation motion of RB in RA, so [⃗vRBt(pt)]| ⃗A = [⃗vRBt]| ⃗A is independent of pt; And, +e.g. with ( ⃗Bit) = λ( ⃗Ait) (e.g. ⃗Ai in foot and ⃗Bi in meter give λ ≃ 3.28), dΘt = λI, hence [⃗vt(pt)]| ⃗A = +λ[⃗vt(pt)] ⃗B + [⃗vRBt]| ⃗A, which is the expected relation (“sum of the velocities with the good units”). +Example 10.2 Motion of the Earth around the Sun: See § 10.11. +10.1 +Referentials and “matrix motions” +10.1.1 +Absolute and relative referentials +Classical mechanics framework: Time and space are decoupled, all the observers share the same time +unit (e.g. the second) and live in “our” Universe modeled as R3 (affine space) with its usual associated +vector space ⃗R3. In the following, the affine space is Rn associated to the vector space ⃗Rn, n ∈ {1, 2, 3}. +An observer A, which we will call the absolute observer, chooses a (rigid body) object ObjRA in the +Universe, chooses one particle in ObjRA, calls OAt its position at t, and chooses three more particles +in ObjRA, calls PAti their positions at t (in the Universe), such that the bi-point vectors ⃗Ait := −−−−−→ +OAtPAti +make a basis in ⃗Rn. He has thus built his (Cartesian) referential RAt = (OAt, ( ⃗Ait)), called the absolute +referential, and written RA = (OA, ( ⃗Ai)) when used by A. E.g. ObjRA is the “Sun extended to infinity”, +and at t, OAt is the position of the center of the Sun in the Universe, ( ⃗Ait) is a Euclidean basis in foot +fixed relative to stars. +61 + +62 +10.1. +Referentials and “matrix motions” +An observer B, which we will call the relative observer, proceeds similarly: He chooses a (rigid body) +object ObjRB in the Universe, builds his Cartesian referential RBt = (OBt, ( ⃗Bit)), called the relative +referential, written RB = (OB, ( ⃗Bi)) when used by B. E.g. ObjRB is the “Earth extended to infinity”, and +at t, OBt is the position of the center of the Earth and ( ⃗Bit) is a Euclidean basis in metre fixed relative +to the Earth. +Mn1 is the vectorial space of n ∗ 1 matrices (column matrices). A and B call Mn1(A) and Mn1(B) the +affine spaces of n ∗ 1 matrices made of the “matrix positions” [−−−→ +OAtpt]| ⃗A and [−−−→ +OBtpt]| ⃗B where pt is the +position at t of a particle in the Universe. +If a function ϕ is given as ϕ(t, x), then ϕt(x) := ϕ(t, x), and conversely. +10.1.2 +Motion of a material object Obj +An object Obj is considered by all observers. Its motion in the Universe is +�Φ : +� +[t1, t2] × Obj → Rn +(t, PObj) → pt = �Φ(t, PObj) = position of the particle PObj at t in the Universe. +(10.5) +At t at pt = �Φ(t, PObj), the Eulerian velocities and accelerations of PObj are +⃗v(t, pt) = ∂�Φ +∂t (t, PObj) +and +⃗γ(t, pt) = ∂2�Φ +∂2t (t, PObj) +(∈ ⃗Rn). +(10.6) +10.1.3 +Quantification: Absolute and relative “motion” of Obj +At t, the position pt = �Φ(t, PObj) of a particle PObj ∈ Obj is spotted by A, resp. B, with the bi-point +vectors −−−→ +OAtpt, resp. −−−→ +OBtpt in ⃗Rn, which components is stored by A, resp. B, in his referentials: With +−−−→ +OAtpt = +n +� +i=1 +xAti ⃗Ait +and +−−−→ +OBtpt = +n +� +i=1 +xBti ⃗Bit, +(10.7) +and with ( ⃗Ei) the canonical basis in Mn1, the n ∗ 1 matrices +⃗xAt := [−−−→ +OApt]| ⃗A = +� +� +xAt1 +... +xAtn +� +� = +n +� +i=1 +xAti ⃗Ei, +and +⃗xBt := [−−−→ +OBpt]| ⃗B = +� +� +xBt1 +... +xBtn +� +� = +n +� +i=1 +xBti ⃗Ei, +(10.8) +are stored by A and B. (Initial notation: ⃗xAt := [−−−→ +OAtpt]|( ⃗Ait), but here OAt and ( ⃗Ait) are fixed in RA, +idem for B.) +Mind the notations: pt is a point, −−−→ +OAtpt is a vector, ⃗xAt is a column matrix (components). +This defines the “absolute motion” ⃗ϕA and “relative motion” ⃗ϕB of Obj (matrix valued): +⃗ϕA : +� +� +� +� +� +[t1, t2]×Obj → Mn1(A) +(t, PObj) → ⃗ϕA(t, PObj) := [ +−−−−−−−−→ +OA�Φ(t, PObj)]| ⃗A = +n +� +i=1 +xAi(t) ⃗Ei +noted += +⃗xA(t) = [−−−−→ +OAp(t)]| ⃗A, +(10.9) +⃗ϕB : +� +� +� +� +� +[t1, t2]×Obj → Mn1(B) +(t, PObj) → ⃗ϕB(t, PObj) := [ +−−−−−−−−→ +OB �Φ(t, PObj)]| ⃗B = +n +� +i=1 +xBi(t) ⃗Ei +noted += +⃗xB(t) = [−−−−→ +OBp(t)]| ⃗B. +(10.10) +And the “absolute” and “relative” velocities and accelerations of PObj are (matrix valued in Mn1): +⃗vA(t, ⃗xAt) := [⃗v(t, pt)]| ⃗A +and +⃗γA(t, ⃗xAt) := [⃗γ(t, pt)]| ⃗A, +when +⃗xAt := [−−−→ +OApt]| ⃗A, +(10.11) +⃗vB(t, ⃗xBt) := [⃗v(t, pt)]| ⃗B +and +⃗γB(t, ⃗xBt) := [⃗γ(t, pt)]| ⃗B, +when +⃗xBt := [−−−→ +OBpt]| ⃗B. +(10.12) +62 + +63 +10.1. +Referentials and “matrix motions” +Exercice 10.3 Prove: ⃗vA(t, ⃗xAt) = ∂ ⃗ϕA +∂t (t, PObj). +Answer. +−−−−−−−→ +OAt�ΦPObj (t) = � +i xAi(t) ⃗Ai(t) gives ⃗v(t, pt) = � +i xAi +′(t) ⃗Ai(t) + xAi(t) ⃗Ai +′(t), thus [⃗v(t, pt)]| ⃗ +A = +� +i xAi +′(t)[ ⃗Ai(t)]| ⃗ +A + xAi(t)[ ⃗Ai +′(t)]| ⃗ +A (since Mn1 is a vector space) = � +i xAi +′(t) ⃗Ei + [⃗0] (the ⃗Ai(t) are static +in RA: +[ ⃗Ai +′(t)]| ⃗ +A = [limh→0 +⃗ +Ai(t+h)− ⃗ +Ai(t) +h +]| ⃗ +A = limh→0 +[ ⃗ +Ai(t+h)]| ⃗ +A−[ ⃗ +Ai(t)]| ⃗ +A +h += limh→0 +⃗ +Ei− ⃗ +Ei +h += 0). +And +⃗ϕAPObj (t) = [ +−−−−−−−→ +OA�ΦPObj (t)]| ⃗ +A = � +i xAi(t)[ ⃗Ai(t)]| ⃗ +A = � +i xAi(t) ⃗Ei, thus ⃗ϕAPObj +′(t) = � +i xAi +′(t) ⃗Ei. +Exercice 10.4 ⃗u is a C1 vector field, p is a point, ⃗xA := [−−→ +OAp]| ⃗A and ⃗uA(⃗xA) := [⃗u(p)]| ⃗A (matrices). +Prove: d⃗uA(⃗xA) = [d⃗u(p)]| ⃗A (endomorphism in Mn1), i.e. d⃗uA(⃗xA).[⃗w]| ⃗A = [d⃗u(p).⃗w]| ⃗A for all ⃗w ∈ ⃗Rn. +Answer. +The point p+h⃗w ∈ Rn is referenced by A as [−−→ +OAp + h⃗w]| ⃗ +A = [−−→ +OAp]| ⃗ +A + h[⃗w]| ⃗ +A = ⃗xA + h[⃗w]| ⃗ +A. +Thus d⃗uA(⃗xA).[⃗w]| ⃗ +A = limh→0 +⃗uA(⃗xA+h[ ⃗w]| ⃗ +A)−⃗uA(⃗xA) +h += limh→0 +[⃗u(p+h ⃗w)]| ⃗ +A−[⃗u(p)]| ⃗ +A +h += limh→0 +[⃗u(p+h ⃗w)− ⃗w(p)]| ⃗ +A +h += +[limh→0 +⃗u(p+h ⃗w)− ⃗w(p) +h +]| ⃗ +A = [d⃗u(p).⃗w]| ⃗ +A = [d⃗u(p)]| ⃗ +A.[⃗w]| ⃗ +A, true for all ⃗w. +Exercice 10.5 Call Qt the transition matrix from ( ⃗Ait) to ( ⃗Bit) at t. Prove ⃗xAt = [−−−−→ +OAOBt]| ⃗A + Qt.⃗xBt. +Answer. ⃗xAt = [−−−→ +OApt]| ⃗ +A = [−−−−→ +OAOBt + −−−→ +OBtpt]| ⃗ +A = [−−−−→ +OAOBt]| ⃗ +A + [−−−→ +OBtpt]| ⃗ +A, and the change of basis formula gives +[−−−→ +OBtpt]| ⃗ +B = Q−1 +t .[−−−→ +OBtpt]| ⃗ +A. +10.1.4 +Motion of RB +Particular case Obj = ObjRB: Its motion in the Universe, also called the motion of RB, is noted +�ΦRB : +� +[t1, t2] × ObjRB → Rn +(t, QRB) → qt = �ΦRB(t, QRB). +(10.13) +At t at qt = �ΦRB(t, QRB), the Eulerian velocities and accelerations of QRB are +⃗vRB(t, qt) = ∂�ΦRB +∂t (t, QRB) +and +⃗γRB(t, qt) = ∂2�ΦRB +∂2t (t, QRB). +(10.14) +10.1.5 +Quantification: Drive and static “motion” of RB +The “drive motion” ⃗ϕD, also called the motion of RB in RA, and the “static motion” ⃗ϕS is the quantification +of �ΦRB by A and by B: +⃗ϕD : +� +� +� +[t1, t2] × ObjRB → Mn1(A) +(t, QRB) → ⃗ϕD(t, QRB) := [ +−−−−−−−−−−→ +OA�ΦRB(t, QRB)]| ⃗A +noted += +⃗yD(t) = [−−−−→ +OAq(t)]| ⃗A, +(10.15) +⃗ϕS : +� +� +� +ObjRB → Mn1(B) +QRB → ⃗ϕS(QRB) := [ +−−−−−−−−−−→ +OB �ΦRB(t, QRB)]| ⃗B +noted += +⃗yS = [−−−−→ +OBq(t)]| ⃗B. +(10.16) +(⃗ϕS is independent of t since ObjRB is fixed in RB.) +The drive velocity, also called the velocity of RB in RA, and static velocity of QRB are +⃗vD(t, ⃗yDt) := [⃗vRB(t, qt)]| ⃗A +when +⃗yDt := [−−→ +OAqt]| ⃗A, +(10.17) +⃗vS(t, ⃗yS) := [⃗vRB(t, qt)]| ⃗B = [⃗0] noted += +⃗0 (null matrix). +(10.18) +And the drive and static accelerations are ⃗γD(t, ⃗yDt) = [⃗γRB(t, qt)]| ⃗A and ⃗γS(t, ⃗yS) = ⃗0. +Exercice 10.6 Why introduce ⃗ϕS (static)? +Answer. You can’t confuse a particle QRB with its stored positions ⃗yS or ⃗yDt at t. And see (10.19). +63 + +64 +10.2. +The translator Θt +10.2 +The translator Θt +10.2.1 +Definition of Θt +Definition 10.7 At t, the translator Θt : Mn1(B) → Mn1(A) is defined with (10.15)-(10.16) by: +� +� +� +[−−−−→ +OBq(t)]| ⃗B = ⃗ϕS(QRB) = ⃗yS position of QRB in RB (static), and +[−−−−→ +OAq(t)]| ⃗A = ⃗ϕDt(QRB) = ⃗yDt position of QRB at t in RA (moving) +� +� +� =⇒ ⃗yDt = Θt(⃗yS), +(10.19) +i.e. Θt is the “inter-referential function at t” which translates the “matrix position” ⃗yS = ⃗ϕS(QRB) = +[ +−−−−−−−−−−→ +OB �ΦRB(t, QRB)]| ⃗B ∈ Mn1(B) (position of QRB as stored by B) to the “matrix position” ⃗yDt = ⃗ϕD(t, QRB) = +[ +−−−−−−−−−−→ +OA�ΦRB(t, QRB)]| ⃗A ∈ Mn1(A) (position of QRB as stored by A). So Θt is defined by +⃗ϕDt = Θt ◦ ⃗ϕS : +� +ObjRB → Mn1(A) +QRB → ⃗ϕDt(QRB) := Θt(⃗ϕS(QRB)), +(10.20) +i.e. defined by +Θt := ⃗ϕDt ◦ ⃗ϕ −1 +S +: +� +Mn1(B) → Mn1(A) +⃗yS → ⃗yDt = Θt(⃗yS) := ⃗ϕDt(⃗ϕ −1 +S +(⃗yS)) +(10.21) +(stored position by B to stored position by A). +E.g., for QOB the particle in ObjRB at t at OBt (chosen by B to locate its origin), (10.20) gives, with +⃗0 the null matrix in Mn1, +[−−−−→ +OAOBt]| ⃗A = Θt(⃗0). +(10.22) +So, Θt is defined such that the following diagram commutes: +⃗yS = ⃗ϕS(QRB) = localization of QRB by B +Θt +� +QRB ∈ ObjRB +⃗ϕS +� +⃗ϕDt +� +⃗yDt = ⃗ϕDt(QRB) = Θt(⃗yS) = localization at t of QRB by A. +(10.23) +10.2.2 +Translation at t for the motion �Φ +t is fixed, the position pt = �Φ(t, PObj) of a particle PObj ∈ Obj is also the position qt = �ΦRB(t, QRB) of +a particle QRB ∈ ObjRB, so ⃗ϕAt(PObj) = [−−−→ +OApt]| ⃗A = [−−→ +OAqt]| ⃗A = ⃗ϕDt(QRB), and ⃗ϕBt(PObj) = [−−−→ +OBpt]| ⃗B = +[−−−→ +OBqt]| ⃗B = ⃗ϕS(QRB), thus (10.20) gives +⃗ϕAt = Θt ◦ ⃗ϕBt . +(10.24) +10.3 +dΘt +10.3.1 +Push-forward +If ⃗yS ∈ Mn1(B), ⃗wS ∈ Mn1 and ⃗yDt = Θt(⃗yS) , then +⃗wSt∗(⃗yDt) := dΘt(⃗yS).⃗wS +(= lim +h→0 +Θt(⃗yS + h⃗wB) − Θt(⃗yS) +h +) +(10.25) +is the push-forward of the matrix ⃗wS ∈ Mn1 by Θt. +So, ⃗wSt∗([−−→ +OAqt]| ⃗A) = dΘt([−−−→ +OBqt]| ⃗B).[⃗wt(qt)]| ⃗B for all qt ∈ Rn and all ⃗wt : Rn → ⃗Rn (vector field). +64 + +65 +10.4. +Translated velocities +10.3.2 +Θt is affine in classical mechanics +Proposition 10.8 In R3 (in classical mechanics), Θt is affine: For all QB0, QB1 ∈ ObjRB and all t, u ∈ R, +with qti = �ΦRB(t, QBi) ∈ Rn (positions at t in our Universe), +Θt([−−−→ +OBqt0]| ⃗B + u [−−−→ +qt0qt1]| ⃗B) = [−−−→ +OAqt0]| ⃗A + u [−−−→ +qt0qt1]| ⃗A, +and +[−−−→ +qt0qt1]| ⃗A = dΘt.[−−−→ +qt0qt1]| ⃗B +(10.26) +the differential dΘt(⃗yS0) =noted dΘt being independent of ⃗yS0. In particular [ ⃗Bit] ⃗A = dΘt.[ ⃗Bi]| ⃗B. In other +words, for all ⃗yS0, ⃗yS1 ∈ Mn1(B) and all t, u ∈ R, +Θt((1−u)⃗yS0 + u ⃗yS1) = (1−u)Θt(⃗yS0) + u Θt(⃗yS1), +and +Θt(⃗yS1) = Θt(⃗yS0) + dΘt.(⃗yS1−⃗yS0). (10.27) +Proof. Consider the straight line (possible in classical mechanics in R3) qt : u → qt(u) = qt0 +u −−−→ +qt0qt1 ∈ +Rn (fixed in RB), in particular, qt(0) = qt0 and qt(1) = qt1. Let ⃗yS(u) = [−−−−−→ +OBqt(u)]| ⃗B (positions stored +by B), so ⃗yS(u) = [−−−→ +OBqt0 + u −−−→ +qt0qt1]| ⃗B = [(1−u)−−−→ +OBqt0 + u −−−→ +OBqt1]| ⃗B = (1−u)[−−−→ +OBqt0]| ⃗B + u [−−−→ +OBqt1]| ⃗B = +(1−u)⃗yS0 + u ⃗yS1, where ⃗yS0 = [−−−→ +OBqt0]| ⃗B = ⃗yS(0) and ⃗yS1 = [−−−→ +OBqt1]| ⃗B = ⃗yS(1). Idem for A: ⃗yDt(u) = +[−−−−−→ +OAqt(u)]| ⃗A = (1−u)⃗yDt0 + u⃗yDt1 (positions stored by A). Thus +(1−u)Θt(⃗yS0) + uΘt(⃗yS1) +(10.19) += +(1−u)⃗yDt0 + u⃗yDt1 = ⃗yDt(u) +(10.19) += +Θt(⃗yS(u)) = Θt((1−u)⃗yS0 + u⃗yS1), +thus (10.27)1, thus (10.26)1. Hence (derivation in u): −Θt(⃗yS0) + Θt(⃗yS1) = dΘt(⃗yS(u)).(−⃗yS0 + ⃗yS1), +true for all u, thus dΘt(⃗yS(u)) is independent of u, dΘt(⃗yS(u)) = dΘt(⃗yS0), true for all ⃗yS0, so +dΘt(⃗yS0) =noted dΘt, thus (10.27)2, thus (10.26)2. +Thus [−−−−−→ +OBtPBti]| ⃗A = dΘt.[−−−−−→ +OBtPBti]| ⃗B where PBti +is s.t. ⃗Bit = −−−−−→ +OBtPBti, thus [ ⃗Bit] ⃗A = dΘt.[ ⃗Bi]| ⃗B. +Exercice 10.9 Call Qt = [Qt,ij] the transition matrix from ( ⃗Ait) to ( ⃗Bit) in ⃗Rn, and ( ⃗Ei) the canonical +basis in Mn1. Prove +[dΘt]| ⃗E = Qt, +i.e. +dΘt. ⃗Ej = +n +� +i=1 +Qt,ij ⃗Ei, ∀j. +(10.28) +Answer. ⃗Bjt = �n +i=1Qt,ij ⃗Ait gives [ ⃗Bjt]| ⃗ +A = �n +i=1Qt,ij ⃗Ei, and [ ⃗Bjt]| ⃗ +A =(10.26) dΘt.[ ⃗Bjt]| ⃗ +B = dΘt. ⃗Ej. +10.4 +Translated velocities +t is fixed, ⃗vt(pt) = +∂�Φ +∂t (t, PObj) is the velocity of a particle PObj ∈ Obj at t at pt, ⃗xAt := [−−−→ +OAtpt]| ⃗A, +⃗xBt := [−−−→ +OBtpt]| ⃗B, and Θt affine. +Definition 10.10 The translated relative velocity and acceleration from B to A at t at pt are the matrices +⃗vBt∗(⃗xAt) := dΘt.⃗vBt(⃗xBt) +and +⃗γBt∗(⃗xAt) = dΘt.⃗γBt(⃗xBt) . +(10.29) +I.e., ⃗vBt∗(⃗xAt) = dΘt.[⃗vt(pt)] ⃗B and ⃗γBt∗(⃗xAt) = dΘt.[⃗γt(pt)] ⃗B. +Interpretation: Let qt0 and qt1 be particles in ObjRB s.t. ⃗vBt(⃗xBt) = [−−−→ +qt0qt1]| ⃗B where ⃗xBt = [−−−→ +OBqt0]| ⃗B +(here −−−→ +qt0qt1 is a tangent vector at qt0 to the curve qt : u → qt(u) = qt0 +u −−−→ +qt0qt1 in the proof of prop. 10.8); +Then [−−−→ +qt0qt1]| ⃗A =(10.26) dΘt.[−−−→ +qt0qt1]| ⃗B gives [−−−→ +qt0qt1]| ⃗A = dΘt.⃗vBt(⃗xBt) = ⃗vBt∗(⃗xAt). Similarly for ⃗γBt∗(⃗xAt). +Exercice 10.11 ( ⃗Ai) and ( ⃗Bi) are Euclidean basis (e.g. in foot and metre), (·, ·)A and (·, ·)B are the +associated Euclidean dot products, λ = || ⃗Bi||A (e.g. ≃ 3.28), ( ⃗Ei) is the canonical basis in Mn1, and +(·, ·)M is the canonical inner dot product in Mn1. Call ⃗Eit∗ := dΘt. ⃗Ei and prove: +∀i, j, ( ⃗Eit∗, ⃗Ejt∗)M = λ2δij, +and +dΘt +T .dΘt = λ2I. +(10.30) +Answer. ( ⃗Bit) is a Euclidean basis for B, thus is a Euclidean orthogonal basis for all observers, in particular +for A, with || ⃗Bit||A = λ for all i. And ⃗Eit∗ = dΘt.[ ⃗Bjt]| ⃗ +B =(10.26) [ ⃗Bit]| ⃗ +A. Thus ( ⃗Eit∗, ⃗Ejt∗)M = [ ⃗Eit∗]T .[ ⃗Ejt∗] = +[ ⃗Bit]T +| ⃗ +A.[ ⃗Bjt]| ⃗ +A = ( ⃗Bit, ⃗Bjt)A = λ2( ⃗Bit, ⃗Bjt)B = λ2δij, thus (10.30)1; Then λ2δij = (dΘt. ⃗Ei, dΘt. ⃗Ej)M = +(dΘt +T .dΘt. ⃗Ei, ⃗Ej)M, true for all i, j, thus dΘt +T .dΘt = λ2I, thus (10.30)2. +65 + +66 +10.5. +Definition of Θ +10.5 +Definition of Θ +Definition 10.12 The translator from B to A is the function Θ defined with (10.20) by +Θ : +� +� +� +� +� +� +t∈[t1,t2] +({t} × Mn1(B)) → Mn1(A) +(t, ⃗yS) → Θ(t, ⃗yS) := Θt(⃗yS) , +(10.31) +i.e., for all QRB ∈ ObjRB and all t, +Θ(t, [ +−−−−−−−−−−→ +OB �ΦRB(t, QRB)]| ⃗B) = [ +−−−−−−−−−−→ +OA�ΦRB(t, QRB)]| ⃗A, +i.e. +Θ(t, ⃗ϕS(QRB)) = ⃗ϕD(t, QRB). +(10.32) +E.g., (10.22) gives Θ(t,⃗0) = [−−−−−→ +OAOB(t)]| ⃗A. +Remark 10.13 The translator Θ looks like a motion, but is not: A “usual” motion is defined by one +observer and connects one particle to its position; While Θ connects two “matrix positions” of one particle +relative to two referentials: Θ is an “inter-referential” function. +10.6 +The “Θ-velocity” is the drive velocity +Definition 10.14 The “Θ-velocity” and “Θ-acceleration” are defined by (Eulerian type definition) +with +⃗yDt = Θ(t, ⃗yS), +� +� +� +� +� +⃗vΘ(t, ⃗yDt) := ∂Θ +∂t (t, ⃗yS) (∈ Mn1), +⃗γΘ(t, ⃗yDt) = ∂2Θ +∂t2 (t, ⃗yS) (∈ Mn1). +(10.33) +Proposition 10.15 +⃗vΘ = ⃗vD +and +⃗γΘ = ⃗γD , +(10.34) +i.e. ⃗vΘ(t, ⃗y) = ⃗vD(t, ⃗y) and ⃗γΘ(t, ⃗y) = ⃗γD(t, ⃗y) in Mn1, for all t and all ⃗y ∈ Mn1(A). +Proof. Θ(t, ⃗ϕS(QRB)) = ⃗ϕD(t, QRB) gives +∂Θ +∂t (t, ⃗ϕS(QRB)) = ∂⃗ϕD +∂t (t, QRB), +i.e. +⃗vΘ(t, Θ(t, ⃗ϕS(QRB))) = ⃗vD(t, ⃗ϕD(t, QRB)), +(10.35) +thus ⃗vΘ(t, ⃗ϕD(t, QRB)) = ⃗vD(t, ⃗ϕD(t, QRB)), thus ⃗vΘ(t, ⃗y) = ⃗vD(t, ⃗y) for all ⃗y ∈ Mn1(A). Idem with +∂2 +∂t2 . +10.7 +The velocity-addition formula +(10.24) gives +⃗ϕA(t, PObj) = Θ(t, ⃗ϕB(t, PObj)), +(10.36) +thus +∂⃗ϕA +∂t (t, PObj) = ∂Θ +∂t (t, ⃗ϕB(t, PObj)) + dΘ(t, ⃗ϕB(t, PObj)).∂⃗ϕB +∂t (t, PObj). +(10.37) +Thus +⃗vA(t, ⃗xAt) = ⃗vΘ(t, ⃗xAt) + dΘ(t, ⃗xBt).⃗vB(t, ⃗xBt), +(10.38) +where ⃗xBt = ⃗ϕB(t, PObj) and ⃗xAt = ⃗ϕA(t, PObj) = Θt(⃗xBt). Thus, with ⃗vΘ =(10.34) ⃗vD, +⃗vAt = ⃗vBt∗ + ⃗vDt +where +⃗vBt∗(⃗xAt) := dΘt(⃗xBt).⃗vBt(⃗xBt), +(10.39) +which is the velocity-addition formula in RA: +⃗vAt the absolute velocity = ⃗vBt∗ the translated relative velocity from B to A ++ ⃗vDt the drive velocity. +(10.40) +In other words (relation between the numerical values of the velocities stored by A and B), +[⃗vt(pt)]| ⃗A = dΘt.[⃗vt(pt)] ⃗B + [⃗vRBt(pt)]| ⃗A. +(10.41) +66 + +67 +10.8. +Coriolis acceleration, and the acceleration-addition formula +10.8 +Coriolis acceleration, and the acceleration-addition formula +(10.37) gives +∂2⃗ϕA +∂t2 +(t, PObj) = ∂2Θ +∂t2 (t, ⃗xBt) + d∂Θ +∂t (t, ⃗xBt).∂⃗ϕB +∂t (t, PObj) ++ +�∂(dΘ) +∂t +(t, ⃗xBt) + d2Θ(t, ⃗xBt).∂⃗ϕB +∂t (t, PObj) +� +.∂⃗ϕB +∂t (t, PObj) + dΘ(t, ⃗xBt).∂2⃗ϕB +∂t2 (t, PObj). +(10.42) +And d2Θt = 0 in our classical framework (Θt is affine); And ∂Θ +∂t (t, ⃗yS) = ⃗vΘt(Θt(⃗yS)) gives ∂(dΘ) +∂t (t, ⃗yS) = +d( ∂Θ +∂t )(t, ⃗yS) = d⃗vΘt(Θt(⃗yS)).dΘt(⃗yS); And ∂ ⃗ϕB +∂t (t, PObj) = ⃗vBt(⃗xBt) where ⃗xBt = ⃗ϕB(t, PObj); Thus +⃗γAt(⃗xAt) = ⃗γΘt(⃗xAt) + 2d⃗vΘt(⃗xAt).dΘt(⃗xBt).⃗vBt(⃗xBt) + dΘt(⃗xBt).⃗γBt(⃗xBt) += ⃗γDt(⃗xAt) + 2d⃗vDt(⃗xAt).⃗vBt∗(⃗xAt) + ⃗γBt∗(⃗xAt). +(10.43) +Definition 10.16 At t, the Coriolis acceleration ⃗γCt at ⃗xAt is +⃗γCt(⃗xAt) = 2d⃗vDt(⃗xAt).⃗vBt∗(⃗xAt), +i.e. +⃗γCt = 2d⃗vDt.⃗vBt∗ . +(10.44) +And the Coriolis acceleration ⃗γC at t at ⃗xAt is ⃗γC(t, ⃗xAt) := ⃗γCt(⃗xAt). +Thus (10.43) gives the acceleration-addition formula in RA: +⃗γAt = ⃗γBt∗ + ⃗γDt + ⃗γCt , +i.e. +(10.45) +⃗γAt the absolute acceleration = ⃗γBt∗ the translated relative acceleration from B to A ++ ⃗γDt the drive acceleration + ⃗γCt the Coriolis acceleration. +(10.46) +In other words (relation between the numerical values of the acceletations stored by A and B), +[⃗γt(pt)]| ⃗A = dΘt.[⃗γt(pt)] ⃗B + [⃗γRBt(pt)]| ⃗A + 2[d⃗vRBt]| ⃗A.dΘt.[⃗vt(pt)]| ⃗B. +(10.47) +10.9 +With an initial time +Let t0, t ∈ R. Consider the Lagrangian associated function Φt0 +t with the motion �Φ of Obj: +Φt0 +t : +� +Ωt0 → Ωt +pt0=�Φ(t0, PObj) → pt = Φt0 +t (pt0) := �Φ(t, PObj). +(10.48) +And, with ⃗xAt = ⃗ϕA(t, PObj) = [−−−→ +OApt]| ⃗A and ⃗xBt = ⃗ϕB(t, PObj) = [−−−→ +OBpt]| ⃗B, define the “matrix motions” +⃗ϕt0 +At : Mn1(A) → Mn1(A) and ⃗ϕt0 +Bt : Mn1(B) → Mn1(B) by +� +� +� +⃗ϕt0 +At(⃗xAt0) := ⃗xAt +(= [ +−−−−−−−−→ +OA�Φ(t, PObj)]| ⃗A = [−−−−−−−→ +OAΦt0 +t (pt0)]| ⃗A = ⃗ϕAt(PObj)), +⃗ϕt0 +Bt(⃗xBt0) := ⃗xBt +(= [ +−−−−−−−−→ +OB �Φ(t, PObj)]| ⃗B = [−−−−−−−→ +OBΦt0 +t (pt0)]| ⃗B = ⃗ϕBt(PObj)). +(10.49) +And Θt(⃗xBt) = ⃗xAt, i.e. Θt(⃗ϕt0 +Bt(⃗xBt0)) = ⃗ϕt0 +At(⃗xAt0) with ⃗xAt0 = Θt0(⃗xBt0), thus +Θt ◦ ⃗ϕt0 +Bt = ⃗ϕt0 +At ◦ Θt0 : Mn1(B) → Mn1(A). +(10.50) +In other words, the following diagram commutes: +⃗xBt0 = ⃗ϕB(t0, PObj) +Θt0 +� +⃗ϕt0 +Bt +� ⃗xBt = ⃗ϕt0 +Bt(⃗xBt0) +Θt +� +PObj ∈ Obj +⃗ϕBt0 +� +⃗ϕAt0 +� +⃗xAt0 = ⃗ϕA(t0, PObj) = Θt0(⃗xBt0) +⃗ϕt0 +At +� ⃗xAt = ⃗ϕt0 +At(⃗xAt0) = Θt(⃗xBt). +(10.51) +Thus, for any vector field ⃗uBt0 in RB, +dΘt(⃗xBt) +� +�� +� +(translation at t) +. d⃗ϕt0 +Bt(⃗xBt0).⃗uBt0(⃗xBt0) +� +�� +� +(deformation from t0 to t) += +d⃗ϕt0 +At(⃗xAt0) +� +�� +� +(deformation from t0 to t) +. dΘt0(⃗xBt0).⃗uBt0(⃗xBt0) +� +�� +� +(translation at t0) +. +(10.52) +67 + +68 +10.10. +Drive and Coriolis forces +Exercice 10.17 Redo the above steps with ObjRB instead of Obj. +Answer. Consider the Lagrangian associated function Φt0 +RBt with the motion �ΦRB of ObjRB: +Φt0 +RBt : +� +ΩRBt0 = Rn → ΩRBt = Rn +qt0 = �ΦRB(t0, QRB) → qt = Φt0 +RBt(qt0) := �ΦRB(t, QRB), +� +(10.53) +then define the “matrix motions” ⃗ϕt0 +Dt : Mn1(A) → Mn1(A) and ⃗ϕt0 +St : Mn1(B) → Mn1(B) by +� +� +� +⃗ϕt0 +Dt(⃗yDt0) := ⃗yDt +(= [ +−−−−−−−−−−→ +OA�ΦRB(t, QRB)]| ⃗ +A = [−−−−−−−−→ +OAΦt0 +RBt(pt0)]| ⃗ +A = ⃗ϕDt(QRB)), +⃗ϕt0 +St(⃗yS) := ⃗yS +(= [ +−−−−−−−−−−→ +OB �ΦRB(t, QRB)]| ⃗ +B = [−−−−−−−−→ +OBΦt0 +RBt(qt0)]| ⃗ +B = ⃗ϕS(QRB)), +(10.54) +Thus ⃗ϕS is a time-shift, which is also abusively noted ⃗ϕt0 +St = I (algebraic identity). So with Θt(⃗yS) = ⃗yDt we get +Θt(⃗ϕt0 +Dt(⃗yS)) = ⃗ϕt0 +Dt(⃗yDt0), with ⃗yDt0 = Θt0(⃗yS), thus +Θt ◦ ⃗ϕt0 +St = ⃗ϕt0 +Dt ◦ Θt0 : Mn1(B) → Mn1(A) +(10.55) +(also abusively written Θt = ⃗ϕt0 +Dt ◦ Θt0). In other words, the following diagram commutes: +⃗yS = ⃗ϕS(QRB) +Θt0 +� +⃗ϕt0 +St = time shift +� ⃗yS = ⃗ϕS(QRB) +Θt +� +QRB ∈ ObjRB +⃗ϕS +� +⃗ϕt0 +D +� +⃗yDt0 = ⃗ϕDt0(QRB) = Θt0(⃗yS) +⃗ϕt0 +Dt � ⃗yDt = ⃗ϕDt(QRB) = ⃗ϕt0 +Dt(⃗yDt0) = Θt(⃗yS). +(10.56) +And (10.55) gives, for any ⃗yS = ⃗ϕS(QRB) and all vector field ⃗uS (static in RB), with ⃗yDt0 = Θt0(⃗yS), +dΘt(⃗yS) +� +�� +� +(translation at t) +. +d⃗ϕt0 +St(⃗yS).⃗uS(⃗yS) +� +�� +� +(time shift from t0 to t) += +d⃗ϕt0 +Dt(⃗yDt0) +� +�� +� +(Drive motion from t0 to t) +. dΘt0(⃗yS).⃗uS(⃗yS) +� +�� +� +(translation at t0) +. +(10.57) +10.10 +Drive and Coriolis forces +10.10.1 +Fundamental principal: requires a Galilean referential +Second Newton’s law of motion (fundamental principle of dynamics): In a Galilean referential, the sum +of the external forces ⃗f on an object is equal to its mass multiplied by its acceleration: +� +external⃗f = m⃗γ +(in a Galilean referential). +(10.58) +Question: And in a Non Galilean referential? +Answer: Then you have to add “observer dependent forces”, i.e. you have to add “apparent forces” +due to the motion of the non Galilean observer. Indeed, the motion of an object in our Universe does +not care about the observer motion (his accelerations and velocities). +See e.g. https://www.youtube.com/watch?v=_36MiCUS1ro for a carousel (a merry-go-round), +See e.g. https://www.youtube.com/watch?v=aeY9tY9vKgs for tornadoes. +10.10.2 +Drive + Coriolis forces = the inertial force +Consider ⃗f(t, pt) = the sum of the external forces acting on PObj at t at pt = �Φ(t, PObj). +In a Galilean referential RA, Newton laws (10.58) means +[⃗ft(pt)]| ⃗A = m [⃗γt(pt)]| ⃗A, +written +⃗fAt(⃗xAt) = m⃗γAt(⃗xAt) +∈ Mn1, +(10.59) +with ⃗xAt := [−−−→ +OApt]| ⃗A, ⃗fAt(⃗xAt) := [⃗ft(pt)]| ⃗A and ⃗γAt(⃗xAt) = [⃗γt(pt)]| ⃗A. With ⃗xAt = Θt(⃗xBt), the accelera- +tion addition formula gives ⃗fAt(⃗xAt) = m(dΘt.⃗γB(⃗xBt) + ⃗γDt(⃗xAt) + ⃗γCt(⃗xAt)) ∈ RA, thus, in RB, +dΘt +−1.⃗fAt(⃗xAt) +� +�� +� +⃗fAt∗(⃗xBt)= ⃗fBt(⃗xBt) += m⃗γB(⃗xBt) + m dΘt +−1.⃗γDt(⃗xAt) +� +�� +� +m ⃗γDt∗(⃗xBt) ++ m dΘt +−1.⃗γCt(⃗xAt) +� +�� +� +m ⃗γCt∗(⃗xBt) +, +(10.60) +and dΘt +−1.[⃗ft(pt)]| ⃗A = dΘt +−1.⃗fAt(⃗xAt) =(10.26) [⃗ft(pt)]| ⃗B =noted ⃗fBt(⃗xBt) is the external forces as quanti- +fied by B at t, cf. (10.26) (with Θt supposed to be affine). And with the pull-back notation, cf. (10.26): +68 + +69 +10.11. +Summary for “Sun and Earth” +Definition 10.18 At t on pt, define +• The drive force ⃗fBDt(⃗xBt) := −m dΘt +−1.⃗γDt(⃗xAt) +(= −m⃗γDt +∗(⃗xBt)). +• The Coriolis force ⃗fBCt(⃗xBt) := −m dΘt +−1.⃗γCt(⃗xAt) +(= −m⃗γCt +∗(⃗xBt)). +• (The inertial, or fictitious, force := ⃗fBDt(⃗xBt) + ⃗fBCt(⃗xBt) = −m dΘt +−1.(⃗γDt + ⃗γCt)(⃗xAt).) +(10.61) +Then (10.60) gives the fundamental principle quantified in RB (non Galilean referential): +⃗fBt(⃗xBt) + ⃗fBDt(⃗xBt) + ⃗fBCt(⃗xBt) = m⃗γB(⃗xBt) , +(10.62) +i.e., at t, in RB: The external force + the Drive and Coriolis forces = m times the acceleration. +10.11 +Summary for “Sun and Earth” +Illustation with a simplified (circular) motion of the Earth around the Sun. +10.11.1 +Coriolis forces on the Earth +1. Referentials. +1.1. Relative referential RB = (OB, ( ⃗B1, ⃗B2, ⃗B3)) chosen by the observer B fixed on the Earth, where OBt = +�ΦRB(t, QOB) is the position of the particle QOB at the center of the Earth, written OB by B (fixed for B), +and ( ⃗B1t, ⃗B2t, ⃗B3t) is a Euclidean basis (e.g. built with the metre) fixed in the Earth, written ( ⃗B1, ⃗B2, ⃗B3) +by B (fixed for B), with ⃗B3 chosen to be along the rotation axis of the Earth and oriented from the south +pole to the north pole; And (·, ·)B is the associated Euclidean dot product. So, a fixed particle QRB in +the Earth at longitude θQRB ∈] − π, π] and latitude ϕQRB ∈ [− π +2 , π +2 ] is referenced by observer B as the +matrix ⃗yS = ⃗ϕS(QRB) = [ +−−−−−−−−−−→ +OB �ΦRB(t, QRB)]| ⃗B = RB +� +� +cos(θQRB ) cos(ϕQRB ) +sin(θQRB ) cos(ϕQRB ) +sin(ϕQRB ) +� +� where RB = || +−−−−−−−−−−→ +OB �ΦRB(t, QRB)||B +is the distance between QOB and QRB (e.g. if QRB is on the surface of the Earth then RB ≃ 6371 km). +1.2. Initial Galilean referential RA0 = (OA0, ( ⃗A1, ⃗A2, ⃗A3)): OA0 is at the center of the Sun and ( ⃗A1, ⃗A2, ⃗A3) is +a Euclidean basis (e.g. built with the foot) fixed relative to the stars, such that ⃗A3 = µ ⃗B3 with µ > 0 +(e.g. µ = 0.3048 and λ = 1 +µ ≃ 3.28); And (·, ·)A is the associated Euclidean dot product. +1.3. Deduced absolute Galilean referential RA = (OAt, ( ⃗A1, ⃗A2, ⃗A3)) chosen by observer A fixed on Earth, +where OAt = OBt, written OA by A (fixed for A). Since it takes more that 365 days for QOB to complete a +rotation around the Sun, the motion of QOB will be considered to be rectilinear at constant velocity “in a +short interval of time” sufficient for the computation of the Coriolis acceleration with “sufficient accuracy” +(simplifies the calculations). +(If A prefers to work with the initial Galilean referential RA0, then the absolute matrix motion +⃗ϕA(t, PObj) = [ +−−−−−−−−→ +OA�Φ(t, PObj)]| ⃗A has to be replaced by ⃗ϕA(t, PObj) = [−−−−−−→ +OA0OB(t)]| ⃗A + [ +−−−−−−−−−−→ +OB(t)�Φ(t, PObj)]| ⃗A, +idem for the drive motion ⃗ϕD.) +2. Drive motion. +2.1. The motion t → qt = �ΦRB(t, QRB) of a particle QRB in the Earth is stored by A as the drive motion ⃗ϕD +given by (matrix valued), with ω the angular velocity of the Earth in RA, +⃗yD(t) = ⃗ϕD(t, QRB) = RA(QRB) +� +� +cos(ωt) cos ϕQRB +sin(ωt) cos ϕQRB +sin ϕQRB +� +� = [−−−−→ +OAq(t)]| ⃗A = +� +� +yD1(t) +yD2(t) +yD3 +� +� , +(10.63) +where RA(QRB) = ||−−−−−→ +QOBQRB||| ⃗A is the distance between QOB and QRB for A (e.g. RA ≃ 20902231 foot if +QRB is on the surface of the Earth). (And (ωt) by replaced by (α0+ω(t−t0)) to be more general.) +2.2. Drive velocity: With ⃗ωD := ω ⃗A3, +⃗vD(t, ⃗yD(t)) = ⃗yD +′(t) = ωRA +� +� +− sin(ωt) cos ϕQRB +cos(ωt) cos ϕQRB +0 +� +� = ω +� +� +−y2(t) +y1(t) +0 +� +� = ω +� +� +0 +−1 +0 +1 +0 +0 +0 +0 +0 +� +� .⃗yD(t) = ⃗ωD∧⃗yD(t). +(10.64) +69 + +70 +10.11. +Summary for “Sun and Earth” +2.3. Drive acceleration: +⃗γD(t, ⃗yDt) = ⃗yD +′′(t) = ⃗ωD ∧ ⃗yD +′(t) = ⃗ωD ∧ ⃗vD(t, ⃗yDt) = ⃗ωD ∧ (⃗ωD ∧ ⃗yD(t)) = −ω2 +� +� +yD1(t) +yD2(t) +0 +� +� +(10.65) += the usual centrifugal acceleration (in a plane parallel to the equatorial plane, drawing). +2.4. Differential of the drive velocity (time and space independent here): (10.64) gives +d⃗vD(t, ⃗yDt) = d⃗vD = +� +� +0 +−ω +0 +ω +0 +0 +0 +0 +0 +� +� = ⃗ωD ∧ . +(10.66) +3. Translator. +3.1. Here OAt = OBt, thus Θt(⃗0) = ⃗0 (with [⃗0] =noted ⃗0 = the null matrix), cf. (10.22). +3.2. Calculation of dΘt. With Θt affine, dΘt.[ ⃗Bit]| ⃗B = [ ⃗Bit]| ⃗A. Thus ⃗B3 = λ ⃗A3 (hypothesis) and dΘt.[ ⃗B3]| ⃗B = +[ ⃗B3t]| ⃗A give dΘt. ⃗E3 = λ ⃗E3 where ( ⃗Ei) is the canonical basis in Mn1. Then let QBi ∈ ObjRB be the +Earth particle which position qti = �ΦRB(t, QBi) makes ⃗Bit := −−−−→ +OBtqti. +So, ⃗B1 and ⃗B2 being in the +equatorial plane, (10.63) gives dΘt. ⃗E1 = dΘt.[ ⃗B1]| ⃗B = [ ⃗B1]| ⃗A = [−−−→ +OAqt1]| ⃗A = λ +� +� +cos(ωt) +sin(ωt) +0 +� +�, and dΘt. ⃗E2 = +dΘt.[ ⃗B2]| ⃗B = [ ⃗B2]| ⃗A = [−−−→ +OAqt2]| ⃗A = λ +� +� +− sin(ωt) +cos(ωt) +0 +� +�. Thus [dΘt]| ⃗E = λ +� +� +cos(ωt) +− sin(ωt) +0 +sin(ωt) +cos(ωt) +0 +0 +0 +1 +� +� = the +expected rotation matrix expanded by λ (change of unit of measurement). +3.3. Calculation of Θt (affine): Θt(⃗yS) = Θt(⃗0) + dΘt.⃗yS, so, with OAt = OBt here, +⃗yDt := Θt(⃗yS) = dΘt.⃗yS +(10.67) +4. Motions of Obj. +4.1. B quantifies the motion �Φ of Obj, i.e. he stores the relative motion ⃗ϕB of Obj, and the relative velocities +and accelerations ⃗vBt and ⃗γB (matrices), cf. (10.10)-(10.12). +4.2. Translations for A: With ⃗xAt = Θt(⃗xBt), +⃗vBt∗(⃗xAt) = dΘt(⃗xBt).⃗vBt(⃗xBt) +and +⃗γBt∗(⃗xAt) = dΘt(⃗xBt).⃗γBt(⃗xBt). +(10.68) +5. Drive force (apparent force in RB due to the motion of B): +⃗fBDt(⃗xBt) = −m dΘt +−1.⃗γDt(⃗xAt) +(10.65) += +λmω2dΘt +−1. +� +� +xA1(t) +xA2(t) +0 +� +� (10.67) += +λmω2 +� +� +xB1(t) +xB2(t) +0 +� +� , +(10.69) +centrifugal force (in a “parallel plane” at latitude of PObj). +6. Coriolis acceleration (apparent acceleration due to the motion of B): +⃗γCt(⃗xAt) = 2 d⃗vDt.(dΘt.⃗vBt(⃗xBt)) = 2 dΘt.d⃗vDt.⃗vBt(⃗xBt) +(10.70) +because dΘt commutes with d⃗vDt (composition of “rotations along the same south-north axis” which reads +as eiωt.ei π +2 = ei π +2 eiωt = ei( π +2 +ωt) in the equatorial plane). +7. Coriolis force (apparent force due to the motion of B): +⃗fBCt(⃗xBt) = −m dΘt +−1.⃗γCt(⃗xAt) = −2m d⃗vDt.⃗vBt(⃗xBt) = −2m⃗ω ∧ ⃗vBt(⃗xBt). +(10.71) +70 + +71 +11.1. +“Isometric objectivity” and “Frame Invariance Principle” +11 +Objectivities +Goal: To give an objective expression of the laws of mechanics; As Maxwell [13] said: “The formula at +which we arrive must be such that a person of any nation, by substituting for the different symbols the +numerical value of the quantities as measured by his own national units, would arrive at a true result”. +Generic notation: if a function z is given as z(t, x), then zt(x) := z(t, x), and conversely. +11.1 +“Isometric objectivity” and “Frame Invariance Principle” +This manuscript is not intended to describe “isometric objectivity”: +“Isometric objectivity” is the framework in which the “principle of material frame-indifference” (“frame +invariance principle”) is settled, principle which states that “Rigid body motions should not affect the +stress constitutive law of a material”. E.g., Truesdell–Noll [19] p. 41: +« Constitutive equations must be invariant under changes of frame of reference. » +Or Germain [9] : +« Axiom of power of internal forces. The virtual power of the "internal forces" acting on a +system S for a given virtual motion is an objective quantity; i.e., it has the same value whatever be the +frame in which the motion is observed. » +NB: Both of these affirmations are limited to “isometric changes of frame” (the same metric for all), as +Truesdell–Noll [19] page 42-43 explain: The “isometric objectivity” concern one observer who defines his +Euclidean dot product and consider only orthonormal change of bases to validate a constitutive law. +If you want to interpret “isometric objectivity” in the “covariant objectivity” framework, then “isometric +objectivity” corresponds to a dictatorial management: One observer with his Euclidean referential (e.g. +based on the English foot), imposes his unit of length to all other users (isometry hypothesis). (Note: +The metre was not adopted by the scientific community until after 1875.) +Moreover, isometric objectivity leads to despise the difference between covariance and contravariance, +due to the uncontrolled use of the Riesz representation theorem. +Remark 11.1 Marsden and Hughes [12] p. 8 use this isometric framework to begin with. But, pages 22 +and 163, they write that a “good modelization” has to be “covariant objective” (observer independent) to +begin with; And they propose a covariant modelization for elasticity at § 3.3. +11.2 +Definition and characterization of the covariant objectivity +11.2.1 +Framework of classical mechanics +Framework of classical mechanics to simplify. Consider two observers A and B and their referentials +RA = (OA, ( ⃗Ai)) and RB = (OB, ( ⃗Bi)). E.g., ( ⃗Ai) and ( ⃗Bi) are Euclidean bases in foot and metre, (·, ·)A +and (·, ·)B is their associated Euclidean dot products. And Θ is the translator, cf. (10.19). +Consider a regular motion �Φ of an object Obj, pt = �Φ(t, PObj) ∈ Rn the position at t of a particle in +our Universe, Ωt = �Φ(t, Obj) the configuration at t, and C = � +t∈[a,b]({t} × Ωt) the set of configurations. +And ⃗xAt := [−−−→ +OApt]| ⃗A ∈ Mn1(A) and ⃗xBt := [−−−→ +OBpt]| ⃗B ∈ Mn1(B) are the stored components of pt relative to +the chosen referentials, Mn1(A) and Mn1(B) being the spaces of n ∗ 1 matrices as referred to by A and B. +11.2.2 +Covariant objectivity of a scalar function +Let f +: +� +C → R +(t, pt) → f(t, pt) +� +be a Eulerian scalar function (e.g., a temperature field). +f is +quantified by A and B as the functions fA +: +� +R×Mn1(A) → R +(t, ⃗xAt) → fA(t, ⃗xAt) := f(t, pt) +� +and fB +: +� +R×Mn1(B) → R +(t, ⃗xBt) → fB(t, ⃗xBt) := f(t, pt) +� +. +Definition 11.2 f is objective covariant iff, for all referentials RA and RB and for all t, +fAt(⃗xAt) = fBt(⃗xBt) +when +⃗xAt = Θt(⃗xBt), +(11.1) +i.e. fAt = fBt∗ is the push-forward of fBt by Θt cf. (6.8). +71 + +72 +11.2. +Definition and characterization of the covariant objectivity +11.2.3 +Covariant objectivity of a vector field +Let +⃗w +: +� +C → ⃗Rn +(t, pt) → ⃗w(t, pt) +� +be a Eulerian vector field (e.g., +a force field). +⃗w is quan- +tified by A and B as the functions +⃗wA +: +� +R×Mn1(A) → Mn1(A) +(t, ⃗xAt) → ⃗wA(t, ⃗xAt) := [⃗w(t, pt)] ⃗A +� +and +⃗wB +: +� +R×Mn1(B) → Mn1(B) +(t, ⃗xBt) → ⃗wB(t, ⃗xBt) := [⃗w(t, pt)] ⃗B +� +. So ⃗wA(t, ⃗xAt) and ⃗wB(t, ⃗xBt) are the column matrices of the +components of ⃗w(t, pt) in RA and RB. +Definition 11.3 ⃗w is objective covariant iff, for all referentials RA and RB and for all t, +⃗wAt(⃗xAt) = dΘt(⃗xBt).⃗wBt(⃗xBt) +when +⃗xAt = Θt(⃗xBt), +(11.2) +i.e. ⃗wAt = ⃗wBt∗ is the push-forward of ⃗wBt by Θt cf. (6.20). +Example 11.4 Fundamental counter-example: A Eulerian velocity field is not objective, cf. (10.39), +because of the drive velocity ⃗vD ̸= ⃗0 in general. Neither is a Eulerian acceleration field, cf. (10.45). +Example 11.5 The field of gravitational forces (external forces) is objective covariant. +11.2.4 +Covariant objectivity of differential forms +Let α : +� +C → Rn∗ +(t, pt) → α(t, pt) +� +be a Eulerian differential form (e.g. a measuring device used to get the inter- +nal power). α is quantified by A and B as the functions αA : +� +R×Mn1(A) → Mn1(A) +(t, ⃗xAt) → αA(t, ⃗xAt) := [α(t, pt)] ⃗A +� +and αB : +� +R×Mn1(B) → Mn1(B) +(t, ⃗xBt) → αB(t, ⃗xBt) := [α(t, pt)] ⃗B +� +. So αA(t, ⃗xAt) and αB(t, ⃗xBt) are the row matrices +of the components of α(t, pt) in RA and RB. +Definition 11.6 α is objective covariant iff, for all referentials RA and RB and for all t, +αAt(⃗xAt) = αBt(⃗xBt).dΘt(⃗xBt)−1 +when +⃗xAt = Θt(⃗xBt). +(11.3) +i.e. αAt = αBt∗ is the push-forward of αBt by Θt cf. (7.3). +NB: (11.3) and (11.2) are compatible: If ⃗w is an objective vector field and if α is an objective differential +form, then the scalar function α.⃗w is objective: +αAt(⃗xAt).⃗wAt(⃗xAt) = αBt(⃗xBt).⃗wBt(⃗xBt) +(= (α(t, pt).⃗w(t, pt)), +(11.4) +since αAt(⃗xAt).⃗wAt(⃗xAt) = (αBt(⃗xBt).dΘt(⃗xBt)−1).(dΘt(⃗xBt).⃗wBt(⃗xBt)) = αBt(⃗xBt).⃗wBt(⃗xBt). +11.2.5 +Covariant objectivity of tensors +A tensor acts on both vector fields and differential forms, and its objectivity is deduced from the previous §. +So, let T be a (Eulerian) tensor corresponding to a “physical quantity”. +The observers A and B +describe T as being the functions TA and TB. +Definition 11.7 T is objective covariant iff, for all referentials RA and RB and for all t, +TAt(⃗xAt) = TBt∗(⃗xAt) +(11.5) +i.e. TAt is the push-forward of TBt by Θt. +(Recall: TBt∗(⃗xAt)(α1(⃗xAt), ..., ⃗w1(⃗xAt)) := TBt(⃗xBt)(α1∗(⃗xBt), ..., ⃗w1∗(⃗xBt)).) +72 + +73 +11.3. +Non objectivity of the velocities +Example 11.8 (Non covariant objectivity of a differential d⃗w) Let ⃗w be an objective vector field, +seen as ⃗wA by A and ⃗wB by B; So ⃗wAt(⃗xAt) =(11.2) dΘt(⃗xBt).⃗wBt(⃗xBt) when ⃗xAt = Θt(⃗xBt), thus +d⃗wAt(⃗xAt).dΘt(⃗xBt) = dΘt(⃗xBt).d⃗wBt(⃗xBt) + (d2Θt(⃗xBt).⃗wBt(⃗xBt)), +(11.6) +hence +d⃗wAt(⃗xAt) = dΘt(⃗xBt).d⃗wBt(⃗xBt).dΘt(⃗xBt)−1 + (d2Θt(⃗xBt).⃗wBt(⃗xBt)).dΘt(⃗xBt)−1 +̸= dΘt(⃗xBt).d⃗wBt(⃗xBt).dΘt(⃗xBt)−1 +when +d2Θt ̸= 0. +(11.7) +Thus d⃗w is not covariant objective in general. However in classical mechanics for “change of Cartesian +referentials” Θt is affine, so d2Θt = 0, and in particular d⃗w is objective when ⃗w is. And +(d2 ⃗wAt(⃗xAt).dΘt(⃗xBt)).dΘt(⃗xBt) + d⃗wAt(⃗xAt).d2Θt(⃗xBt) += dΘt(⃗xBt).d2 ⃗wBt(⃗xBt) + 2 d2Θt(⃗xBt).d⃗wBt(⃗xBt) + d3Θt(⃗xBt).⃗wBt(⃗xBt). +(11.8) +Thus d2 ⃗w is not covariant objective in general (but if Θt is affine then d2 ⃗w is objective if ⃗w is). +11.3 +Non objectivity of the velocities +11.3.1 +Eulerian velocity ⃗v : not covariant (and not isometric) objective +Velocity addition formala: With ⃗vBt∗(⃗xAt) = dΘt(⃗xBt).⃗w(⃗xBt) when ⃗xAt = Θt(⃗xBt), cf. (10.39), +⃗vAt(⃗xAt) = ⃗vBt∗(⃗xAt) + ⃗vDt(⃗xAt) +̸= ⃗vBt∗(⃗xAt) +when +⃗vDt(⃗xAt) ̸= ⃗0, +(11.9) +thus a Eulerian velocity field is not covariant objective (and not isometric objective). +11.3.2 +d⃗v is not objective +The velocity addition formula, (⃗vAt − ⃗vDt)(⃗xAt) = ⃗vBt∗(⃗xAt) = dΘt(⃗xBt).⃗vBt(⃗xBt) when ⃗xAt = Θt(⃗xBt), +gives +d(⃗vAt − ⃗vDt)(⃗xAt).dΘt(⃗xBt) = dΘt(⃗xBt).d⃗vBt(⃗xBt) + d2Θt(⃗xBt).⃗vBt(⃗xBt), +(11.10) +thus d⃗v is neither covariant objective nor isometric objective because of d⃗vD: +d⃗vAt(⃗xAt) = d⃗vBt∗(⃗xAt) + d⃗vDt(⃗xAt) + d2Θt(⃗xBt).⃗vBt(⃗xBt).dΘt(⃗xBt)−1 ̸= d⃗vBt∗(⃗xAt) +in general. (11.11) +Remark 11.9 Recall: “Isometric objective” implies +• The use of the same Euclidean metric in RB and RA, i.e. (·, ·)A = (·, ·)B, +• �ΦRB (motion of RB) is a solid body motion, and +• Θt is affine (so d2Θt = 0 for all t). +Exercice 11.10 Prove, with Qt the (orthonormal) transition matrix from ( ⃗Ai) to ( ⃗Bi): +[d⃗vt]| ⃗B = Qt.[d⃗vt]| ⃗A.Q−1 +t ++ Q′(t).Q−1 +t , +written +[L]| ⃗B = Q.[L]| ⃗A.QT + +• +Q.QT . +(11.12) +(Used in classical mechanics courses, to prove that d⃗v isn’t “isometric objective” because of +• +Q.QT .) +Answer. t0, t ∈ R, pt0 = �Φ(t0, PObj), pt = �Φ(t, PObj) = Φt0 +t (pt0), ⃗v(t, pt) = ∂ �Φ +∂t (t, PObj), and F t0 +t (pt0) = dΦt0 +t (pt0). +So ⃗v(t, Φt0 +t (pt0)) = +∂Φt0 +pt0 +∂t +(t, pt0), thus d⃗v(t, pt).F t0 +pt0 (t) = +∂F t0 +pt0 +∂t +(t). +And (4.30), with F t0 +pt0 =noted F, gives +[F(t)]|⃗at0 , ⃗ +B = Q(t).[F(t)]|⃗at0 , ⃗ +A, thus [F ′(t)]|⃗at0 , ⃗ +B = Q′(t).[F(t)]|⃗at0 , ⃗ +A + Q(t).[F ′(t)]|⃗at0 , ⃗ +A. Thus [d⃗v(t, pt)]| ⃗ +B = +[F t0 +pt0 +′(t).F t0 +pt0 (t)]| ⃗ +B += [F t0 +pt0 +′(t)]| ⃗ +B.[F t0 +pt0 (t)]| ⃗ +B += (Q′(t).[F(t)]|⃗at0 , ⃗ +A + Q(t).[F ′(t)]|⃗at0 , ⃗ +A).[F(t)]−1 +|⃗at0 , ⃗ +A.Q(t)−1 += +Q′(t).Q(t)−1 + Q(t).[F ′(t)]|⃗at0 , ⃗ +A.[F(t)]−1 +|⃗at0 , ⃗ +A.Q(t)−1 = Q′(t).Q(t)−1 + Q(t).[d⃗v(t, pt)]| ⃗ +A.Q(t)−1. And cf. (3.34)- +(3.35). +Exercice 11.11 Prove that d2⃗v is “isometric objective” when �ΦRB is a rigid body motion. +Answer. (11.8) with ⃗vA − ⃗vD instead of ⃗wA, and ⃗vB instead of ⃗wB give, in an “isometric objective” framework, +d2(⃗vAt − ⃗vDt)(⃗xAt).(⃗uBt∗, ⃗wBt∗) = dΘt(⃗xBt).d2⃗vBt(⃗xBt)(⃗uB, ⃗wB). +(11.13) +Here d2⃗vDt = 0 (rigid body motion), thus d2⃗v is “isometric objective”. +73 + +74 +11.4. +The Lie derivatives are covariant objective +11.3.3 +d⃗v + d⃗vT is “isometric objective” +Proposition 11.12 If �ΦRB is a rigid body motion then d⃗vt + d⃗vT +t is “isometric objective” +d⃗vAt + d⃗vT +At = (d⃗vBt + d⃗vT +Bt)∗. +(11.14) +(Isometric framework: The rate of deformation tensor is independent of an added added rigid motion.) +Proof. Q.QT = I gives +• +Q.QT + ( +• +Q.QT )T = 0, then apply (11.12). +Exercice 11.13 Prove that Ω = d⃗v−d⃗vT +2 +is not isometric objective. +Answer. (11.11) gives d⃗vT +At = d⃗vT +Bt∗ + d⃗vT +Dt, thus +d⃗vAt−d⃗vT +At +2 += +d⃗vBt∗−d⃗vT +Bt∗ +2 ++ +d⃗vDt−d⃗vT +Dt +2 +̸= +d⃗vBt∗−d⃗vT +Bt∗ +2 +, even if +�ΦRB is a solid body motion (then +d⃗vDt−d⃗vT +Dt +2 += ⃗ω∧ is a rotation time a dilation). +11.3.4 +Lagrangian velocities +The Lagrangian velocities do not define a vector field, cf. § 3.2.2. Thus asking about the objectivity of +Lagrangian velocities is meaningless. +11.4 +The Lie derivatives are covariant objective +Framework of § 10. In particular we have the velocity-addition formula ⃗vAt = ⃗vBt∗ + ⃗vDt in RA where +⃗vBt∗(⃗xAt) = dΘt(⃗xBt).⃗vBt(⃗xBt) and ⃗xBt = Θt(⃗xAt), cf. (10.39). +The objectivity under concern is the covariant objectivity (no inner dot product or basis required). +The Lie derivatives are also called “objective rates” because they are covariant objectives. Easy proofs. +11.4.1 +Scalar functions +Proposition 11.14 If f be a covariant objective function, cf. (11.1), then its Lie derivative L⃗vf is +covariant objective: +L⃗vAfA = Θ∗(L⃗vBfB), +i.e. +L⃗vAfA(t, ⃗xAt) = L⃗vBfB(t, ⃗xBt) +when +⃗xAt = Θt(⃗xBt), +(11.15) +i.e., DfA +Dt (t, ⃗xAt) = DfB +Dt (t, ⃗xBt), i.e. ( ∂fA +∂t + dfA.⃗vA)(t, ⃗xAt) = ( ∂fB +∂t + dfB.⃗vB)(t, ⃗xBt). +Proof. Consider the motion t → p(t) = �Φ(tPObj) of a particle PObj, and ⃗xA(t) = [−−−−→ +OAp(t)]| ⃗A and ⃗xB(t) = +[−−−−→ +OBp(t)]| ⃗B. With f objective, (11.1) gives fB(t, ⃗xB(t)) = fA(t, Θ(t, ⃗xB(t))) (= fA(t, ⃗xA(t))), thus +DfB +Dt (t, ⃗xB(t)) = ∂fA +∂t (t, ⃗xA(t)) + dfAt(⃗xA(t)).(∂Θ +∂t (t, ⃗xB(t)) +� +�� +� +⃗vDt(⃗xAt) ++ dΘt(⃗xB(t)).⃗vBt(⃗xB(t))) +� +�� +� +⃗vBt∗(⃗xAt) += ∂fA +∂t (t, ⃗xAt) + dfAt(⃗xAt).⃗vAt(⃗xAt) = DfA +Dt (t, ⃗xAt), +(11.16) +thanks to velocity addiction formula ⃗vAt = ⃗vBt∗ + ⃗vDt. +11.4.2 +Vector fields +Proposition 11.15 Let ⃗w be a covariant objective vector field, cf. (11.2). Then its Lie derivative L⃗v ⃗w +is covariant objective: +L⃗vA ⃗wA = Θ∗(L⃗vB ⃗wB), +(11.17) +i.e., when ⃗xAt = Θt(⃗xBt), +L⃗vA ⃗wA(t, ⃗xAt) = dΘt(⃗xBt).L⃗vB ⃗wB(t, ⃗xBt), +(11.18) +i.e., +(D ⃗wA +Dt +− d⃗vA.⃗wA)(t, ⃗xAt) = dΘ(t, ⃗xBt).(D ⃗wB +Dt +− d⃗vB.⃗wB)(t, ⃗xBt), +(11.19) +i.e., +(∂ ⃗wA +∂t ++ d⃗wA.⃗vA − d⃗vA.⃗wA)(t, ⃗xAt) = dΘ(t, ⃗xBt).(∂ ⃗wB +∂t ++ d⃗wB.⃗vB − d⃗vB.⃗wB)(t, ⃗xBt). +(11.20) +But the partial, convected, material, and Lie autonomous derivatives are not covariant objective (not +74 + +75 +11.4. +The Lie derivatives are covariant objective +even isometric objective because of the drive velocity ⃗vD): We have +(d⃗wAt.(⃗vAt−⃗vDt))(⃗xAt) = (dΘt.(d⃗wBt.⃗vBt) + (d2Θt.⃗wBt).⃗vBt)(⃗xBt), +(11.21) +(d(⃗vAt−⃗vDt).⃗wAt)(⃗xAt) = (dΘt.(d⃗vBt.⃗wBt) + (d2Θt.⃗vBt).⃗wBt)(⃗xBt), +(11.22) +(d(⃗vAt−⃗vDt).(⃗vAt−⃗vDt))(⃗xAt) = (dΘt.(d⃗vBt.⃗vBt) + d2Θt(⃗vBt,⃗vBt))(⃗xBt), +(11.23) +L0 +(⃗vAt−⃗vDt) ⃗wAt(⃗xAt) = dΘt(⃗xBt).L0 +⃗vBt ⃗wBt(⃗xBt), +(11.24) +∂ ⃗wA +∂t (t, ⃗xAt) + L0 +⃗vD ⃗wAt(⃗xAt) = dΘt(⃗xBt).∂ ⃗wB +∂t (t, ⃗xBt), +(11.25) +D ⃗wA +Dt (t, ⃗xAt) − d⃗vDt.⃗wAt(⃗xAt) = dΘt.(⃗xBt).D ⃗wB +Dt (t, ⃗xBt) + d2Θt(⃗vBt, ⃗wBt)(⃗xBt), +(11.26) +∂(⃗vA−⃗vD) +∂t +(t, ⃗xAt) + L0 +⃗vD(⃗vA−⃗vD)(t, ⃗xAt) = dΘt(⃗xBt).∂⃗vB +∂t (t, ⃗xBt). +(11.27) +Proof. • ⃗wAt(Θt(⃗xBt)) = dΘt(⃗xBt).⃗wBt(⃗xBt) gives +d⃗wAt(⃗xAt).dΘt(⃗xBt) = d2Θt(⃗xBt).⃗wBt(⃗xBt) + dΘt(⃗xBt).d⃗wB(⃗xBt), +(11.28) +thus, with dΘt(⃗xBt).⃗vBt(⃗xBt) = (⃗vAt−⃗vDt)(⃗xAt) = ⃗vBt∗(⃗xAt) (velocity-addition formula), +d⃗wAt(⃗xAt).(⃗vAt−⃗vDt)(⃗xAt) = (d2Θt(⃗xBt).⃗vBt(⃗xBt)).⃗wBt(⃗xBt) + dΘt(⃗xBt).d⃗wBt(⃗xBt).⃗vBt(⃗xBt), +hence (11.21). In particular d⃗wAt(⃗xAt).⃗vAt(⃗xAt) ̸= dΘt(⃗xBt).(d⃗wBt(⃗xBt).⃗vBt(⃗xBt)) (the vector field d⃗w.⃗v is +not objective). +• (⃗vAt−⃗vDt)(Θt(⃗xBt)) = dΘt(⃗xBt).⃗vBt(⃗xBt) gives +d(⃗vAt−⃗vDt)(⃗xAt).dΘt(⃗xBt) = d2Θt(⃗xBt).⃗vBt(⃗xBt) + dΘt(⃗xBt).d⃗vBt(⃗xBt), +so, applied to ⃗wBt (resp. ⃗vBt), we get (11.22) (resp. (11.23)). Hence (11.24). +• If ⃗xAt = Θt(⃗xB), then ⃗wA(t, Θ(t, ⃗xB)) = dΘ(t, ⃗xB).⃗wB(t, ⃗xB), so, with ∂Θ +∂t (t, ⃗xB) = ⃗vΘt(⃗xAt), we get +∂ ⃗wA +∂t (t, ⃗xAt) + d⃗wAt(⃗xAt).⃗vΘt(⃗xAt) = d∂Θ +∂t (t, ⃗xB).⃗wBt(⃗xB) + dΘt(⃗xB).∂ ⃗wB +∂t (t, ⃗xB) += (d⃗vΘt(⃗xAt).dΘt(⃗xB)).⃗wBt(⃗xB) + dΘt(⃗xB).∂ ⃗wB +∂t (t, ⃗xB), +Thus (11.25) since ⃗vΘ = ⃗vD; Then (11.21) gives (11.26). +• ⃗vB∗(t, Θ(t, ⃗xB)) = dΘ(t, ⃗xB).⃗vB(t, ⃗xB) gives +∂⃗vB∗ +∂t (t, ⃗xAt) + d⃗vB∗(⃗xAt).⃗vΘ(t, ⃗xAt) = +∂dΘ +∂t (t, ⃗xB) +� +�� +� +d⃗vΘt(⃗xAt).dΘt(⃗xB) +.⃗vBt(⃗xB) + dΘ(t, ⃗xB).∂⃗vB +∂t (t, ⃗xB, ) +since ∂dΘ +∂t (t, ⃗xB) = d( ∂Θ +∂t )(t, ⃗xB) and ∂Θ +∂t (t, ⃗xB) = ⃗vΘ(t, ⃗xAt) = ⃗vΘt(Θt(⃗xB)); hence (11.27). +11.4.3 +Tensors +Proposition 11.16 It T is a covariant objective tensor, then its Lie derivatives are covariant objectives: +L⃗vATA = Θ∗(L⃗vBTB). +(11.29) +Proof. Corollary of (11.15) and (11.18) to get L⃗v(α.⃗w) = (L⃗vα).⃗w + α.(L⃗v ⃗w); Then use L⃗v(t1 ⊗ t2) = +(L⃗vt1) ⊗ t2 + t1 ⊗ (L⃗vt2). +75 + +76 +11.5. +Taylor expansions and ubiquity gift +11.5 +Taylor expansions and ubiquity gift +11.5.1 +In Rn with ubiquity +Generic formula: +f(t) = f(t0) + (t−t0) f ′(t0) + (t−t0)2 +2 +f ′(t0)2 + o((t−t0)2). +(11.30) +In particular f(t) = ⃗w(t, p(t)) gives +⃗w(t, p(t)) = ⃗w(t0, p(t0)) + (t−t0) D ⃗w +Dt (t0, p(t0)) + (t−t0)2 +2 +D ⃗w +Dt (t0, p(t0))2 + o((t−t0)2). +(11.31) +Problem : +⃗w(t, p(t)) is a vecteur at t at p(t) while ⃗w(t0, p(t0)) is a vecteur at t0 at p(t0), so (11.31) +cannot be written +⃗w(t, p(t)) − +� +⃗w(t0, p(t0)) + (t−t0) D ⃗w +Dt (t0, p(t0)) + (t−t0)2 +2 +D ⃗w +Dt (t0, p(t0))2� += o((t−t0)2), +(11.32) +since the left-hand side supposes the ubiquity gift. +E.g. in a non-planar manifold (e.g. a surface in R3 considered on its own), ⃗w(t, pt) ∈ Tpt(Ωt) = the +linear tangent space at p(t) = pt, whereas ⃗w(t0, pt0) ∈ Tpt0(Ωt0) = the linear tangent space at p(t0) = pt0, +and the tangent spaces Tpt(Ωt) and Tpt0(Ωt0) are distinct at two distinct points in general; Thus the +left-hand side of (11.32) is meaningless. +In R3 our affine space (our Universe), Tpt(Ωt) and Tpt0(Ωt0) are identified with ⃗R3, and (11.31) is well +defined, and very useful! +11.5.2 +General case +By definition, cf. (9.10), with p(t) = Φt0(t, pt0) = Φt0 +t (pt0), +L⃗v ⃗w(t0, pt0) = dΦt0 +t (pt0)−1.⃗w(t, p(t)) − ⃗w(t0, pt0) +t − t0 ++ o(1). +(11.33) +Thus, +dΦt0 +t (pt0)−1.⃗w(t, p(t)) = ⃗w(t0, pt0) + (t−t0) L⃗v ⃗w(t0, pt0) + o(t−t0). +(11.34) +Hence we get the first order Taylor expansion without ubiquity gift: +⃗w(t, p(t)) = dΦt0 +t (pt0). +� +⃗w + (t−t0) L⃗v ⃗w +� +(t0, pt0) + o(t−t0), +(11.35) +both side of the equality being in Tpt(Ωt) (meaningful in any manifold). +Proposition 11.17 In Rn, with the gift of ubiquity, (11.35) gives (11.31). +Proof. dΦt0(t0+h, pt0)).⃗w(t0, pt0) =(4.35) ⃗w(t0, pt0) + h d⃗v(t0, pt0).⃗w(t0, pt0) + o(h), thus +⃗w(t, pt) +(11.35) += +(I + h d⃗v(t0, pt0) + o(h)). +� +(⃗w + h D ⃗w +Dt − h d⃗v.⃗w)(t0, pt0) + o(h) +� += (⃗w + h d⃗v.⃗w + h D ⃗w +Dt − h d⃗v.⃗w)(t0, pt0) + o(h) = (⃗w + h D ⃗w +Dt )(t0, pt0) + o(h), +which is (11.31). +Proposition 11.18 In Rn, at second order, +⃗w(t, p(t)) = dΦt0 +t (pt0). +� +(⃗w + hL⃗v ⃗w + h2 +2 L⃗v(L⃗v ⃗w))(t0, pt0) + o(h2) +� +. +(11.36) +So if the values ⃗w(t0, pt0), L⃗v ⃗w(t0, pt0) and L⃗v(L⃗v ⃗w)(t0, pt0) are known, then ⃗w(t, p(t)) is estimated at +second order thanks to the push-forward of (⃗w + hL⃗v ⃗w + h2 +2 L⃗v(L⃗v ⃗w))(t0, pt0) by Φt0 +t . +76 + +77 +11.5. +Taylor expansions and ubiquity gift +Proof. Let dΦt0(t, pt0) = F t0 +pt0 (t). +Let ⃗g(t) = dΦt0(t, pt0)−1.⃗w(t, p(t)) when p(t) = Φt0(t, pt0). +So +L⃗v ⃗w(t0, pt0) = ⃗g ′(t0), cf. (9.11). And D ⃗w +Du (u, p(u)) = d⃗v(u, p(u)).⃗w(u, p(u)) + F t +u(pt).⃗g ′(u), cf. (9.16). +Thus +D2 ⃗w +Du2 (u, p(u)) = +D(d⃗v) +Du (u, p(u)).⃗w(u, p(u)) + d⃗v(u, p(u)). D ⃗w +Du (u, p(u)) + d⃗v(u, p(u).F t +u(pt).⃗g ′(u) + +F t +u(pt).⃗g ′′(u). Thus D2 ⃗w +Dt2 (t, p(t)) = ( D(d⃗v) +Dt .⃗w + d⃗v. D ⃗w +Dt + d⃗v.L⃗v ⃗w)(t, p(t)) + ⃗g ′′(t). With (9.39) we get +⃗g ′′(t) = L⃗v(L⃗v ⃗w)(t, p(t)), cf. (9.39). +Alternate proof (calculation): (4.34) gives F t0 +pt0 (t) = It0 + h d⃗v(t0, pt0) + h2 +2 d⃗γ(t0, pt0) + o(h2). Thus, +omitting the reference to (t0, pt0) to lighten the writing, +dΦt0 +t (pt0).(⃗w + hL⃗v ⃗w + h2 +2 L⃗vL⃗v ⃗w + o(h2)) += +� +I + h d⃗v + h2 +2 d(D⃗v +Dt ) + o(h2) +� +. +� +⃗w + hL⃗v ⃗w + h2 +2 L⃗vL⃗v ⃗w + o(h2) +� +(11.37) +The h0 term is I.⃗w = ⃗w. The h term is L⃗v ⃗w + d⃗v.⃗w = D ⃗w +Dt . The h2 term is the sum of +• 1 +2L⃗vL⃗v ⃗w = 1 +2(D2 ⃗w +Dt2 − 2 d⃗v.D ⃗w +Dt − D(d⃗v) +Dt +.⃗w + d⃗v.d⃗v.⃗w), cf.(9.39), +• d⃗v.L⃗v ⃗w = d⃗v.D ⃗w +Dt − d⃗v.d⃗v.⃗w = 1 +2(2d⃗v.D ⃗w +Dt − 2d⃗v.d⃗v.⃗w), +• 1 +2d(D⃗v +Dt ).⃗w = 1 +2(D(d⃗v) +Dt +.⃗w + d⃗v.d⃗v.⃗w), cf.(2.36). +And the sum gives D2 ⃗w +Dt2 . +77 + +78 +Part V +Appendix +In this appendix, we tried to give standard results useful in mechanics, results that are scattered in the +existing literature, and sometimes difficult to find except in math books (differential geometry). +The definitions, notations and results are detailed, so that no ambiguity is possible (some notations +can be nightmarish when not understood, or misused, or come like a bull in a china-shop). +All the +results presented apply to solids, fluids, thermodynamics, general relativity, electromagnetism, quantum +mechanics, chemistry... (the same math applies to all... even applies to mechanical engineers...). +A +Classical and duality notations +A.1 +Contravariant vector and basis +A.1.1 +Contravariant vector +Let (E, +, .) =noted E be a real vector space (= a linear space on the field R). +Definition A.1 An element ⃗x ∈ E is called a vector, or a “contravariant vector”. +A vector is a vector... So why this name contravariant? Historical answer: Because of the change of +basis formula [⃗x]|new = P −1.[⃗x]|old, see (A.28), which uses P −1. +So, what is a covariant vector? Answer: From the vector space E, you can build the vector space (an +overlay) L(E; R) =noted E∗ = the space of linear forms on E (a linear form is a measuring instrument +that gives values to vectors). Then an element ℓ ∈ E∗ will be called a covariant vector, because of the +change of basis formula [ℓ]|new = [ℓ]|old.P. See § A.5 for details. +A.1.2 +Basis +Definitions: • n vectors ⃗e1, ...,⃗en ∈ E are linearly independent iff, for all λ, ..., λn ∈ R, the equality +�n +i=1λi⃗ei = ⃗0 implies λi = 0 for all i = 1, ..., n. +• n vectors ⃗e1, ...,⃗en ∈ E span E iff, for all ⃗x ∈ E, ∃λ1, ..., λn ∈ R such that ⃗x = �n +i=1λi⃗ei. +• A basis in E is a set {⃗e1, ...,⃗en} ⊂ E made of n linearly independent vectors which span E, in which +case the dimension of E is n. +A.1.3 +Canonical basis +Consider the field R of reals and the Cartesian product ⃗Rn = R × ... × R, n times. The canonical basis is +⃗e1 = (1, 0, ..., 0), ..., ⃗en = (0, ..., 0, 1), +(A.1) +with 0 = the addition identity element used n−1 times, and 1 = the multiplication identity element used +once. +Remark A.2 The 3-D geometric space we live in has no canonical basis: What would the identity +element 1 mean? 1 metre? 1 foot? And there is no “intrinsic” preferred direction to define ⃗e1. So the +Cartesian product ⃗Rn = R × ... × R and its canonical basis form an abstract mathematical model. +A.1.4 +Cartesian basis +(René Descartes 1596-1650.) Let n = 1, 2, 3, let Rn be the usual affine space (space of points), and let +⃗Rn = (⃗Rn, +, .) be the usual real vector space of bipoint vectors with its usual algebraic operations. +Let p ∈ Rn, and let (⃗ei(p)) be a basis at p. +A Cartesian basis in ⃗Rn is a basis independent of p (the same at all p), and then (⃗ei(p)) =noted (⃗ei). +Example of a non Cartesian basis: The polar basis, see example 6.11 (polar coordinate system). +And a Euclidean basis is a particular Cartesian basis described in § B.1. +More generally, a Cartesian basis refers to En = E ×...×E (n-times) where E is a dimension 1 vector +space. +78 + +79 +A.2. +Representation of a vector relative to a basis +A.2 +Representation of a vector relative to a basis +We give: +• the classical notation (non ambiguous), e.g. used by Arnold [3] and Germain [8], and +• the duality notation (can be ambiguous because of misuses), e.g. used by Marsden and Hughes [12]. +Both classical and duality notation are equally good, but if you have any doubt, use the classical notations. +Definition A.3 Let ⃗x ∈ E. Let (⃗ei) be a basis in E. The components of ⃗x relative to the basis (⃗ei) are +the n real numbers x1, ..., xn (classical notation) also named x1, ..., xn (duality notation) such that +⃗x = x1⃗e1 + ... + xn⃗en +� +�� +� +clas. += x1⃗e1 + ... + xn⃗en +� +�� +� +dual +, +i.e. +[⃗x]|⃗e = +� +� +x1 +... +xn +� +� +� �� � +clas. += +� +� +x1 +... +xn +� +� +� �� � +dual +, +(A.2) +[⃗x]|⃗e being the column matrix representing ⃗x relative to the basis (⃗ei). (Of course xi = xi for all i.) And +the column matrix [⃗x]|⃗e is simply named [⃗x] if one chosen basis is imposed to all. With the sum sign: +⃗x = +n +� +i=1 +xi⃗ei +� �� � +clas. += +n +� +i=1 +xi⃗ei +� �� � +dual +(= +n +� +J=1 +xJ⃗eJ = +n +� +α=1 +xα⃗eα). +(A.3) +(The index in a summation is a dummy index, even if you do not write the sum sign � as can be done +with Enstein’s convention: ⃗x = �n +j=1xj⃗ej =noted xj⃗ej = xi⃗ei = xJ⃗eJ = xα⃗eα.) +Example A.4 In ⃗R2 with ⃗x = 3⃗e1 + 4⃗e2 = �2 +i=1 xi⃗ei = �2 +i=1 xi⃗ei: We have x1=x1=3 and x2=x2=4. +And [⃗x]|⃗e = 3[⃗e1]|⃗e +4[⃗e2]|⃗e = �2 +i=1 xi[⃗ei]|⃗e = �2 +i=1 xi[⃗ei]|⃗e. In particular, with δi +j = δij := +� += 1 if i=j += 0 if i̸=j +� +the Kronecker symbols, +⃗ej = +n +� +i=1 +δij⃗ei +� �� � +clas. += +n +� +i=1 +δi +j⃗ei +� �� � +dual +, +i.e. +[⃗e1]|⃗e = +� +� +� +� +1 +0 +... +0 +� +� +� +� , ..., [⃗en]|⃗e = +� +� +� +� +0 +... +0 +1 +� +� +� +� , +(A.4) +that is, the components of ⃗ej are δij with classical notations, and δi +j with duality notations. And the +matrices [⃗ej]|⃗e mimic the use of theoretical Cartesian space ⃗Rn = R × ... × R and its canonical basis. +Remark A.5 The column matrix [⃗x]|⃗e is also called a “column vector”. NB: A “column vector” is not a +vector, but just a matrix (a collection of real numbers). See the change of basis formula (A.28) where +the same vector is represented by two “column vectors” (two column matrices). +A.3 +Dual basis +Recall: Let E and F be vector spaces and (F(E; F), +, .) =noted F(E; F) be the usual real vector space +of functions with the internal addition (f, g) → f +g defined by (f +g)(x) := f(x)+g(x) and the external +multiplication (λ, f) → λ.f defined by (λ.f)(x) := λ(f(x)), for all f, g ∈ F(E; F), x ∈ E, λ ∈ R. And +λ.f =noted λf for all f ∈ F(E; F) and λ ∈ R. +A.3.1 +Linear forms = “Covariant vectors” +Definition A.6 The set E∗ := L(E; R) of linear scalar valued functions is called the dual of E: +E∗ := L(E; R) = the dual of E. +(A.5) +And a linear scalar valued function ℓ ∈ E∗ is called a linear form. +More precisely, E∗ as defined in (A.5) is the algebraic dual of E; To define the topological dual usually +needed with L2 functions in mechanics, E needs to be a Banach space (a vector space equipped with +a norm with which E is complete), and E∗ is then the set of continuous linear forms. (If E is finite +dimensional then any linear form is continuous relative to any norm since all norms are equivalent in +finite dimension.) +79 + +80 +A.3. +Dual basis +E∗ is a vector space: sub-space of (F(R; R), +, .) (trivial). +Interpretation: It answers the question: What does a function E → R do? Answer: Like any function, +it gives values to vectors: ℓ(⃗u) = the value of ⃗u through ℓ. That is, a ℓ ∈ E∗ is a measuring tool for +vectors: If ⃗u ∈ E then ℓ(⃗u) = real value given by ℓ. +Notation: If ℓ ∈ E∗ then +∀⃗u ∈ E, +ℓ(⃗u) noted += +ℓ.⃗u, +(A.6) +also written ⟨ℓ, ⃗u⟩E∗,E where ⟨., .⟩E∗,E is the duality bracket: The dot in ℓ.⃗u is “the distributivity dot” +since linearity ℓ(⃗u + λ⃗v) = ℓ(⃗u) + λℓ(⃗v) = distributivity ℓ.(⃗u + λ⃗v) = ℓ.⃗u + λℓ.⃗v. +NB: The dot in ℓ.⃗u is not an inner dot product (since ℓ /∈ E while ⃗u ∈ E). +Definition A.7 A linear form ℓ in E∗ is also called a “covariant vector”; Co-variant refers to: +1- The action of a function on a vector, cf. (A.6) (co-variant calculation), and +2- The change of coordinate formula [ℓ]new = [ℓ]|old.P, see (A.28) (covariant formula). +NB: E∗ being a vector space, an element ℓ ∈ E∗ is indeed a vector. But E∗ has no existence if E has +not been specified first since E∗ := L(E; R). And ℓ ∈ E∗ can’t be confused with a vector ⃗u ∈ E since +there is no natural canonical isomorphism between E and E∗ (no “intrinsic representation”), see § T.2. +Remark A.8 Misner–Thorne–Wheeler [14], box 2.1, insist: “Without it [the distinction between covari- +ance and contravariance], one cannot know whether a vector is meant or the very different object that is +a linear form.” +A.3.2 +Covariant dual basis (= the functions that give the components of a vector) +Notation: If ⃗u1, ..., ⃗uk are vectors in E, then Vect{⃗u1, ..., ⃗uk} := the vector space spanned by ⃗u1, ..., ⃗uk. +Let E be a finite dimensional vector space, and let (⃗ei)i=1,...,n be a basis in E +Definition A.9 Let i ∈ [1, n]N. The scalar projection on Vect{⃗ei} parallel to Vect{⃗e1, ...,⃗ei−1,⃗ei+1, ...,⃗en} +is the linear form named πei ∈ E∗ with the classical notation, named ei ∈ E∗ with the duality notation, +defined by, for all i, j, +� +clas. not. : +πei(⃗ej) = δij, +i.e. +πei.⃗ej = δij, +dual not. : +ei(⃗ej) = δi +j, +i.e. +ei.⃗ej = δi +j. +(A.7) +Thus, πei = ei being linear, if ⃗x =clas. �n +i=1xi⃗ei =dual �n +i=1xi⃗ei (classical or duality notations), then +(A.7) gives +πei.⃗x clas. += xi += +xi dual += ei.⃗x, +(A.8) +i.e. πei = ei gives the i-th component of a vector ⃗x relative to the basis (⃗ei), see figure A.1. +Figure A.1: +Parallel projections: πe1(⃗x) = x1 and πe2(⃗x) = x2 (dual not.: e1(⃗x) = x1 and e2(⃗x) = x2). +NB: The dual basis (πei) is intrinsic to (⃗ei); And there can’t be any notion of orthogonality in E here +since we can’t use a inner dot product: The functions πei = ei and vectors ⃗x do not belong to a same +vector space. +Proposition A.10 and definition of the dual basis. (πei)i=1,...,n = (ei)i=1,...,n is a basis in E∗, +called the (covariant) dual basis of the basis (⃗ei). +80 + +2 +f +1 +1 +2 +1 +er81 +A.3. +Dual basis +Proof. If �n +i=1λiπei = 0, then 0 = (�n +i=1λiπei)(⃗ej) = �n +i=1λiπei(⃗ej) = �n +i=1λiδij = λj for all j, +thus (πei)i=1,...,n is a family of n independent vectors in E∗. Then let ℓ ∈ E∗ and m = � +i(ℓ.⃗ei)πei. +Thus m ∈ E∗ (since E∗ is a vector space), and m(⃗ej) = � +i(ℓ.⃗ei)(πei.⃗ej) = � +i(ℓ.⃗ei)δij = (ℓ.⃗ej), thus +m = ℓ, thus ℓ = � +i(ℓ.⃗ei)πei, thus Vect{(πei)i=1,...,n} span E∗; Thus (πei)i=1,...,n is a basis in E∗; Thus +dim E∗ = n. (Use duality notations if you prefer.) +Example A.11 Following example 1.1. The size of a child is represented on a wall by a bipoint vector ⃗u. +And English observer chooses the foot as unit of length, represented by a vertical bipoint vector which he +names ⃗e. And then defines the linear form πe : ⃗R → R by πe.⃗e = 1. Thus πe is a measuring instrument, +which gives s = πe.⃗u = the size of the child in foot, i.e. ⃗u = s⃗e. +Exercice A.12 Let (⃗ai) and (⃗bi) be bases and let (πai) and (πbi) be the dual bases. Let λ ̸= 0. Prove: +If ∀i = 1, ..., n, ⃗bi = λ⃗ai, +then +∀i = 1, ..., n, πbi = 1 +λ πai. +(A.9) +(With duality notations, bi = 1 +λ ai.) +Answer. πbi.⃗bj = δij = πai.⃗aj = πai. +⃗bj +λ = 1 +λ πai.⃗bj for all j (since πai is linear), thus πbi = 1 +λ πai, true for all i. +A.3.3 +Example: aeronautical units +Example A.13 International aeronautical units: Horizontal length = nautical mile (NM), altitude = +English foot (ft). Application: An air traffic controller chooses the point O = the position of its control +tower, and a plane p is located thanks to the bipoint vector ⃗x = −→ +Op. And the traffic controller chooses ⃗e1 = +the vector of length 1 NM oriented South (first runway), ⃗e2 = the vector of length 1 NM oriented Southwest +(second runway), ⃗e3 = the vertical vector of length 1 ft. Thus his referential is R = (O, (⃗e1,⃗e2,⃗e3)), and +his dual basis (πe1, πe2, πe3) is defined by πei(⃗ej) = δij for all i, j, cf. (A.7). He writes ⃗x = �n +i=1xi⃗ei ∈ ⃗Rn, +so that x1 = πe1(⃗x) = the distance to the south in NM, x2 = πe2(⃗x) = the distance to the southwest +in NM, x3 = πe3(⃗x) = the altitude in ft. +Here the basis (⃗ei) is not a Euclidean basis. This non Euclidean basis (⃗ei) is however vital if you take +a plane. (A Euclidean basis is not essential to life...). See next remark A.14. +Remark A.14 The metre is the international unit for NASA that launched the Mars Climate Orbiter +probe, and the foot is the international vertical unit for aviation; And for the Mars Climate Orbiter landing +procedure, NASA (uses the metre) asked Lockheed Martin (uses the foot) to do the computation. Result? +The Mars Climate Orbiter space probe burned in the Martian atmosphere because of λ ∼ 3 times too +high a speed during the landing procedure: One metre is λ ∼ 3 times one foot, and someone forgot it... +Although NASA and Lockheed Martin used a Euclidean dot product... But not the same (one based on +a metre, and one based on the foot). Objectivity and covariance can be useful... +A.3.4 +Matrix representation of a linear form +Let ℓ ∈ E∗, let (⃗ei) be a basis: The components of ℓ are the n reals +ℓi := ℓ(⃗ei) = ℓ.⃗ei, +and +[ℓ]|⃗e = ( ℓ1 +... +ℓn ) +(A.10) +is the row matrix of ℓ, +called the matrix of ℓ relative to (⃗ei). +Thus, +if ⃗x +∈ +E +and +⃗x =clas. �n +i=1xi⃗ei =dual �n +i=1xi⃗ei, then +ℓ.⃗x clas. += +n +� +i=1 +ℓixi +dual += +n +� +i=1 +ℓixi = [ℓ]|⃗e.[⃗x]|⃗e +(A.11) +with usual matrix computation rules (a 1 ∗ n matrix times a n ∗ 1 matrix). +In particular for the dual basis (πei) = (ei) (classical and duality notations), +[πej]|⃗e = [ej]|⃗e = (0 ... 0 +1 +���� +jth position +0 ... 0) +(= row matrix = [⃗ej]T +|⃗e). +(A.12) +Thus we have, with classical and duality notations, +ℓ clas. += +n +� +i=1 +ℓi πei +dual += +n +� +i=1 +ℓi ei. +(A.13) +81 + +82 +A.4. +Einstein convention +Remark A.15 Relative to a basis, a vector is represented by a column matrix, cf. (A.2), and a linear +form by a row matrix, cf. (A.10). This enables: +• The use of matrix calculation to compute ℓ.⃗x = [ℓ]|⃗e.[⃗x]|⃗e, cf. (A.11), not to be confused with an +inner dot product calculation ⃗x • ⃗y relative to an inner dot product in E for ⃗x, ⃗y ∈ E. +• Not to confuse the “nature of objects”: Relative to a basis, a (contravariant) vector is a mathematical +object represented by a column matrix, while a linear form (covariant vector) is a mathematical object +represented by a row matrix. Cf. remark A.8. +A.3.5 +Example: Thermodynamic +Consider the Cartesian space ⃗R2 = {(T, P) ∈ R×R} = {(temperature,pressure)}. There is no meaningful +inner dot product in this ⃗R2: What would +√ +T 2+P 2 mean (Pythagoras: Can you add Kelvin degrees and +kg/(m·s2)? Thus, in thermodynamics, the (covariant) dual bases are the main ingredient for calculations. +E.g., in the Cartesian product ⃗R2 = R × R consider the basis ( ⃗E1=(1, 0), ⃗E2=(0, 1)) (after a choice +of temperature and pressure units); Let ⃗X ∈ ⃗R2, ⃗X = T ⃗E1 + P ⃗E2 =noted (T, P), and let (πE1, πE2) = +(E1, E2) =noted (dT, dP) be the (covariant) dual basis. The first principle of thermodynamics tells that +the density α of internal energy is an exact differential form: ∃U ∈ C1( ⃗R2; R) s.t. α = dU. So, at any +⃗X0 = (T0, P0), +α( ⃗X0) = dU( ⃗X0) = ∂U +∂T ( ⃗X0) dT + ∂U +∂P ( ⃗X0) dP +and +[dU( ⃗X0)]| ⃗E = +� ∂U +∂T ( ⃗X0) +∂U +∂P ( ⃗X0) +� +(A.14) +(row matrix). And we have the first order Taylor expansion in the vicinity of ⃗X0, +U( ⃗X0 + δ ⃗X) = U( ⃗X0) + dU( ⃗X0).δ ⃗X + o(δ ⃗X) += U(T0, P0) + δT ∂U +∂T (T0, P0) + δP ∂U +∂T (T0, P0) + o((δT, δP)). +(A.15) +Matrix computation: Column matrices for vectors, row matrices for linear forms: +[ ⃗E1]| ⃗E = +� +1 +0 +� +, +[ ⃗E2]| ⃗E = +� +0 +1 +� +, +[ ⃗X0]| ⃗E = +� +T0 +P0 +� +, +[δ ⃗X]| ⃗E = +� +δT +δP +� +, +(A.16) +[E1]| ⃗E = [dT]| ⃗E = ( 1 +0 ) , +[E2]| ⃗E = [dP]| ⃗E = ( 0 +1 ) , +[dU]| ⃗E = ( ∂U +∂T +∂U +∂P ) +(A.17) +give +dU( ⃗X0).δ ⃗X = +� ∂U +∂T ( ⃗X0) +∂U +∂P ( ⃗X0) +� +. +� +δT +δP +� += ∂U +∂T ( ⃗X0)δT + ∂U +∂P ( ⃗X0)δP. +(A.18) +This is a “covariant calculation” (in particular no inner dot product has been used). +A.4 +Einstein convention +A.4.1 +Definition +When you work with components (after a choice of a basis), the goal is to visually differentiate a lin- +ear form from a vector (to visually differentiate covariance from contravariance). Framework: a finite +dimension vector space E, dim E = n, and duality notations. +Einstein Convention: +1. A basis in E (contravariant) is written with bottom indices: E.g., (⃗ei) is a basis in E. +2. A vector ⃗x ∈ E (contravariant) has its components relative to (⃗ei) (quantification) written with top +indices: ⃗x = �n +i=1xi⃗ei, and is represented by the column matrix [⃗x]|⃗e = +� +� +x1 +... +xn +� +�. (Classical notations: +⃗x = �n +i=1xi⃗ei, and column matrix of xi.) +3. A basis in E∗ = L(E; R) (covariant) is written with top indices: E.g., (ei) ∈ E∗n is the dual basis of +the basis (⃗ei). (Classical notations: (πei).) +4. A linear form ℓ ∈ E∗ (covariant) has its components relative to (ei) (quantification) written with bottom +indices: ℓ = �n +i=1ℓiei, and its matrix representation is the row matrix [ℓ]|⃗e = ( ℓ1 +... +ℓn ). +82 + +83 +A.5. +Change of basis formulas +5. You can also omit the sum sign � when there are repeated indices at a different position; E.g. +�n +i=1xi⃗ei =noted xi⃗ei, and �n +i=1Lij⃗ei =noted Li +j⃗ei. In fact, before computers and word processors, to +print �n +i=1 was not easy. But with LATEX this is no more a problem, so in this manuscript the sum +sign � is not omitted (and some confusions are avoided). +Remark A.16 Einstein’s convention is not mandatory. +E.g. Arnold doesn’t use it when he doesn’t +need it, or when it makes reading difficult, or when it induces misunderstandings. In classical mechanics, +Einstein’s convention may induce more confusion than understandings, and may be misused... so it is +better not to use it: Golden rule: Use classical notations when in doubt. +A.4.2 +Do not mistake yourself +1. Einstein’s convention is just meant not to confuse a linear function with a vector. +2. It only deals with quantification relative to a basis. +3. Classical notations are as good as duality notations, even you are told that classical notations cannot +detect obvious errors in component manipulations... But duality notations can be misused in classical +mechanics (cf. the paradigmatic example of the vectorial dual basis, correctly treated at § F.7); And thus +add confusion to the confusion. +4. The convention does not admit shortcuts; E.g. with a metric: g(⃗u,⃗v) = �n +i,j=1gijuivj shows the observer +dependence on a choice of a basis thanks to the gij; And even if gij = δij you cannot write g(⃗u,⃗v) = +�n +i,j=1uivj: You have to write g(⃗u,⃗v) = �n +i,j=1gijuivj: Unmissable in physics because you need to see +the metric and bases in use. +5. Golden rule: Return to classical notations if in doubt. (Einstein’s convention can add confusions, un- +truths, misinterpretations, absurdities, misuses...) +A.5 +Change of basis formulas +E being a finite dimension vector space, dim E = n, let (⃗eold,i) and (⃗enew,i) be two bases in E, and let +(πold,i) and (πnew,i) be the dual bases in E∗, written (ei +old) and (ei +new) with duality notations. +A.5.1 +Change of basis endomorphism and transition matrix +Definition A.17 The change of basis endomorphism P ∈ L(E; E) from (⃗eold,i) to (⃗enew,i) is the endo- +morphism (= the linear map E → E) defined by, for all j ∈ [1, n]N, +P.⃗eold,j = ⃗enew,j . +(A.19) +Definition A.18 The transition matrix from (⃗eold,i) to (⃗enew,i) is the matrix P := [P]|⃗eold = [Pij] of the +endomorphism P relative to the basis (⃗eold,i), i.e. defined by, for all j, +⃗enew,j = P.⃗eold,j = +n +� +i=1 +Pij ⃗eold,i, +i.e. +[⃗enew,j]|⃗eold = P.[⃗eold,j]|⃗eold = +� +� +� +P1j +... +Pnj +� +� +� , +(A.20) +i.e., [⃗enew,j]|⃗eold is the j-th column of P = [P]|⃗eold. +Duality notations: ⃗enew,j = �n +i=1P ij ⃗eold,i and +P := [P]|⃗eold = [P ij]. +Apart from the classical and notations, you may find other “component type” notations: +⃗enew,j = +n +� +i=1 +Pij ⃗eold,i = +n +� +i=1 +(Pj)i ⃗eold,i = +n +� +i=1 +P i +j ⃗eold,i = +n +� +i=1 +(Pj)i ⃗eold,i, +(A.21) +i.e. Pij = (Pj)i = P ij = (Pj)i are four notations for the i-th component of ⃗ej, i.e. +[⃗enew,j]|⃗eold = +� +� +� +P1j +... +Pnj +� +� +� = +� +� +� +(Pj)1 +... +(Pj)n +� +� +� = +� +� +� +P 1 +j +... +P n +j +� +� +� = +� +� +� +(Pj)1 +... +(Pj)n +� +� +� +(= the j-th column of P). +(A.22) +83 + +84 +A.5. +Change of basis formulas +A.5.2 +Inverse of the transition matrix +The inverse endomorphism Q := P−1 ∈ L(E; E), cf. (A.19), is given by, for all j ∈ [1, n]N, +⃗eold,j = Q.⃗enew,j +(= P−1.⃗enew,j), +(A.23) +i.e. Q is change of basis endomorphism from (⃗enew,i) to (⃗eold,i). And Q := [Q]|⃗enew = [Qij] is the transition +matrix from (⃗enew,i) to (⃗eold,i): +⃗eold,j = +n +� +i=1 +Qij⃗enew,i, +[⃗eold,j]|⃗enew = +� +� +� +Q1j +... +Qnj +� +� +� . +(A.24) +i.e. Q is change of basis endomorphism from (⃗enew,i) to (⃗eold,i). +Use other notation if you prefer: Qij = (Qj)i = Qij = (Qj)i +Proposition A.19 +Q = P −1. +(A.25) +Proof. ⃗enew,j = P.⃗eold,j = �n +i=1Pij⃗eold,i = �n +i=1Pij(�n +k=1Qki⃗enew,k) = �n +k=1(�n +i=1QkiPij)⃗enew,k = +�n +k=1(Q.P)kj⃗enew,k for all j, thus (Q.P)kj = δkj for all j, k. Hence Q.P = I, i.e. (A.25). +Exercice A.20 Prove +� +[P]|⃗eold = [P]|⃗enew = P, +[Q]|⃗enew = [Q]|⃗eold = Q, +� +, i.e. +� +P.⃗enew,j = �n +i,j=1Pij⃗enew,i +(= �n +i,j=1P ij⃗enew,i = �n +i,j=1(Pj)i⃗enew,i), +Q.⃗eold,j = �n +i,j=1Qij⃗eold,i +(= �n +i,j=1Qij⃗eold,i = �n +i,j=1(Qj)i⃗eold,i). +(A.26) +Answer. +Z += [Zij] = [P]|⃗enew means P.⃗enew,j += � +i Zij⃗enew,i, i.e. ⃗enew,j += Q.(�n +i=1Zij⃗enew,i) = +�n +i=1ZijQ.⃗enew,i = �n +i=1Zij(�n +k=1Qki⃗enew,k) = �n +k=1(�n +i=1QkiZij)⃗enew,k = �n +k=1(Q.Z)kj⃗enew,k for all j, thus +(Q.Z)kj = δkj for all j, k, thus Q.Z = I, thus Z = P. Idem for Q, thus (A.26). +Remark A.21 P T ̸= P −1 in general. E.g., (⃗eold,i) = (⃗ai) is a foot-built Euclidean basis, and (⃗enew,i) = +(⃗bi) is a metre-built Euclidean basis, and ⃗bi = λ⃗ai for all i (the basis are “aligned”), so P = λI; Thus +P T = λI and P −1 = 1 +λI ̸= P T , since λ = +1 +0.3048 ̸= 1. Thus it is essential not to confuse P T and P −1 (not +to confuse covariance with contravariance), cf. e.g. the Mars Climate Orbiter crash (remark A.14). +A.5.3 +Change of dual basis +Proposition A.22 (πnew,i) and (πold,i) being the dual bases of (⃗enew,i) and (⃗eold,i), for all i ∈ [1, n]N, +πnew,i = +n +� +j=1 +Qijπold,i, +and +[πnew,i]|⃗eold = ( Qi1 +... +Qin ) +(i-th row of Q), +(A.27) +to compare with (A.20) (matrices of linear forms are row matrices). +Duality notations: ei +new = �n +j=1Qijej +old and [ei +new]|⃗eold = ( Qi1 +... +Qin ). +Proof. πnew,i(⃗eold,k) =(A.24) πnew,i(� +j Qjk⃗enew,j) = � +j Qjk πnew,i(⃗enew,j) = � +j Qjk δij = Qik, and +� +j Qijπold,j(⃗eold,k) = � +j Qijδjk = Qik, true for all i, k, thus πnew,i = � +j Qij, i.e. (A.27) +A.5.4 +Change of coordinate system for vectors and linear forms +Proposition A.23 Let ⃗x ∈ E and ℓ ∈ E∗. Then +• [⃗x]|⃗enew = P −1.[⃗x]|⃗eold +(contravariance formula for vectors: between column matrices), +• [ℓ]|⃗enew = [ℓ]|⃗eold.P +(covariance formula for linear forms: between row matrices). +(A.28) +And the scalar value ℓ.⃗x is computed indifferently with one or the other basis (objective result): +ℓ.⃗x = [ℓ]|⃗eold.[⃗x]|⃗eold = [ℓ]|⃗enew.[⃗x]|⃗enew. +(A.29) +84 + +85 +A.6. +Bidual basis (and contravariance) +Proof. Let ⃗x = � +j xj⃗eold,j = � +i yi⃗enew,i. +We have ⃗x = � +j xj⃗eold,j = � +j xj(�n +i=1Qij⃗enew,i) = +� +ij Qijxj⃗enew,i, thus yi = � +j Qijxj for all i, thus (A.28)1. +And ℓ = � +j mjπnew,j = � +i ℓiπold,i =(A.27) � +ij ℓiPijπnew,j gives mj = � +i ℓiPij for all j, thus (A.28)2. +(Use duality notations if you prefer.) +Thus [ℓ]|⃗enew.[⃗x]|⃗enew = ([ℓ]|⃗eold.P).(P −1.[⃗x]|⃗eold) = [ℓ]|⃗eold.[⃗x]|⃗eold, hence (A.29). +Notation: (A.28) and ⃗x = � +j xj⃗eold,j = � +i yi⃗enew,i give yi = �n +j=1Qijxj, which means: yi is the +function defined by yi(x1, ..., xn) = �n +j=1Qijxj, thus Qij = ∂yi +∂xj (x1, ..., xn); Similarly with Pij; Which is +written +Qij = ∂yi +∂xj +, +and +Pij = ∂xi +∂yj +. +(A.30) +(Use duality notations if you prefer, e.g. Qij = ∂yi +∂xj .) +A.6 +Bidual basis (and contravariance) +Definition A.24 The dual of E∗ is E∗∗ := (E∗)∗ = L(E∗; R) and is named the bidual of E. +E∗∗ is also called the space of contravariant vectors = the space of directional derivatives. +Definition A.25 Let (⃗ei) be a basis in E, let (πei) be its dual basis (basis in E∗). The dual basis (∂i) +of (πei) is called the bidual basis of (⃗ei). (Duality notations: (πei) = (ei).) +(The notation ∂i refers to the derivation in the direction ⃗ei: ∂i(df(⃗x)) = df(⃗x).⃗ei = ∂f +∂xi (⃗x), see § S.1.) +Thus, the linear form ∂i ∈ E∗∗ = L(E∗; R) are characterized by, for all j, +∂i.πej = δij = πej.⃗ei, +so +ℓ = +n +� +i=1 +ℓiπei +iff +ℓi = ∂i.ℓ (= ℓ.⃗ei). +(A.31) +Indeed, ∂i(ℓ) = ∂i(�n +j=1ℓjπej) = �n +j=1ℓj∂i(πej) = �n +j=1ℓjδij = ℓi. (Duality notation: ∂i.ej = δj +i = ej.⃗ei +and ℓ = �n +i=1ℓiei.) +Remark: With the natural canonical isomorphism J : +� +E → E∗∗ = L(E∗; R) +⃗u → J (⃗u), where J (⃗u).ℓ := ℓ.⃗u, ∀ℓ ∈ E∗ +� +see (T.9), we can identify ⃗u and J (⃗u) (observer independent identification), thus ∂i = J (⃗ei) =noted ⃗ei, +and (A.31) reads (usual notation in differential geometry) ⃗ei.πej = δij and ℓi = ⃗ei.ℓ. +A.7 +Bilinear forms +A.7.1 +Definition +Let E and F be vector spaces. +Definition A.26 • A bilinear form is a 2-multilinear form β(·, ·) : +� +E × F → R +(⃗u, ⃗w) → β(⃗u, ⃗w) +� +. +So, β(⃗u1 +λ⃗u2, ⃗w) = β(⃗u1, ⃗w)+λβ(⃗u2, ⃗w) and β(⃗u, ⃗w1 +λ⃗w2) = β(⃗u, ⃗w1)+λβ(⃗u, ⃗w2) for all ⃗u, ⃗u1, ⃗u2 ∈ E, +⃗w, ⃗w1, ⃗w2 ∈ F, λ ∈ R. +• L(E, F; R) is the set of bilinear forms E × F → R. +• If (ℓ, m) ∈ E∗ × F ∗, then the bilinear form ℓ ⊗ m ∈ L(E, F; R) is defined by +(ℓ ⊗ m)(⃗u, ⃗w) = ℓ(⃗u)m(⃗w) +(= (ℓ.⃗u)(m.⃗w)) +(A.32) +for all (⃗u, ⃗w) ∈ E × F, and is called an elementary bilinear form. +A.7.2 +The transposed of a bilinear form +(Warning: Not to be confused with the subjective definition of a transposed of a linear map which requires +inner dot products to be defined, see e.g. (A.54).) +Definition A.27 If β ∈ L(E, F; R) then its transposed is the bilinear form βT ∈ L(F, E; R) defined by, +for all (⃗w, ⃗u) ∈ F × E, +βT (⃗w, ⃗u) = β(⃗u, ⃗w). +(A.33) +(This definition is observer independent: no basis or inner dot product is required in this definition.) +85 + +86 +A.7. +Bilinear forms +A.7.3 +Symmetric and definite positive bilinear forms +Definition A.28 Here F = E (no choice), and β ∈ L(E, E; R). +• β is semi-positive, iff for all ⃗u ∈ E, +β(⃗u, ⃗u) ≥ 0. +(A.34) +• β is definite positive, iff for all ⃗u ̸= ⃗0, +β(⃗u, ⃗u) > 0. +(A.35) +• β is symmetric iff βT = β, i.e., for all ⃗u,⃗v ∈ E, +β(⃗u,⃗v) = β(⃗v, ⃗u). +(A.36) +A.7.4 +Inner dot product, and metric +Definition A.29 • An “inner dot product” (or “scalar inner dot product”, or “inner scalar product”, or +“inner product”) in a vector space E is a bilinear form β =noted g =noted g(·, ·) ∈ L(E, E; R) which is +symmetric and definite positive. And then (for inner dot products) +g(·, ·) noted += +(·, ·)g +noted += +· +•g ·, +i.e. +g(⃗u, ⃗w) = (⃗u, ⃗w)g +noted += +⃗u •g ⃗w, ∀⃗u, ⃗w ∈ E. +(A.37) +• Then two vectors ⃗u, ⃗w ∈ E are (·, ·)g-orthogonal iff (⃗u, ⃗w)g = 0. +• And the associated norm with (·, ·)g is the function ||.||g : E → R+ given by, for all ⃗u ∈ E, +||⃗u||g = +� +(⃗u, ⃗u)g. +(A.38) +(To prove that it is a norm, use the Cauchy–Schwarz inequality (A.39).) +• An “semi-inner dot product” (·, ·)g (or “semi-scalar inner dot product”) in a vector space E is a +bilinear form β =noted g(·, ·) ∈ L(E, E; R) which is symmetric and semi-positive. And the associated +semi-norm is given by (A.38). +Proposition A.30 (Cauchy–Schwarz inequality.) (·, ·)g being an inner dot product in E, +∀⃗u, ⃗w ∈ E, +|(⃗u, ⃗w)g| ≤ ||⃗u||g||⃗w||g. +(A.39) +And |(⃗u, ⃗w)g| = ||⃗u||g||⃗w||g iff ⃗u and ⃗w are parallel. +Proof. Let p(λ) = ||⃗u+λ⃗w||2 +g = (⃗u+λ⃗w, ⃗u+λ⃗w)g, so p(λ) = aλ2 + bλ + c where a = ||⃗w||2 +g, b = 2(⃗u, ⃗w)g +and c = ||⃗u||2 +g. With p(λ) ≥ 0 (since(·, ·)g is positive), we get b2 − 4ac ≥ 0, thus (A.39), and p(λ) = 0 iff +⃗u+λ⃗w = 0. +Definition A.31 (Metric.) With Rn our usual affine geometric space, n = 1, 2 or 3, and ⃗Rn = the usual +associated vector space made of bipoint vectors. Let Ω ⊂ Rn be open in Rn. A metric in Ω is a C∞ +function g : +� Ω → L(⃗Rn, ⃗Rn; R) +p → g(p) noted += +gp +� +such that gp is an inner dot product in ⃗Rn at each p ∈ Ω. +Particular Case: When the gp is independent of p (general case in continuum mechanics), a metric is +simply called a inner dot product (e.g. a Euclidean metric is called a Euclidean dot product). +(In a differentiable manifold Ω, a metric is a C∞ �0 +2 +� +tensor g s.t. g(p) is an inner dot product at each +p ∈ Ω. A Riemannian metric is a metric s.t. g(p) is a Euclidean dot product at each p ∈ Ω.) +A.7.5 +Quantification: Matrice [βij] and tensorial representation +dim E = n, dim F = m, β ∈ L(E, F; R), (⃗ai) is a basis in E which dual basis is (πai), (⃗bi) is a basis in F +which dual basis is (πbi). (With duality notations, (πai) = (ai) and (πbi) = (bi).) +Definition A.32 The components of β ∈ L(E, F; R) relative to the bases (⃗ai) and (⃗bi) are the nm reals +βij := β(⃗ai,⃗bj), +and +[β]|⃗a,⃗b = [βij] i=1,...,n +j=1,...,m +(A.40) +is the matrix of β relative to the bases (⃗ai) and (⃗bi), simply written [βij] if the bases are implicit. +And if F = E and (⃗bi) = (⃗ai) then [β]|⃗a,⃗b =noted [β]|⃗a. +86 + +87 +A.8. +Linear maps +Proposition A.33 A bilinear form β ∈ L(E, F; R) is known as soon as the nm scalars βij = β(⃗ai,⃗bj) +are known, and, for all (⃗u, ⃗w) ∈ E × F, +β(⃗u, ⃗w) = [⃗u]|⃗a +T .[β]|⃗a,⃗b.[⃗w]|⃗b, +written +β(⃗u, ⃗w) = [⃗u]T .[β].[⃗w] , +(A.41) +so +β = +n +� +i=1 +m +� +j=1 +βijπai ⊗ πbj, +(A.42) +and a basis in L(E, F; R) is made of the nm functions πai ⊗ πbj, and dim L(E, F; R) = nm. (Duality +notations: β = �n +i=1 +�m +j=1βijai ⊗ bj.) +Proof. β being bilinear, ⃗u = �n +i=1ui⃗ai and ⃗w = �n +j=1wj⃗bj give β(⃗u, ⃗w) = �n +i,j=1uiwjβ(⃗ai,⃗bj) = +�n +i,j=1uiβijwj = ([⃗u]|⃗a)T .[β]|⃗a,⃗b.[⃗w]|⃗b, thus (A.41). In particular, if the βij are known, then b is known. +And (πai ⊗ πbj)(⃗ak,⃗bℓ) =(A.32) (πai.⃗ak)(πbj.⃗bℓ) = δikδjℓ (all the elements of the matrix [πai ⊗ πbj]|⃗a,⃗b are +zero except the element at the intersection of row i and column j which is equal to 1). And (πai ⊗ +πbj)(⃗u, ⃗w) =(A.32) (πai.⃗u)(πbj.⃗w) = uiwj, thus β(⃗u, ⃗w) = �n +i,j=1βijuiwj = �n +i,j=1βij(πai ⊗ πbj)(⃗u, ⃗w), +thus β := �n +i,j=1βij(πai ⊗ πbj), thus the πai ⊗ πbj span L(E, F; R). And � +ij λij(πai ⊗ πbj) = 0 implies +0 = (� +ij λij(πai ⊗ πbj))(⃗ak,⃗bℓ) = � +ij λij(πai ⊗ πbj)(⃗ak,⃗bℓ) = λkℓ = 0 for all k, ℓ; Thus the πai ⊗ πbj are +independent. Thus (πai ⊗ πbj) is a basis in L(E, F; R) and dim(L(E, F; R)) = nm. (Duality notations: +β(⃗u, ⃗w) = �n +i,j=1βijuiwj and β := �n +i,j=1βij ai ⊗ bj.) +Example A.34 dim E = dim F = 2. [β]|⃗a,⃗b = +� +1 +2 +0 +3 +� +means β(⃗a1,⃗b1) = β11 = 1, β(⃗a1,⃗b2) = β12 = 2, +β(⃗a2,⃗b1) = β21 = 0, β(⃗a2,⃗b2) = β22 = 3. And β12 = [⃗a1]T +|⃗a.[β]|⃗a,⃗b.[⃗b2]|⃗b = ( 1 +0 ) . +� +1 +2 +0 +3 +� +. +� +0 +1 +� += 2. +Exercice A.35 Let β ∈ L(E, E; R), let (⃗ai) and (⃗bi) be two bases in A, and let λ ∈ R∗. Prove: +if, ∀i ∈ [1, n]N, ⃗bi = λ⃗ai, +then +[β]|⃗b = λ2[β]|⃗a. +(A.43) +(A change of unit, e.g. from foot to metre, has a “big” influence on the matrix.) +Answer. ⃗bi = λ⃗ai give β(⃗bi,⃗bj) = β(λ⃗ai, λ⃗aj) = λ2β(⃗ai,⃗aj) (bilinearity), thus [β]|⃗b = λ2[β]|⃗a. +Exercice A.36 Prove +[βT ]⃗b,⃗a = ([β]⃗a,⃗b)T , +written +[βT ] = [β]T . +(A.44) +Answer. Let[β]⃗a,⃗b = [βij] i=1,...,n +j=1,...,m and [βT ]⃗b,⃗a = [γij] i=1,...,m +j=1,...,n . We have γij = βT (⃗bi,⃗aj) = β(⃗aj,⃗bi) = βji, qed. +A.8 +Linear maps +A.8.1 +Definition +Let E and F be vector spaces. +Definition A.37 • A function L : E → F is linear iff L(⃗u1 + λ⃗u2) = L(⃗u1) + λL(⃗u2) for all ⃗u1, ⃗u2 ∈ E +and all λ ∈ R (distributivity type relation). And (distributivity notation): +L(⃗u) noted += +L.⃗u, +so +L(⃗u1 + λ⃗u2) = L.(⃗u1 + λ⃗u2) = L.⃗u1 + λL.⃗u2. +(A.45) +NB: This dot notation L.⃗u is a linearity notation (distributivity type notation); It is an “outer” dot product +between a (linear) function and a vector; It is not an “inner” dot product since L and ⃗u don’t belong to +a same space. It is not a matrix product since no basis has been introduced yet (no quantification has +been done yet). +• L(E; F) is the set of linear maps E → F (vector space, subspace of (F(E; F), +, .)). +• If F = E then a linear map L ∈ L(E; E) is called an endomorphism in E. +(If F = R then a linear map E → R is called a linear form, and E∗ := L(E; R) is the dual of E.) +87 + +88 +A.8. +Linear maps +Vocabulary: Let Li(E; E) be the space of linear invertible linear maps. If E is a finite dimension vector +space, dim E = n, then, in algebra, the set (Li(E; E), ◦) of linear maps equipped with the composition +rule is named GLn(E) = “the linear group” (it is indeed a group, easy check). And the “linear group” of +n ∗ n invertible matrices is GLn(Mn) := (Li(Mn; Mn), .), i.e. Li(Mn; Mn) with the matrix product rule. +Exercice A.38 (Math exercise.) +Let E = (E, ||.||E) and F = (F, ||.||F ) be Banach spaces, and let +Lic(E; F) be the space of invertible linear continuous maps E → F, with its usual norm ||L|| = +sup||⃗x||E=1 ||L.⃗x||F . Let Z : +� +Lic(E; F) → Lic(E; F) +L → L−1 +� +. Prove dZ(L).M = −L−1 ◦ M ◦ L−1, for all +M ∈ Lic(E; F). (Recall: In finite dimension, a linear map is always continuous.) +Answer. Consider limh→0 +Z(L+hM)−Z(L) +h += limh→0 +(L+hM)−1−L−1 +h +( =noted dZ(L).M if the limit exists). With +N = L−1.M we have L + hM = L(I + hN), and (I + hN) is invertible as soon as ||hN|| < 1, i.e. h < +1 +||N|| = +1 +||L−1.M||, its inverse being I − hN + h2N − ... (Neumann serie); Thus I + hN = I − hN + o(h), and +(L + hM)−1 = (I + hN)−1.L−1 = (I − hN + o(h)).L−1 = L−1 − hN.L−1 + o(h). +Thus +(L+hM)−1−L−1 +h += +L−1−hN.L−1+o(h)−L−1 +h += −N.L−1 + o(1) −→h→0 −N.L−1. +A.8.2 +Quantification: Matrices [Lij] = [Lij] +dim E = n, dim F = m, L ∈ L(E; F), (⃗ai) is a basis in E which dual basis is (πai), (⃗bi) is a basis in F +which dual basis is (πbi). (With duality notations, (πai) = (ai) and (πbi) = (bi).) +Definition A.39 The components of a linear map L ∈ L(E; F) relative to the bases (⃗ai) and (⃗bi) are +the nm reals named Lij (classical notation) = Lij (duality notation), which are the components of the +vectors L.⃗aj relative to the basis (⃗bi). That is: +� +� +� +� +� +� +� +� +� +� +� +clas. not. : L.⃗aj = +m +� +i=1 +Lij⃗bi, +dual not. +L.⃗aj = +m +� +i=1 +Li +j⃗bi, +� +� +� +� +� +� +� +� +� +� +� +, +i.e. +[L.⃗aj]|⃗b +clas. += +� +� +� +L1j +... +Lmj +� +� +� +dual += +� +� +� +L1j +... +Lmj +� +� +� . +(A.46) +And +[L]|⃗a,⃗b +clas. += [Lij] i=1,...,m +j=1,...,n +dual += [Li +j] i=1,...,m +j=1,...,n +(A.47) +is the matrix of L relative to the bases (⃗ai) and (⃗bi) (so [L.⃗aj]|⃗b is the j-th column of [L]|⃗a,⃗b). +Particular case: If E = F (so L is an endomorphism) and if (⃗bi) = (⃗ai) then [L]|⃗a,⃗a =noted [L]|⃗a. +Example A.40 n = m = 2. [L]|⃗a,⃗b = +� +1 +2 +0 +3 +� +means L.⃗a1 = ⃗b1 and L.⃗a2 = 2⃗b1 + 3⃗b2 (column reading). +Here L11=1, L12=2, L21=0, L22=3 (duality notations: L11=1, L12=2, L21=0, L22=3). +And L being linear, for all ⃗u ∈ E, ⃗u = �n +j=1uj⃗aj = �n +j=1uj⃗aj, we get, thanks to linearity, +L.⃗u clas. += +m +� +i=1 +n +� +j=1 +Lijuj⃗bi +dual += +m +� +i=1 +n +� +j=1 +Li +juj⃗bi, +i.e. +[L.⃗u]|⃗b = [L]|⃗a,⃗b.[⃗u]|⃗a . +(A.48) +Shortened notation: [L.⃗u] = [L].[⃗u] when the bases are implicit. +Proposition A.41 A linear map L ∈ L(E; F) is known as soon as the n vectors L.⃗aj are known, +j ∈ [1, n]N. And the linear maps Lij ∈ L(E; F) defined by Lij.⃗aℓ = δjℓ⃗bi (all the elements of the matrix +[Lij]|⃗a,⃗b vanish except the element at the intersection of row i and column j which is equal to 1), for +i, ℓ = 1, ..., n and j = 1, ..., m, constitute a basis ∈ L(E; F). (Duality notations: Lij =noted Lij, and +Lij.⃗aℓ = δj +ℓ⃗bi.) So, dim(L(E; F)) = nm. +Proof. ⃗u ∈ E and ⃗u = � +k uj⃗aj give L.⃗u = � +j ujL.⃗aj, since L is linear. Thus L is known iff the n vectors +L.⃗aj are known for all j = 1, ..., n; And L.⃗ak = � +i Lik⃗bi together with � +ij LijLij.⃗ak = � +ij Lijδjk⃗bi = +� +i Lik⃗bi, for all k, thus L = � +ij LijLij, thus the Lij span L(E; F). And �m +i=1 +�n +j=1λijLij = 0 implies, +for all ℓ, ⃗0 = �m +i=1 +�n +j=1λijLij.⃗aℓ = �m +i=1 +�n +j=1λijδjℓ⃗bi = �m +i=1λiℓ⃗bi, thus λiℓ = 0, for all i and ℓ. Thus +the Lij are independent. Thus (Lij) i=1,...,n +j=1,...,m is a basis in L(E; F). +88 + +89 +A.9. +Transposed matrix +Exercice A.42 If L ∈ L(E; E) (endomorphism), if (⃗ai) is a basis in E, prove: +if λ ∈ R∗ and ⃗bi = λ⃗ai ∀i ∈ [1, n]N, +then +[L]|⃗b = [L]|⃗a, +(A.49) +i.e., a change of unit has not influence on the matrix of an endomorphism. Check with the change of +basis formulas. NB: To compare with (A.43): Covariance and contravariance should not be confused. +Answer. +Let L.⃗aj = �n +i=1Laij⃗ai and L.⃗bj = �n +i=1Lbij⃗bi. +Then �n +i=1Lbij⃗bi = L.⃗bj = L.(λ⃗aj) = λL.⃗aj = +λ�n +i=1Laij⃗ai = λ�n +i=1Laij +⃗bi +λ = �n +i=1Laij⃗bi, thus Lbij = Laij. +Change of basis formula: [L]|⃗b = P −1.[L]|⃗a.P with P = λI here. +A.8.3 +Trace of an endomorphism +Let E be a vector space, dim E = n. Let ⃗u ∈ E and ℓ ∈ E∗ and call L ⃗w,ℓ ∈ L(E; E) the endomorphism, +called an elementary endomorphism, defined by +L ⃗w,ℓ.⃗u := ⃗w(ℓ.⃗u) = (ℓ.⃗u)⃗w. +(A.50) +Definition A.43 The trace of the endomorphism L ⃗w,ℓ is the real +Tr(L ⃗w,ℓ) := ℓ.⃗w. +(A.51) +And the trace operator is the linear map Tr : +� +L(E; E) → R +L → Tr(L) +� +defined on elementary endomor- +phisms ℓ ⊗ ⃗w by (A.51). +Proposition A.44 Let L ∈ L(E; E). The real Tr(L) is objective (is intrinsic to L), i.e. is independent +of any basis in E. And (quantification) if (⃗ei) is a basis and L.⃗ej = �n +i=1Lij⃗ei for all j, then +Tr(L) = +n +� +i=1 +Lii (∈ R), +(A.52) +i.e., Tr(L) is the trace of the matrix [L]|⃗e. (Duality notations L.⃗ej = �n +i=1Lij⃗ei and Tr(L) = �n +i=1Lii.) +Proof. Tr(L ⃗w,ℓ) := ℓ.⃗w is a real that can be considered by any observer, and which value is the same for +all observers, cf. (A.29): It is objective. +Let L ∈ L(E; E). Let (⃗ai) be a basis and (πai) be its (covariant) dual basis, and L.⃗aj = � +i Lij⃗ai, +i.e. [L][⃗a = [Lij]. +And we have (� +ik LikL⃗ai,πak).⃗aj = � +ik LikL⃗ai,πak.⃗aj =(A.50) � +ik Lik⃗ai(πak.⃗aj) = +� +ik Lik⃗aiδkj = � +i Lij⃗ai, thus L = � +ij LijL⃗ai,πaj (sum of elementary endomorphisms), thus, Tr being +linear Tr(L) = � +ij LijTrL⃗ai,πaj = � +ij Lijδji = � +i Lii, thus (A.52). +Exercice A.45 Check with the change of basis formula that Tr(L) is an invariant (the same value for +all observers). +Answer. Let (⃗ai) and (⃗bi) be two bases, P = [Pij] be the transition matrix from (⃗ai) to (⃗bi), Q = P −1, [L][⃗a = +[(La)ij], [L][⃗b = [(Lb)ij]. We have [L][⃗b =(A.104) P −1.[L][⃗a.P, i.e. (Lb)ij = � +kℓ Qik(La)kℓPℓj, thus � +i(Lb)ii = +� +ikℓ Qik(La)kℓPℓi = � +kℓ(P.Q)ℓk(La)kℓ = � +kℓ δℓk(La)kℓ = � +k(La)kk, qed. +Alternative definition with one-one tensors: see § Q.6. +A.9 +Transposed matrix +The definition can be found in any elementary books, e.g., Strang [18]: If M = [Mij] i=1,...,m +j=1,...,n is an m ∗ n +matrix then its transposed is the n ∗ m matrix M T = [(M T )ij] i=1,...,n +j=1,...,m defined by +(M T )ij := Mji +(A.53) +(exchange rows and columns). E.g., M = +� +1 +2 +3 +4 +� +gives M T = +� +1 +3 +2 +4 +� +, and (M T )12=M21=3. +And M is symmetric iff M T = M (this requires m=n). +And M.N = [� +k MikNkj] i +j gives (M.N)T = [� +k MjkNki] i +j = [� +k(N T )ik(M T )kj] i +j = N T .M T . +Exercice A.46 Prove: If M is an n ∗ n invertible matrix then M T is invertible and (M T )−1 = (M −1)T +( =noted M −T ); And if moreover M is symmetric, then M −1 is symmetric. +Answer. M.M −1 = I gives (M −1)T .M T = IT = I, thus M T is invertible with (M T )−1 = (M −1)T . Moreover if +M = M T then M −1 = (M −1)T . +89 + +90 +A.10. +A transposed endomorphism: depends on a chosen inner dot product +A.10 +A transposed endomorphism: depends on a chosen inner dot product +Not to be confused with the transposed of a matrix, cf. (A.53). +And not to be confused with the +transposed of a bilinear form (observer independent), cf. (A.33); In particular, a transposed of a linear +map depends on the observer who use it (depends on the choice of an inner dot product). +A.10.1 +Definition (requires an inner dot product: Not objective) +Let E be a finite dimensional vector space equipped with an inner dot product g(·, ·) = (·, ·)g. +Definition A.47 The transpose of an endomorphism L ∈ L(E; E) relative to (·, ·)g is the endomorphism +LT +g ∈ L(E; E) defined by +∀⃗x, ⃗y ∈ E, +(LT +g .⃗y, ⃗x)g = (⃗y, L.⃗x)g, +i.e. +(LT +g .⃗y) •g ⃗x = ⃗y •g (L.⃗x). +(A.54) +(It depends on (·, ·)g, see (A.59).) If (·, ·)g is an imposed Euclidean dot product (isometric framework) +then LT +g =noted LT , thus (LT .⃗y, ⃗x)g = (⃗y, L.⃗x)g, i.e. (LT .⃗y) • ⃗x = ⃗y • (L.⃗x). +Exercice A.48 (Math exercise.) The existence and uniqueness of LT +g is e.g. proved with a basis in E +when E is finite dimensional, see next § A.10.2. More general proof: Prove: If (E, (·, ·)g) is an infinite +dimensional Hilbert space and if L ∈ L(E; E) is continuous, then LT +g exists, is unique, and is continuous +(apply the Riesz representation theorem F.1). +Answer. Let ⃗y ∈ E, then let ℓ⃗yg : ⃗x ∈ E → ℓ⃗yg(⃗x) := (⃗y, L.⃗x)g ∈ R. ℓ⃗yg is linear (trivial since L is linear and (·, ·)g +is bilinear) and continuous: |ℓ⃗yg.⃗x| ≤ ||⃗y||g||L.⃗x||g ≤ ||⃗y||g||L|| ||⃗x||g gives ||ℓ⃗yg||E∗ ≤ ||L|| ||⃗y||g < ∞; Let ⃗ℓ⃗yg ∈ E +be the (·, ·)g-Riesz representation of ℓ⃗yg ∈ E∗: ℓ⃗yg.⃗x = (⃗ℓ⃗yg, ⃗x)g for all ⃗x, with ||⃗ℓ⃗yg||g = ||ℓ⃗yg||E∗; We have thus +defined LT +g : ⃗y ∈ E → LT +g (⃗y) := ⃗ℓ⃗yg ∈ E; with (LT +g (⃗y), ⃗x)g = (⃗ℓ⃗yg, ⃗x)g = ℓ⃗yg.⃗x = (⃗y, L.⃗x)g, thus LT +g is linear (since +(·, ·)g is bilinear) and continuous: ||LT +g .⃗y||g = ||⃗ℓ⃗yg||g = ||ℓ⃗yg||E∗ ≤ ||L|| ||⃗y||g gives ||LT +g || ≤ ||L||L(E;E) < ∞. +Remark A.49 Recall: The transposed βT of a bilinear form β is objective, cf. (A.33): We don’t need +any tool like an inner dot product to define βT . Not to be confused with: The transposed LT +g =noted LT +of a linear map L is subjective: It depends on a choice of an inner dot products (·, ·)g by an observer. In +particular it is dangerous to represent a linear map in a basis with its “bilinear tensorial representation” +when dealing with the transposed: L ∈ L(E; F) is naturally canonically represented by the bilinear form +βL ∈ L(F ∗, E; R), and thus (βL)T ∈ L(E, F ∗; R); And +L.⃗aj = +n +� +i=1 +Li +j⃗bi gives βL = +n +� +i,j=1 +Li +j⃗bi ⊗ aj, thus (βL)T (A.33) += +n +� +i,j=1 +Lj +iai ⊗⃗bj, +(A.55) +while LT ∈ L(F; E) is naturally canonically represented by the bilinear form β(LT ) ∈ L(E∗, F; R), and +LT .⃗bj = +n +� +i=1 +(LT )i +j⃗ai gives b(LT ) = +n +� +i,j=1 +(LT )i +j⃗ai ⊗ bj, thus +β(LT ) ̸= (βL)T +(A.56) +for two reasons: 1- ⃗ai ⊗ bj ̸= ai ⊗ ⃗bj, and 2- LT := LT +gh depends on chosen inner dot products (·, ·)g +and (·, ·)h by observers in E and F, see (A.73): (LT )ij =(A.73) �n +k,ℓ=1([g]−1)ikLℓk hℓj; While (βL)T is +independent of any inner dot products. (In fact (βL)T ∈ L(F ∗, E; R) is the tensorial representation of the +adjoint L∗ ∈ L(F ∗; E∗) of L: With L∗.bj = �n +i=1(L∗)ijai we get � +(L∗) = �n +i,j=1(L∗)ijai ⊗⃗bj = (βL)T , +see (A.83).) +So in continuum mechanics it is strongly advised not to use the tensorial notation for linear maps +when dealing with transposed (e.g. when using F T the transposed of the deformation gradient). +A.10.2 +Quantification with bases +Let (⃗ei) be a basis in E, let gij := g(⃗ei,⃗ej), so [g]|⃗e := [gij] =noted [g], and let (classical notation) +L.⃗ej = +n +� +i=1 +Lij⃗ei, +LT +g .⃗ej = +n +� +i=1 +(LT +g )ij, +i.e. +[L]|⃗e = [Lij] noted += +[L], +[LT +g ]|⃗e = [(LT +g )ij] noted += +[LT +g ]. +(A.57) +90 + +91 +A.10. +A transposed endomorphism: depends on a chosen inner dot product +(A.54) gives [⃗x]T .[g].[LT +g .⃗y] = [L.⃗x]T .[g].[⃗y] for all ⃗x, ⃗y, thus +[g].[LT +g ] = [L]T .[g], +i.e. +n +� +k=1 +gik(LT +g )kj = +n +� +k=1 +Lki gkj +(A.58) +i.e., +[LT +g ] = [g]−1.[L]T .[g] , +i.e. +(LT +g )ij = +n +� +k,ℓ=1 +([g]−1)ikLℓkgℓj. +(A.59) +To compare with (A.56). If and only if (⃗ei) is (·, ·)g-orthonormal then [g] = [δij] and (LT +g )ij = Lji. With +duality notations, L.⃗ej = �n +i=1Lij⃗ei, LT +g .⃗ej = �n +i=1(LT +g )ij, [L]|⃗e = [Lij], [LT +g ]|⃗e = [(LT +g )ij], and +n +� +k=1 +gik(LT +g )k +j = +n +� +k=1 +Lk +i gkj, +i.e. +(LT +g )i +j = +n +� +k,ℓ=1 +([g]−1)ikLℓ +k gℓj. +(A.60) +Remark A.50 The last equation (A.60)2 is also written +(LT +g )i +j = +n +� +k,ℓ=1 +gikLℓ +k gℓj +when +([g]⃗e)−1 = [gij]−1 noted += +[gij]. +(A.61) +Don’t be fooled by the notation gij, defined by [gij] := [gij]−1. (It is also the short notation for (g♯)ij, +see (F.32).) Use classical notations to avoid misuses and misinterpretations. +Remark A.51 A bilinear form β ∈ L(E, E; R) satisfies [βT ] = [β]T . +A linear endomorphism L ∈ +L(E; E) satisfies [LT +g ] = [g]−1.[L]T .[g] ̸= [L]T in general (e.g. take [L] = +� +0 +1 +1 +0 +� +and [g] = +� +1 +0 +0 +2 +� +). +So do not confuse a bilinear on E (objective) with a linear endomorphism on E (subjective). +Exercice A.52 In ⃗R2, let (⃗e1,⃗e2) be a basis. Let L ∈ L( ⃗R2; ⃗R2) be defined by [L]|⃗e = +� +0 +1 +1 +0 +� +. Find +two inner dot products (·, ·)g and (·, ·)h in ⃗R2 such that LT +g ̸= LT +h (a transposed endomorphism is not +unique, is not intrinsic to L, since it depends on a choice of an inner dot product by an observer). +Answer. Calculations with (A.58): +Choose (·, ·)g given by [g]|⃗e = +� 1 +0 +0 +1 +� += [I]. Thus [LT +g ]|⃗e = [I].[L]|⃗e.[I] = +� 0 +1 +1 +0 +� +; So LT +g = L. +Choose (·, ·)h given by [h]|⃗e = +� 1 +0 +0 +2 +� +. Thus [LT +h ]|⃗e = [h]−1 +|⃗e .[L]|⃗e.[h]|⃗e = +� 0 +2 +1 +2 +0 +� +; So LT +h ̸= L. +Thus LT +h ̸= LT +g , e.g., ⃗e2 = LT +g .⃗e1 ̸= LT +h .⃗e1 = 1 +2⃗e2. +Exercice A.53 Prove: If L is invertible then LT +g is invertible, and (LT +g )−1 = (L−1)T +g (written L−T +g +). +Answer. Suppose: ∃⃗y ∈ E, ⃗y ̸= ⃗0, s.t. LT +g .⃗y = 0. L being invertible, ∃!⃗x ∈ E s.t. L.⃗x = ⃗y, with ⃗x ̸= ⃗0 since ⃗y ̸= ⃗0 +and L is linear; And LT +g .⃗y = 0 gives LT +g .L.⃗x = 0, thus (LT +g .L.⃗x, ⃗x)g = 0, thus ||L.⃗x||2 +g = 0, thus L.⃗x = 0, thus +⃗x = 0 since L is linear bijective; Absurd. Thus Ker(LT +g ) = {⃗0}, thus LT +g is invertible since it is an endomorphism. +And (LT +g .(L−1)T +g .⃗x, ⃗y)g +(A.54) += ((L−1)T +g .⃗x, L.⃗y)g +(A.54) += (⃗x, (L−1).L.⃗y)g = (⃗x, ⃗y)g = (LT +g .(LT +g )−1.⃗x, ⃗y)g, true ∀⃗x, ⃗y, thus +LT +g .(L−1)T +g = LT +g .(LT +g )−1, thus (L−1)T +g = (LT +g )−1 since LT +g is invertible. +Exercice A.54 Special case of proportional inner dot products (·, ·)a and (·, ·)b: ∃λ > 0 s.t. (·, ·)a = +λ2(·, ·)b. Prove: LT +a = LT +b : Two proportional inner dot products give the same transposed endomorphism. +Answer. (LT +b .⃗y, ⃗x)b = (⃗y, L.⃗x)b = λ2(⃗y, L.⃗x)a = λ2(LT +a .⃗y, ⃗x)a = (LT +a .⃗y, ⃗x)b, for all ⃗x, ⃗y, so LT +b = LT +a . +A.10.3 +Symmetric endomorphism +Definition A.55 An endomorphism L ∈ L(E; E) is (·, ·)g-symmetric iff LT +g = L: +L (·, ·)g-symmetric +⇐⇒ +LT +g = L +⇐⇒ +(L.⃗x, ⃗y)g = (⃗x, L.⃗y)g, +∀⃗x, ⃗y ∈ E. +(A.62) +Remark A.56 The symmetric character of an endomorphism L is not intrinsic to the endomorphism: +It depends on (·, ·)g; See exercise A.52 where L is (·, ·)g-symmetric while it is not (·, ·)h-symmetric. +91 + +92 +A.11. +A transposed of a linear map: depends on chosen inner dot products +A.10.4 +The general flat ♭ notation for an endomorphism: Relative to a (·, ·)g +Let (·, ·)g be an inner dot product in a vector space E, and let L ∈ L(E; E) (a C0 endomorphism). +Definition A.57 The bilinear form L♭ +g ∈ L(E, E; R) which is (·, ·)g-associated to the endomorphism +L ∈ L(E; E) is defined by, for all ⃗u, ⃗w ∈ E, +L♭ +g(⃗u, ⃗w) := (⃗u, L.⃗w)g. +(A.63) +(L♭ +g depends on a choice of a (·, ·)g.) We have thus defined the (·, ·)g-dependent operator: +(.)♭ +g = Jg(.) : +� +L(E; E) → L(E, E; R) +L → Jg(L) := L♭ +g, +(A.64) +If (·, ·)g is imposed, then L♭ +g =noted L♭. +(The bilinearity of L♭ +g is trivial since L is linear and (·, ·)g is bilinear, and the bilinear form L♭ +g +continuous as soon as L and (·, ·)g are since |L♭ +g(⃗u,⃗v)| ≤ ||g|| ||L.⃗u|| ||⃗v|| ≤ (||g|| ||L||) ||⃗u|| ||⃗v||.) +Proposition A.58 With the natural canonical isomorphism L ∈ L(E; E) ≃ TL ∈ L(E∗, E; R) given by +TL(ℓ, ⃗w) = ℓ.L.⃗w, and with TL =noted L, the function (.)♭ +g is the change of contravariance to covariance +mapping given by +L♭ +g = g.L. +(A.65) +Proof. Recall: The contraction of an elementary +�0 +2 +� +tensor ℓ1 ⊗ ℓ2 with an elementary +�1 +1 +� +tensor ⃗v ⊗ ℓ3 +is the +�0 +2 +� +tensor (ℓ1 ⊗ℓ2).(⃗v ⊗ℓ3) := (ℓ2.⃗v)ℓ1 ⊗ℓ3. And the contraction on any tensors is the bilinear map +defined on elementaty tensors. So, with a basis (⃗ei) in E and its dual basis (ei) in E∗, if g = � +ij gijei⊗ej +and L = � +ij Lij⃗ei ⊗ ej then g.L = � +ijk gikLkj⃗ei ⊗ ej. Thus (g.L)(⃗u, ⃗w) = � +ij uiwj(g.L)(⃗ei,⃗ej) = +� +ijk uiwjgikLkj = � +ik uigik(L.⃗w)k = � +ik uigik(L.⃗w)k = g(⃗u, L.⃗w) = L♭ +g(⃗u, ⃗w). +Quantification: Let (⃗ei) be a basis in E, and, with duality notations motivated by the flat notation +“i top changed into i bottom” in the components Lij of L, let gij := g(⃗ei,⃗ej), L.⃗ej = �n +i=1Lij⃗ei and +L♭ +g,ij = L♭ +g(⃗ei,⃗ej), i.e. with tensorial notations for calculations +g = +� +ij +gijei ⊗ ej, +L = +� +ij +Li +j⃗ei ⊗ ej, +L♭ +g = +� +ij +L♭ +g,ijei ⊗ ej. +(A.66) +So [g]|⃗e = [gij], [L]|⃗e = [Lij] and [L♭ +g]|⃗e = [L♭ +g,ij]. Then (A.65) gives (or see next exercise) +[L♭ +g] = [g].[L] . +(A.67) +Exercice A.59 Prove (A.67) with components. +Answer. With (A.63) we get L♭ +g,ij = L♭ +g(⃗ei,⃗ej) = (⃗ei, L.⃗ej)g = (⃗ei, +� +k +Lk +j⃗ek)g = +� +k +Lk +jgik = ([g].[L])ij. +Remark A.60 A change of variance, here from the +�1 +1 +� +type tensor L to the +�0 +2 +� +tensor L♭ +g, is necessarily +observer dependent: There is no natural canonical isomorphism between a vector space E and its dual E∗, +see § T.2. Details: Here fix ⃗w and write ℓg,⃗w(⃗u) = (⃗u, L.⃗w)g (= L♭ +g(⃗u, ⃗w)); Thus ℓg,⃗w ∈ E∗ is the (·, ·)g- +representation function (linear form) of the vector L.⃗w, i.e. ℓg,⃗w = ⃗Rg(L.⃗w) where ⃗Rg is the (·, ·)g-Riesz- +representation operator (the change of variance operator, see (F.3). +A.11 +A transposed of a linear map: depends on chosen inner dot products +This paragraph is needed to define the transposed of the deformation gradient. +Not to be confused with the transposed of a matrix, cf. (A.53). And not to be confused with the +objective transposed of a bilinear form, cf. (A.33); E.g., a transposed of a linear map is not objective. +92 + +93 +A.11. +A transposed of a linear map: depends on chosen inner dot products +A.11.1 +Definition (subjective) +(E, (·, ·)g) and (F, (·, ·)h) are Hilbert spaces, and L ∈ L(E; F) (which is supposed to be continuous if E +and F are infinite dimensional). E.g., E = ⃗Rn +t0, F = ⃗Rn +t , L = dΦt0 +t (P) ∈ L(⃗Rn +t0; ⃗Rn +t ) = the deformation +gradient, cf. (4.1), (·, ·)g is the foot built Euclidean dot product chosen by the observer who made the +measurements at t0, (·, ·)h is the metre built Euclidean dot product chosen by the observer who makes +the measurements at t. +Definition A.61 The transposed of L ∈ L(E; F) relative to (·, ·)g and (·, ·)h is the linear map LT +gh ∈ +L(F; E) defined by, for all (⃗x, ⃗y) ∈ E × F, +(LT +gh.⃗y, ⃗x)g = (⃗y, L.⃗x)h, +(A.68) +where we used the dot notation LT +gh(⃗y) =noted LT +gh.⃗y since LT +gh is linear. This defines the map +(.)T +gh : +� +L(E; F) → L(F; E) +L → (.)T +gh(L) := LT +gh +(A.69) +NB: So a linear map has an infinite number of transposed (it depends on inner dot products). +Notation: If (·, ·)g and (·, ·)h are imposed then LT +gh =noted LT . +And if F = E and (·, ·)h = (·, ·)g then LT +gh = LT +g , see § A.10. +A.11.2 +Quantification with bases +Let (⃗ai) and (⃗bi) be bases in E and F, let gij := g(⃗ai,⃗aj), hij := h(⃗bi,⃗bj), [g]|⃗a = [gij], [h]|⃗b = [hij], and +let (classical notation) +L.⃗aj = +m +� +i=1 +Lij⃗bi, +i.e. +[L]|⃗a,⃗b = [Lij] noted += +[L], +LT +gh.⃗bj = +n +� +i=1 +(LT +gh)ij⃗ai, +i.e. +[LT +gh]|⃗b,⃗a = [(LT +gh)ij] noted += +[LT +gh]. +(A.70) +(A.68) gives [⃗x]T +|⃗a.[g]|⃗a.[LT +gh.⃗y]|⃗y = ([L.⃗x]|⃗b)T .[h]|⃗b.[⃗y]|⃗b for all ⃗x, ⃗y, thus, [g]|⃗a.[LT +gh]|⃗b,⃗a = ([L]|⃗a,⃗b)T .[h]|⃗b and +[LT +gh]|⃗b,⃗a = [g]−1 +|⃗a .([L]|⃗a,⃗b)T .[h]|⃗b. Shortened notation: +[g].[LT ] = [L]T .[h], +i.e. +n +� +k=1 +gik(LT +gh)kj = +m +� +k=1 +Lki hkj, +(A.71) +i.e. +[LT ] = [g]−1.[L]T .[h] , +i.e. +(LT +gh)ij = +n +� +k=1 +m +� +ℓ=1 +([g]−1)ikLℓkhℓj. +(A.72) +With duality notations, L.⃗ej = �n +i=1Lij⃗ei, [L]|⃗e = [Lij], LT +gh.⃗ej = �n +i=1(LT +gh)ij, [LT +gh]|⃗e = [(LT +gh)ij], and +n +� +k=1 +gik(LT +gh)k +j = +n +� +k=1 +Lk +i hkj, +i.e. +(LT +gh)i +j = +n +� +k,ℓ=1 +([g]−1)ikLℓ +k hℓj +( noted += +n +� +k,ℓ=1 +(gikLℓ +k hℓj). +(A.73) +(Be careful with the notation ([g]−1)ik =noted gij, see remark A.50.) +Exercice A.62 Prove: If L is invertible then (LT +gh)−1 = (L−1)T +hg. +Answer. (LT +gh.(L−1)T +hg.⃗x, ⃗y)g = ((L−1)T +hg.⃗x, L.⃗y)h = (⃗x, L−1.L.⃗y)g = (⃗x, ⃗y)g = (LT +gh.(LT +gh)−1.⃗x, ⃗y)g, true ∀⃗x, ⃗y. +A.11.3 +Deformation gradient symmetric: Absurd +The symmetry of a linear map L ∈ L(E; F) is a nonsense if E ̸= F. +E.g.: The gradient of deformation F t0 +t (pt0) = dΦt0 +t (pt0) =noted F ∈ L(⃗Rn +t0; ⃗Rn +t ) cannot be symmetric +since F T ∈ L(⃗Rn +t ; ⃗Rn +t0). Idem for the first Piola–Kirchhoff tensor PKt0 +t , which motivates the introduction +of the symmetric second Piola–Kirchhoff tensor SKt0 +t , see Marsden–Hughes [12] or § M.2.3. +93 + +94 +A.12. +The adjoint of a linear map (objective) +A.11.4 +Isometry +Definition A.63 A linear map L ∈ L(E; F) is an isometry relative to (·, ·)g and (·, ·)h iff +∀⃗x, ⃗y ∈ E, +(L.⃗x, L.⃗y)h = (⃗x, ⃗y)g, +i.e. +LT +gh ◦ L = IE (identity in E). +(A.74) +In particular, an endomorphism L ∈ L(E; E) is a (·, ·)g-isometry iff +∀⃗x, ⃗y ∈ E, +(L.⃗x, L.⃗y)g = (⃗x, ⃗y)g, +i.e. +LT +g ◦ L = IE. +(A.75) +Thus, if L ∈ L(E; F) is an isometry and (⃗ei) is a (·, ·)g-orthonormal basis, then (L.⃗ei) is a (·, ·)h- +orthonormal basis, since (L.⃗ei, L.⃗ej)h = (⃗ei,⃗ej)g = δij for all i, j. +Exercice A.64 Let ⃗f : E → F. Prove: +if +⃗f is an isometry then ⃗f is linear. +(A.76) +Answer. Let (⃗ei) be a (·, ·)g-orthonormal basis; Thus (⃗f(⃗ei)) is a (·, ·)h-orthonormal basis (since ⃗f is an isometry). +Thus, if ⃗x = �n +i=1xi⃗ei then ⃗f(⃗x) +b.o.n. += +n +� +i=1 +(⃗f(⃗x), ⃗f(⃗ei))h ⃗f(⃗ei) +hyp. += +n +� +i=1 +(⃗x,⃗ei)g ⃗f(⃗ei) +b.o.n. += +n +� +i=1 +xi ⃗f(⃗ei), thus ⃗f(⃗x+λ⃗y) = +n +� +i=1 +(xi + λyi)⃗f(⃗ei) = +n +� +i=1 +xi ⃗f(⃗ei) + λ +n +� +i=1 +yi ⃗f(⃗ei) = ⃗f(⃗x) + λ⃗f(⃗y), thus ⃗f is linear. +Exercice A.65 Rn is an affine space, ⃗Rn is the usual associated vector space, and (·, ·)g is an inner dot +product in ⃗Rn. Definition: A distance-preserving function f : p ∈ Rn → f(p) ∈ Rn is a function s.t. +||−−−−−→ +f(p)f(q)||g = ||−→ +pq||g, +∀p, q ∈ Rn. +(A.77) +Prove: If f is a distance-preserving function, then f is affine. +Answer. Let O ∈ Rn (an origin) and ⃗f : ⃗x = −→ +Op ∈ ⃗Rn → ⃗f(⃗x) := −−−−−−→ +f(O)f(p) (vectorial associated function). Let +⃗x = −→ +Op and ⃗y = −→ +Oq. Then the remarkable identity 2(⃗f(⃗x), ⃗f(⃗y))g = ||⃗f(⃗x)||2 +g + ||⃗f(⃗y)||2 +g − ||⃗f(⃗x)−⃗f(⃗y)||2 +g gives +2(⃗f(⃗x), ⃗f(⃗y))g = ||⃗f(⃗x)||2 +g+||⃗f(⃗y)||2 +g−||−−−−−→ +f(q)f(p)||2 +g = ||⃗f(⃗x)||2 +g+||⃗f(⃗y)||2 +g−||−→ +qp||2 +g = ||⃗x||2 +g+||⃗y||2 +g−||⃗x−⃗y||2 +g = 2(⃗x, ⃗y)g, +thus ⃗f is an isometry, thus ⃗f is linear cf. (A.76), thus f is affine since f(p) = f(O) + ⃗f(−→ +Op). +A.12 +The adjoint of a linear map (objective) +(For mathematicians; May produce misunderstandings, misuses, problematic mechanical interpretations.) +No inner dot product is required here: A linear map L has only one adjoint L∗ (intrinsic to L); While +L has many transposed LT = LT +gh which depend on inner dot products. +A.12.1 +Definition +E and F are vector spaces, and E∗ = L(E; R) and F ∗ = L(F; R) are the dual spaces (made of linear +continuous forms). (If E and F are finite dimensional, the continuity is always satisfied.) +Definition A.66 Let L ∈ L(E; F) (linear and continuous); Its adjoint is the linear map L∗ ∈ L(F ∗; E∗) +canonically defined by +L∗ : +� +F ∗ → E∗ +m → L∗(m) := m ◦ L, +(A.78) +i.e., for all (⃗x, m) ∈ E × F ∗, +(L∗(m))(⃗x) := m(L(⃗x)). +(A.79) +(The adjoint L∗ cannot be confused with a transposed LT which requires inner dot products, cf. (A.68).) +The linearity of L∗ is trivial, thus, together with the linearity of m and L, we can use the dot notation: +L∗.m := m.L, +and +(L∗.m).⃗x := m.L.⃗x. +(A.80) +And ||L∗.m||E∗ = ||m.L||E∗ ≤ ||m||F ∗||L||L(E;F ) gives ||L∗||L(F ∗;E∗) ≤ ||L||L(E;F ) < ∞, thus L∗ is +continuous (when L is). +94 + +95 +A.13. +Tensorial representation of a linear map +A.12.2 +Quantification +E and F are finite dimensional, dim E = n, dim F = m, and (⃗ai) and (⃗bi) are bases in E and F. Let +[L]|⃗a,⃗b =noted [L], [L∗]|b,a =noted [L∗], [m]|b =noted [m] and [⃗x]|⃗a =noted [⃗x] be the matrices relative to the +chosen bases: (A.80) gives ([L∗].[m].[⃗x] = [m].[L].[⃗x] for all ⃗x ∈ E and m ∈ F ∗, thus, for all m ∈ F ∗ +(recall that [m] is a line matrix), thus [L∗].[m]T = ([L]T .[m]T , thus +[L∗] = [L]T +(transposed matrix). +(A.81) +(Full notation: [L∗]|b,a = ([L]|⃗a,⃗b)T .) +Details: With the dual bases (πai) and (πbi), with L.⃗aj = �m +i=1Lij⃗bi, i.e. [L]|⃗a,⃗b = [Lij] i=1,...,m +j=1,...,n , and +with L∗.πbj = �n +i=1(L∗)ijπai, i.e. [L∗]|b,a = [(L∗)ij] i=1,...,n +j=1,...,m , (A.80) gives, for all (i, j) ∈ [1, n]N × [1, m]N, +(L∗.πbj).⃗ai = πbj.(L.⃗ai), +thus +(L∗)ij = Lji +and +[L∗] = [L]T . +(A.82) +Duality notations (warning: can be misused): L.⃗aj = �m +i=1Lij⃗bi, i.e. [L]|⃗a,⃗b = [Lij] i=1,...,m +j=1,...,n , and L∗.bj = +�n +i=1(L∗)i jai, i.e. [L∗]|b,a = [((L∗)i j] i=1,...,n +j=1,...,m , thus, for all (i, j) ∈ [1, n]N × [1, m]N, +(L∗.bj).⃗ai = bj.(L.⃗ai), +thus +(L∗)i +j = Lj +i +and +[L∗] = [L]T . +(A.83) +(Recall: Use classical notations if in doubt, or, preferably, don’t use duality notations here.) +Remark A.67 Reminder: The transposed bT of a bilinear b form is intrinsic to b, and the adjoint L∗ of +a linear map L is intrinsic to L; But a transposed LT of a linear form L is not intrinsic to the linear form +(it depends on chosen inner dot products): Watch out for the (unfortunate) vocabulary “transpose”! +A.12.3 +Relation with the transposed when inner dot products are introduced +let L ∈ L(E; F). We need inner dot products (·, ·)g and (·, ·)h in E and F to define LT = LT +gh. To +have a functional relation between L∗ and LT +gh, we use the (·, ·)g-Riesz representation mapping ⃗Rg : +� +E∗ → E +ℓ → ⃗Rg(ℓ) = ⃗ℓg +� +, where ℓ.⃗x = (⃗ℓg, ⃗x)g for all ⃗x ∈ E, see (F.3); idem with F. +Let L ∈ L(E; F) (continuous). For all ⃗x ∈ E and all m ∈ F ∗ we have +(L∗.m).⃗x +(A.79) += +m.(L.⃗x), +thus +(⃗Rg(L∗.m), ⃗x)g = (⃗Rh(m), L.⃗x)h, +(A.84) +thus ((⃗Rg ◦ L∗).m), ⃗x)g = ((LT +gh ◦ ⃗Rh).m, L.⃗x)g. Thus ⃗Rg ◦ L∗ = LT +gh ◦ ⃗Rh, i.e. +LT +gh = ⃗Rg ◦ L∗ ◦ (⃗Rh)−1 +i.e. +E +LT +gh +←− +F +⃗Rg ↑ +↑ ⃗Rh +E∗ ←− +L∗ F ∗ +is a commutative diagram. +(A.85) +Exercice A.68 From (A.85), recover (A.71), i.e. [LT +gh] = [g]−1.[L]T .[h]. +Answer. [LT +gh] =(A.85) [⃗Rg].[L∗].[⃗Rh]−1 =(F.6) [g]−1.[L]T .[h]. +A.13 +Tensorial representation of a linear map +A.13.1 +A tensorial representation +Consider the natural canonical isomorphism (between linear maps E → F and bilinear forms F ∗×E → R) +� +J : +� +L(E; F) → L(F ∗, E; R) +L → βL = � +J (L) +� +where +βL(m, ⃗u) := m.(L.⃗u), +∀(m, ⃗u) ∈ F ∗ × E, +(A.86) +see § T.4. And βL is also named L for calculations purposes, see (A.89). +(NB: It can be dangerous to substitute L with βL, see e.g. § A.7.2.) +95 + +96 +A.14. +Change of basis formulas for bilinear forms and linear maps +Quantification: Let (⃗ai)i=1,...,n be a basis in E, (⃗bi)i=1,...,m be a basis in F which dual basis is (πbi), +L ∈ L(E; F). Then +βL(πbi,⃗ai) = πbi.L.⃗ai. +(A.87) +Thus, if +L.⃗aj = +m +� +i=1 +Lij⃗bi +then +βL = +m +� +i=1 +n +� +j=1 +Lij⃗bi ⊗ πaj +(A.88) +Indeed, +(� +ij Lij⃗bi ⊗ πaj)(πbk,⃗aℓ) += +� +ij Lij(⃗bi ⊗ πaj)(πbk,⃗aℓ) += +� +ij Lij(⃗bi.πbk)(πaj.⃗aℓ) += +� +ij Lij(⃗bi.πbk)(πaj.⃗aℓ) = � +ij Lijδkiδjℓ = Lkℓ = πbk.L.⃗aℓ, so (A.87) gives (A.88). +Duality notations: L.⃗aj = �m +i=1Lij⃗bi and βL = �m +i=1 +�n +j=1Lij⃗bi ⊗ aj. +Contraction rule. If you write L = �m +i=1 +�n +j=1Lij⃗bi ⊗πaj (≃ βL), then the vector L.⃗u ∈ F is computed +thanks to the “contraction rule”: +L.⃗u = ( +m +� +i=1 +n +� +j=1 +Lij⃗bi ⊗ πaj).⃗u +� �� � +contraction +:= +m +� +i=1 +n +� +j=1 +Lij⃗bi(πaj.⃗u) = +m +� +i=1 +n +� +j=1 +Lijuj⃗bi. +(A.89) +(With duality notations: L.⃗u = ( +m +� +i=1 +n +� +j=1 +Li +j⃗bi ⊗ aj).⃗u +� �� � +contraction += +m +� +i=1 +n +� +j=1 +Li +j⃗bi(aj.⃗u) = +m +� +i=1 +n +� +j=1 +Li +juj⃗bi.) +Remark A.69 Warning: The bilinear form βL should not be confused with the linear map L: The +domain of definition of βL is F ∗ × E, and βL acts on the two objects ℓ (linear form) and ⃗u (vector) to +get a scalar result; While the domain of definition of L is E, and L acts one object ⃗u to get a vector +result. However, you can use the tensorial notation for L... only to calculate L.⃗u with (A.89). +A.13.2 +Warning: Confusion between transposed and adjoint +The transposed LT ∈ L(F; E) of a linear map L ∈ L(E; F) needs inner dot products to be defined, +cf (A.68): It is not intrinsic to L, not objective ; While the transposed bT ∈ L(B, A; R) of a bilinear +form b ∈ L(A, B; R) is intrinsic to L (it does not need inner dot products to be defined). +So if you represent a linear map L ∈ L(E; F) by its tensorial representation βL ∈ L(F ∗, E; R), +cf. (A.88), then +1- you know the transposed (βL)T (given by (βL)T (⃗w, ⃗u) = βL(⃗u, ⃗w)), +2- but you cannot deduce the transposed LT ∈ L(F; E) from (βL)T (i.e., to start with (βL)T is +misleading): You need to choose inner dot products, and then use the formula (LT .⃗y, ⃗x)g = (⃗y, L.⃗x)h +where LT := LT +gh to get [LT ] = [g]−1.[L]T .[h] (and [LT ] ̸= [L]T in general). +3- In particular: If L ∈ L(E; E) is symmetric (relative to the chosen inner dot products), then +βL ∈ L(E∗, E; R) is never symmetric because E∗ ̸= E ! (Recall: there is no natural canonical isomorphism +between E and E∗.) +A.14 +Change of basis formulas for bilinear forms and linear maps +A.14.1 +Notations for transitions matrices for bilinear forms and linear maps +Let A and B be finite dimension vector spaces, dim A = n, dim B = m. (E.g. application to the change +of basis formula for the deformation gradient A=⃗Rn +t0 → B=⃗Rn +t .) +Let (⃗aold,i) and (⃗anew,i) be two bases in A, and (⃗bold,i) and (⃗bnew,i) be two bases in B. +Let PA +and PB be the change of basis endomorphisms from old to new bases, and PA := [PA]|⃗aold = [PAij] and +PB := [PB]|⃗bold = [PBij] be the associated transition matrices, and QA = PA +−1 and QB = PB +−1: +⃗anew,j = PA.⃗aold,i = +n +� +i,j=1 +PAij⃗aold,i, +πanew,j = +n +� +i=1 +QAijπaold,i, +⃗bnew,j = PB.⃗bold,i = +m +� +i,j=1 +PBij⃗bold,i, +πbnew,j = +n +� +i,j=1 +QBijπbold,i. +(A.90) +Duality notations: ⃗anew,j = �n +i=1PA +i +j⃗aold,i and ai +new = �n +j=1QA +i +jaj +old and ⃗bnew,j = �n +i=1PB +i +j⃗bold,i and +bi +new = �n +j=1QB +i +jbj +old. +96 + +97 +A.14. +Change of basis formulas for bilinear forms and linear maps +A.14.2 +Change of coordinate system for bilinear forms ∈ L(A, B; R) +Let g ∈ L(A, B; R), and, for all (i, j) ∈ [1, n]N × [1, m]N, +g(⃗aold,i,⃗bold,j) = Mij, +g(⃗anew,i,⃗bnew,j) = Nij, +i.e. +� +� +� +[g]|olds = M = [Mij] i=1,...,n +j=1,...,m , +[g]|news = N = [Nij] i=1,...,n +j=1,...,m . +(A.91) +Proposition A.70 Change of basis formula: +[g]|news = PA +T .[g]|olds.PB, +i.e. +N = PA +T .M.PB. +(A.92) +In particular, if A = B and (⃗aold,i) = (⃗bold,i) and (⃗anew,i) = (⃗bnew,i), then PA = PB =noted P, and +[g]|new = P T .[g]old.P , +i.e. +N = P T .M.P. +(A.93) +Proof. Nij = g(⃗anew,i,⃗bnew,j) = � +kℓ PA +k +iPB +ℓ +jg(⃗aold,k,⃗bold,ℓ) = � +kℓ PA +k +iMkℓPB +ℓ +j = � +kℓ(PA +T )ikMkℓPB +ℓ +j. +Exercice A.71 Prove (objective result): +g(⃗u, ⃗w) = [⃗u]T +|⃗anew.[g]|news.[⃗w]|⃗bnew = [⃗u]T +|⃗aold.[g]|olds.[⃗w]|⃗bold. +(A.94) +Answer. [⃗u]T +|⃗anew.[g]|news.[⃗w]|⃗bnew = (PA +−1.[⃗u]|⃗aold)T .(PA +T .[g]|olds.PB).(PB +−1.[⃗w]|⃗bold). +A.14.3 +Change of coordinate system for bilinear forms ∈ L(A∗, B∗; R) +Let z ∈ L(A∗, B∗; R), and, for all (i, j) ∈ [1, n]N × [1, m]N, +z(ai +old, bj +old) = M ij, +z(ai +new, bj +new) = N ij, +i.e. +� +� +� +[z]|olds = M = [M ij] i=1,...,n +j=1,...,m , +[z]|news = N = [N ij] i=1,...,n +j=1,...,m . +(A.95) +Proposition A.72 Change of basis formula: +[z]|news = PA +−T .[z]|olds.PB +−1, +i.e. +N = PA +−T .M.PB +−1. +(A.96) +In particular, if A = B and (⃗aold,i) = (⃗bold,i) and (⃗anew,i) = (⃗bnew,i), then PA = PB =noted P, and +[z]|new = P −T .[z]old.P −1 , +i.e. +N = P −T .M.P −1. +(A.97) +Proof. Nij = z(ai +new, bj +new) = � +kℓ QA +k +iQB +ℓ +jz(ak +old, bℓ +old) = � +kℓ QA +k +iM kℓQB +ℓ +j = � +kℓ(QA +T )ikM kℓQB +ℓ +j. +A.14.4 +Change of coordinate system for bilinear forms ∈ L(B∗, A; R) +(Toward linear maps L ∈ L(A; B) ≃ L(B∗, A; R) thanks to the natural canonical isomorphism.) +Let T ∈ L(B∗, A; R), and, for all (i, j) ∈ [1, n]N × [1, m]N, +T(bi +old,⃗aold,j) = M i +j, +T(bi +new,⃗anew,j) = N i +j, +i.e. +� +� +� +[T]|olds = M = [M i +j] i=1,...,n +j=1,...,m , +[T]|news = N = [N i +j] i=1,...,n +j=1,...,m . +(A.98) +Proposition A.73 Change of basis formula: +[T]|news = PB +−1.[T]|olds.PA, +i.e. +N = QA.M.PB. +(A.99) +In particular, if A = B and (⃗aold,i) = (⃗bold,i) and (⃗anew,i) = (⃗bnew,i), then PA = PB =noted P, and +[T]|new = P −1.[T]old.P , +i.e. +N = P −1.M.P, +i.e. +N i +j = +n +� +k,ℓ=1 +Qi +kM k +ℓP ℓ +j. +(A.100) +Proof. N ij = T(bi +new,⃗anew,j) = � +kℓ QB +i +kPA +ℓ +jT(bi +old,⃗aold,j) = � +kℓ QB +i +kM ijPA +ℓ +j +97 + +98 +A.14. +Change of basis formulas for bilinear forms and linear maps +A.14.5 +Change of coordinate system for tri-linear forms ∈ L(A∗, A, A; R) +(Toward d2⃗u: +For a vector field ⃗u ∈ Γ(U) ≃ T 1 +0 (U), ⃗u(p) ∈ ⃗Rn, its differential satisfies d⃗u(p) ∈ +L(⃗Rn; ⃗Rn) ≃ L(Rn∗, ⃗Rn; R), and d2⃗u(p) ∈ L(⃗Rn; L(⃗Rn; ⃗Rn)) ≃ L(Rn∗, ⃗Rn, ⃗Rn; R), see § S.1.3.) +Consider a tri-linear form T ∈ L(A∗, A, A; R), and +M i +jk = T(ai +old,⃗aold,j,⃗aold,k), +N i +jk = T(ai +new,⃗anew,j,⃗anew,k), +i.e. +[T]|⃗aold = [M i +jk], +[T]|⃗anew = [N i +jk]. +(A.101) +Then +N i +jk = +n +� +λ,µ,ν=1 +Qi +λP µ +j P ν +k M λ +µν. +(A.102) +Indeed � +λµν M λ +µν⃗aold,λ ⊗ aµ +old ⊗ aν +old = � +λµνijk M λ +µνQi +λP µ +j P ν +k⃗anew,i ⊗ aj +new ⊗ ak +new. +A.14.6 +Change of coordinate system for linear maps ∈ L(A; B) +Notation of § A.14.1. Let L ∈ L(A; B) be a linear map, and let, for all j = 1, ..., n, +� +� +� +� +� +� +� +� +� +� +� +L.⃗aold,j = +m +� +i=1 +Mij⃗bold,i = +m +� +i=1 +M i +j⃗bold,i +i.e. +[L]|olds = M = [Mij] = [M i +j] i=1,...,m +j=1,...,n , +L.⃗anew,j = +m +� +i=1 +Nij⃗bnew,i = +m +� +i=1 +N i +j⃗bnew,i +i.e. +[L]|news = N = [Nij] = [N i +j] i=1,...,m +j=1,...,n , +(A.103) +with classical and duality notations. +Proposition A.74 Change of bases formula: +[L]|news = PB +−1.[L]|olds.PA, +i.e. +N = PB +−1.M.PA. +(A.104) +In particular, if A = B, if (⃗aold,i) = (⃗bold,i), (⃗anew,i) = (⃗bnew,i), then PA = PB =noted P and +[L]|new = P −1.[L]|old.P , +i.e. +N = P −1.M.P, +i.e. +Nij = +n +� +k,ℓ=1 +QikMkℓPℓj, +(A.105) +with Q = P −1, and with duality notations N ij = � +kℓ QikM kℓP ℓj. +Proof. +L.⃗anew,j += +� +i N ij⃗bnew,i += +� +ik N ijPB +k +i⃗bold,k += +� +k(PB.N)kj⃗bold,k +and L.⃗anew,j += +L.(� +i PA +i +j⃗aold,i) = � +i PA +i +j +� +k M ki⃗bold,k = � +k(M.PA)kj⃗bold,k, for all j, thus PB.N = M.PA. +Exercice A.75 Prove: +ℓ.L.⃗u = [ℓ]|⃗bnew.[L]|news.[⃗u]|⃗anew = [ℓ]|⃗bold.[L]|olds.[⃗u]|⃗aold +(objective result). +(A.106) +Answer. [ℓ]|⃗bnew.[L]|news.[⃗u]|⃗anew = ([ℓ]|⃗bold.PB).(PB +−1.[L]|olds.PA).(PA +−1.[⃗u]|⃗aold). +Remark A.76 Bilinear forms in L(A, A; R) and endomorphisms in L(A; A) behave differently: The +formulas (A.93) and (A.105) should not be confused since P −1 ̸= P T in general. E.g., if an English +observer uses a Euclidean (old) basis (⃗ai) = (⃗aold,i) in foot, if a French observer uses a Euclidean (new) +basis (⃗bi) = (⃗anew,i) in metre, and if (simple case) ⃗bi = λ⃗ai for all i (change of unit), then +[L]|new = [L]|old, +while +[g]|new = λ2 +���� +>10 +[g]|old. +(A.107) +Quite different results! I.e. P −1.[L]|old.P ̸= P T .[L]|old.P for a general change of basis. See the Mars +Climate Orbiter crash, remark A.14, where someone forgot that 1 foot ̸= 1 metre. +B +Euclidean Frameworks +Time and space are decoupled (classical mechanics). Rn is the geometric affine space, n = 1, 2, 3, and ⃗Rn +is the associated vector space made of “bi-point vectors”. +98 + +99 +B.1. +Euclidean basis +B.1 +Euclidean basis +Manufacturing of a Euclidean basis. +An observer chooses a unit of measure (foot, metre, a unit of length used by Euclid, the diameter a +of pipe...) and makes a “unit rod” of length 1 in this unit. +Postulate: The length of the rod does not depend on its direction in space. +• Space dimension n = 1: This rod models a vector ⃗e1 which makes a basis (⃗e1) called the Euclidean +basis relative to the chosen unit of measure. +• Space dimension n ≥ 2: +- The observers makes three rods of length 3, 4 and 5, and makes a triangle (A, B, C) with A, B and +C are the vertices and A not on the side on length 5. +- Pythagoras: 32 + 42 = 52 gives: The triangle (A, B, C) is said to have a right angle at A. +- Two vectors ⃗u and ⃗w in ⃗Rn are orthogonal iff the triangle (A, B, C) can be positioned such that ⃗ +AB +and ⃗ +AC are parallel to ⃗u and ⃗w. +- A basis (⃗ei)i=1,...,n is Euclidean relative to the chosen unit of measurement iff the ⃗ei are two to two +orthogonal and their length is 1 (relative to the chosen unit). +Example B.1 An English observer defines a Euclidean basis (⃗ai) using the foot. A French observer +defines a Euclidean basis (⃗bi) using the metre. We have +1 foot = µ metre, +µ = 0.3048, +and +1 metre = λ foot, +λ = 1 +µ ≃ 3.28. +(B.1) +(µ = 0, 3048 is the official length in metre for the English foot.) E.g., the bases are “aligned” iff, for all i, +⃗bi = λ⃗ai +(change of measurement unit), +(B.2) +thus the transition matrix from (⃗ai) to (⃗bi) is P = λI, thus P T = P, P −1 = 1 +λI and P T .P = λ2I. +Remark B.2 The bases used in practice are not all Euclidean. See example A.13, especially if you fly. +B.2 +Euclidean dot product +Definition B.3 An observer who has built) his Euclidean basis (⃗ei), cf. § B.1. The associated Euclidean +dot product is the bilinear form g(·, ·) = (·, ·)g ∈ L(⃗Rn, ⃗Rn; R) defined by +(gij =) +g(⃗ei,⃗ej) = δij, +∀i, j, +i.e. +[g]|⃗e = [δij] = I. +(B.3) +In other words, +(·, ·)g := +n +� +i=1 +πei ⊗ πei = +n +� +i=1 +ei ⊗ ei, +(B.4) +with classical and duality notations, (πei) = (ei) being the dual basis of (⃗ei). And if you want to use the +Einstein convention you have to write (·, ·)g := �n +i,j=1δijei ⊗ ej: You cannot avoid writing δij = gij. +Thus, for all ⃗x, ⃗y ∈ ⃗Rn, with ⃗x = �n +i=1xi⃗ei and ⃗y = �n +i=1yi⃗ei (classical notations), +(⃗x, ⃗y)g = +n +� +i=1 +xiyi = [⃗x]T +|⃗e.[⃗y]|⃗e. +(B.5) +With duality notations, ⃗x = �n +i=1xi⃗ei, ⃗y = �n +i=1yi⃗ei and (⃗x, ⃗y)g = �n +i=1xiyi; And if you want to use +the Einstein convention then write (⃗x, ⃗y)g := �n +i,j=1δijxiyj: You cannot avoid writing δij. +Definition B.4 The associated norm is ||.||g := +� +(·, ·)g, and the length of a vector ⃗x relative to the +chosen Euclidean unit of measurement is ||⃗x||g := +� +(⃗x, ⃗x)g. +Thus with the Euclidean basis (⃗ei) (used to build (·, ·)g), if ⃗x = �n +i=1xi⃗ei, then ||⃗x||g = +��n +i=1x2 +i is +the length of ⃗x relative to the chosen Euclidean unit of measure (Pythagoras). (With duality notations +||⃗x||g = +��n +i=1(xi)2, and if you want to use the Einstein convention: ||⃗x||g = +��n +i,j=1δijxixj.) +Definition B.5 The angle θ(⃗x, ⃗y) between two vectors ⃗x, ⃗y ∈ ⃗Rn − {⃗0} is defined by +cos(θ(⃗x, ⃗y)) = ( +⃗x +||⃗x||g +, +⃗y +||⃗y||g +)g. +(B.6) +(With a calculator, this formula gives θ(⃗x, ⃗y) = arccos(( +⃗x +||⃗x||g , +⃗y +||⃗y||g )g) a value in [0, π].) +99 + +100 +B.3. +Change of Euclidean basis +B.3 +Change of Euclidean basis +Let (⃗ai) (e.g. English observer basis built with the foot) and (⃗bi) (e.g. French observer basis built with +the metre) be Euclidean bases in ⃗Rn, and let (·, ·)g and (·, ·)h be the associated Euclidean dot products. +B.3.1 +Two Euclidean dot products are proportional +Proposition B.6 If λ = ||⃗b1||g, then ||⃗bi||g = λ for all i = 1, ..., n (change of unit) and +(·, ·)g = λ2(·, ·)h, +and +||.||g = λ||.||h. +(B.7) +Proof. By definition of a Euclidean basis, the length of the rod that enabled to define (⃗bi) is independent +of i, cf. § B.1, thus ||⃗bi||g = ||⃗b1||g for all i, and here ||⃗bi||g =noted λ. Thus ||⃗bi||2 +g = λ2 = λ2||⃗bi||2 +h for all i, +since ||⃗bi||2 +h = 1. And if i ̸= j then (⃗bi,⃗bj)g = 0 = (⃗bi,⃗bj)h since ⃗bi and ⃗bj form a right angle (Pythagoras), +cf. (B.4). Hence (⃗bi,⃗bj)g = λ2(⃗bi,⃗bj)h for all i, j, thus (B.7). +Example B.7 Continuation of example B.1: (·, ·)a = �n +i=1ai ⊗ ai is the English Euclidean dot product +(foot), and (·, ·)b = �n +i=1bi ⊗ bi is the French Euclidean dot product (metre). (B.7) and (B.1) give: +(·, ·)a = λ2(·, ·)b +and +||.||a = λ||.||b, +with +λ ≃ 3.28 +and +λ2 ≃ 10.76. +(B.8) +In particular, if ⃗w is s.t. ||⃗w||b = 1 (its length is 1 metre), then ||⃗w||a = λ (its length is λ ≃ 3.28 foot). +B.3.2 +Counterexample : non existence of a Euclidean dot product +1- Thermodynamic: Let T be the temperature and P the pressure, and consider the Cartesian vector +space {(T, P)} = {(temperature,pressure)} = R × R. There is no associated Euclidean dot product: An +associated norm would give ||(T, P)|| = +√ +T 2 + P 2 ∈ R which is meaningless (incompatible dimensions). +See § A.3.5. +2- Polar coordinate system ⃗q = (r, θ) ∈ R × R: There is no Euclidean norm +√ +r2 + θ2 for ⃗q that is +physically meaningful (incompatible dimensions), see example 6.11. +B.4 +Euclidean transposed of the deformation gradient +Let n ∈ {1, 2, 3} and consider a linear map L ∈ L(⃗Rn +t0; ⃗Rn +t ) (e.g., L = F t0 +t (P)). +Let (·, ·)G be a Euclidean dot product in ⃗Rn +t0 (used in the past by someone), and let (·, ·)g and (·, ·)h +be Euclidean dot products in ⃗Rn +t (the actual space where the results are obtained by two observers, e.g., +(·, ·)g built with a foot and (·, ·)h built with a metre). Let LT +Gg and LT +Gh be the transposed of L relative +to the dot products, that is, LT +Gg and LT +Gh in L(⃗Rn +t ; ⃗Rn +t0) are characterized by, for all ( ⃗X, ⃗y) ∈ ⃗Rn +t0 × ⃗Rn +t , +cf. (A.68), +(LT +Gg.⃗y, ⃗X)G = (L. ⃗X, ⃗y)g +and +(LT +Gh.⃗y, ⃗X)G = (L. ⃗X, ⃗y)h. +(B.9) +Corollary B.8 +if +(·, ·)g = λ2(·, ·)h +then +LT +Gg = λ2LT +Gh. +(B.10) +NB: Do not forget λ2, cf. remark A.14 (Mars Climate Orbiter crash). +Proof. (LT +Gg.⃗y, ⃗X)G +(B.9) += (L. ⃗X, ⃗y)g +(B.10)1 += +λ2(L. ⃗X, ⃗y)h +(B.9) += λ2(LT +Gh.⃗y, ⃗X)G for all ⃗X ∈ ⃗Rn +t0 and all ⃗y ∈ ⃗Rn +t , +thus LT +Gg.⃗y = λ2LT +Gh.⃗y for all ⃗y ∈ ⃗Rn +t , thus (B.10)2. +B.5 +The Euclidean transposed for endomorphisms +Let n ∈ {1, 2, 3} and consider an endomorphism L ∈ L(⃗Rn +t ; ⃗Rn +t ) (e.g. L = d⃗vt(p) ∈ L(⃗Rn +t ; ⃗Rn +t ) the +differential of the Eulerian velocity). Let (·, ·)g and (·, ·)h be dot products in ⃗Rn. Let LT +g and LT +h be the +transposed of L relative to (·, ·)g and (·, ·)h, that is, LT +g and LT +h in L(⃗Rn +t ; ⃗Rn +t ) are the endomorphisms +defined by, for all ⃗x, ⃗y ∈ ⃗Rn +t , cf. (A.54), +(LT +g .⃗y, ⃗x)g = (L.⃗x, ⃗y)g, +and +(LT +h .⃗y, ⃗x)h = (L.⃗x, ⃗y)h. +(B.11) +100 + +101 +B.6. +Unit normal vector, unit normal form +Corollary B.9 +if +(·, ·)g = λ2(·, ·)h +then +LT +g = LT +h +noted += +LT +∈ L(⃗Rn +t ; ⃗Rn +t ) +(B.12) +(an endomorphism type relation): Thus we can speak of “the Euclidean transposed of an endomorphism”. +Proof. (LT +g .⃗y, ⃗x)g +(B.11) += (L.⃗x, ⃗y)g +hyp += λ2(L.⃗x, ⃗y)h +(B.11) += +λ2(LT +h .⃗y, ⃗x)h +hyp += (LT +h .⃗y, ⃗x)g for all ⃗x, ⃗y ∈ ⃗Rn, thus +LT +g .⃗y = LT +h .⃗y for all ⃗y ∈ ⃗Rn. +B.6 +Unit normal vector, unit normal form +The results in this § are not objective: We need a Euclidean dot product (need a unit of length: Foot? +Meter?) to get a unit (Euclidean) normal vector. +B.6.1 +Framework +(·, ·)g is a Euclidean dot product (needed to define Euclidean orthonormality) and, for all ⃗u, ⃗w ∈ ⃗Rn, +(⃗u, ⃗w)g +noted += +⃗u •g ⃗w +(B.13) +(or =noted ⃗u • ⃗w when one chosen Euclidean dot product is imposed to all). +Ω is a regular open bounded set in Rn, n = 2 or 3, and Γ := ∂Ω is its regular surface (dimension n−1). +If p ∈ Γ then TpΓ is the tangent plane at p to Γ, and a basis (⃗β1(p), ..., ⃗βn−1(p)) in TpΓ is known (usually +obtained thanks to a coordinate system describing Γ). And, to lighten the writings, (⃗β1(p), ..., ⃗βn−1(p)) +is written (⃗β1, ..., ⃗βn−1). +B.6.2 +Unit normal vector +Call ⃗ng(p) the unit outward normal vector at p ∈ Γ at TpΓ relative to (·, ·)g; So ⃗ng(p) •g ⃗βi(p) = 0 for all +i = 1, ..., n−1, and ||⃗ng(p)||g = 1, i.e. ⃗ng is defined on Γ by (up to its sign) +∀i = 1, ..., n−1, ⃗βi •g ⃗ng = 0, +and +⃗ng •g ⃗ng = 1 +(= ||⃗ng||2 +g), +(B.14) +i.e., at any p ∈ Γ, ⃗ng(p) is orthogonal to the hyperplane Vect{⃗β1(p), ..., ⃗βn−1(p)} and ⃗ng(p) is unitary. So +(⃗β1(p), ..., ⃗βn−1(p),⃗ng(p)) is a basis at p in ⃗Rn, written in short (⃗β1, ..., ⃗βn−1,⃗ng). Drawing. +Thus, for all ⃗w ∈ ⃗Rn, if ⃗w = �n−1 +i=1 wi⃗βi + wn⃗ng (classical notations) then +wn = ⃗w •g ⃗ng = the normal component of ⃗w at p at Γ. +(B.15) +(wn depends on (·, ·)g.) (Duality notations: ⃗w = �n−1 +i=1 wi⃗βi + wn⃗ng and wn = ⃗w •g ⃗ng.) +Exercice B.10 Let (⃗ai) be a basis in ⃗Rn, ⃗βj = �n +i=1Bij⃗ai for j = 1, ..., n−1, and ⃗ng = �n +i=1ni⃗ai, and +gij = g(⃗ai,⃗aj) for all i, j. What equations satisfy the nj? And particular case (⃗ai) is (·, ·)g-orthonormal? +Answer. (B.14) gives [⃗βi]T +|⃗a.[g]|⃗a.[⃗ng]|⃗a = 0 for i = 1, ..., n−1 (so n−1 equations), with [⃗ng]T +|⃗a.[g]|⃗a.[⃗ng]|⃗a = 1 (so 1 +equation), and ⃗ng is obtained up to its sign. +If (⃗ai) is (·, ·)g-orthonormal, then �n +j=1Bijnj = 0 for j = 1, ..., n−1, with �n +i=1n2 +i = 1. +Exercice B.11 Let (⃗ai) be a Euclidean basis in foot, (⃗bi) a Euclidean basis in metre, (·, ·)a and (·, ·)b +the associated Euclidean dot products, so (·, ·)a = λ2(·, ·)b with λ ≃ 3.28, cf. (B.7). Let ⃗na(p) and ⃗nb(p) +be the corresponding unit outward normal vectors, cf. (B.14). 1- Prove (up to the sign): +⃗nb = λ⃗na, +and +(⃗w,⃗na)a = λ(⃗w,⃗nb)b +∀⃗w ∈ ⃗Rn +(B.16) +2- Then let ⃗na = �m +i=1nai⃗ai and ⃗nb = �m +i=1nbi⃗bi; Prove: +If, ∀i = 1, ..., n, ⃗bi = λ⃗ai +then +∀i = 1, ..., n, nai = nbi. +(B.17) +So the vectors ⃗na and ⃗nb are different (λ > 1), and their respective components are equal... relative to +different bases! And of course 1 = ||⃗na||2 +a = �n +i=1(nai)2 = �n +i=1(nbi)2 = ||⃗nb||2 +b = 1. +Answer. ⃗na(p) ∥ ⃗nb(p), since the vectors are Euclidean and orthogonal to TpΓ cf. (B.14). And ||.||a = λ||.||b +cf. (B.8), thus ||⃗nb||b = 1 = ||⃗na||a = λ||⃗na||b = ||λ⃗na||b, so ⃗nb = ±λ⃗na. And they both are outward vectors, so +⃗nb = +λ⃗na. Thus (⃗w,⃗na)a = λ2(⃗w,⃗na)b = λ2(⃗w, ⃗nb +λ )b = λ(⃗w,⃗nb)b. +And if ⃗bi = λ⃗ai (B.16) gives �n +i=1ni +b⃗bi = λ�n +i=1ni +a⃗ai = �n +i=1ni +a(λ⃗ai) = �n +i=1ni +a⃗bi, then ni +a = ni +b. +101 + +102 +B.7. +Integration by parts (Green–Gauss–Ostrogradsky) +B.6.3 +Unit normal form n♭ associated to ⃗n +(For mathematicians; May produce misunderstandings and lack of mechanical interpretations; Don’t +forget: n♭ is obtained after ⃗n has been defined.) +At p ∈ Γ, once you have computed ⃗ng(p), you can define the associated unit normal form n♭ +g(p) ∈ Rn∗: +It is the linear form defined by n♭ +g(p).⃗w := ⃗ng(p) •g ⃗w for all ⃗w ∈ ⃗Rn, i.e. on Γ, for all ⃗w ∈ ⃗Rn, +n♭ +g.⃗w := ⃗ng •g ⃗w +(B.18) +( =noted ⃗n • ⃗w if one chosen Euclidean dot product is imposed to all). Thus [n♭ +g].[⃗w] = [⃗ng]T .[g].[⃗w]. +Quantification: Let (⃗ei) be a basis in ⃗Rn; Then (B.18) gives [n♭ +g]|⃗e.[⃗w]|⃗e = [⃗ng]T +|⃗e.[g]|⃗e.[⃗w]|⃗e simply +written [n♭ +g].[⃗w] = [⃗ng]T .[g].[⃗w] if the basis (⃗ei) is imposed. +So, with duality notations to justify the ♭ notation, with (ei) the dual basis of (⃗ei), let +⃗ng = +n +� +i=1 +ni +g⃗ei +and +n♭ +g = +n +� +i=1 +ngiei. +(B.19) +i.e. ni +g and ngi are the components of ⃗ng and n♭ +g relative to the basis (⃗ei) and (ei). Since (B.18) gives +n♭ +g.⃗ei := ⃗ng •g ⃗ei for all i, we get, for all i, +nig = +n +� +j=1 +gijnj +g +(B.20) +Particular case (⃗ei) is a (·, ·)g-Euclidean basis, then nig = ni +g. +Classical notations: ⃗ng = �n +i=1(⃗ng)i⃗ei, dual basis (πei), n♭ +g = �n +i=1(n♭ +g)iπei, (n♭ +g)i = �n +j=1gij(⃗ng)j. +NB: In physics don’t forget to write the gij in (B.20) even if gij = δij, since you need to see the chosen +metric and basis (and verify the Einstein convention), although (B.20) is simply written ni = �n +j=1gijnj... +B.7 +Integration by parts (Green–Gauss–Ostrogradsky) +Let Ω be a regular bounded open set in Rn and Γ = ∂Ω its frontier, let ϕ ∈ C1(Ω; R), let (⃗ei) be a +Euclidean basis and (·, ·)g ites associated Euclidean dot product, let +∂ϕ +∂xi (p) := dϕ(p).⃗ei (usual notation), +let ⃗ng(p) = ⃗n(p) = �n +i=1ni(p)⃗ei (classical notations) be the unit outward normal at p ∈ Γ. Then, for +i = 1, ..., n, +� +p∈Ω +∂ϕ +∂xi +(p) dΩ = +� +p∈Γ +ϕ(p)ni(p) dΓ, +in short +� +Ω +∂ϕ +∂xi +dΩ = +� +Γ +ϕni dΓ. +(B.21) +Thus, for any v ∈ C1(Ω; R), with ϕv instead of ϕ in (B.21), we get the integration by parts formula +(Green formula): +� +Ω +∂ϕ +∂xi +v dΩ = − +� +Ω +ϕ ∂v +∂xi +dΩ + +� +Γ +ϕvni dΓ. +(B.22) +Thus, for any ⃗v ∈ C1(Ω; ⃗Rn) (vector field), with ⃗v(p) = �n +i=1vi(p)⃗ei) we get +� +Ω +∂ϕ +∂xi +vi dΩ = − +� +Ω +ϕ ∂vi +∂xi +dΩ + +� +Γ +ϕvini dΓ. +(B.23) +Thus, with the gradient vector +⃗ +gradϕ(p) = �n +i=1 +∂ϕ +∂xi⃗ei and with div⃗v = �n +i=1 +∂vi +∂xi , we get the Gauss– +Ostrogradsky formula: +� +Ω +⃗ +gradϕ • ⃗v dΩ = − +� +Ω +ϕ div⃗v dΩ + +� +Γ +ϕ⃗v • ⃗n dΓ. +(B.24) +(And +� +Γ ϕ⃗v • ⃗n dΓ gives the flux through Γ.) +Exercice B.12 Use the differential dϕ instead of the gradient +⃗ +gradϕ (which is the (·, ·)g-Riesz represen- +tation vector of dϕ) to express (B.23). Is the use of n♭ useful in that case? +Answer. +� +Ω dϕ.⃗v dΩ = − +� +Ω ϕ div⃗v dΩ + +� +Γ ϕ⃗v • ⃗n dΓ. Since n♭ depends on ⃗n (definition), there is no reason that +justifies the use of n♭ (unless you want to introduce useless notations here). +102 + +103 +C.1. +The symmetric and antisymmetric parts of d⃗v +C +Rate of deformation tensor and spin tensor +Let �Φ : [t1, t2] × Obj → Rn be a regular motion, cf. (1.5), and let ⃗v : C → ⃗Rn be the Eulerian velocity +field, cf. (2.4), that is, ⃗v(t, p) = ∂Φ +∂t (t, PObj) when p = �Φ(t, PObj). Its differential d⃗v is given in (2.8). +At t, an observer chooses a unit of measurement (foot, metre...) and builds the associated Euclidean +dot product (·, ·)g in ⃗Rn +t , cf. § B.2. (We loose the objective point of view here). And the same (·, ·)g is +used at all t. +C.1 +The symmetric and antisymmetric parts of d⃗v +With the imposed chosen Euclidean dot product (·, ·)g in ⃗Rn +t , we can consider the transposed endomor- +phism d⃗vt(p)T +g =noted d⃗vt(p)T ∈ L(⃗Rn +t ; ⃗Rn +t ), which is defined by, for all ⃗w1, ⃗w2 ∈ ⃗Rn +t vectors at p, +(d⃗vt(p)T .⃗w1, ⃗w2)g = (⃗w1, d⃗vt(p).⃗w2)g +(C.1) +cf. § A.11. We have thus defined +d⃗vT +t : +� +Ωt → L(⃗Rn +t ; ⃗Rn +t ) +p → d⃗vT +t (p) := d⃗vt(p)T +(C.2) +Other usual notations (definitions): d⃗vt(p)T =noted d⃗v(t, p)T =noted d⃗vT (t, p). +Definition C.1 The (Eulerian) rate of deformation tensor, or stretching tensor, is the (·, ·)g-symmetric +part of d⃗v: +D = d⃗v + d⃗vT +2 +, +i.e., +∀(t, p) ∈ +� +t∈R +({t} × Ωt), +D(t, p) = d⃗v(t, p) + d⃗v(t, p)T +2 +. +(C.3) +The (Eulerian) spin tensor is the (·, ·)g-antisymmetric part of d⃗v: +Ω = d⃗v − d⃗vT +2 +, +i.e., +∀(t, p) ∈ +� +t∈R +({t} × Ωt), +Ω(t, p) = d⃗v(t, p) − d⃗v(t, p)T +2 +. +(C.4) +(So d⃗v = D + Ω with D the rate of deformation tensor and Ω = ⃗ω∧ a rotation times a dilation, see the +following.) +NB: The same notation is used for the set of points Ωt = Φt0 +t (Ωt0) ⊂ Rn and for the spin tensor +Ωt = d⃗vt−d⃗vT +t +2 +: The context removes ambiguities. +C.2 +Quantification with a basis +With a basis (⃗ei) in ⃗Rn +t , (C.1) gives +[g]|⃗e.[d⃗vT ]|⃗e = [d⃗v]T +|⃗e.[g]|⃗e, +and +[d⃗vT ]|⃗e = [g]−1 +|⃗e .[d⃗v]T +|⃗e.[g]|⃗e. +(C.5) +In particular, if (⃗ei) is a (·, ·)g-orthonormal basis, then [d⃗vT ]|⃗e = [d⃗v]T +|⃗e (orthonormal basis case). Thus for +the endomorphisms D and Ω, and with the above Euclidean framework and its Euclidean orthonormal +basis, we have D.⃗ej = �n +i=1Dij⃗ei and Ω.⃗ej = �n +i=1Ωij⃗ei with Dij = 1 +2( ∂vi +∂xj + ∂vj +∂xi ) and Ωij = 1 +2( ∂vi +∂xj − ∂vj +∂xi ), +that is, +[D]|⃗e = +[d⃗v]|⃗e + [d⃗v]T +|⃗e +2 +and +[Ω]|⃗e = +[d⃗v]|⃗e − [d⃗v]T +|⃗e +2 +(Euclidean framework). +(C.6) +Duality notations: D.⃗ej = �n +i=1Di +j⃗ei, Di +j = 1 +2( ∂vi +∂xj + ∂vj +∂xi ) and Ω.⃗ej = �n +i=1Ωij⃗ei, Ωij = 1 +2( ∂vi +∂xj − ∂vj +∂xi ), so +with Di +j = Dj +i and Ωij = −Ωji. +103 + +104 +E.1. +Affine motions and rigid body motions +D +Interpretation of the rate of deformation tensor +We are interested in the evolution of the deformation gradient F(t) := F t0 +pt0 (t) along the trajectory of a +particle PObj which was at pt0 at t0. So: +Let ⃗A = ⃗a(t0, pt0) and ⃗B = ⃗b(t0, pt0) be vectors at t0 at pt0 in Ωt0, and consider their push-forwards +by the flow Φt0 +t (the transported vectors), i.e. the vectors at t at p(t) = Φt0 +pt0 (t) given by +⃗a(t, p(t)) := F(t). ⃗A +and +⃗b(t, p(t)) := F(t). ⃗B. +(D.1) +see (4.3) and figure 4.1. They define the function +(⃗a,⃗b)g : +� +C → R +(t, pt) → (⃗a,⃗b)g(t, pt) := (⃗a(t, pt),⃗b(t, pt))g. +(D.2) +Proposition D.1 The rate of deformation tensor D = d⃗v+d⃗vT +2 +gives (half) the evolution rate between +two vectors deformed by the flow, that is, along trajectories, +D(⃗a,⃗b)g +Dt += 2(D.⃗a,⃗b)g. +(D.3) +Proof. f(t) := (⃗a(t, p(t)),⃗b(t, p(t)))g = (F(t). ⃗A, F(t). ⃗B)g gives +f ′(t) = (F ′(t). ⃗A, F(t). ⃗B)g + (F(t). ⃗A, F ′(t). ⃗B)g. +(D.4) +And F ′(t) = d⃗v(t, p(t)).F(t), cf. (3.33). Thus, with ⃗a(t, p(t)) = F(t). ⃗A and ⃗b(t, p(t)) = F(t). ⃗B, +f ′(t) = (d⃗v(t, p(t)).F(t). ⃗A, F(t). ⃗B)g + (F(t). ⃗A, d⃗v(t, p(t)).F(t). ⃗B)g += (d⃗v(t, p(t)).⃗a(t, p(t)),⃗b(t, p(t)))g + (⃗a(t, p(t)), d⃗v(t, p(t)).⃗b(t, p(t)))g += ((d⃗v(t, p(t)) + d⃗v(t, p(t))T ).⃗a(t, p(t)),⃗b(t, p(t)))g, +(D.5) +i.e. (D.3), since f(t) = (⃗a,⃗b)g(t, p(t)) gives f ′(t) = D(⃗a,⃗b)g +Dt +(t, p(t)). +E +Rigid body motions and the spin tensor +Choose a Euclidean dot product (·, ·)g (required to characterize a rigid body motion). +Result: A rigid body motion is a motion whose Eulerian velocity satisfies d⃗v + d⃗vT = 0, i.e., D = 0 +(Eulerian approach independent of any initial time t0 chosen by some observer). +But the usual classical introduction to rigid body motion relies on some initial time t0 (Lagrangian +approach). So, to begin with, let us do it with the Lagrangian approach. Recall: T the first order Taylor +expansion of Φt0 +t in the vicinity of a pt0 ∈ Ωt0 is +Φt0 +t (qt0) = Φt0 +t (pt0) + F t0 +t (pt0).−−−→ +pt0qt0 + o(−−−→ +pt0qt0). +(E.1) +E.1 +Affine motions and rigid body motions +E.1.1 +Affine motions +Definition E.1 Φt0 is an affine motion (understood “affine motion in space”) iff Φt0 +t is an “affine motion”, +i.e. iff Φt0 +t is a C1 diffeomorphism (in space), and (E.1) reads, for all pt0, qt0 ∈ Ωt0 and all t ∈ [t1, t2], +Φt0 +t (qt0) = Φt0 +t (pt0) + F t0 +t (pt0).−−−→ +pt0qt0. +(E.2) +Marsden–Hughes notations: Φ(Q) = Φ(P) + F(P).−−→ +PQ. +Proposition E.2 and definition. If Φt0 is an affine motion, then F t0 +t (pt0) is independent of pt0, i.e., +for all t ∈]t1, t2[ and all pt0 ∈ Ωt0 and all qt0 ∈ Ωt0, +F t0 +t (pt0) = F t0 +t (qt0) noted += +F t0 +t . +(E.3) +And then dF t0 +t (pt0) = 0, i.e. d2Φt0 +t (pt0) = 0. And for all t ∈]t1, t2[, Φt is an affine motion: For all +τ ∈]t1, t2[ and all pt, qt ∈ Ωt, +Φt +τ(qt) = Φt +τ(pt) + F t +τ.−−→ +ptqt. +(E.4) +And �Φ is said to be an affine motion (understood “affine motion in space”). +104 + +105 +E.1. +Affine motions and rigid body motions +Proof. qt0 = pt0 + −−−→ +pt0qt0 gives Φt0 +t (qt0) = Φt0 +t (pt0 + −−−→ +pt0qt0) = Φt0 +t (pt0) + dΦt0 +t (pt0).−−−→ +pt0qt0, and, similarly, +Φt0 +t (pt0) = Φt0 +t (qt0 +−−−→ +qt0pt0) = Φt0 +t (qt0)+dΦt0 +t (qt0).−−−→ +qt0pt0. Thus (addition) Φt0 +t (qt0)+Φt0 +t (pt0) = Φt0 +t (pt0)+ +Φt0 +t (qt0) + (dΦt0 +t (pt0) − dΦt0 +t (qt0)).−−−→ +pt0qt0, thus (dΦt0 +t (pt0) − dΦt0 +t (qt0)).−−−→ +pt0qt0 = 0, true for all pt0, qt0, thus +dΦt0 +t (pt0) − dΦt0 +t (qt0) = 0, i.e. (E.3). +Thus d2Φt0 +t (pt0).⃗ut0 = limh→0 +dΦt0 +t (pt0+h⃗ut0)−dΦt0 +t (pt0) +h += limh→0 +dΦt0 +t −dΦt0 +t +h += 0 for all pt0 and all ⃗ut0, +thus d2Φt0 +t (pt0) = 0 for all pt0, thus d2Φt0 +t = 0. +And (5.17) gives (Φt +τ ◦ Φt0 +t )(pt0) = Φt0 +τ (pt0), thus, with pt = Φt0 +t (pt0), we get dΦt +τ(pt).dΦt0 +t (pt0) = +dΦt0 +τ (pt0), thus dΦt +τ(pt) = dΦt0 +τ (pt0).dΦt0 +t (pt0)−1, and (E.2) gives +dΦt +τ(pt) = dΦt0 +τ .dΦt0 +t +−1 noted += +dΦt +τ +(independent of pt), +(E.5) +thus (E.4). +Corollary E.3 If �Φ is affine then, ⃗vt is affine for all t, and ⃗V t0 +t +is affine for all t0, t, i.e., for all pt ∈ Ωt we +have d⃗vt(pt) = d⃗vt (independent of pt), and for all pt0 ∈ Ωt0 we have d⃗V t0 +t (pt0) =noted d⃗V t0 +t +(independent +of pt0): For all qt ∈ Ωt and all qt0 ∈ Ωt0, +� +• ⃗vt(qt) = ⃗vt(pt) + d⃗vt.−−→ +ptqt, +• ⃗V t0 +t (qt0) = ⃗V t0 +t (pt0) + d⃗V t0 +t .−−−→ +pt0qt0. +(E.6) +Proof. (E.2) gives Φt0(t, qt0) = Φt0(t, pt0) + F t0(t).−−−→ +pt0qt0, and the derivation in time gives (E.6)2, +then (E.6)1 thanks to pt = Φt0 +t (pt0), qt = Φt0 +t (qt0) and −−−→ +pt0qt0 = (F t0 +t )−1.−−→ +ptqt, cf. (E.2). +Example E.4 In R2, with a basis ( ⃗E1, ⃗E2) in ⃗Rn +t0 and a basis (⃗e1,⃗e2) ∈ ⃗Rn +t , then F t0 +t +given by [F t0 +t ]| ⃗E,⃗e = +� +1 + t +2t2 +3t3 +et +� +derives from the affine motion [−−−−−−−−−−−→ +Φt0 +t (pt0)Φt0 +t (qt0)]|⃗e = +� +1 + t +2t2 +3t3 +et +� +.[−−−→ +pt0qt0]| ⃗E. +E.1.2 +Rigid body motion +A Euclidean dot product (·, ·)g in ⃗Rn +t is chosen, the same at all time t. Let Φ := Φt0 +t +and F := F t0 +t . +Recall: If P ∈ Ωt0 and p = Φ(P) (∈ Ωt) then the transposed of the linear map F(P) ∈ L(⃗Rn +t0; ⃗Rn +t ) relative +to (·, ·)g is the linear map F T (p) := F(P)T ∈ L(⃗Rn +t ; ⃗Rn +t0) defined by +F T (p) := F(P)T : +� ⃗Rn +t +→ ⃗Rn +t0 +⃗wp → F T (p).⃗wp +s.t. +(F T (p).⃗wp, ⃗UP )g = (⃗wp, F(P).⃗UP )g, ∀⃗UP ∈ ⃗Rn +t0. +(E.7) +We have thus defined the function F T : Ωt → L(⃗Rn +t ; ⃗Rn +t0). +Particular case: For an affine motion, since F is independent of P, we get F T is independent of p. +Definition E.5 A rigid body motion is an affine motion �Φ such that, for all t0, t ∈ R, P ∈ Ωt0, ⃗UP , ⃗WP ∈ +⃗Rn +t0, and with p = Φt0 +t (P), +(F.⃗UP , F. ⃗WP )g = (⃗UP , ⃗WP )g, +i.e. +(F T .F.⃗UP , ⃗WP )g = (⃗UP , ⃗WP )g, +i.e. +F T .F = I . +(E.8) +(Angles and lengths are unchanged.) +In other words, with the Cauchy strain tensor C ∈ L(⃗Rn +t0; ⃗Rn +t0) defined by C = F T .F, the motion is +rigid iff it is affine and +C = I , +i.e. +F −1 = F T . +(E.9) +Proposition E.6 If Φt0 is a rigid body motion, if ( ⃗Ai) is a (·, ·)g-Euclidean basis in ⃗Rn +t0, if P ∈ Ωt0, if +t ∈ [t0, T] and p = Φt0 +t (P), and if ⃗ai(t, p) = F t0(t, P). ⃗Ai for all i, then ⃗ai(t, p) =noted ⃗ai,t is independent +of p, and (⃗ai,t) is a (·, ·)g-Euclidean basis with the same orientation than ( ⃗Ai) for all t. +105 + +106 +E.2. +Representation of the spin tensor Ω: vectors, and pseudo-vectors +Proof. Φt0 +t +is affine, thus, for all t, P, F t0 +t (P) = F t0 +t +(independent of P), thus ⃗ai,t(p) = F t0 +t . ⃗Ai ∈ ⃗Rn +t +is independent of p, this at all t. +Let t be fixed and ⃗ai,t =noted ⃗ai (= F. ⃗Aj). +We get (⃗ai,⃗aj)g = +(F. ⃗Ai, F. ⃗Aj)g = (F T .F. ⃗Ai, ⃗Aj)g = (I. ⃗Ai, ⃗Aj)g = ( ⃗Ai, ⃗Aj)g = δij for all i, j, thus (⃗ai) is (·, ·)g-orthonormal +basis. And det(⃗a1, ...,⃗an) = det(F. ⃗A1, ..., F. ⃗An) = det(F) det( ⃗A1, ..., ⃗An) = det(F) since ( ⃗Ai) is a (·, ·)g- +orthonormal basis. +And, Φt0 +t +being a diffeomorphism, t → det(F t0 +t ) is continuous, does not vanish, +moreover with det(F t0 +t0 ) = det(I) = 1 > 0; Thus det(F t0 +t ) > 0 for all t, hence det(⃗a1, ...,⃗an) > 0: The +bases have the same orientation. +Example E.7 In R2, a rigid body motion is given by F t0 +t += +� +cos(θ(t)) +− sin(θ(t)) +sin(θ(t)) +cos(θ(t)) +� +with θ a regular +function s.t. θ(t0) = 0. +Exercice E.8 Let �Φ be a rigid body motion. Prove +(F T )′(t) = (F ′(t))T , +and +F T .F ′ is antisymmetric. +(E.10) +Answer. +Let t ∈ R, p(t) = Φt0 +t (P), ⃗U, ⃗W ∈ ⃗Rn +t0 and ⃗w(t, p(t)) = F(t). ⃗W. +And recall that the function +F T : t → F T (t) is defined (as usual) by F T (t) := (F(t))T . We have (F(t)T .⃗w(t, p(t)), ⃗U)g = (⃗w(t, p(t)), F(t).⃗U)g. +Thus ((F T )′(t).⃗w(t, p(t))+F T (t). D ⃗w +Dt (t, p(t)), ⃗U)g = ( D ⃗w +Dt (t, p(t)), F(t).⃗U)g+(⃗w(t, p(t)), F ′(t).⃗U)g, which simplifies +into ((F T )′(t).⃗w(t, p(t)), ⃗U)g = (⃗w(t, p(t)), F ′(t).⃗U)g = ((F ′(t))T .⃗w(t, p(t)), ⃗U)g, thus (F T )′(t) = (F ′(t))T . +And (E.8) reads F T (t).F(t) = It0, thus (F T )′(t).F(t)+F T (t).F ′(t) = 0, thus (F ′)T (t).F(t)+F T (t).F ′(t) = 0, +thus F T .F ′ is antisymmetric. +E.1.3 +Alternative definition of a rigid body motion: d⃗v + d⃗vT = 0 +The stretching tensor Dt = d⃗vt+d⃗vT +t +2 +and the spin tensor Ωt = d⃗vt−d⃗vT +t +2 +have been defined in (C.3)-(C.4). +Proposition E.9 If �Φ is a rigid body motion, cf. (E.8), then the endomorphism d⃗vt ∈ L( ⃗ +Rn +t ; ⃗ +Rn +t ) is +antisymmetric at all t: +d⃗vt = Ωt, +i.e. +Dt = 0. +(E.11) +Conversely, if d⃗vt + d⃗vT +t = 0 at all t, then �Φ is a rigid body motion (here no initial time is required). +So the relation « d⃗vt + d⃗vT +t = 0 for all t » gives an equivalent definition to the definition E.5. +Proof. +Let F(t) +:= +F t0 +pt0 (t) and F T (t) +:= +F(t)T +and V (t) +:= +⃗V t0 +pt0 (t) += +(Φt0 +pt0 )′(t) += +⃗v(t, pt) (the Lagrangian and Eulerian velocities). +(E.8) gives (F.F T )′(t) = 0 = F ′(t).F(t)T + +F(t).(F T )′(t) +(E.10) += +F ′(t).F(t)T + (F ′(t).F(t)T )T += +dV (t).F(t)−1 + (dV (t).F(t)−1)T (3.27) += +d⃗v(t, pt) + +d⃗v(t, pt)T . Thus (E.11). +Conversely, suppose d⃗v + d⃗vT = 0. Then (D.3) gives D(⃗a,⃗b)g +Dt += 0, thus (⃗a,⃗b)g(t, pt) = (⃗a,⃗b)g(t0, pt0) +for all t, t0 and all pt0 = Φt0 +t (pt), i.e. (F t0 +t (pt0). ⃗A, F t0 +t (pt0). ⃗B)g = ( ⃗A, ⃗B)g for all t, t0, all pt0 and all +⃗A, ⃗B ∈ ⃗Rn +t0: Thus �Φ is a rigid body motion, cf (E.8). +E.2 +Representation of the spin tensor Ω: vectors, and pseudo-vectors +We are dealing here with concepts that are sometimes misunderstood. Framework: Rn = R3. +E.2.1 +Reminder +• The determinant det|⃗e associated with a basis (⃗ei) in R3 is the alternating multilinear form defined +by det|⃗e(⃗e1,⃗e2,⃗e3) = 1; The algebraic volume (or signed volume) limited by three vectors ⃗u1, ⃗u2, ⃗u3 is +det|⃗e(⃗u1, ⃗u2, ⃗u3); And the (positive) volume is | det|⃗e(⃗u1, ⃗u2, ⃗u3)|, see § K. +• Let A and B be two observers (e.g. A=English and B=French), let (⃗ai) be a Euclidean basis chosen +by A (e.g. based on the foot), let (⃗bi) be a Euclidean basis chosen by B (e.g. based on the metre), see § B.1. +106 + +107 +E.2. +Representation of the spin tensor Ω: vectors, and pseudo-vectors +Let λ = ||⃗b1||a > 0 (change of unit of length coefficient). The relation between the determinants is: +det +|⃗a = ±λ3 det +|⃗b +with +� +� +� +� +� ++ if det +|⃗a (⃗b1,⃗b2,⃗b3) > 0 +(i.e. if the bases have the same orientation), +− if det +|⃗a (⃗b1,⃗b2,⃗b3) < 0 +(i.e. if the bases have opposite orientation). +(E.12) +In particular, if A and B use the same unit of length (or if A uses two (·, ·)g-Euclidean basis (⃗ai) and (⃗bi)), +then λ = 1 and det|⃗a = ± det|⃗b. +• With an imposed Euclidean dot product (·, ·)g: An endomorphism L is (·, ·)g-antisymmetric iff +∀⃗u,⃗v, (L.⃗u,⃗v)g + (⃗u, L.⃗v)g = 0, +i.e. +LT = −L. +(E.13) +E.2.2 +Definition of the vector product (cross product) +Let (⃗ei) be a (·, ·)g-orthonormal basis, let ⃗u,⃗v ∈ ⃗R3, and let ℓ⃗e,⃗u,⃗v ∈ L( ⃗R3, R) be the linear form defined +by +ℓ⃗e,⃗u,⃗v : +� +� +� +⃗R3 → R +⃗z → ℓ⃗e,⃗u,⃗v(⃗z) := det +|⃗e (⃗u,⃗v, ⃗z) +(E.14) +(the algebraic volume of the parallelepiped limited by ⃗u,⃗v, ⃗z in the Euclidean chosen unit). +Definition E.10 The vector product, or cross product, ⃗u ∧e ⃗v of two vectors ⃗u and ⃗v is the (·, ·)g-Riesz +representation vector of ℓ⃗e,⃗u,⃗v, that is, ⃗u ∧e ⃗v ∈ ⃗R3 is characterized by ℓ⃗e,⃗u,⃗v(⃗z) = (⃗u ∧e ⃗v, ⃗z)g for all +⃗z ∈ ⃗R3, cf. (F.2), i.e. +∀⃗z ∈ ⃗R3, +(⃗u ∧e ⃗v, ⃗z)g = det +|⃗e (⃗u,⃗v, ⃗z) . +(E.15) +NB: ⃗u ∧e ⃗v depends on (·, ·)g since we need a (·, ·)g-Euclidean basis (⃗ei) (and depends on the orientation +of (⃗ei). +We have thus defined the bilinear cross product operator +∧e : +� ⃗R3 × ⃗R3 → ⃗R3 +(⃗u,⃗v) → ∧e(⃗u,⃗v) := ⃗u ∧e ⃗v. +(E.16) +(The bilinearity is trivial thanks to the multilinearity of the determinant.) If one Euclidean basis is +imposed by one observer to all the other observers, then ⃗u ∧e ⃗v is written ⃗u ∧ ⃗v (non objective). +E.2.3 +Calculation of the vector product +⃗u = �3 +i=1 ui⃗ei, ⃗v = �3 +i=1 vi⃗ei and (E.15) give +(⃗u ∧e ⃗v,⃗e1)g = det +|⃗e (⃗u,⃗v,⃗e1) = det +� +� +u1 +v1 +1 +u2 +v2 +0 +u3 +v3 +0 +� +� = det +� +u2 +v2 +u3 +v3 +� += u2v3 − u3v2. +(E.17) +Similar calculation for (⃗u ∧e ⃗v,⃗e2)e and (⃗u ∧e ⃗v,⃗e3)e, thus +⃗u ∧e ⃗v = +3 +� +i=1 +(ui+1vi+2 − ui+2vi+1)⃗ei, +i.e. +[⃗u ∧e ⃗v]|⃗e = +� +� +u2v3 − u3v2 +u3v1 − u1v3 +u1v2 − u2v1 +� +� . +(E.18) +with the generic notation w4 := w1 and w5 = w2. (In particular ⃗ei ∧e ⃗ei+1 = ⃗ei+2.) +Proposition E.11 1- ⃗u ∧e ⃗v = −⃗v ∧e ⃗u. +2- ⃗u ∥ ⃗v iff ⃗u ∧e ⃗v = 0. +3- If ⃗u and ⃗v are independent then ⃗u ∧e ⃗v is orthogonal to the linear space Vect{⃗u,⃗v} generated by ⃗u +and ⃗v. +4- ⃗u ∧e ⃗v depends on the unit of measurement and on the orientation of (⃗ei): If (·, ·)a and (·, ·)b are +two Euclidean dot products, let λ > 0 such that (·, ·)a = λ2(·, ·)b, and then +⃗u ∧a ⃗v = ±λ⃗u ∧b ⃗v. +(E.19) +107 + +108 +E.2. +Representation of the spin tensor Ω: vectors, and pseudo-vectors +Proof. 1- det|⃗e(⃗u,⃗v, ⃗z) = − det|⃗e(⃗v, ⃗u, ⃗z) (since det|⃗e is alternated). +2- If ⃗u ∥ ⃗v then det|⃗e(⃗u,⃗v, ⃗z) = 0 = (⃗u ∧e ⃗v, ⃗z)e, so ⃗u ∧e ⃗v ⊥g ⃗z, for all ⃗z. And if ⃗u ∧e ⃗v = 0 then (E.18) +gives ⃗u ∥ ⃗v. +3- If ⃗z ∈ Vect{⃗u,⃗v} then det|⃗e(⃗u,⃗v, ⃗z) = 0 = (⃗u ∧e ⃗v,⃗z)g thus ⃗u ∧e ⃗v ⊥g ⃗z. +4- (⃗u ∧a ⃗v, ⃗z)a +(E.15) += +det +|⃗a (⃗u,⃗v, ⃗z) +(E.12) += +±λ3 det +|⃗b +(⃗u,⃗v, ⃗z) +(E.15) += +±λ3(⃗u ∧b ⃗v, ⃗z)b = ±λ3 1 +λ2 (⃗u ∧b ⃗v, ⃗z)a, true +for all ⃗z, thus (E.19). +Exercice E.12 Prove that ⃗u ∧e ⃗v is a contravariant vector. +Answer. It is a vector (Riesz representation vector) in ⃗R3, so it is contravariant; Or calculation: It satisfies the +contravariance change of basis formula, see (F.17). +E.2.4 +Antisymmetric endomorphism represented by a vector +Proposition E.13 Let (⃗ei) be a chosen (·, ·)g-Euclidean basis. If an endomorphism Ω ∈ L( ⃗R3; ⃗R3) is +(·, ·)g-antisymmetric then there exists a unique vector ⃗ωe ∈ ⃗R3 s.t., for all ⃗y, ⃗z ∈ ⃗R3, +(Ω.⃗y, ⃗z)g = det +|⃗e (⃗ωe, ⃗y,⃗z), +(E.20) +i.e., there exists a unique vector ⃗ωe ∈ ⃗R3 s.t., for all ⃗y, ⃗z ∈ ⃗R3, +Ω.⃗y = ⃗ωe ∧e ⃗y , +(E.21) +And +[Ω]|⃗e = +� +� +0 +−c +b +c +0 +−a +−b +a +0 +� +� +iff +[⃗ωe]|⃗e = +� +� +a +b +c +� +� . +(E.22) +In particular Ω.⃗ωe = ⃗0 (= ⃗ωe ∧e ⃗ωe), i.e. ⃗ωe is an eigenvector associated with the eigenvalue 0. +Proof. Ω is antisymmetric, thus [Ω]|⃗e is given as in (E.22). In particular [Ω.⃗e1]|⃗e = [Ω]|⃗e.[⃗e1]|⃗e = +� +� +0 +c +−b +� +�. +Calculation of the components of ⃗ωe if it exists: Let ⃗ω = ω1⃗e1 + ω2⃗e2 + ω3⃗e3; thus [⃗ω ∧ ⃗e1]|⃗e = +� +� +0 +ω3 +−ω2 +� +�, +cf. (E.18), thus ω3 = c and ω2 = b; Idem with ⃗e2 so that ω1 = a. Thus if it exists ⃗ω is unique. And ⃗ωe +given in (E.22) satisfies (E.21): It exists. +Proposition E.14 Let (·, ·)a and (·, ·)b be two Euclidean dot products (e.g. in foot and metre), let (⃗ai) +and (⃗bi) be Euclidean associated bases, let ||⃗b1||a = λ (change of unit coefficient), so (·, ·)a = λ2(·, ·)b). +Suppose [Ω]|⃗a = +� +� +0 +−c +b +c +0 +−a +−b +a +0 +� +�, thus [⃗ωa]|⃗a = +� +� +a +b +c +� +�, cf. (E.22). +Then (change of representation +vector for Ω): +• If (⃗bi) and (⃗ai) have the same orientation, then +⃗ωb = λ⃗ωa, +• If (⃗bi) and (⃗ai) have opposite orientation, then +⃗ωb = −λ⃗ωa, +(E.23) +E.g., if ⃗bi = λ⃗ai for all i (change of unit, same orientation) then ⃗ωb = λ⃗ωa, and if ⃗b1 = −λ⃗a1, ⃗b2 = λ⃗a2, +⃗b3 = λ⃗a3 (change of unit, opposite orientation) then ⃗ωb = −λ⃗ωa. +NB: The formula ⃗ωb = ±λ⃗ωa is a change of vector formula, not a change of basis formula. +Proof. Apply (E.19). +Interpretation of ⃗ωe: Suppose [Ω]|⃗e = α +� +� +0 +−1 +0 +1 +0 +0 +0 +0 +0 +� +�. So Ω is the rotation with angle +π +2 in the +horizontal plane composed with the dilation with ratio α. And [⃗ωe]|⃗e = α +� +� +0 +0 +1 +� +� = α⃗e3 is orthogonal to +the horizontal plane and gives the rotation axis and the dilation coefficient. +108 + +109 +E.3. +Pseudo-cross product, and pseudo-vector +Exercice E.15 Let Ω s.t. [Ω]|⃗e = +� +� +0 +−c +b +c +0 +−a +−b +a +0 +� +� (see (E.22)). Find a direct orthonormal basis (⃗bi) +(relative to (⃗ei)) s.t. [Ω]|⃗b = +√ +a2+b2+c2 +� +� +0 +−1 +0 +1 +0 +0 +0 +0 +0 +� +�. +Answer. Let ⃗b3 = +⃗ωe +||⃗ωe||e , that is, [⃗b3]|⃗e = +1 +√ +a2+b2+c2 +� +� +a +b +c +� +�. Then choose ⃗b1 ⊥ ⃗b3, e.g. [⃗b1]|⃗e = +1 +√ +a2+b2 +� +� +−b +a +0 +� +�. +Then choose ⃗b2 = ⃗b3 ∧e ⃗b1, that is, [⃗b2]|⃗e = +1 +√ +a2+b2 +1 +√ +a2+b2+c2 +� +� +−ac +−bc +a2 + b2 +� +�. Thus (⃗bi) is a direct orthonormal +basis, and the transition matrix from (⃗ei) to (⃗bi) is P = +� +[⃗b1]|⃗e +[⃗b2]|⃗e +[⃗b3]|⃗e +� +. And [Ω]|⃗b = P −1.[Ω]|⃗e.P (change +of basis formula), with P −1 = P T (change of orthonormal basis), thus [Ω]|⃗b = P T .[Ω]|⃗e.P +With [Ω]|⃗e.[⃗b1]|⃗e = +1 +√ +b2+c2 +� +� +0 +−c +b +c +0 +−a +−b +a +0 +� +� . +� +� +−b +a +0 +� +� = +1 +√ +b2+c2 +� +� +−ac +−bc +a2 + b2 +� +� = +√ +a2+b2+c2[⃗b2]|⃗e (expected), +[Ω]|⃗e.[⃗b2]|⃗e = +1 +√ +b2+c2 +1 +√ +a2+b2+c2 +� +� +0 +−c +b +c +0 +−a +−b +a +0 +� +� . +� +� +−ac +−bc +a2 + b2 +� +� = +1 +√ +b2+c2 +1 +√ +a2+b2+c2 +� +� +bc2 + b(a2 + b2) +−ac2 − a(a2 + b2) +abc − abc +� +� = +− +√ +a2+b2+c2[⃗b1]|⃗e +(expected), +and +[Ω]|⃗e.[⃗b3]|⃗e += +[⃗0] +(expected +since ⃗b3 +∥ +⃗ωe). +Thus +[Ω]|⃗e.P += +√ +a2+b2+c2 � +[⃗b2]|⃗e +−[⃗b1]|⃗e +[⃗0]|⃗e +� +. And (P T .[Ω]|⃗e.P)ij = [⃗bi]T +|⃗e.[Ω]|⃗e.[⃗bj]|⃗e gives the result. +E.2.5 +Curl +Definition E.16 If ⃗v is a C1 vector field, if (⃗ei) is a Euclidean basis in ⃗R3, and if ⃗v = �3 +i=1 vi⃗ei, then +the curl (or rotational) of ⃗v relative to (⃗ei) is the vector field +⃗ +curle⃗v = ⃗ +rote⃗v given by +⃗ +curle⃗v = +3 +� +i=1 +( ∂vi+2 +∂xi+1 +− ∂vi+1 +∂xi+2 +)⃗ei, +i.e. +[ ⃗ +curle⃗v]|⃗e = +� +� +∂v3 +∂x2 − ∂v2 +∂x3 +∂v1 +∂x3 − ∂v3 +∂x1 +∂v2 +∂x1 − ∂v1 +∂x2 +� +� . +(E.24) +Proposition E.17 Let Ω(t, pt) = d⃗v(t,pt)−d⃗v(t,pt)T +2 +, and let ⃗ωe(t, pt) be the associated vector relative to +the Euclidean basis (⃗ei), cf. (E.21). Then +⃗ωe = 1 +2 +⃗ +curle⃗v. +(E.25) +Proof. (C.6) gives [Ω]|⃗e = +1 +2 +� +� +0 +∂v1 +∂x2 − ∂v2 +∂x1 +∂v1 +∂x3 − ∂v3 +∂x1 +· +0 +∂v2 +∂x3 − ∂v3 +∂x2 +· +· +0 +� +�, with [Ω]|⃗e antisymmetric. +Thus (E.22), +(E.18) and (E.24) gives (E.25). +E.3 +Pseudo-cross product, and pseudo-vector +Framework: M31 the space of 3∗1 matrices, so we leave the vector framework to enter the matrix world. +E.3.1 +Definition +Definition E.18 A column matrice is also called a pseudo-vector, or a column vector. +Definition E.19 +� +� +x1 +x2 +x3 +� +� noted += +[⃗x] and +� +� +y1 +y2 +y3 +� +� noted += +[⃗y] being two matrices in M31, their pseudo-cross +product is +� +� +x1 +x2 +x3 +� +� ⟲∧ +� +� +y1 +y2 +y3 +� +� := +� +� +x2y3 − x3y2 +x3y1 − x1y3 +x1y2 − x2y1 +� +� noted += +[⃗x] +⟲∧[⃗y]. +(E.26) +Thus the pseudo-cross product of two pseudo-vectors is a pseudo-vector (is a matrix). +109 + +110 +E.4. +Examples +E.3.2 +Antisymmetric matrix represented by a pseudo-vector +Definition E.20 Let A = [Aij] = +� +� +0 +−c +b +c +0 +−a +−b +a +0 +� +� be an antisymmetric matrix (Aji = −Aij for all +i, j). The pseudo-vecteur +⟲ω associated to A is the column matrix +⟲ω := +� +� +a +b +c +� +� . So, +A.[⃗y] = +⟲ω +⟲∧[⃗y] , +i.e. +A. +� +� +y1 +y2 +y3 +� +� = +⟲ω +⟲∧ +� +� +y1 +y2 +y3 +� +� , +for all matrix [⃗y] = +� +� +y1 +y2 +y3 +� +� . +(E.27) +E.3.3 +Antisymmetric endomorphism and its pseudo-vectors representations +Let R3 be our usual affine space, (·, ·)g be a Euclidean dot product, and (⃗ei) be a (·, ·)g-Euclidean +associated basis. Let Ω be an antisymmetric endomorphism relative to (·, ·)g, so ΩT = −Ω, cf. (E.13). +Thus [Ω]|⃗e is an antisymmetric matrix. Call +⟲ω the associated pseudo-vector, i.e., cf. (E.27), for all ⃗y ∈ ⃗R3, +[Ω]|⃗e.[⃗y]|⃗e = +⟲ω +⟲∧[⃗y]|⃗e. +(E.28) +This formula is widely used in mechanics, and unfortunately sometimes noted Ω.⃗y = ⃗ω ∧ ⃗y (!): +Be careful: (E.28) is not a vectorial formula; This is just a formula for matrix calculations which +gives false result if a change of basis is considered; E.g., with (⃗a1,⃗a2,⃗a3) be a (·, ·)g-Euclidean basis, and +(⃗b1,⃗b2,⃗b3) = (−⃗a1,⃗a2,⃗a3). So (⃗bi) is also a (·, ·)g-Euclidean basis, but with a different orientation. +1- Vector approach: Let P be the transition matrix from (⃗ai) to (⃗bi), so P = +� +� +−1 +0 +0 +0 +1 +0 +0 +0 +1 +� +�. Let +[Ω]|⃗a = +� +� +0 +−c +b +c +0 +−a +−b +a +0 +� +�. Thus, Ω being an endomorphism, the change of basis formula gives +[Ω]|⃗b = P −1.[Ω]|⃗a.P = +� +� +−1 +0 +0 +0 +1 +0 +0 +0 +1 +� +� . +� +� +0 +−c +b +c +0 +−a +−b +a +0 +� +� . +� +� +−1 +0 +0 +0 +1 +0 +0 +0 +1 +� +� = +� +� +0 +c +−b +−c +0 +−a +b +a +0 +� +� . +(E.29) +Thus the vectors ⃗ωa and ⃗ωb are given by (E.22): +[⃗ωa]|⃗a = +� +� +a +b +c +� +� , +[⃗ωb]|⃗b = +� +� +a +−b +−c +� +� , +i.e. +� +⃗ωa = a⃗a1 + b⃗a2 + c⃗a3, +⃗ωb = a⃗b1 − b⃗b2 − c⃗b3, +� +thus +⃗ωb = −⃗ωa . +(E.30) +2- Matrix approach (E.27) gives [Ω]|⃗a.[⃗y] = +⟲ωa +⟲∧[⃗y] and [Ω]|⃗b.[⃗y] = +⟲ωb +⟲∧[⃗y], with +⟲ωa = +� +� +a +b +c +� +� +and +⟲ωb = +� +� +a +−b +−c +� +� , +so +⟲ωa ̸= − +⟲ωb . +(E.31) +And +⟲ω does not represent a single vector either, since it does not satisfy the vector change of basis formula +⟲ωb ̸= P −1. +⟲ωa. Thus +⟲ω is not a vector (is not tensorial): It is just a matrix (called a “pseudo-vector”). +E.4 +Examples +E.4.1 +Rectilinear motion +Let �Φ : [t1, t2] × Obj → Rn be a C1 motion. Let t0 ∈]t1, t2[ and PObj ∈ Obj. +110 + +111 +E.4. +Examples +Definition E.21 The motion of PObj is rectilinear iff, for all t0, t ∈ [t1, t2], +�ΦPObj (t) − �ΦPObj (t0) +t−t0 +∥ �ΦPObj +′(t0). +(E.32) +And the motion is rectilinear uniform iff, for all t0, t ∈ [t1, t2], +�ΦPObj (t) = �ΦPObj (t0) + (t−t0) �ΦPObj +′(t0), +i.e. +p(t) = p(t0) + (t−t0) ⃗V t0(t0, p(t0)) +(E.33) +when p(t) = �Φ(t, PObj), that is, the trajectory is traveled at constant velocity. +E.4.2 +Circular motion +Let ( ⃗E1, ⃗E2) be a Euclidean basis. Let t0 ∈ [t1, t2]. A motion Φt0 is a circular motion iff +−−−−−→ +OΦt0 +P (t) = x(t) ⃗E1 + y(t) ⃗E2, +[−−−−−→ +OΦt0 +P (t)]| ⃗E = +� +x(t) = a + R cos(θ(t)) +y(t) = b + R sin(θ(t)) +� +, +(E.34) +for some R > 0 (called the radius), some a, b ∈ R, and some function θ : R → R. And +� +a +b +� += OC ∈ R2 +is the center of the circle and θ(t) is the angle at t. And the particle PObj (s.t. �Φ(t0, PObj) = P) stays on +the circle with center OC and radius R. +The circular motion is uniforme iff, for all t, θ′′(t) = 0, that is, ∃ω0 ∈ R, ∀t ∈ [t1, t2], θ(t) = ω0t. +Notation: +⃗ϕt0 +P (t) = R cos(θ(t) ⃗E1 + R sin(θ(t)) ⃗E2, +so +[⃗ϕt0 +P (t)]| ⃗E = +� +R cos(θ(t)) +R sin(θ(t)) +� +. +(E.35) +Thus the Lagrangian velocity of a circular motion is +⃗V t0 +P (t) = (Φt0 +t )′(t) = (⃗ϕt0 +P )′(t), +so +[⃗V t0 +P (t)]| ⃗E = Rθ′(t) +� +− sin(θ(t)) +cos(θ(t)) +� +, +(E.36) +and ⃗V t0 +P (t) is orthogonal to ⃗ϕt0 +P (t) (the radius vector). And the Lagrangian acceleration is +⃗Γ t0 +P (t) = Rθ′′(t) +� +− sin(θ(t)) +cos(θ(t)) +� ++ R(θ′(t))2 +� +− cos(θ(t)) +− sin(θ(t)) +� +. +(E.37) +Consider +⃗er(t) = +⃗ϕt0 +P (t) +||⃗ϕt0 +P (t)|| = +� +cos(θ(t)) +sin(θ(t)) +� +, +and +⃗eθ(t) = +� +− sin(θ(t)) +cos(θ(t)) +� +, +(E.38) +thus (⃗er(t),⃗eθ(t)) is an orthonormal basis. Then : +⃗V t0 +P (t) = Rθ′(t)⃗eθ(t), +(E.39) +and : +⃗Γ t0 +P (t) = −R(θ′(t))2 ⃗er(t) + Rθ′′(t)⃗eθ(t). +(E.40) +E.g., in R3 and a motion in he “horizontal” plane given by (⃗e1,⃗e2), the vertical line being given by ⃗E3. +Here +⃗V t0 +P (t) = ⃗ω(t) ∧ ⃗ϕt0 +P (t), +where +⃗ω(t) = ω(t)⃗e3 +and +ω(t) = θ′(t). +(E.41) +And +⃗Γ t0(t) = d⃗ω +dt (t) ∧ ⃗ϕt0 +P (t) + ⃗ω(t) ∧ ⃗V t0 +P (t) +(= Rdω +dt (t)⃗eθ(t) − ω2(t)R⃗er(t)). +(E.42) +111 + +112 +E.4. +Examples +E.4.3 +Motion of a planet (centripetal acceleration) +Illustration: Obj is e.g. a planet from the solar system. +Let (⃗e1,⃗e2,⃗e3) be a Euclidean basis (e.g. fixed relative to stars an (⃗e1,⃗e2) define the ecliptic plane), +(·, ·)g be the Euclidean associated dot product, ||.|| the Euclidean associated norm, O an origin in R3 +(e.g. the center of the Sun), and R = (O, (⃗ei)). +Consider a motion �Φ of Obj in R, cf. (1.5). Let t0 ∈ [t1, t1], and consider Φt0 =noted Φ or ⃗ϕ t0 =noted ⃗ϕ, +cf. (3.1)-(3.4). +Definition E.22 The motion of a particle PObj is a centripetal acceleration motion iff the particle is not +static and, at all time, its acceleration vector ⃗A(t) points to a fixed point F (focus). +We will take the focus F as the origin of the referential, that is, O := F. +Thus, for all t ∈ [t1, t2], −−−−−→ +OΦP (t) ∥ ⃗AP (t), that is, +−−−−−→ +OΦP (t) ∧ ⃗AP (t) = ⃗0. +(E.43) +Remark E.23 A rectilinear motion is a centripetal acceleration motion, but such a motion is usually +excluded in the definition E.22. +Example E.24 The motion of a planet from the solar system is a centripetal acceleration motion: An +elliptical motion of focus the center of the Sun. +Example E.25 The second Newton’s law of motion � ⃗f = m⃗γ (Galilean referential) gives: If � ⃗f is, +at all time, directed to a unique point F, then the motion is a centripetal acceleration motion. +Let Φ be a centripetal acceleration motion, let O be the focus, and let ⃗ϕP (t) := −−−−−→ +OΦP (t). So the +Lagrangian velocity and acceleration are +⃗VP (t) = dΦP +dt (t) = d⃗ϕP +dt (t), +and +⃗AP (t) = d2ΦP +dt2 (t) = d2⃗ϕP +dt2 (t), +(E.44) +and ⃗ϕP (t) ∧ ⃗AP (t) = ⃗0, cf. (E.43). +Definition E.26 The areolar velocity at t is the vector +⃗Z(t) = 1 +2 ⃗ϕP (t) ∧ ⃗VP (t). +(E.45) +Proposition E.27 If Φ is a centripetal acceleration motion, then the areolar velocity is contant, that is, +d⃗Z +dt (t) = ⃗0 pour tout t, so +⃗Z(t) = ⃗Z(t0), +∀t. +(E.46) +That is, the position vectors sweep equal areas in equal times. And ⃗Z(t0) = ⃗0 iff Φ is a rectilinear motion. +If ⃗Z(t0) ̸= ⃗0 then : +- ⃗ϕP (t) and ⃗VP (t) are orthogonal to ⃗Z(t0) at all time t, +- The motion of the particle PObj takes place in the affine plane orthogonal to ⃗Z(t0) passing through O. +- ⃗VP (t) never vanishes. +Proof. (E.45) and (E.43) give 2 d⃗Z +dt (t) = d⃗ϕP +dt (t)∧⃗VP (t)+⃗ϕ(t)∧ d⃗VP +dt (t) = ⃗VP (t)∧⃗VP (t)+⃗ϕ(t)∧ ⃗AP (t) = ⃗0+⃗0. +Thus ⃗Z is constant, ⃗Z(t) = ⃗Z(t0) for all t. Then, if ⃗Z(t0) ̸= ⃗0 then ⃗Z(t) ̸= ⃗0 pour tout t, and +• ⃗Z(t) = 1 +2 ⃗ϕP (t) ∧ ⃗VP (t) gives that ⃗ϕP (t) et ⃗VP (t) are orthogonal to ⃗Z(t0) for all t, thus ⃗AP (t) is +orthogonal to ⃗Z(t0), cf. (E.43). +• The Taylor expansion reads ⃗ϕP (t) = ⃗ϕP (t0) + ⃗VP (t0)(t−t0) + +� t +τ=t0 ⃗AP (τ)(t−τ)2 dτ, with ⃗VP (t0) +and ⃗AP (τ) ⊥ ⃗Z(t0) for all τ, thus ⃗ϕP (t) − ⃗ϕP (t0) ⊥ ⃗Z(t0) for all τ, that is −−−→ +Op(t) − −−→ +OP = −−−→ +Pp(t) ⊥ ⃗Z(t0) +for all τ, Thus p(t) belongs to the affine plane containing P orthogonal to ⃗Z(t0), for all t. And −−→ +OP = +⃗ϕP (t0) ⊥ ⃗Z(t0), thus O belong to the same plane. +• ⃗Z(t) = ⃗Z(t0) ̸= ⃗0 implies ⃗VP (t) ̸= ⃗0 for all t, and (E.45) gives: (⃗ϕP (t), ⃗VP (t), ⃗Z(t0)) is a positively- +oriented basis. Since ⃗ϕP and ⃗V are continuous and do not vanish, since ⃗Z(t0) ̸= ⃗0, we get: PObj “turns +around ⃗Z(t0)” and keeps its direction. +112 + +113 +E.4. +Examples +If ⃗Z(t) = ⃗0 then ⃗ϕP (t) ∥ ⃗VP (t) for all t, cf. (E.45), so ⃗VP (t) = f(t)⃗ϕP (t) where f is some scalar +function. And ⃗VP (t) = ⃗ϕP ′(t) gives ⃗ϕP ′(t) = f(t)⃗ϕP (t), thus ⃗ϕP (t) = ⃗ϕP (t0)eF (t) where F is a primitive +of f s.t. F(t0) = 0, thus ⃗ϕP (t) ∥ ⃗ϕP (t0), so −−−−−→ +OΦP (t) ∥ −−−−−−→ +OΦP (t0), for all t: The motion is rectilinear. +Interpretation. (Non rectilinear motion.) +The area swept by ⃗ϕP (t) is, at first order, the area of the +triangle whose sides are ⃗ϕP (t) and ⃗ϕP (t + τ) (“anglular sector”). So, with τ close to 0, let +⃗St(τ) = 1 +2 ⃗ϕP (t) ∧ ⃗ϕP (t + τ), +and +St(τ) = ||⃗St(τ)||, +(E.47) +the vectorial an scalar area. With ⃗ϕP (t+τ) = ⃗ϕP (t) + ⃗VP (t)τ + o(τ) we get +⃗St(τ) = 1 +2 ⃗ϕP (t) ∧ (⃗VP (t)τ + o(τ)), +(E.48) +Since ⃗St(0) = 0 we get +⃗St(τ)−⃗S(0) +τ += 1 +2 ⃗ϕP (t) ∧ ⃗VP (t) + o(1), then +d⃗St +dτ (0) = 1 +2 ⃗ϕP (t) ∧ ⃗VP (t) = ⃗Z(t) = ⃗Z(t0), +(E.49) +thanks to (E.46), thus +d⃗St +dτ (0) = d⃗St0 +dτ (0), +∀t ∈ [t0, T], +(E.50) +that is, the rate of variation of ⃗St is constant. And with ||⃗St(∆τ)||2 = (⃗St(∆τ), ⃗St(∆τ)) we get +d||⃗St||2 +dτ +(∆τ) = 2(d⃗St +dτ (∆τ), ⃗St(∆τ)), +(E.51) +so, since ⃗St(0) = 0, +d||⃗St||2 +dτ +(0) = 0. +(E.52) +Therefore the function t → ||⃗St(0)||2 = St(0)2 is constant, thus t → St(0) est constant, and dSt +dτ (0) is +constant. +Exercice E.28 Give a parametrization of the swept area, and redo the calculations. +Answer. Let +r(t) = ||⃗ϕP (t)||, +θ(t) = � +p(t)OP +(angle), +(E.53) +then +⃗ϕP (t) = +� +� +r(t) cos(θ(t)) +r(t) sin(θ(t)) +0 +� +� . +(E.54) +Thus +⃗VP (t) = +� +� +r′(t) cos(θ(t) − r(t))θ′(t) sin(θ(t)) +r′(t) sin(θ(t) + r(t))θ′(t) cos(θ(t)) +0 +� +� . +(E.55) +With (E.45) we get +⃗Z(t) = 1 +2 +� +� +0 +0 +r2(t)θ′(t) +� +� , +with +r2(t)θ′(t) = r2(t0)θ′(t0) +(constant), +(E.56) +cf. (E.46). A parametrization of the swept area is then +⃗A : +� +[0, 1] × [t0, T] → R3 +(ρ, t) → ⃗A(ρ, t) +� +, +⃗A(ρ, t) = +� +� +ρ r(t) cos(θ(t)) +ρ r(t) sin(θ(t)) +0 +� +� . +(E.57) +Therefore, the tangent associated vectors are +∂ ⃗A +∂ρ (ρ, t) = +� +� +r(t) cos(θ(t)) +r(t) sin(θ(t)) +0 +� +� , +∂ ⃗A +∂t (ρ, t) = +� +� +ρr′(t) cos(θ(t) − ρr(t))θ′(t) sin(θ(t)) +ρr′(t) sin(θ(t) + ρr(t))θ′(t) cos(θ(t)) +0 +� +� , +(E.58) +113 + +114 +E.4. +Examples +hence the vectorial and scalare element areas are +d⃗σ = (∂ ⃗A +∂ρ ∧ ∂ ⃗A +∂t )dρdt = +� +� +0 +0 +ρr2θ′ dρdt +� +� , +dσ = ρr2θ′ dρdθ. +(E.59) +Therefore the area between t0 and t is +A(t) = A(t0) + +� 1 +ρ=0 +� t +τ=t0 +ρr2(τ)θ′(τ) dρdτ = 1 +2 +� t +τ=t0 +r(τ)2θ′(τ) dτ. +(E.60) +Hence +A′(t) = r(t)2θ′(t) = r(t0)2θ′(t0) +(= constant = ||⃗Z(t0)||), +(E.61) +cf. (E.56). +Exercice E.29 Prove the Binet formulas (non rectilinear central motion): +VP (t)2 = Z2 +0 +� 1 +r2 + (d 1 +r +dθ )2� +(t), +⃗ΓP (t) = −Z2 +0 +r2 +�1 +r + d2 1 +r +dθ2 +� +(t)⃗er(t), +(E.62) +for the energy and the acceleration. +Answer. +Proposition E.27 tells that Φ is a planar motion. +With (E.53) and ⃗er(t) = +� cos(θ(t)) +sin(θ(t)) +� +we have +⃗ϕ(t) = r(t)⃗er(t) (in the plane). Let ⃗eθ(t) = +� − sin(θ(t)) +cos(θ(t)) +� +, thus +⃗V (t) = dr +dt (t)⃗er(t) + r(t)d⃗er +dt (t) = r′(t)⃗er(t) + r(t)θ′(t)⃗eθ(t). +And ⃗er(t) ⊥ ⃗eθ(t) gives +V 2(t) = (r′(t))2 + (r(t)θ′(t))2. +Since θ′(t) ̸= 0 for all t (non rectilinear central motion) Let s(θ(t)) = r(t). Let us suppose that θ is C1, thus +θ′ > 0 or θ′ < 0, and θ : t → θ(t) defines a change of variable. And +r′(t) = s′(θ(t))θ′(t). +And (E.61) and θ′(t) = +Z0 +r2(t) give +V 2(t(θ)) = (s′(θ))2 Z2 +0 +r4(t) + r2(t) Z2 +0 +r4(t) = Z2 +0((s′(θ))2 +s4(θ) ++ +1 +s2(θ)) = Z2 +0[ +�d 1 +s +dθ (θ) +�2 ++ +1 +s2(θ)]. +Thus r(t) = s(θ) and dr +dθ := ds +dθ give the first Binet formula. Then +⃗Γ(t) = r′′(t)⃗er(t) + r′(t)d⃗er +dt (t) + (r′(t)θ′(t) + r(t)θ′′(t))⃗eθ(t) + r(t)θ′(t)d⃗eθ +dt (t), +with d⃗er +dt ∥ ⃗eθ, and d⃗eθ +dt (t) = −θ′(t)⃗er(t), and ⃗eθ ⊥ ⃗Γ (central motion), we get +⃗Γ(t) = (r′′(t) − r(t)(θ′(t))2)⃗er(t). +And +r′(t) = s′(θ)θ′(t) = s′(θ) Z0 +r2(t) = Z0 s′(θ) +s2(θ) = −Z0 +d 1 +s +dθ (θ), +thus +r′′(t) = −Z0 +d2 1 +s +dθ2 (θ) θ′(t) = − Z2 +0 +r2(t) +d2 1 +s +dθ2 (θ), +which is the second Binet formula. +114 + +115 +F.1. +The Riesz representation theorem +F +Riesz representation theorem +F.1 +The Riesz representation theorem +Framework: (E, (·, ·)g) is Hilbert space, i.e. E is a vector space equipped with an inner dot product (·, ·)g +such that, with the associated norm defined by||⃗v||g := +� +(⃗v,⃗v)g, (E, ||.||g) is a complete space. And +E∗ = L(E; R) is the space of the linear and continuous forms on E (the space of linear “measuring tools”) +equipped with its norm ||ℓ||E∗ := +sup +||⃗x||g=1 +|ℓ.⃗x| < ∞. +• We have the easy statement: +∀⃗v ∈ E (vector), ∃!vg ∈ E∗ (linear continuous form) +s.t. +vg.⃗x = (⃗v, ⃗x)g, ∀⃗x ∈ E, +(F.1) +and ||vg||E∗ = ||⃗v||g. +(Usual notation in finite dimension: vg.⃗x = ⃗v •g ⃗x, or simply v.⃗x = ⃗v • ⃗x if a +chosen (·, ·)g is imposed to all observers.) +Indeed: Define vg : E → R by vg(⃗x) = (⃗v, ⃗x)g for all ⃗x ∈ E; The definition domain of vg is E and +vg is trivially linear; And the Cauchy–Schwarz inequality gives |vg(⃗x)| = |(⃗v, ⃗x)g| ≤ ||⃗v||g ||⃗x||g for all +⃗x ∈ E, thus ||vg||E∗ ≤ ||⃗v||g < ∞, thus vg is continuous; And |vg(⃗v)| = |(⃗v,⃗v)g| = ||⃗v||g ||⃗v||g, thus +||vg||E∗ ≥ ||⃗v||g, thus ||vg||E∗ = ||⃗v||g. +• The Riesz representation theorem concerns the converse: If you choose an inner dot product (·, ·)g +in E (e.g. English of French), then you can represent a “measuring instrument” ℓ ∈ E∗ by a vector ⃗ℓg ∈ E: +Theorem F.1 (Riesz representation theorem, and definition) (E, (·, ·)g) being a Hilbert space, +∀ℓ ∈ E∗ (linear continuous form), ∃!⃗ℓg ∈ E (vector) +s.t. +ℓ.⃗x = (⃗ℓg, ⃗x)g, ∀⃗x ∈ E, +(F.2) +and ||⃗ℓg||g = ||ℓ||E∗. And ⃗ℓg is called the (·, ·)g-Riesz representation vector of ℓ (depends on g). +(Usual notation in finite dimension: vg.⃗x = ⃗v •g ⃗x, or simply v.⃗x = ⃗v • ⃗x if a chosen (·, ·)g is imposed +to all observers.) +Proof. Easy in finite dimension: With a basis (⃗ei), if [ℓ]|⃗e = ( ℓ1 +. . . +ℓn ) (row matrix since ℓ is a linear +form) then (F.2) gives [ℓ]|⃗e.[⃗x]|⃗e = [⃗ℓg]T +⃗e .[g]|⃗e.[⃗x]|⃗e, thus [⃗ℓg]⃗e = [g]−1 +|⃗e .[ℓ]T +|⃗e (column matrix), thus ⃗ℓg. +General case (e.g. with E = L2(Ω) and the finite element method): If ℓ = 0 then ⃗ℓg = ⃗0 (trivial). +Suppose ℓ ̸= 0: Thus Kerℓ = ℓ−1({0}) ̸= {⃗0} (the kernel). If dim E = 1, it is trivial (exercise). Suppose +dim E ≥ 2. Since ℓ is continuous, its kernel Kerℓ = ℓ−1({0}) is closed in E. Thus, if ⃗x ∈ E, then its (·, ·)g- +orthogonal projection ⃗x0 ∈ Kerℓ on Kerℓ exists, is unique, and is given by: ∀⃗y0 ∈ Kerℓ, (⃗x −⃗x0, ⃗y0)g = 0. +(So ⃗x − ⃗x0 ⊥g Kerℓ.) Choose a ⃗x /∈ Kerℓ (possible since ℓ ̸= 0), and let ⃗n := +⃗x−⃗x0 +||⃗x−⃗x0||g ; So ⃗n is a (·, ·)g- +orthonormal vector to Kerℓ, and (Kerℓ)⊥ = Vect{⃗n} since dim(Kerℓ)⊥ = 1 (in finite dimension cf. the +Dimension Formula which states that the dimension of the domain of a linear map is the sum of the +dimension of its range and the dimension of its kernel, and in infinite dimension see next exercise F.2). +And E = Kerℓ ⊕ (Kerℓ)⊥ since both vector spaces are closed (an orthogonal is always closed in a Hilbert +space). Thus if ⃗x ∈ E then ⃗x = ⃗x0 + λ⃗n ∈ Kerℓ ⊕ (Kerℓ)⊥; Thus (⃗x,⃗n)g = λ and ℓ(⃗x) = 0 + λℓ(⃗n) = +(⃗x,⃗n)gℓ(⃗n) = (⃗x, ℓ(⃗n)⃗n)g (bilinearity of (·, ·)g); Thus ⃗ℓg := ℓ(⃗n)⃗n satisfies (F.2). +And if ⃗ℓg1 and ⃗ℓg2 +satisfy (F.2) then (⃗ℓg1 − ⃗ℓg2, ⃗x)g = 0 for all ⃗x ∈ E, thus ⃗ℓg1 − ⃗ℓg2 = 0. Thus ⃗ℓg is unique. +And the Cauchy–Schwarz theorem give ||ℓ||E∗ := sup||⃗x||g=1 |ℓ(⃗x)| = sup||⃗x||g=1 |(⃗ℓg, ⃗x)g| = ||⃗ℓg||g. +⃗Rg is an isomorphism between Banach spaces: linearity since (⃗Rg(ℓ + λm), ⃗x)g = (ℓ + λm).⃗x = ℓ.⃗x + +λm.⃗x = (⃗Rg(ℓ), ⃗x)g +λ(⃗Rg(m), ⃗x)g = (⃗Rg(ℓ)+λ⃗Rg(m), ⃗x)g for all ⃗x gives ⃗Rg(ℓ+λm) = ⃗Rg(ℓ)+λ⃗Rg(m), +bijectivity thanks to (F.1) and (F.2), and the norm is kept since ||⃗ℓg||g = ||ℓ||E∗. +Exercice F.2 Prove: If ℓ ∈ E∗−{0} then dim(Kerℓ)⊥ = 1 (= dim(Im(ℓ)) = dim R). +Answer. Consider the restriction ℓ|Kerℓ⊥ : +� +(Kerℓ)⊥ → R +⃗x → ℓ|Kerℓ⊥.⃗x = ℓ.⃗x +� +. It is linear (since ℓ is), and thus one to +one: Indeed it is onto since ℓ ̸= 0, and it is one to one since if ℓ|Kerℓ⊥(⃗x) = 0 = ℓ(⃗x) then ⃗x ∈ (Kerℓ)⊥ � Kerℓ = {⃗0}, +thus ⃗x = 0. Thus dim(Kerℓ)⊥ ≤ dim(Im(ℓ)) = 1: Indeed, if ⃗z1, ⃗z2 ∈ (Kerℓ)⊥−{⃗0} then ℓ|Kerℓ⊥(⃗z1) ∈ R and +ℓ|Kerℓ⊥(⃗z2) ∈ R, thus ∃λ ∈ R s.t. ℓ|Kerℓ⊥(⃗z2) = λℓ|Kerℓ⊥(⃗z1), thus ℓ|Kerℓ⊥(⃗z2 − λ⃗z1) = 0, thus ⃗z2 − λ⃗z1 = ⃗0 since +ℓ|Kerℓ⊥ is one to one. And ⃗n ∈ (Kerℓ)⊥ gives dim(Kerℓ)⊥ ≥ 1 (above proof). Thus dim(Kerℓ)⊥ = 1 = Vect{⃗n}. +115 + +116 +F.2. +The Riesz representation operator +F.2 +The Riesz representation operator +The Riesz representation theorem F.1 gives the (·, ·)g-Riesz representation operator +⃗Rg : +� +E∗ → E +ℓ → ⃗Rg(ℓ) := ⃗ℓg, +i.e. +(⃗Rg(ℓ),⃗v)g = ℓ.⃗v, ∀⃗v ∈ E. +(F.3) +So ⃗Rg transforms a « covariant ℓ » into a « contravariant ⃗ℓg » thanks to the tool (·, ·)g. +NB (fundamental): ⃗Rg is a isomorphism between the Banach spaces (E, ||.||g) and (E∗, ||.||E∗), but ⃗Rg +is not canonical since it requires a man made tool (an inner dot product chosen by some observer) to be +defined. (An isomorphism E ↔ E∗ can never be canonical, see § T.2.) +And with G the set of inner dot products in E, we have thus defined the Riesz representation mapping +⃗R : +� +G × E∗ → E +(g, ℓ) → ⃗R(g, ℓ) := ⃗ℓg = ⃗Rg(ℓ) = ⃗ℓ(g). +(F.4) +So ⃗R has two inputs: A choice (·, ·)g by an observer for the first slot, a linear form for the second slot. +F.3 +Quantification with a basis +Here E is finite dimensional, dim E = n, ℓ ∈ E∗ (a linear form), (·, ·)g is an inner dot product, (⃗ei) is a +basis, (πei) is the dual basis (classical notations). Let +gij = g(⃗ei,⃗ej), +ℓ = +n +� +i=1 +ℓiπei, +⃗ℓg = +n +� +i=1 +(⃗ℓg)i⃗ei, +⃗Rg.πej = +n +� +i=1 +Rij⃗ei, +(F.5) +i.e. [g]|⃗e = [gij], [ℓ]|πe = ( ℓ1 +... +ℓn ) (row matrix), [⃗ℓg]|⃗e = +� +� +� +(⃗ℓg)1 +... +(⃗ℓg)n +� +� +� (column matrix), [⃗Rg]πe,⃗e = [Rij]. +(Duality notations: ℓ = �n +i=1ℓiei, ⃗ℓg = �n +i=1ℓi +g⃗ei, ⃗Rg.ej = �n +i=1Rij⃗ei, [⃗Rg]e,⃗e = [Rij].) +Proposition F.3 +[⃗ℓg] = [g]−1.[ℓ]T +and +[⃗Rg] = [g]−1 , +i.e. +(⃗ℓg)i = +n +� +j=1 +([g]−1)ij(ℓ)j = +n +� +j=1 +(⃗Rg)ij(ℓ)j. +(F.6) +Full matrix notation: [⃗ℓg]|⃗e = ([g]|⃗e)−1.([ℓ]|πe)T , and [⃗Rg]|πe,⃗e = ([g]|⃗e)−1. +Duality notation to see the change of variance induced by (·, ·)g (bottom index for ℓ, top index for ⃗ℓg): +ℓi +g = +n +� +j=1 +Rijℓj, +(F.7) +i.e. ℓi +g = �n +j=1gijℓj when ([g]−1)ij =noted [gij]. +(In particular, if (⃗ei) is a (·, ·)g-orthonormal basis, then [⃗Rg] = [g]−1 = I and ℓi +g = ℓi.) +Proof. (F.2) gives [ℓ]|⃗e.[⃗x]|⃗e = [⃗ℓg]T +|⃗e.[g]|⃗e.[⃗x]|⃗e for all ⃗x, thus [ℓ]|⃗e = [⃗ℓg]T +|⃗e.[g]|⃗e, thus [g]|⃗e.[⃗ℓg]|⃗e = [ℓ]T +|⃗e (since +[g]|⃗e = [g]T +|⃗e), thus [⃗ℓg] = [g]−1.[ℓ]T . +And ⃗Rg.ℓ =(F.3) ⃗ℓg gives �n +j=1(ℓ)j ⃗Rg.πj = �n +i=1(⃗ℓg)i⃗ei, thus �n +i,j=1(ℓ)jRij⃗ei = �n +i=1(⃗ℓg)i⃗ei, thus +�n +j=1Rij(ℓ)j = (⃗ℓg)i for all i, thus [⃗Rg].[ℓ]T = [⃗ℓg]. Thus [⃗Rg] = [g]−1. +Remark F.4 If a chosen inner dot product (·, ·)g is imposed (e.g. Euclidean foot based) and if duality +notations are used, then a usual notation for ⃗ℓg is ℓ♯, since ⃗ℓg = ⃗Rg(ℓ) = �n +i=1ℓi⃗ei with a top index for +ℓi: the index i has been raised through ⃗Rg. Then (F.2) and (F.6) read (isometric framework) +ℓ.⃗x = ℓ♯ • ⃗x +and +[ℓ♯]|⃗e = [g]−1 +|⃗e .[ℓ]T +|⃗e. +(F.8) +We won’t use this notation (we deal with objectivity). +116 + +117 +F.4. +Change of Riesz representation vector, and Euclidean case +F.4 +Change of Riesz representation vector, and Euclidean case +For one linear form ℓ ∈ E∗, two observers with their inner dot products (·, ·)g and (·, ·)h get two Riesz +representation vectors ⃗ℓg = ⃗Rg(ℓ) and ⃗ℓh = ⃗Rh(ℓ) given by, cf. (F.2): +∀⃗x ∈ E, +(⃗ℓg, ⃗x)g = ℓ.⃗x = (⃗ℓh, ⃗x)h. +(F.9) +Proposition F.5 For any basis (⃗ei) in E, we have the change of representation vector formula: +[h]|⃗e.[⃗ℓh]|⃗e = [g]|⃗e.[⃗ℓg]|⃗e, +i.e. +[⃗ℓh]|⃗e = [h]−1 +|⃗e .[g]|⃗e.[⃗ℓg]|⃗e. +(F.10) +In particular (for the Euclidean case), with λ > 0: +If (·, ·)g = λ2(·, ·)h +then +⃗ℓh = λ2⃗ℓg. +(F.11) +Conversely, if ⃗ℓh = λ2⃗ℓg for all linear forms ℓ ∈ E∗, then (·, ·)g = λ2(·, ·)h. +So, a linear form ℓ cannot be identified with a Riesz representation vector (which one: ⃗ℓg? ⃗ℓh?); +In other words, a Riesz representation vector ⃗Rg(ℓ) is not objective, is not intrinsic to a linear form ℓ. +NB: (F.10)-(F.11) is a “change of vector formula” (one linear form gives two vectors relative to two +inner dot products); It is not a “change of basis formula” (for one vector and its two sets of components). +Proof. (F.9) gives [⃗x]T +|⃗e.[g]|⃗e.[⃗ℓg]|⃗e = [⃗x]T +|⃗e.[h]|⃗e.[⃗ℓh]|⃗e for all ⃗x, hence [g]|⃗e.[⃗ℓg]|⃗e = [h]|⃗e.[⃗ℓh]|⃗e, i.e. (F.10). +In particular λ2(·, ·)h = (·, ·)g give λ2(⃗ℓg, ⃗x)h = (⃗ℓg, ⃗x)g =(F.9)(⃗ℓh, ⃗x)h for all ⃗x, hence λ2⃗ℓg = ⃗ℓh. +Converse: λ2⃗ℓg = ⃗ℓh for all ℓ gives λ2(⃗ℓg, ⃗x)h = (⃗ℓh, ⃗x)h +(F.9) += (⃗ℓg, ⃗x)g, for all ⃗x and for all ℓ, thus for +all ⃗ℓg thanks to the isomorphism ⃗Rg : E∗ → E, cf. (F.3), thus λ2(·, ·)h = (·, ·)g. +Example F.6 If (·, ·)g and (·, ·)h are the Euclidean dot products made with the foot and the metre then, +with (F.9), +(·, ·)g = λ2(·, ·)h +=⇒ +⃗ℓh = λ2⃗ℓg, +with +λ2 > 10 : +(F.12) +⃗ℓg (English) and ⃗ℓh (French) are quite different! A Riesz representation vector is subjective, and certainly +not “canonical” (a word that you may find in books where... nothing is defined...). +E.g., aviation: If you do want to use a Riesz representation vector to represent a ℓ ∈ Rn∗, it is vital to +know which Euclidean dot product is in use, see also remark A.14 (Mars Climate Orbiter Crash). Recall: +The foot is the international unit of altitude for aviation. +Example F.7 If f ∈ C1(Rn; R) and p ∈ Rn, the differential of f at p is the linear form df(p) ∈ Rn∗ +defined by, for all ⃗w ∈ ⃗Rn, +df(p).⃗w := lim +h→0 +f(p + h⃗w) − f(p) +h +(definition independent of any inner dot product), +(F.13) +see (S.5). If you can choose an inner dot product (·, ·)g then the gradient +⃗ +gradgf(p) is the (·, ·)g-Riesz +representation vector of df(p): +⃗ +gradgf(p) := ⃗Rg(df(p)), +i.e. +df(p).⃗w = +⃗ +gradgf(p) •g ⃗w, ∀⃗w ∈ ⃗Rn. +(F.14) +And (F.12) gives +⃗ +gradhf(p) = λ2 ⃗ +gradgf(p) +with +λ2 > 10 (English vs French) : +(F.15) +The gradient is very dependent on the observer (the gradient is subjective, the differential is objective). +Remark F.8 The “gradient” is observer dependent; We already had this observer dependence for the +usual derivative in the 1-D case f : x ∈ R → f(x) ∈ R; Question: What does f ′(x) = 3 mean? +Answer. 11- For one observer, it means f ′(x) = limh→0 +f(x+h)−f(x) +h +... but... where in the departure +space this observer has chosen a basis vector ⃗a of length 1 for him (e.g. length 1 foot) which he calls 1; +So, with no abusive notations, his derivative f ′(x) is in fact f ′ +a(x) = limh→0 +f(x+h⃗a)−f(x) +h +. +117 + +118 +F.5. +A Riesz representation vector is contravariant +12- For some other observer, it means f ′(x) = limh→0 +f(x+h)−f(x) +h +... but... where in the departure +space this observer has chosen a basis vector ⃗b of length 1 for him (e.g. length 1 metre) which he calls 1; +So, with no abusive notations, his derivative f ′(x) is in fact f ′ +b(x) = limh→0 +f(x+h⃗b)−f(x) +h +. +13- Both observer use the same formula f ′(x) = limh→0 +f(x+h)−f(x) +h +but get different results! In- +deed, if ⃗b = λ⃗a, then = lim +h→0 +f(x + h⃗b) − f(x) +h += lim +h→0 +f(x + hλ⃗a) − f(x) +h += λ lim +h→0 +f(x + (hλ)⃗a) − f(x) +(hλ) += +λ lim +k→0 +f(x + k⃗a) − f(x) +k +thus +f ′ +b(x) = λf ′ +a(x), +with +λ ≃ 3.28 +(F.16) +with foot and metre... Quite different results! (In fact f ′(x) = opposite side +adjacent side depends on the length of +the adjacent side...) +Remark F.9 We insist on the subjectivity of the gradient: +20- The differential of f at a point x along a vector ⃗w ∈ ⃗R is df(x).⃗w = limh→0 +f(x+h⃗w)−f(x) +h +and is +objective: The observers all use this same formula. +21- An observer chooses a Euclidean dot product (·, ·)g (e.g. based on the foot), then represent df(x) +by its (·, ·)g-Riesz representation vector ⃗Rg(df(x)) =noted +⃗ +gradgf(x) called the gradient of f at x relative +to (·, ·)g. +22- Another observer chooses a Euclidean dot product (·, ·)h (e.g. based on the metre), then represent +df(x) by its (·, ·)h-Riesz representation vector ⃗Rh(df(x)) =noted +⃗ +gradhf(x) called the gradient of f at x +relative to (·, ·)h. +23- Both observer use the same formula df(x).⃗w = limh→0 +f(x+h⃗w)−f(x) +h +to get a different result: +⃗ +gradhf = λ2 ⃗ +gradgf, because they use different measuring tools (one based on the foot, the other on the +metre). +24- Recall: The gradient depends on a choice of a Euclidean unit. +Exercice F.10 In (F.16) we have f ′ +b(x) = λf ′ +a(x). And the 1-D gradient gives gradbf(x) = λ2gradaf(x). +Why? +Answer. To define a gradient gradaf we need a Euclidean dots products (·, ·)a built from a basis (⃗a) in ⃗R, while +to define f ′ +a we need a unit of length. Details: (⃗a) and (⃗b) are two bases in ⃗R with ⃗b = λ⃗a, thus (·, ·)a = λ2(·, ·)b +(since 1 = (⃗a,⃗a)a = (⃗b,⃗b)b = (λ⃗a, λ⃗a)b = λ2(⃗a,⃗a)b gives (⃗a,⃗a)a = λ2(⃗a,⃗a)b and (⃗a) is a basis). +And we +have f ′ +b(x) =(F.16) λf ′ +a(x), i.e. df(x).⃗b = λdf(x).⃗a, thus (gradfb(x),⃗b)b = λ(gradfa(x),⃗a)a, thus (gradfb(x),⃗b)b = +λλ2(gradfa(x), +⃗b +λ)b = (λ2gradfa(x),⃗b)b, thus gradfb(x) = λ2gradfa(x). +Exercice F.11 1- Prove that (·, ·)g = λ2(·, ·)h gives ||⃗ℓh||g = λ||⃗ℓh||h. 2- Does it contradict the Riesz +representation theorem which gives ||ℓ||Rn∗ = ||⃗ℓg||Rn? +Answer. 1- ⃗ℓh =(F.11) λ2⃗ℓg gives ||⃗ℓh||h = λ2||⃗ℓg||h = λ||⃗ℓg||g since ||.||h = λ||.||g. +2- No, since ||ℓ||Rn∗ := sup||⃗x||Rn =1 |ℓ.⃗x| depends on the norm ||.||Rn chosen; Here ||.||Rn is either ||.||g or ||.||h. +Thus if you write ||ℓ||Rn∗ =noted ||ℓ||g∗ if you use the norme ||.||g, then ||ℓ||h∗ = sup⃗v∈⃗Rn +|ℓ.⃗v| +||⃗v||h = sup⃗v∈⃗Rn +|ℓ.⃗v| +1 +λ ||⃗v||g = +λ sup⃗v∈⃗Rn +|ℓ.⃗v| +||⃗v||g = λ||ℓ||g∗. +F.5 +A Riesz representation vector is contravariant +⃗ℓg is a vector in E, cf. (F.2), so it is contravariant. To be convinced: +Exercice F.12 Check: +[⃗ℓg]|new = P −1.[⃗ℓg]|old +(contravariance formula). +(F.17) +Answer. Consider two bases (⃗eold,i) and (⃗enew,i) in E. With the change of basis formulas [⃗x]|new = P −1.[⃗x]|old +and [g]|new = P T .[g]|old.P, (F.2) gives (with (A.94)), for all ⃗x, +[⃗x]T +|old.[g]|old.[⃗ℓg]|old = ℓ.⃗x = [⃗x]T +|new.[g]|new.[⃗ℓg]|new += ([⃗x]T +|old.P −T ).(P T .[g]|old.P).[⃗ℓg]|new = [⃗x]T +|old.[g]|old.(P.[⃗ℓg]|new), +(F.18) +thus [⃗ℓg]|old = P.[⃗ℓg]|new since [g] is invertible (an inner dot product is positive definite), thus (F.17). +118 + +119 +F.6. +What is a vector versus a (·, ·)g-vector? +Remark F.13 • Dont forget: A representation vector ⃗ℓg is not intrinsic to the linear form ℓ because it +depends on a (·, ·)g (depends on a observer: foot? metre?)). Reminder: there is no natural canonical +isomorphism between E and E∗, i.e. it is impossible to identify a linear form with a vector, see § T.2. +• ⃗ℓg is incompatible with the use of push-forwards, cf. § 7.2. +• ⃗ℓg is incompatible with the use of Lie derivatives, cf. (9.51). +F.6 +What is a vector versus a (·, ·)g-vector? +1- Originally, a vector was a bipoint vector ⃗v = −−→ +AB in ⃗R3 used to represent of a “material object”. +E.g. the height of a child is represented on a wall by a vertical bipoint vector ⃗x starting from the ground +up to a pencil line. The vector ⃗x is objective: Any observer uses this same vector to get the height of the +child... and then use “their subjective unit” (foot, metre...) to give a value. +2- Then (mid 19th century), the concept of vector space was introduced: It is a quadruplet (E, +, K, .) +where + is an inner law, (E, +) is a group, K is a field, . is a external law on E (called a scalar +multiplication) compatible with + (see any math book). +And then the concept of scalar inner dot product (in a vector space) was introduced. +3- We can then get non “material” vectors (“subjectively built vectors”). E.g.: start with our usual +vector space ⃗Rn of bi-point vectors, then consider its dual (Rn∗, +, R, .) =noted Rn∗. Then, for a given +ℓ ∈ Rn∗ (a given measuring device), consider two observers: An English observer with his foot built +Euclidean dot product (·, ·)g, and a French observer with with his metre built Euclidean dot product (·, ·)h. +These observers build their own artificial Riesz representation vectors ⃗ℓg = ⃗Rg(ℓ) ∈ ⃗Rn and ⃗ℓh = ⃗Rh(ℓ), +cf (F.12); They remark that ⃗ℓg ̸= ⃗ℓh: These constructions are very subjective. +4- Then, with differential geometry, a vector ⃗v has been redefined: It is a “tangent vector”, which +means that there exists a C1 curve c : s ∈ [a, b] → c(s) ∈ E such that ⃗v is defined at a p = c(s) ∈ Im(c) +by ⃗v(p) := ⃗c ′(s) (so a vector is part of a vector field, here defined along the range of c). (This definition +of a tangent vector is applicable to “tangent vectors to a surface” i.e. tangent vectors to a manifold, +see e.g. § 9.1.1,2-.) +Then it is shown that ⃗v is equivalent to +∂ +∂⃗v = the directional derivative in the +direction ⃗v (natural canonical isomorphism between E and E∗∗). +For other equivalent definitions of +vectors, see Abraham–Marsden [1]. +F.7 +The “(·, ·)g-dual vectorial bases” of one basis (and warnings) +Framework: E is a finite dimensional vector space, dim E = n (e.g. E = ⃗R3). An observer chooses +an inner dot product (·, ·)g (e.g., in ⃗R3, a foot-built Euclidean dot product). Hence the results will be +subjective. And (⃗ei) is some basis in E. +F.7.1 +A basis and its many associated “dual vectorial basis” +Definition F.14 The (·, ·)g-dual vectorial basis (or (·, ·)g-vectorial dual basis, or (·, ·)g-dual basis) of the +basis (⃗ei) is the basis (⃗eig) in E defined by +∀j = 1, ..., n, +(⃗eig,⃗ej)g = δij, +i.e. +⃗eig •g ⃗ej = δij . +(F.19) +NB: A vectorial dual basis is not unique: It depends on the chosen inner dot product, see e.g. (F.22). +NB: Pay attention to the notations: ⃗eig is a contravariant vector (⃗eig ∈ E), so, even if you use the +Einstein convention, the index i in ⃗eig must be a bottom index. +Let (πei) be the (covariant) dual basis of the basis (⃗ei), i.e. the πei ∈ E∗ are the objective (the same +for all observers) linear forms defined by πei.⃗ej = δij for all j, cf. (A.7). +Definition F.15 (Equivalent definition.) The (·, ·)g-dual vectorial basis of the basis (⃗ei) is the basis +(⃗eig) in E made of the (·, ·)g-Riesz representative vectors of the πei, i.e. +⃗eig := ⃗Rg(πei) , +i.e. +⃗eig •g ⃗v = πei.⃗v, ∀⃗v ∈ E. +(F.20) +where ⃗Rg is the (·, ·)g-Riesz operator, see (F.3). +With duality notations, (ei) is the dual basis and ⃗eig := ⃗Rg(ei), i.e. (⃗eig,⃗v)g = ei.⃗v for all ⃗v ∈ E where +here the position of the index i is bottom on the left and up on the right, since ⃗Rg changes a covariant +vector (a linear form) into a contravariant vector. +119 + +120 +F.7. +The “(·, ·)g-dual vectorial bases” of one basis (and warnings) +Exercice F.16 Prove that the vectors ⃗eig satisfy the contravariant change of basis formula +[⃗eig]|new = P −1.[⃗eig]|old +(F.21) +(the ⃗ejg are indeed “contravariant vectors”). +Answer. • First answer: ⃗eig is a vector in E, thus it is contravariant. +• Second answer: Apply (F.17) since ⃗eig is a Riesz-representation vector. +• Third answer = direct computation: Consider two bases (⃗ai) and (⃗bi), and the transition matrix P from +(⃗ai) to (⃗bi), i.e., ⃗bj = �n +i=1Pij⃗ai for all j. (F.19) and the change of basis formulas for the vectors ⃗ei and the +bilinear form (·, ·)g give [⃗ej]T +|⃗a.[g]|⃗a.[⃗eig]|⃗a = (⃗eig,⃗ej)g = [⃗ej]T +|⃗b.[g]|⃗b.[⃗eig]|⃗b = (P −1.[⃗ej]|⃗a)T .(P T .[g]|⃗a.P).[⃗eig]|⃗a = +[⃗ej]T +|⃗a.[g]|⃗a.P.[⃗eig]|⃗a for all i, j, thus [⃗eig]|⃗a = P.[⃗eig]|⃗b, thus (F.21). +Exercice F.17 Consider two inner dot products (·, ·)a and (·, ·)b (e.g., a foot-built and a metre-built +Euclidean dot product), and a basis (⃗ei) in E. Call (⃗eia) and (⃗eib) the (·, ·)a and (·, ·)b-dual vectorial +bases. Prove: +(·, ·)a = λ2(·, ·)b +=⇒ +⃗eib = λ2⃗eia, +∀i. +(F.22) +E.g., λ2 > 10 with foot and metre built Euclidean bases: ⃗eib is very different from ⃗eia ! A dual vectorial +basis highly depends on an observer: A vectorial dual basis is not intrinsic to (⃗ei) (not objective). +Answer. (F.19) gives (⃗eib,⃗ej)b = δij = (⃗eia,⃗ej)a = λ2(⃗eia,⃗ej)b, thus (⃗eib − λ2⃗eia,⃗ej)b = δij, for all i, j. +Example F.18 If (⃗ei) is a (·, ·)g-orthonormal basis we trivially get ⃗eig = ⃗ei for all i, i.e., (⃗eig) = (⃗ei).This +particular case is not compatible with joint work by an English (foot) and French (metre) observer. +F.7.2 +Components of ⃗ejg in the basis (⃗ei) +Proposition F.19 The components of ⃗ejg in the basis (⃗ei) are given by, for any j ∈ [1, n]N, +[⃗ejg]|⃗e = [⃗Rg]|⃗e.[⃗ej]|⃗e = ([g]|⃗e)−1.[⃗ej]|⃗e = the j-th column of ([g]|⃗e)−1, +(F.23) +i.e. the i-th component of ⃗ejg is ([g]−1 +|⃗e )ij. +Thus the matrix of g(·, ·) in the basis (⃗eig) is the inverse of the matrix of g(·, ·) in the basis (⃗ei): +([g(⃗eig,⃗ejg)] =) +[g]|(⃗eig) = [g]|(⃗ei) +−1 +(= ([g(⃗ei,⃗ej)])−1). +(F.24) +Proof. First proof of (F.23) (straight forward calculation): (F.19) gives, for all i, j, +[⃗ej]T +|⃗e.[g]|⃗e.[⃗eig]|⃗e = δij = [⃗ej]T +|⃗e.[⃗ei]|⃗e, +thus +[g]|⃗e.[⃗eig]|⃗e = [⃗ei]|⃗e. +(F.25) +Second proof of (F.23): Apply (F.6) (generic Riesz representation result) to get (F.23). +Thus, [g]|⃗e being symmetric we have [g]|⃗e−1 symmetric, and g(⃗eig,⃗ejg) = [⃗eig]T +|⃗e.[g]|⃗e.[⃗ejg]|⃗e = +[⃗ei]T +|⃗e.[g]|⃗e−1.[g]|⃗e.[g]|⃗e−1.[⃗ej]|⃗e = [⃗ei]T +|⃗e.[g]|⃗e−1.[⃗ej]|⃗e = ([g]|⃗e−1)ij, thus (F.24). +Example F.20 ⃗R2, [g]|⃗e = +� +1 +0 +0 +2 +� +, thus [g]−1 +|⃗e = +� +1 +0 +0 +1 +2 +� +. Thus ⃗e1g = ⃗e1, ⃗e2g = 1 +2⃗e2. +Remark F.21 Warning: When ([g]−1 +|⃗e )ij =noted gij then (F.23) reads +⃗ejg = +n +� +i=1 +gij⃗ei, +(F.26) +where the Einstein convention is not satisfied: The Einstein convention is satisfied with +⃗ejg = +n +� +i=1 +(⃗ejg)i⃗ei +noted += +n +� +i=1 +(Pj)i⃗ei +(F.27) +(the components of vectors have up indices), and this can be verified with (⃗eig,⃗ej)g =(F.19) δij which gives +�n +k,ℓ=1(Pi)kgkj = δij. And in (F.26) the scalars gij is just another name for (Pj)i, nothing more (nothing +to do with the Einstein convention). +120 + +121 +F.7. +The “(·, ·)g-dual vectorial bases” of one basis (and warnings) +We insist:In other words: M = [g]|⃗e = [Mij] is a matrix, and its inverse is the matrix M −1 = [Mij]−1: +A matrix is just a collection of scalars (has nothing to do with the Einstein convention), and its inverse +is also a collection of scalars, and you do not change this fact by calling M −1 =noted [M ij] (the use of up +indices is irrelevant for matrices). See remark A.50. +And because (Pj)i equals ([g]−1 +|⃗e )ij =noted gij, some people rename ⃗ejg as ⃗e j... to get ⃗e j = �n +i=1gij⃗ei... +But doing so they despise Einstein’s convention, despite eventual claims: They confuse covariance and +contravariance... and add confusion to the confusion. +NB: Recall: If in trouble with a notation which comes as a surprise (the notation gij here), use classical +notations: Then no misuse of Einstein’s convention and no possible misinterpretation. In particular here +⃗ejg is a (contravariant) vector. +F.7.3 +Multiple admissible notations for the components of ⃗ejg +Let P ∈ L(E; E) be the change of basis endomorphism from (⃗ei) to (⃗eig): defined by P.⃗ej = ⃗ejg. And +let P = [P]|⃗e (the associated transition matrix). It gives multiple admissible (non confusing) notations +for the components of ⃗ejg relative to the basis (⃗ei): +⃗ejg = P.⃗ej = +n +� +j=1 +Pij⃗ei = +n +� +j=1 +(Pj)i⃗ei +� +�� +� +clas. += +n +� +j=1 +(Pj)i⃗ei = +n +� +j=1 +P i +j⃗ei +� +�� +� +dual +, +(F.28) +i.e. the i-th component of the vector ⃗ejg has the names Pij = (Pj)i = (Pj)i = P ij, i.e. P = [P]|⃗e = +[Pij] = [(Pj)i] = [(Pj)i] = [P ij] (four different notations for the same matrix), i.e. +∀j, +[⃗ejg]|⃗e = [P]|⃗e.[⃗ej]|⃗e = +� +� +� +P1j +... +Pnj +� +� +� = +� +� +� +(Pj)1 +... +(Pj)n +� +� +� = +� +� +� +P 1j +... +P nj +� +� +� = +� +� +� +(Pj)1 +... +(Pj)n +� +� +� +(F.29) += the j-th column of [P]|⃗e. You can choose any notation, depending on your current need or mood... +F.7.4 +(Huge) differences between “the (covariant) dual basis” and “a dual vectorial basis” +1. A basis (⃗ei) has an infinite number of vectorial dual bases (⃗eig), as many as the number of inner +dot products (·, ·)g (as many as observers), see (F.23). +While a basis (⃗ei) has a unique intrinsic (covariant) dual basis (πei) noted += +(ei), cf. (A.7): Two +observers who consider the same basis (⃗ei) have the same (covariant) dual basis. +2. πei = ei is covariant, while ⃗ei and ⃗eig are contravariant. And there is no transition matrix between +(⃗ei) and (πei) = (ei), since ⃗ei ∈ E and πei = ei ∈ E∗ don’t live in the same vector space. +3. If you fly, it is vital to use the dual basis (πei) = (ei): It is possibly fatal if you confuse foot and +metre at takeoff and at landing (if you survived takeoff) because of the choice of different Euclidean +dot product (·, ·)g or (·, ·)h; See e.g. the Mars Climate Orbiter crash, remark A.14. +Einstein’s +convention can help... only if it is really followed. +F.7.5 +About the notation gij = shorthand notation for (g♯)ij +Definition F.22 The Riesz associated inner dot product g♯ ∈ L(E∗, E∗; R) is the bilinear form defined +by, for all ℓ, m ∈ E∗, +g♯(ℓ, m) := g(⃗ℓg, ⃗mg), +i.e. +(ℓ, m)g♯ := (⃗ℓg, ⃗mg)g. +(F.30) +where ⃗ℓg = ⃗Rg(ℓ) and ⃗mg = ⃗Rg(m). +Thus g♯(·, ·) =noted (·, ·)g♯ is indeed an inner dot product in E∗: trivial check. +Quantification: +(⃗ei) is a basis in E and (ei) is its dual basis (duality notations). (F.30) gives: +(g♯)ij := g♯(ei, ej) = g(⃗eig,⃗ejg), +thus +[g♯]|e = [(g♯)ij] = [gij]−1 = [g]−1 +|⃗e , +(F.31) +cf. (F.24). And +[(g♯)ij] +shorthand += +notation [gij] . +(F.32) +Classical notations: [g♯]|e = [(g♯)ij] = [g♯(πei, πej)] = [g(⃗eig,⃗ejg)] = [gij]−1 = ([g]|⃗e)−1. +121 + +122 +G.0. +Goal +Exercice F.23 How do we compute g♯(ℓ, m) with matrix computations? +Answer. +ℓ = �n +i=1ℓiei and m = �n +j=1mjej give g♯(ℓ, m) = �n +i,j=1ℓimjg♯(ei, ej) = �n +i,j=1ℓi(g♯)ijmj = +[ℓ]|⃗e.[g♯]|⃗e.[m]T +|⃗e = [ℓ]|⃗e.[g]−1 +|⃗e .[m]T +|⃗e (a linear form is represented by a row matrix,). +Exercice F.24 (F.30) tells that the +�2 +0 +� +tensor g♯ ∈ L(E∗, E∗; R) was created from the +�0 +2 +� +tensor g = +(·, ·)g ∈ L(E, E; R) using twice the (·, ·)g-Riesz representation theorem. +1- Show that if you use the (·, ·)g-Riesz representation theorem just once you get the +�1 +1 +� +tensor +g♮ ∈ L(E∗, E; R) ≃ L(E; E) given by +g♮ = I. +(F.33) +2- Reciprocal: What is the +�0 +2 +� +tensor g♭ ∈ L(E, E; R) that you create from the identity I ∈ L(E; E) +when using the (·, ·)g-Riesz representation theorem once? +3- Summary: �I = g♮ gives (�I)♭ = g♭ = g and (�I)♯ = g♯ +Answer. 1- g♮ ∈ L(E∗, E; R) is defined by g♮(ℓ, ⃗w) = (⃗ℓg, ⃗w)g for all (ℓ, ⃗w) ∈ E∗ × E, where ⃗ℓg is the (·, ·)g-Riesz +representation vector of ℓ. Thus g♮(ℓ, ⃗w) = ℓ.⃗w = ℓ.I.⃗w, for all (ℓ, ⃗w) ∈ E∗×E, hence g♮ ∈ L(E∗, E; R) is naturally +canonically associated with the identity I ∈ L(E; E). +2- The identity operator I ∈ L(E; E) (observer independent) is naturally canonically associated with the +�1 +1 +� +tensor �I ∈ L(E∗, E; R) defined by �I(ℓ, ⃗w) = ℓ.I.⃗w = ℓ.⃗w for all (ℓ, ⃗w) ∈ E∗ × E, thus �I = g♮. +G +Cauchy–Green deformation tensor C = F T.F +Framework: �Φ : +� +R × Obj → Rn +(t, PObj) → �Φ(t, PObj) +� +is a motion of Obj, Ωτ = �Φ(τ, PObj) is the configuration of Obj +at any τ, t0 and t are fixed, Φ := Φt0 +t : +� +Ωt0 → Ωt +P → p = Φ(P) +� +is the associated motion between t0 and t, +and F(P) := dΦ(P) : +� +� +� +� +� +⃗ +Rn +t0 → ⃗Rn +t +⃗W → ⃗w = F(P). ⃗W := lim +h→0 +Φ(P+h ⃗W) − Φ(P) +h +� +� +� +� +� +is the deformation gradient +at P between t0 and t, cf. (4.2). +G.0 +Goal +Construction of C (summary of Cauchy’s approach): +1- At t0, consider two vectors ⃗W1 and ⃗W2, +2- at t, they are distorted by the motion and become the vectors F. ⃗W1 and F. ⃗W2; +3- Then choose a Euclidean dot product (·, ·)g =noted · +• ·, the same at all t (to simplify); +4- Then, by definition of the transposed, (F. ⃗W1) • (F. ⃗W2) = (F T .F. ⃗W1) • ⃗W2: You have got the +Cauchy strain tensor C := F T .F; +5- Then (F. ⃗W1) • (F. ⃗W2) − ⃗W1 • ⃗W2 = ((C−I). ⃗W1) • ⃗W2 gives a measure of the deformation with ⃗W2 +as a reference, measure that is used to build first order constitutive laws for the stress (Cauchy). +G.1 +Transposed F T: Inner dot products required +We first give the functional definition of F T (qualitative); Then we get the usual matrix representation +of F T relative to observers (quantification). +G.1.1 +Definition of the function F T +At t0, a past observer chose an inner dot product (·, ·)G in ⃗Rn +t0, and at t the present observer chooses an +inner dot product (·, ·)g in ⃗Rn +t . +By definition, the transposed of the linear map F(P) ∈ L(⃗Rn +t0; ⃗Rn +t ) relative to (·, ·)G and (·, ·)g is the +linear map F(P)T +Gg ∈ L(⃗Rn +t ; ⃗Rn +t0) defined by, for all ⃗UP ∈ ⃗Rn +t0 (vector at P) and ⃗wp ∈ ⃗Rn +t (vector at p), +(F(P)T +Gg.⃗wp, ⃗UP )G = (F(P).⃗UP , ⃗wp)g, +in short +(F T +Gg.⃗w, ⃗U)G = (F.⃗U, ⃗w)g , +(G.1) +122 + +123 +G.1. +Transposed F T : Inner dot products required +see (A.68). This defines F T +Gg(p) := F(P)T +Gg when p = Φ(P): +F T +Gg : +� +� +� +Ωt → L(⃗Rn +t ; ⃗Rn +t0) +p → F T +Gg(p) := F(P)T +Gg +� +� +� , +so in short +(F T .⃗w) •G ⃗U = ⃗w •g (F.⃗U) , +(G.2) +without forgetting that F T := F T +Gg depends on (·, ·)G and (·, ·)g. +Exercice G.1 1. In F T .⃗z. ⃗W = ⃗z.F. ⃗W = F. ⃗W.⃗z = ⃗W.F T .⃗z, which dots are inner dot products? +2. What does F. ⃗W1.F. ⃗W2 = ⃗W1.F T .F. ⃗W2 mean? +Answer. 1. No choice: ( ⃗W, ⃗z) ∈ ⃗Rn +t0 × ⃗Rn +t , so (F T .⃗z) •G ⃗W = ⃗z •g (F. ⃗W) = (F. ⃗W) •g ⃗z = ⃗W •G (F T .⃗z). +2. No choice: ⃗W1, ⃗W2 ∈ ⃗Rn +t0, so (F. ⃗W1) •g (F. ⃗W2) = ⃗W1 •G (F T .(F. ⃗W2)). +Remark G.2 More generally, on a surface Ω (a manifold), (G.1) is defined for all (⃗UP , ⃗wp) ∈ TP Ωt0 × +TpΩt, where Tpτ Ωτ is the tangent space at Ωτ at pτ. +G.1.2 +Quantification with bases (matrix representation) +Classical notations: (⃗ai) is a basis in ⃗Rn +t0, and (⃗bi) is a basis in ⃗Rn +t . Marsden–Hughes duality notations: +( ⃗EI) is a basis in ⃗Rn +t0 and (⃗ei) is a basis in ⃗Rn +t . And the reference to the points P and p is omitted to +lighten the writings (use the full notation of § G.1.1 if in doubt). +Let [G] := [(⃗ai,⃗aj)G], [g] := [(⃗bi,⃗bj)g], [F]|⃗a,⃗b = [Fij] =noted [F], [F T ]|⃗b,⃗a = [(F T )ij] =noted [F T ]. +(G.1) gives [⃗U]T .[G].[F T .⃗w] = [F.⃗U]T .[G].[⃗w], thus [⃗U]T .[G].[F T ].[⃗w] = [⃗U]T .[F]T .[g].[⃗w], for all ⃗U, ⃗w, +thus +[G].[F T ] = [F]T .[g], +i.e. +[F T ] = [G]−1.[F]T .[g] . +(G.3) +Remark G.3 If (⃗ai) and (⃗bi) are (·, ·)G and (·, ·)g-orthonormal bases, then [C] = [F]T .[F]. But recall: If +you need to work with a coordinate system, then the bases in use are the coordinate system bases which +are not orthonormal in general, i.e. [G]−1 ̸= I and [g]−1 ̸= I in general. +Exercice G.4 Use classical notation, then Marsden duality notations, to express (G.3) with components. +Answer. Classical notations: +Gij = G(⃗ai,⃗aj), +gij = g(⃗bi,⃗bj), +i.e. +[G]⃗a = [Gij], +[g]|⃗b = [gij], +and +F.⃗aj = +n +� +i=1 +Fij⃗bi, +F T .⃗bj = +n +� +i=1 +(F T )ij⃗ai, +i.e. +[F]|⃗a,⃗b = [Fij], +[F T ]|⃗b,⃗a = [(F T )ij]. +(G.4) +Then (F T .⃗bj,⃗ai)G =(G.1)(⃗bj, F.⃗ai)g gives (�n +k=1(F T )kj⃗ak,⃗ai)G = (⃗bj, �n +k=1Fki⃗bk)g, thus �n +k=1(F T )kj(⃗ak,⃗ai)G = +�n +k=1Fki(⃗bj,⃗bk)g with Fki = ([F]T )ik, thus +n +� +k=1 +Gik(F T )kj = +n +� +k=1 +([F]T )ikgkj, +i.e. +(F T )ij = +n +� +k,ℓ=1 +([G]−1)ikFℓkgℓj, +(G.5) +for all i, j, thus (G.3). +Marsden notations: GIJ = G( ⃗EI, ⃗Ej), gij = g(⃗ei,⃗ej), F. ⃗EJ = �n +i=1F i +J⃗ei, F T .⃗ej = �n +I=1(F T )I +j ⃗EI, thus +n +� +K=1 +GIK(F T )K +j = +n +� +k=1 +F k +Igkj, +i.e. +(F T )I +j = +n +� +K,k=1 +GIKF k +Kgkj +where +[GIJ] := [GIJ]−1. +G.1.3 +Remark: F ∗ +(For mathematicians: F ∗ doesn’t seem to be very useful in mechanics, apart from making simple things +difficult, and playing games with components and duality notations...). +Definition G.5 The adjoint of the linear map F ∈ L(⃗Rn +t0; ⃗Rn +t ) (acting on vectors) is the linear map +F ∗ ∈ L(⃗Rn∗ +t ; ⃗Rn∗ +t0 ) (acting on functions) canonically defined by, for all m ∈ ⃗Rn∗ +t , +F ∗(m) := m ◦ F, +written +F ∗.m = m.F (∈ ⃗Rn∗ +t0 ). +(G.6) +123 + +124 +G.2. +Cauchy–Green deformation tensor C +So, for all (m, ⃗W) ∈ ⃗Rn∗ +t +× ⃗Rn +t0, +(F ∗.m). ⃗W = m.F. ⃗W (∈ R). +(G.7) +Quantification (matrix representation): (πai) and (πbi) are the covariant dual bases of (⃗ai) and (⃗bi). +Let (F ∗)ij be the components of F ∗ relative to these dual bases: +F ∗.πbj = +n +� +I=1 +(F ∗)ijπai, +i.e. +[F ∗]|πb,πa = [(F ∗)ij]. +(G.8) +(G.7) gives (F ∗.πbj).⃗ai = πbj.F.⃗ai, thus +∀i, j, (F ∗)ij = Fji , +i.e. +[F ∗]|πb,πa = ([F]|⃗a,⃗b)T , +in short +[F ∗] = [F]T . +(G.9) +Marsden duality notations: F ∗.ej = �n +I=1(F ∗)IjEI gives (F ∗)Ij = F jI for all I, j. +Interpretation of F ∗. As usual in classical mechanics, we use Euclidean dot products, here (·, ·)G in ⃗Rn +t0 +and (·, ·)g in ⃗Rn +t . Then we use the (·, ·)G-Riesz representation vector ⃗RG(F ∗.m) ∈ ⃗Rn +t0 of F ∗.m ∈ ⃗Rn∗ +t0 , +and the (·, ·)g-Riesz representation vector ⃗Rg(m) ∈ ⃗Rn +t of m ∈ ⃗Rn∗ +t , so, for all m ∈ ⃗Rn∗ +t +and ⃗W ∈ ⃗Rn +t0, +(F ∗.m). ⃗W = ⃗RG(F ∗.m) •G ⃗W, +and +m.(F. ⃗W) = ⃗Rg(m) •g F. ⃗W = (F T .⃗Rg(m)) •G ⃗W. +(G.10) +Thus (G.7) gives ⃗RG(F ∗.m) = F T .⃗Rg(m), thus +⃗RG.F ∗ = F T .⃗Rg, +i.e. +F ∗ = ⃗RG +−1.F T .⃗Rg. +(G.11) +Remark G.6 The definition of F ∗ is intrinsic to F (objective), while the definition of F T is not intrinsic +to F (not objective) since it needs inner dot products (observer choices) to be defined. +G.2 +Cauchy–Green deformation tensor C +G.2.1 +Definition of C +Consider vectors ⃗Wi ∈ ⃗Rn +t0 at P, i = 1, 2, and their push forwards ⃗wi toward p = Φ(P), i.e. +⃗wi = F. ⃗Wi, +(G.12) +short notation for ⃗wi(p) = F(P). ⃗Wi(P). With the chosen inner dot products (·, ·)G in ⃗Rn +t0 and (·, ·)g +in ⃗Rn +t , we get (⃗w1(p), ⃗w2(p))g = (F(P). ⃗W1(P), F(P). ⃗W2(P))g=(G.2)(F T +Gg(p).F(P). ⃗W1(P), ⃗W2(P))G when +p = Φ(P), written in short: +(⃗w1, ⃗w2)g = (F. ⃗W1, F. ⃗W2)g = (F T .F +� �� � +C +. ⃗W1, ⃗W2)G, +(G.13) +Definition G.7 The (right) Cauchy–Green deformation tensor at P ∈ Ωt0 relative to (·, ·)G and (·, ·)g, +is the endomorphism CGg(P) ∈ L(⃗Rn +t0; ⃗Rn +t0) defined by +CGg(P) := F T +Gg(p) ◦ F(P), +in short +C := F T .F . +(G.14) +So +C = F T ◦ F : ⃗W +F +−→ F( ⃗W) +F T +−→ F T (F( ⃗W)) = C( ⃗W), +(G.15) +with F and F T linear, thus C is linear and C( ⃗W) is written C. ⃗W = F T .F. ⃗W. And (G.13) tells that C +is characterized by, for all ⃗W1, ⃗W2 ∈ ⃗Rn +t0, +⃗w1 •g ⃗w2 = (C. ⃗W1) •G ⃗W2 = (F. ⃗W1) •g (F. ⃗W2) . +(G.16) +Moreover, (·, ·)g being symmetric (inner dot product), C is a (·, ·)G-symmetric endomorphism in ⃗Rn +t0, i.e., +for all ⃗W1, ⃗W2 ∈ ⃗Rn +t0, +(C. ⃗W1, ⃗W2)G = ( ⃗W1, C. ⃗W2)G, +i.e. +(C. ⃗W1) •G ⃗W2 = ⃗W1 •G (C. ⃗W2), +(G.17) +since (F T .F. ⃗W1, ⃗W2)G = (F. ⃗W1, F. ⃗W2)g = ( ⃗W1, F T .F. ⃗W2)G. +124 + +125 +G.3. +Time Taylor expansion of C +G.2.2 +Quantification +(G.14) gives [C] = [F T ].[F], with [F T ] =(G.3)[G]−1.[F]T .[g], thus +[C] = [G]−1.[F]T .[g].[F] , +(G.18) +short notation for [CGg]|⃗a = [G]−1 +|⃗a .([F]|⃗a,⃗b)T .[g]|⃗b.[F]|⃗a,⃗b. +Exercice G.8 Use classical notation, then duality notations, to express (G.18) with components. +Answer. Classical notations: +F.⃗aj = +n +� +i=1 +Fij⃗bi +and +C.⃗aj = +n +� +i=1 +Cij⃗ai, +i.e. +[F]|⃗a,⃗b = [Fij] +and +[C]|⃗a = [Cij]. +(G.19) +(G.16)-(G.17) +give +(⃗ai, C.⃗aj)G += +(F.⃗ai, F.⃗aj)g, +so +(⃗ai, � +k Ckj⃗ak)G += +(� +k Fki⃗bk, � +ℓ Fℓj⃗bℓ)g, +thus +� +k Ckj(⃗ai,⃗ak)G = � +kℓ Fki(⃗bk,⃗bℓ)gFℓj, i.e. +n +� +k=1 +GikCkj = +n +� +k,ℓ=1 +Fki gkℓFℓj = +n +� +k,ℓ=1 +([F]T )ik gkℓFℓj, +so +[G].[C] = [F]T .[g].[F] , +(G.20) +so Cij = �n +k,ℓ,m=1([G]−1)imFkm gkℓFℓj = �n +k,ℓ,m=1([G]−1)im([F]T )mk gkℓFℓj. Duality notations: +F. ⃗EJ = +n +� +i=1 +F i +J⃗ei +and +C. ⃗EJ = +n +� +I=1 +CI +J ⃗EI, +i.e. +[F]| ⃗ +E,⃗e = [F i +J] +and +[C]| ⃗ +E = [CI +J], and +n +� +K=1 +GIKCK +J = +n +� +k,ℓ=1 +F k +I gkℓF ℓ +J, +and +CI +J = +n +� +k,ℓ,M=1 +GIMF k +M gkℓF ℓ +J +when +[GIJ] := [GIJ]−1. +(G.21) +Exercice G.9 (·, ·)G is a Euclidean dot product in foot, (·, ·)g is a Euclidean dot product in metre, so +(·, ·)g = µ2(·, ·)G with µ ≃ 0.3048; And (⃗ai) = (⃗bi) is a (·, ·)G-orthonormal basis. Prove +[C] = µ2[F]T .[F]. +(G.22) +Answer. [C]|⃗a =(G.18) [G]−1 +|⃗a .[F]T +|⃗a,⃗a.[g]|⃗a.[F]|⃗a,⃗a gives [C]|⃗a = I.[F]T +|⃗a,⃗a.µ2I.[F]|⃗a,⃗a. Shorten notation = (G.22). +G.3 +Time Taylor expansion of C +Here we use a unique inner dot product (·, ·)G = (·, ·)g at all time (to compare results in the vicinity +of t0). Moreover we use an orthonormal basis (to lighten the notations), thus, in short, [C] = [F]T .[F]. +P is fixed, Ct0 +t (P) =noted C(t), and [C(t)] = [F(t)]T .[F(t)] (since [G] = [g] = I here), and +⃗V t0 +t (P) =noted ⃗V (t) and ⃗At0 +t (P) =noted ⃗A(t) are the Lagrangian velocities and accelerations, and ⃗v(t, p) +and ⃗γ(t, p) are the Eulerian velocities and accelerations at t at p = Φt0 +t (t, P). +With Lagrangian variables (used to define C): F(t+h) = F(t) + h d⃗V (t) + h2 +2 d ⃗A(t) + o(h2) gives +[C(t+h)] = [F(t+h)]T .[F(t+h)] += [F T + h d⃗V T + h2 +2 d ⃗AT + o(h2)](t)[F + h d⃗V + h2 +2 d ⃗A + o(h2)](t) += [C(t) + h ([F T ].[d⃗V ] + [d⃗V ]T .[F])(t) +� +�� +� +=[(Ct0 +P )′(t)] =noted [C′(t)] ++h2 +2 ([F]T .[d ⃗A] + 2[d⃗V ]T .[d⃗V ] + [d ⃗A]T .[F])(t) +� +�� +� +=[(Ct0 +P )′′(t)] =noted [C′′(t)] +)(t) + o(h2). +(G.23) +(As usual with Lagrangian variables, we have three times involved: t0, t and t+h).) In particular +[Ct0 +P (t0+h)] = I + ([d⃗V ] + [d⃗V ]T )(t0) + h2 +2 ([d ⃗A] + 2[d⃗V ]T .[d⃗V ] + [d ⃗A]T )(t0) + o(h2). +(G.24) +Abusively written Ct0 +P (t0+h) = I + (d⃗V + d⃗V T )(t0) + h2 +2 (d ⃗A + 2d⃗V T .d⃗V + d ⃗AT )(t0) + o(h2), but don’t +forget it is a matrix meaning. +125 + +126 +G.4. +Remark: C♭ +With Eulerian variables: With p(t) = Φt0(t, P), we have d⃗V t0(t, P) = d⃗v(t, p(t)).F(t) and d ⃗At0(t, P) = +d⃗γ(t, p(t)).F(t), thus writing d⃗v := d⃗v(t, p(t)) and d⃗γ := d⃗γ(t, p(t)) (for short), +Ct0 +P (t+h) = Ct0 +P (t) + h (F T (t).(d⃗v + d⃗vT )(t, p(t)).F(t)) ++ h2 +2 (F T (t).(d⃗γ + 2d⃗vT .d⃗v + d⃗γT )(t, p(t)).F(t)) + o(h2). +(G.25) +abusive notation of [Ct0 +P (t+h)] = ... (matrices relative to a basis). +Remark G.10 F ′′ = d ⃗A is easy to interpret, but C′′ = F T .d ⃗A + 2d⃗V T .d⃗V + d ⃗AT .F = (F T .d ⃗A + +d⃗V T .d⃗V ) + (F T .d ⃗A + d⃗V T .d⃗V )T is not that easy to interpret (and in not linear in ⃗V ). +We already had a problem with the composition of flows: The formula F t0 +t2 = F t1 +t2 .F t0 +t1 is simple +(determinism), but the formula Ct0 +t2 = (F t0 +t2 )T .F t0 +t2 = (F t0 +t1 )T .(F t1 +t2 )T .F t1 +t2 .F t0 +t1 = (F t0 +t1 )T .Ct1 +t2 .F t0 +t1 is “not that +simple” (̸= Ct1 +t2 .Ct0 +t1 ). (Indeed, to consider C instead of F amounts to consider the “motion squared”, cf. +(C. ⃗W, ⃗W)g = ||F. ⃗W||2 +g.) +Since C′(t0) = d⃗V (t0) + d⃗V (t0)T this may have little consequences for linear approximation near t0, +but ultimately not small consequences for second-order approximations (and large deformations) if C′′ is +used to make constitutive laws. The consideration of Lie derivatives may be an interesting alternative. +G.4 +Remark: C♭ +For mathematicians: May produce errors, misuses, covariance-contravariance confusion, see next § G.4.2. +For the general ♭ notation see § A.10.4. +G.4.1 +Definition of C♭... +Definition G.11 At P ∈ Ωt0, the bilinear form C♭ +Gg(P) =noted C♭ ∈ L(⃗Rn +t0, ⃗Rn +t0; R) associated with the +linear map CGg(P) =noted C ∈ L(⃗Rn +t0; ⃗Rn +t0) is defined by, for all ⃗W1, ⃗W2 ∈ ⃗Rn +t0 vectors at P, +C♭( ⃗W1, ⃗W2) := ( ⃗W1, C. ⃗W2)G +(= (F. ⃗W1, F. ⃗W2)g). +(G.26) +Then C♭ is a bilinear symmetric form (trivial) and is a metric in ⃗Rn +t0 when F t0 +t +=noted F is a diffeo- +morphism (usual hypothesis), but not a Euclidean one (it is iff C = I i.e. for rigid body motions). +Quantification: (G.26) gives [ ⃗W2]T .[C♭].[ ⃗W1] = [ ⃗W2]T .[G].[C].[ ⃗W1] for all ⃗W1, ⃗W2 since C♭ and (·, ·)G +are symmetric, thus +[C♭] = [G].[C] +(= [F]T .[g].[F]). +(G.27) +Exercice G.12 Use duality notations to express (G.27) with components, and explain the flat ♭ notation. +Answer. (G.26) gives C♭( ⃗EJ, ⃗EI) := (C. ⃗EJ, ⃗EI)G. Thus with C♭( ⃗EI, ⃗EJ) = CIJ and C. ⃗EJ = � +I CI +J ⃗EI we get +CJI = � +K CK +J( ⃗EK, ⃗EI)G = � +K CK +JGKI; And C♭ and (·, ·)G are symmetric, thus +CIJ = +� +K +GIKCK +J, +i.e. +[C♭]| ⃗ +E = [G]| ⃗ +E.[C]| ⃗ +E. +(G.28) +The flat notation C♭ is due to: The top index I in CI +J has been transformed into a bottom index in CIJ in C♭, +which characterizes a change of variance because of the use of an inner dot product. +And (G.27) also gives CIJ = (F. ⃗EI, F. ⃗EJ)g = � +kℓ F k +IF ℓ +J(⃗bk,⃗bℓ)g, thus +CIJ = +� +kℓ +F k +IgkℓF ℓ +J = +� +kℓ +(F T )I +kgkℓF ℓ +J, +i.e. +[C♭]| ⃗ +E = ([F]| ⃗ +E,⃗e)T .[g]|⃗e.[F]| ⃗ +E,⃗e. +(G.29) +G.4.2 +... and remarks about C♭... and Jaumann +C♭ can also be defined only with (·, ·)g by, for all ⃗W1, ⃗W2 ∈ ⃗Rn +t0, +C♭ +g( ⃗W1, ⃗W2) := (F. ⃗W1, F. ⃗W2)g, +(G.30) +i.e., C♭ +g := g∗ =noted C♭. So we can also say that C♭ +g is the pull-back of the metric (·, ·)g by Φ, see (8.9). +126 + +127 +G.5. +Stretch ratio and deformed angle +• However C♭ = C♭ +g is useless in itself: C♭ is not a Euclidean dot product (it is a metric defined at +each P by C♭ +g(P)( ⃗W1, ⃗W2) := (F(P). ⃗W1, F(P). ⃗W2)g for all ⃗W1, ⃗W2 ∈ ⃗Rn +t0 vectors at P). In fact, C♭ is +only useful to characterize a deformation if the value C♭( ⃗W1, ⃗W2) can be compared with the initial value +( ⃗W1, ⃗W2)G, i.e. if a Euclidean dot product (·, ·)G was introduced in ⃗Rn +t0: This is why C♭ is classically +defined from C, cf. (G.26). +• You may want to use the infinitesimal strain tensor ε = F +F T +2 +− I, or the Green–Lagrange defor- +mation tensor E = 1 +2(C − I), obtained from F T := F T +Gg (essential). +• There is no objective “trace” for a +�0 +2 +� +tensor like C♭, while Tr(C) is objective since C is an endo- +morphism (≃ a +�1 +1 +� +tensor). +• The Lie derivatives of a second order tensor depends on the type of the tensor, and the Lie derivative +of the +�1 +1 +� +tensor like C gives the Jaumann derivative, which is usually preferred to the Lie derivative of +the +�0 +2 +� +tensor like C♭ which is the lower convected Lie derivative, see remark G.13. +So the introduction and use of C♭ in mechanics mostly complicate things unnecessarily, and interferes +with basic understandings like the distinction between covariance and contravariance. +Remark G.13 Interpretation issue (with Jaumann). +2D = d⃗v + d⃗vT gives 2 DD +Dt = D(d⃗v) +Dt ++ D(d⃗v)T +Dt += d⃗γ + d⃗γT − d⃗v.d⃗v − d⃗vT .d⃗vT , thus, with (G.23) and +keeping in mind the matrix meaning, +C′′(t) = F(t)T .(2DD +Dt + d⃗v.d⃗v + d⃗vT .d⃗vT + 2d⃗vT .d⃗v)(t, p(t)).F(t) += 2F(t)T .(DD +Dt + D.d⃗v + d⃗vT .D)(t, p(t)).F(t). +(G.31) +The DD +Dt + D.d⃗v + d⃗vT .D term looks like a lower-convected Lie derivative, but with d⃗vT instead of d⃗v∗, +cf. (9.58); So you may find (G.31) written as C′′ = 2F T .L⃗vD.F. But you get disappointing results when +using the the lower convected Lie derivative (Jaumann is usually preferred). In fact, it is L⃗vD♭ (lower +convected Lie derivative) that should be used, where D♭ +g := +d⃗v♭ +g+(d⃗v♭ +g)T +2 +, to get (C♭)′′ = 2F T .L⃗vD♭ +g.F. +G.5 +Stretch ratio and deformed angle +Here (·, ·)g = (·, ·)G, i.e. at t0 and t we use the same Euclidean dot product, to be able to compare the +lengths relative to the same unit of measurement. (If (·, ·)g ̸= (·, ·)G then use (·, ·)g = µ2(·, ·)G.) +G.5.1 +Stretch ratio +The stretch ratio at P ∈ ⃗Rn +t0 between t0 and t for a ⃗WP ∈ ⃗Rn +t0 is defined by +λ( ⃗WP ) := ||⃗wp||G +|| ⃗WP ||G += ||FP . ⃗WP ||G +|| ⃗WP ||G +(= ||FP .( +⃗WP +|| ⃗WP ||G +)||G) +(G.32) +where ⃗wp = FP . ⃗WP is the deformed vector by the motion at p = Φ(P). I.e., in short +∀ ⃗W ∈ ⃗Rn +t0 s.t. || ⃗W|| = 1, +λ( ⃗W) := ||F. ⃗W||. +(G.33) +(You may find: λ(d ⃗X) = ||F.d ⃗X|| with d ⃗X a unit vector(!); This notation should be avoided, see § 4.3.) +G.5.2 +Deformed angle +Recall: The angle θt0 = +� +( ⃗W1, ⃗W2) formed by two vectors ⃗W1 and ⃗W2 in ⃗ +Rn +t0−{⃗0} at P ∈ Ωt0 is given by +cos(θt0) = +⃗ +W1 +|| ⃗ +W1||G +• +⃗ +W2 +|| ⃗ +W2||G (= ( +⃗ +W1 +|| ⃗ +W1||G , +⃗ +W2 +|| ⃗ +W2||G )G). +With the deformed vectors ⃗wi = F. ⃗Wi at p = Φt0 +t (P), the deformed angle is θt defined by +cos(θt) := +� +(⃗w1, ⃗w2) = +⃗w1 +||⃗w1|| +• +⃗w2 +||⃗w2|| = +F. ⃗W1 +||F. ⃗W1|| +• +F. ⃗W2 +||F. ⃗W2|| +(= (C. ⃗W1) • ⃗W2 +||⃗w1|| ||⃗w2|| ). +(G.34) +127 + +128 +G.6. +Decompositions of C +G.6 +Decompositions of C +G.6.1 +Spherical and deviatoric tensors +Definition G.14 The deformation spheric tensor is +Csph = 1 +nTr(C) I, +(G.35) +with Tr(C) = the trace of the endomorphism C (there is no “trace” for the +�0 +2 +� +tensor C♭). +Definition G.15 The deviatoric tensor is +Cdev = C − Csph. +(G.36) +(So Tr(Cdev) = 0 , and C = Csph + Cdev.) +G.6.2 +Rigid motion +The deformation is rigid iff, for all t0, t, +(F t0 +t )T .F t0 +t += I, +i.e. +Ct0 +t = I, +written +C = I = F T .F. +(G.37) +Thus, after a rigid body motion, lengths and angles are left unchanged. +G.6.3 +Diagonalization of C +Proposition G.16 C = F T .F being symmetric positive, C is diagonalizable, its eigenvalues are positive, +and ⃗ +Rn +t0 has an orthonormal basis made of eigenvectors of C. +Proof. (C(P). ⃗W1, ⃗W2)G = (F(P). ⃗W1, F(P). ⃗W2)g = ( ⃗W1, C(P). ⃗W2)G, thus C is (·, ·)G-symmetric. +(C. ⃗W1, ⃗W1)G = (F. ⃗W1, F. ⃗W1)g = ||F. ⃗W1||2 +g > 0 when ⃗W1 ̸= ⃗0, since F invertible (Φt0 +t is supposed to +be a diffeomorphism). Thus C est (·, ·)G-symmetric definite positive real endomorphism. +Definition G.17 Let λi be the eigenvalues of C. Then the √λi are called the principal stretches. And +the associated eigenvectors give the principal directions. +G.6.4 +Mohr circle +This § deals with general properties of 3 ∗ 3 symmetric positive endomorphism, like Ct0 +t (P). +Consider ⃗R3 with a Euclidean dot product (·, ·)R3 and a (·, ·)R3-orthonormal basis (⃗ai). +Let M : ⃗R3 → ⃗R3 be a symmetric positive endomorphism. Thus M is diagonalizable in a (·, ·)R3- +orthonormal basis (⃗e1,⃗e2,⃗e3), that is, ∃λ1, λ2, λ3 ∈ R, ∃⃗e1,⃗e2,⃗e3 ∈ ⃗R3 s.t. +M.⃗ei = λi⃗ei +and +(⃗ei,⃗ej)R3 = δij, +so +[M]|⃗e = diag(λ1, λ2, λ3) = +� +� +λ1 +0 +0 +0 +λ2 +0 +0 +0 +λ3 +� +� . +(G.38) +And the orthonormal basis (⃗e1,⃗e2,⃗e3) is ordered s.t. λ1 ≥ λ2 ≥ λ3 (> 0). +Let S be the unit sphere in R3, that is the set {(x, y, z) : x2 + y2 + z2 = 1}. Its image M(S) by M +is the ellipsoid {(x, y, z) : x2 +λ2 +1 + y2 +λ2 +2 + z2 +λ2 +3 = 1}. Then consider ⃗n = � +i ni⃗ei s.t. ||⃗n||R3 = 1: +[⃗n]|⃗e = +� +� +n1 +n2 +n3 +� +� +with +n2 +1 + n2 +2 + n2 +3 = 1. +(G.39) +Thus its image ⃗A = M.⃗n ∈ M(S) satisfies +⃗A = M.⃗n, +[ ⃗A]|⃗e = +� +� +λ1n1 +λ2n2 +λ3n3 +� +� . +(G.40) +Then define +An = ( ⃗A,⃗n)R3, +⃗A⊥ = ⃗A − An⃗n, +A⊥ := || ⃗A⊥||. +(G.41) +So ⃗A = An⃗n + ⃗A⊥ ∈ Vect{⃗n} ⊗ Vect{⃗n}⊥. (Remark: ⃗A⊥ is not orthonormal to the ellipsoid M(S), but +is orthonormal to the initial sphere S.) +128 + +129 +G.7. +Green–Lagrange deformation tensor E +Mohr Circle purpose: To find a relation: +A⊥ = f(An), +(G.42) +relation between “the normal force An” (to the initial sphere) and the “tangent forceA⊥” (to the initial +sphere). +(G.39), (G.40) and An = (M.⃗n,⃗n)R3 give +� +� +� +� +� +n2 +1 + n2 +2 + n2 +3 = 1, +λ1n2 +1 + λ2n2 +2 + λ3n2 +3 = An +λ2 +1n2 +1 + λ2 +2n2 +2 + λ2 +3n2 +3 = || ⃗A||2 = A2 +n + A2 +⊥. +(G.43) +This is linear system with the unknowns n2 +1, n2 +2, n2 +3. The solution is +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +n2 +1 = A2 +⊥ + (An − λ2)(An − λ3) +(λ1 − λ2)(λ1 − λ3) +, +n2 +2 = A2 +⊥ + (An − λ3)(An − λ1) +(λ2 − λ3)(λ2 − λ1) +, +n2 +3 = A2 +⊥ + (An − λ1)(An − λ2) +(λ3 − λ1)(λ3 − λ2) +. +(G.44) +The n2 +i being non negative, and with λ1 > λ2 > λ3 ≥ 0, we get +� +� +� +� +� +A2 +⊥ + (An − λ2)(An − λ3) ≥ 0, +A2 +⊥ + (An − λ3)(An − λ1) ≤ 0, +A2 +⊥ + (An − λ1)(An − λ2) ≥ 0. +(G.45) +Then let x = An and y = A⊥, and consider, for some a, b ∈ R, the equation +y2 + (x − a)(x − b) = 0, +so +(x − a+b +2 )2 + y2 = (a−b)2 +4 +. +This is the equation of a circle centered at ( a+b +2 , 0) with radius |a−b| +2 +. +Thus (G.45)2 tells that An and A⊥ are inside the circle centered at ( λ1+λ3 +2 +, 0) with radius λ1−λ3 +2 +, +and (G.45)1,3 tell that An and A⊥ are outside the other circles (adjacent and included in the first, +drawing). +Exercice G.18 What happens if λ1 = λ2 = λ3 > 0? +Answer. Then +� +� +� +� +� +� +� +� +� +� +� +� +� +n2 +1 + n2 +2 + n2 +3 = 1, +n2 +1 + n2 +2 + n2 +3 = An +λ1 , +n2 +1 + n2 +2 + n2 +3 = A2 +n + A2 +⊥ +λ2 +1 +. +� +� +� +� +� +� +� +� +� +� +� +� +� +Thus An = λ1 and A2 +n + A2 +⊥ = λ2 +1, thus A⊥ = 0. Here C = λ1I, +and we deal with a dilation: A⊥ = 0. +Exercice G.19 What happens if λ1 = λ2 > λ3 > 0? +Answer. Then +� +� +� +� +� +� +� +n2 +1 + n2 +2 + n2 +3 = 1, +λ1(1 − n2 +3) + λ3n2 +3 = An, +λ2 +1(1 − n2 +3) + λ2 +3n2 +3 = A2 +n + A2 +⊥. +� +� +� +� +� +� +� +Thus An = λ1 − (λ1 − λ3)n2 +3 ∈ [λ3, λ1], and A⊥ = ±(λ2 +1 − +(λ2 +1 − λ2 +3)n2 +3 − A2 +n) +1 +2 , with A2 +n + A2 +⊥ a point on the circle with radius λ2 +1(1 − n2 +3) + λ2 +3n2 +3. +G.7 +Green–Lagrange deformation tensor E +(G.13) gives (⃗w1, ⃗w2)g = (F. ⃗W1, F. ⃗W2)g = (C. ⃗W, ⃗W)G at p = Φ(P), thus +(⃗w1, ⃗w2)g − ( ⃗W1, ⃗W2)G = ((C − I). ⃗W1, ⃗W2)G. +(G.46) +129 + +130 +G.8. +Small deformations (linearization): The infinitesimal strain tensor ε +Definition G.20 The Green–Lagrange tensor (or Green–Saint Venant tensor) at P relative to t0 and t +is the endomorphism Et0 +t (P) ∈ L(⃗Rn +t0; ⃗Rn +t0) defined by +Et0 +t (P) := Ct0 +t (P) − It0 +2 +, +in short +E = C − I +2 +(= F T .F − I +2 +). +(G.47) +(In particular E = 0 for rigid body motions.) And Et0 +t : Ωt0 → L(⃗Rn +t0; ⃗Rn +t0) is the Green–Lagrange tensor +relative to t0 and t. +The 1 +2 because (C., .) = (F., F.) corresponds to the “motion squared”, see the following linearization. +And we get the time Taylor expansion of Et0 +P (t) = 1 +2(Ct0 +P (t) − It0) with p(t) = Φt0 +P (t) and (G.25): +Et0 +P (t+h) = F t0 +P (t)T . +� +h d⃗v + d⃗vT +2 ++ h2 +2 (d⃗γ + d⃗γT +2 ++ d⃗vT .d⃗v) +� +(t, p(t)).F t0 +P (t) + o(h2) += F t0 +P (t)T . +� +h D + h2 ((DD +Dt + D.d⃗v + d⃗vT .D)(t, p(t))).F t0 +P (t) + o(h2). +(G.48) +G.8 +Small deformations (linearization): The infinitesimal strain tensor ε +G.8.1 +Landau notations big-O and little-o +Reminder. Let f, g : R → R and x0 ∈ R. +f = O(g) near x0 +⇐⇒ +∃C > 0, ∃η > 0, ∀x s.t. |x − x0| < η, |f(x)| < C|g(x)|. +(G.49) +and f is said to be “comparable with g” near x0. +If |g| > 0 then the conclusion reads |f(x)| +|g(x)| < C; And f = O(xn) near x=0 iff |f(x)| +|xn| < C near x=0. +And +f = o(g) near x0 +⇐⇒ +∀ε > 0, ∃η > 0, ∀x s.t. |x − x0| < η, |f(x)| < ε|g(x). +(G.50) +and f is said to be “negligible compared with g near x0”. +If |g| > 0 then the conclusion reads |f(x)| +|g(x)| −→x→x0 0. And f = o(xn) near x=0 iff |f(x)| +|xn| −→x→0 0. +G.8.2 +Definition of the infinitesimal strain tensor ε +The motion is supposed to be C2. Along a trajectory, with F t0 +P (t0) = I we have, near t0, +F t0 +P (t0+h) = I + O(h), +(G.51) +thus F t0 +P (t0+h). ⃗W = ⃗W + O(h) for all ⃗W ∈ ⃗Rn +t0, i.e., near t0, +||⃗w − ⃗W|| = O(h) +when +⃗w = F t0 +P (t0+h). ⃗W. +(G.52) +This supposes the use of a unique inner dot product (·, ·)G = (·, ·)g at all time, and (G.52) means +||⃗w − ⃗W||g = O(h) near t0. +Definition G.21 If (·, ·)g is an inner dot product, the same at all time, and if (⃗ei) is a (·, ·)g-orthonormal +basis, the same at all time, then the infinitesimal strain tensor at P is the matrix defined by +[ε(P)]|⃗e = +[F(P)]|⃗e + [F(P)]T +|⃗e +2 +− [I], +(G.53) +abusively written in short, +ε := F + F T +2 +− I +(matrix meaning). +(G.54) +(And more precisely, at P ∈ Ωt0 and between t0 and t, [εt0 +t (P)]|⃗e = +[F t0 +t (P )]|⃗e+[F t0 +t (P )]T +|⃗e +2 +− [I].) +So ε. ⃗W = F. ⃗ +W +F T . ⃗ +W +2 +− . ⃗W means [ε]|⃗e.[ ⃗W]|⃗e = +[F ]|⃗e.[ ⃗ +W ]|⃗e+[F ]T +|⃗e.[ ⃗ +W ]|⃗e +2 +− [ ⃗W]|⃗e. +130 + +131 +H.1. +Definition +Remark G.22 ε in (G.54) cannot be a tensor (cannot be a function) since F t0 +t (P) : +⃗ +Rn +t0 → ⃗ +Rn +t and +F t0 +t (P)T : ⃗ +Rn +t → ⃗ +Rn +t0 and It0 : ⃗ +Rn +t0 → ⃗ +Rn +t0 don’t have the same definition domain. +So ε is not a function, is not a tensor: It is a matrix... But is called “the infinitesimal strain tensor”... +Proposition G.23 The Green–Lagrange tensor E = F T .F −I +2 +∈ L(⃗Rn +t0; ⃗Rn +t0) satisfies near t0: +E = ε + o(t−t0) +(= F + F T +2 +− I + o(t−t0)) +(matrix meaning), +(G.55) +which means [E] = [ε] + o(t−t0). Thus, “for small deformations” we write E ≃ ε, i.e. E ≃ F +F T +2 +− I. +Interpretation: (G.55) is a linearization of E, since we keep the linear part of the “quadratic” E = +1 +2(F T .F − I) given by (E. ⃗W, ⃗U)g = 1 +2 +� +(F. ⃗W, F.⃗U)g − ( ⃗W, ⃗U)g +� +for all ⃗U, ⃗W ∈ ⃗Rn +t0 (“motion squared” cf. +the (F·, F·)g term). +Proof. A (·, ·)g-orthonormal basis being chosen, [F T ] =(G.3)[F]T , thus [C] = [F]T .[F], thus +2[E] = [C] − [I] = [F]T .[F] − [I] = ([F]T − [I).([F] − [I) + [F]T + [F] − 2[I]. +(G.56) +Then, near t0 and with h = t−t0, (G.51) gives ([F]T − [I]).([F] − I]) = O(h)O(h) = O(h2), thus +2[E] = [F]T + [F] − 2[I] + O(h), thus (G.55). +H +Finger tensor F.F T (left Cauchy–Green tensor) +Finger’s approach is consistent with the foundations of relativity (Galileo classical relativity or Einstein +general relativity): We can only do measurements at the current time t, and we can refer to the past. +There is a lot of misunderstandings, as was the case for the Cauchy–Green deformation tensor C, due +to the lack of precise definitions: Definition domain? Value domain? Points at stake (p or P)? Euclidean +dot product (English? French?)? Covariance? Contravariance?... +H.1 +Definition +Let �Φ be motion, t0 ∈ R, Φt0 the associated motion, P ∈ Ωt0, t ∈ R, and F t0 +t (P) := dΦt0 +t (P) ∈ L( ⃗ +Rn +t0; ⃗ +Rn +t ). +And let (·, ·)G and (·, ·)g be Euclidean dot products in ⃗ +Rn +t0 and ⃗ +Rn +t . +Definition H.1 The Finger tensor bt0 +t (pt), or left Cauchy–Green deformation tensor, at t at pt relative +to t0 is the endomorphism ∈ L( ⃗ +Rn +t ; ⃗ +Rn +t ) defined by, with P = Φt0 +t +−1(pt), +bt0 +t (pt) := F t0 +t (P).(F t0 +t )T +Gg(pt) +written in short +b = F.F T , +(H.1) +i.e. is defined by (bt0 +t (pt).⃗w1, ⃗w2)g = (F t0 +t (P)T .⃗w1, F t0 +t (P)T .⃗w2)G = ((F t0 +t )T (pt).⃗w1, (F t0 +t )T (pt).⃗w2)G, for +all ⃗w1, ⃗w2 vectors at pt ∈ Ωt, written in short +(b.⃗w1, ⃗w2)g = (F T .⃗w1, F T .⃗w2)G. +(H.2) +(To compare with C = F T .F and (C. ⃗W1, ⃗W2)G = (F. ⃗W1, F. ⃗W2)g.) +And the Finger tensor relative to t0 is +bt0 : +� +� +� +� +� +C = +� +t +({t} × Ωt) → L( ⃗ +Rn +t ; ⃗ +Rn +t ) +(t, pt) → bt0(t, pt) := bt0 +t (pt). +(H.3) +NB: bt0 looks like a Eulerian function, but isn’t, since it depends on a t0. +Other definition found: +Bt0 +t := bt0 +t ◦ (Φt0 +t )−1, +i.e. +Bt0 +t (P) := bt0 +t (pt) = F t0 +t (P).F t0 +t (P)T , +written +B = F.F T . +(H.4) +Pay attention: Bt0 +t (P) ∈ L( ⃗ +Rn +t ; ⃗ +Rn +t ) is an endomorphism at t at pt, not at t0 at P: E.g., Bt0 +t (P).⃗wt(pt) = +bt0 +t (pt).⃗wt(pt) is meaningful, while Bt0 +t (P). ⃗Wt0(P) is absurd. +Remark H.2 The push-forward by Φ := Φt0 +t +of the Cauchy–Green deformation tensor C = F T .F is +Φ∗(C) = F.C.F −1 = F.F T = b, cf. (8.15): It is the Finger tensor. So the endomorphism C in ⃗Rn +t0 is the +pull-back of the endomorphism b in ⃗Rn +t . (However a push-forward and a pull-back don’t depend on any +inner dot product while the transposed F T does...). +131 + +132 +H.2. +b−1 +H.2 +b−1 +With pull-backs (towards the virtual power principle at t). +With pt += Φt0 +t (P) and +⃗Wi(P) = +(F t0 +t (P))−1.⃗wi(pt): +( ⃗W1, ⃗W2)G = (F −1.⃗w1, F −1.⃗w2)G = (F −T .F −1.⃗w1, ⃗w2)g = (b−1.⃗w1, ⃗w2)g. +(H.5) +So b−1 := (bt0 +t )−1 is useful: +(bt0 +t )−1 : +� +Ωt → L(⃗Rn +t ; ⃗Rn +t ) +pt → (bt0 +t )−1(pt) = F t0 +t (P)−T .F t0 +t (P)−1 = Ht0 +t (pt)T .Ht0 +t (pt) +(H.6) +with pt = Φt0 +t (P) and Ht0 +t (pt) = (F t0 +t (P))−1 cf. (4.41). Thus we can define +(bt0)−1 : +� +� +� +� +� +� +t +({t} × Ωt) → L(⃗Rn +t ; ⃗Rn +t ) +(t, pt) → (bt0)−1(t, pt) := (bt0 +t )−1(pt). +(H.7) +Remark: (bt0)−1 looks like a Eulerian function, but isn’t, since it depends on t0. +In short: +b−1 = HT .H, +to compare with +C = F T .F, +(H.8) +and with ⃗w = F. ⃗W, +b−1.⃗w = HT . ⃗W, +to compare with +C. ⃗W = F T .⃗w, +(H.9) +and with ⃗Wi = F −1.⃗wi, i.e. ⃗wi = F. ⃗Wi, +(b−1.⃗w1, ⃗w2)g = ( ⃗W1, ⃗W2)G, +to compare with +(C. ⃗W1, ⃗W2)G = (⃗w1, ⃗w2)g. +(H.10) +Remark H.3 pt = Φt0 +t (P) and b(pt) = F(P).F(P)T and C(P) = F(P)T .F(P) give +b(pt).F(P) = F(P).C(P), +(H.11) +written b = F.C.F −1. Thus b−1 = F.C−1.F −1, so +Φt0∗ +t +b−1 = F −1.b−1.F = F −1.F −T = (F T .F)−1 = C−1, +(H.12) +i.e. the pull-back of b−1 is C−1, i.e. b−1 is the push-forward of C−1. +H.3 +Time derivatives of b−1 +With (H.7) let (bt0)−1 =noted b−1 = HT .H. Thus, along a trajectory, and with (4.45), we get +Db−1 +Dt += DHT +Dt .H + HT .DH +Dt = −d⃗vT .HT .H − HT .H.d⃗v += − b−1.d⃗v − d⃗vT .b−1. +(H.13) +Exercice H.4 Prove (H.13) with (H.10). +Answer. +(H.10) gives +D +Dt(b−1.⃗w1, ⃗w2)g += 0 = ( +Db−1 +Dt .⃗w1, ⃗w2)g + (b−1. D ⃗w1 +Dt , ⃗w2)g + (b−1.⃗w1, D ⃗w2 +Dt )g, and +⃗wi(t, p(t)) = F t0(t, P). ⃗Wt0(P) gives D ⃗wi +Dt = d⃗v.⃗wi, thus ( +Db−1 +Dt .⃗w1, ⃗w2)g +(b−1.d⃗v.⃗w1, ⃗w2)g +(b−1.⃗w1, d⃗v.⃗w2)g = 0, +thus (H.13). +Exercice H.5 Prove (H.13) with F T .b−1.F = It0. +Answer. b−1 = (F.F T )−1 = F −T .F −1 gives F T .b−1.F = It0, thus (F T )′.b−1.F + F T . +Db−1 +Dt .F + F T .b−1.F ′ = 0, +thus F T .d⃗vT .b−1.F + F T . +Db−1 +Dt .F + F T .b−1.d⃗v.F = 0, thus (H.13). +132 + +133 +H.4. +Euler–Almansi tensor a +H.4 +Euler–Almansi tensor a +Euler–Almansi approach is consistent with the foundations of relativity (Galileo relativity or Einstein +general relativity): We can only do measurements at the current time t, and we can refer to the past. +At t in Ωt, consider the Finger tensor b = F.F T and its inverse b−1 = F −T .F T = HT .H cf. (H.8). +Definition H.6 Euler–Almansi tenor at pt ∈ Ωt is the endomorphism at0 +t (pt) ∈ L( ⃗ +Rn +t ; ⃗ +Rn +t ) defined by +at0 +t (pt) = 1 +2(It − bt0 +t (pt)−1) = 1 +2(It − H(pt)T .H(pt)), +(H.14) +written +a = 1 +2(I − b−1) = 1 +2(I − HT .H), +(H.15) +to compare with the Green–Lagrange tensor E = 1 +2(C − I) = 1 +2(F T .F − I) ∈ L( ⃗ +Rn +t0; ⃗ +Rn +t0). +Remark: at0 looks like a Eulerian function, but isn’t, since it depends on t0. +(H.10) gives (⃗wi = F. ⃗Wi) +(⃗w1, ⃗w2)g − ( ⃗W1, ⃗W2)G = 2(a.⃗w1, ⃗w2)g, +(H.16) +to compare with (⃗w1, ⃗w2)g − ( ⃗W1, ⃗W2)G = 2(E. ⃗W1, ⃗W2)G. (This also gives (a.⃗w1, ⃗w2)g = (E. ⃗W1, ⃗W2)G.) +And (H.15) gives +F T .a.F = E, +i.e. +a = F −T .E.F −1, +(H.17) +standing for F t0 +t (P)T .at0 +t (p).F t0 +t (P) = Et0 +t (P) when p = Φt0 +t (P). +Remark H.7 at0 +t is not the push-forward of Et0 +t +by Φt0 +t (the push-forward is F.E.F −1). +H.5 +Time Taylor expansion for a +(H.13) gives +Da +Dt = b−1.d⃗v + d⃗vT .b−1 +2 +. +(H.18) +H.6 +Almansi modified Infinitesimal strain tensor �ε +We are at t (present time) and remember the past: We prefer a definition of a infinitesimal strain tensor +�ε from the Euler–Almansi tensor a, instead of ε from Euler–Lagrange tensor E, cf. § G.8.2. +Same Euclidean framework as in § G.8.2, and matrix meaning again. +We have I − b−1 = I − HT .H = −(I − HT ).(I − H) + 2I − HT − H where H stands for Ht0 +t (pt). +Thus, for small displacement we get I − b−1 = 2I − HT − H + O(h), so +a(t, p(t)) = �ε(t, p(t)) + O(h) +where +�ε := I − H + HT +2 +. +(H.19) +And, with t = t0 + h we have F t0(t, P) = I + (t−t0) d⃗v(t, P) + o(t−t0), cf. (4.35), thus we have +Ht0(t, p(t)) = F t0(t, P)−1 = I − (t−t0) d⃗v(t, P) + o(t−t0) when p(t) = Φt0(t, P). Thus +F t0(t, P) − I = I − Ht0(t, p(t)) + O(t−t0). +(H.20) +Therefore, for small displacements (|t − t0| << 1): +a(t, p(t)) ≃ �ε(t, p(t)) ≃ εt0(t, P) +(matrix meaning). +(H.21) +I +Polar decomposition, elasticity and objectivity +I.1 +Polar decompositions of F (“isometric objectivity”) +The motion is supposed regular, t0, t ∈ R, pt0 ∈ Ωt0, F := F t0 +t (pt0) (= dΦt0 +t (pt0)), (·, ·)G and (·, ·)g are +Euclidean dot products in ⃗Rn +t0 and ⃗Rn +t , and C = F T .F ∈ L(⃗Rn +t0; ⃗Rn +t0). Here the covariant objectivity is +abandoned due to the need for inner dot products. +133 + +134 +I.1. +Polar decompositions of F (“isometric objectivity”) +I.1.1 +F = R.U (right polar decomposition) +The endomorphism C being (·, ·)G-symmetric definite positive (the motion is supposed to be regular), +∃α1, ..., αn ∈ R∗ ++ (the eigenvalues), ∃⃗c1, ...,⃗cn ∈ ⃗Rn +t0 (associated eigenvectors), such that, for all i = 1, ..., n, +C.⃗ci = αi⃗ci +and +(⃗ci) is a (·, ·)G-orthonormal basis in ⃗Rn +t0. +(I.1) +So, if (⃗ai) is a (·, ·)G-Euclidean basis then D = P −1.[C]⃗a.P, where D = diag(α1, ..., αn) = [C]⃗c and +P −1 = P T , P = [Pij] being the transition matrix from (⃗ai) to (⃗ci), i.e. defined by ⃗cj = � +i Pij⃗ai for all j. +Then, define the endomorphism U ∈ L(⃗Rn +t0; ⃗Rn +t0), called the right stretch tensor, by, for all i = 1, ..., n, +U.⃗ci = √αi ⃗ci, +and +U noted += +√ +C, +(I.2) +the √αi being called the principal stretches. Then, define the linear map R ∈ L(⃗Rn +t0; ⃗Rn +t ), called the +rotation map, by +R := F ◦ U −1 noted += +F.U −1, +(I.3) +so that +F = R ◦ U +noted += +R.U, +called the right polar decomposition of F. +(I.4) +Proposition I.1 We have: +C = U ◦ U noted += +U 2, +U is symmetric definite positive, +R−1 = RT . +(I.5) +And +the right polar decomposition +F = R ◦ U +is unique : +(I.6) +If F = �R◦ �U where �U ∈ L(⃗Rn +t0; ⃗Rn +t0) is symmetric definite positive and �R ∈ L(⃗Rn +t0; ⃗Rn +t ) satisfies �R−1 = �RT , +then �U = U and �R = R. +Proof. +(I.2) yields (U ◦ U).⃗cj += λ⃗cj += C.⃗cj for all j, cf. (I.1), thus U ◦ U += C ; Then +(U T .⃗ci,⃗cj)G = (⃗ci, U.⃗cj)G = (⃗ci, √αj⃗cj)G = √αj(⃗ci,⃗cj)G = √αjδij = √αiδij = √αi(⃗ci,⃗cj)G = +(√αi⃗ci,⃗cj)G = (U.⃗ci,⃗cj)G for all i, j, thus U T = U (symmetry). +Then RT ◦ R = U −T ◦ F T ◦ F ◦ U −1 = U −T ◦ C ◦ U −1 = U −1 ◦ (U ◦ U) ◦ U −1 = It0 identity +in ⃗Rn +t0. (Details: (RT .R. ⃗W, ⃗Z)G = (R. ⃗W, R.⃗Z)g = (F.U −1. ⃗W, F.U −1.⃗Z)g = (F T .F.U −1. ⃗W, U −1.⃗Z)G = +(U 2.U −1. ⃗W, U −1.⃗Z)G = (U −1.U. ⃗W, ⃗Z)G = ( ⃗W, ⃗Z)G.) Thus R−1 = RT ∈ L(⃗Rn +t ; ⃗Rn +t0), thus R ◦ RT = +R ◦ R−1 = It identity in ⃗Rn +t . +And F = �R ◦ �U = R ◦ U gives F T ◦ F = �U T ◦ �RT ◦ �R ◦ �U = U T ◦ �RT ◦ �R ◦ U, thus F T ◦ F = �U T ◦ �U, +with F T ◦ F = U T ◦ U, thus �U ◦ �U = U ◦ U = +√ +C, thus �U = U (uniqueness of the positive square root +eigenvalues). Hence �R = R. +I.1.2 +F = S.R0.U (shifted right polar decomposition for covariant objectivity) +In fact we need to be more specific if the gift of ubiquity is prohibited: Since we work with the affine +space Rn, consider the Marsden’s shifter +S := St0 +t (pt0) : +� +Tpt0(Ωt0) noted += +⃗Rn +t0 → Tpt(Ωt) noted += +⃗Rn +t +⃗wt0,pt0 → (S.⃗wt0,pt0 )(t, pt) = ⃗wt0,pt0 +where +pt = Φt0 +t (pt0). +(I.7) +NB: 1- S looks like the algebraic identity if you have time and space ubiquity gift (otherwise it is not), +2- S is not a topological identity since it changes the norms in the general case: You consider ||⃗wt0,pt0 ||G +at t0 and ||S.⃗wt0,pt0 ||g = ||⃗wt0,pt0 ||g at t. +Then, let R0 ∈ L(Tpt0(Ωt0); Tpt0(Ωt0)) =noted L(⃗Rn +t0; ⃗Rn +t0) be the endomorphism defined by, in short, +R0 := S−1 ◦ R noted += +S−1.R, +so +R = S.R0 +(= S ◦ R0). +(I.8) +Full notations: (R0)t0 +t,Gg(pt0) := (St0 +t )−1(Rt0 +t,Gg(pt0)). +134 + +135 +I.1. +Polar decompositions of F (“isometric objectivity”) +Proposition I.2 The endomorphism R0 = S−1 ◦ R is a rotation operator in (⃗Rn +t0, (·, ·)G): +R−1 +0 += RT +0 +in (⃗Rn +t0, (·, ·)G). +(I.9) +And +F = S ◦ R0 ◦ U. +(I.10) +Interpretation: F is composed of: The pure deformation U (endomorphism in ⃗Rn +t0), the rotation R0 +(endomorphism in ⃗Rn +t0), and the shift operator S : ⃗Rn +t0 → ⃗Rn +t (from past to present time and position). +Proof. +(RT +0 . ⃗W2, ⃗W1)G = (R0. ⃗W1, ⃗W2)G +(definition of the transposed) += (S−1.R. ⃗W1, ⃗W2)G +(definition of R0) += (R. ⃗W1, ⃗W2)G +(S is the algebraic identity) += (RT . ⃗W2, ⃗W1)g +(definition of RT ) += (R−1. ⃗W2, ⃗W1)g +(cf. (I.5) ) += (R−1 +0 . ⃗W2, ⃗W1)G +(S is the algebraic identity), +(I.11) +true for all ⃗W1, ⃗W2 ∈ ⃗Rn +t0, thus RT +0 = R−1 +0 +in (⃗Rn +t0, (·, ·)G). And (I.8) and (I.4) give (I.10). +Exercice I.3 Let D = diag(αi), let (⃗ai) be a Euclidean basis in ⃗Rn +t0, let P be the transition matrix +from (⃗ai) to (⃗ci), so [C]|⃗a = P.D.P −1; Prove [U]|⃗a = P. +√ +D.P −1. Case (⃗ai) = ( ⃗Ei) is a (·, ·)g-orthonormal +basis? +Answer. The n equations (I.1) (for j = 1, ..., n), read as the matrix equation [C]|⃗a.P = P.D since [⃗cj]⃗a is the +j-th column of P. And he n equations (I.2) (for j = 1, ..., n), read as the matrix equation [U]|⃗a.P = P. +√ +D since +[⃗cj]⃗a is the j-th column of P. +Remark I.4 Instead of R0 ∈ L(⃗Rn +t0; ⃗Rn +t0), cf. (I.8), you may prefer to consider �R0 ∈ L(⃗Rn +t ; ⃗Rn +t ) defined +by R = �R0 ◦ S, i.e., �R0 = R ◦ S−1. +I.1.3 +F = V.R (left polar decomposition) +Same steps than for the right polar decomposition, but with pull-backs (with F −1 instead of F). +Let pt = Φt0 +t (pt0) ∈ Ωt, let bt0 +t (pt) := F t0 +t (pt0) ◦ (F t0 +t )T (pt) ∈ L(⃗Rn +t ; ⃗Rn +t ), written b = F ◦ F T (the left +Cauchy–Green deformation tensor also called the Finger tensor). The endomorphism b being symmetric +definite positive: ∃β1, ..., βn ∈ R∗ ++ (the eigenvalues), ∃⃗d1, ..., ⃗dn ∈ ⃗Rn +t (associated eigenvectors), such that, +for all i = 1, ..., n, +b.⃗di = βi ⃗di, +and +(⃗di) is a (·, ·)g-orthonormal basis in ⃗Rn +t . +(I.12) +Then, define the unique endomorphism V ∈ L(⃗Rn +t ; ⃗Rn +t ), called the left stretch tensor, by, for all i = 1, ..., n, +V.⃗di = +� +βi ⃗di, +and +V noted += +� +b. +(I.13) +(Full notation: V t0 +t,Gg(pt) = +� +bt0 +t (pt)Gg.) Then define the linear map Rℓ ∈ L(⃗Rn +t0; ⃗Rn +t ) by +Rℓ := V −1 ◦ F noted += +V −1.F, +(I.14) +so that +F = V ◦ Rℓ +noted += +V.Rℓ, +called the left polar decomposition of F. +(I.15) +Proposition I.5 We have: 1- +b = V ◦ V noted += +V 2, +V is symmetric definite positive, +R−1 +ℓ += RT +ℓ . +(I.16) +And the left polar decomposition F = V ◦ R is unique. +2- Rℓ = R and V = R.U.R−1 (so U and V are similar), thus U and V have the same eigenvalues, i.e., +αi = βi for all i, and ⃗di = R.⃗ci for all i gives a relation between eigenvectors. +135 + +136 +I.2. +Linear elasticity: A Classical “tensorial” approach +Proof. 1- Use F −1 and b−1 = (F −1)T .(F −1), instead of F and C = F T .F, to get F −1 = R−1 +ℓ .U −1 +ℓ +, +cf. (I.3); Thus F = Uℓ.Rℓ; Then name Uℓ = V to get (I.15) and (I.16). +2- V.Rℓ = F = R.U = (R.U.R−1).R, thus, by uniqueness of the right polar decomposition, V = +R.U.R−1 (so U and V are similar) and Rℓ = R. Thus, with (I.12), βi ⃗di = V.⃗di = R.U.(R−1.⃗di), thus with +⃗ci = R−1.⃗di, then (⃗ci) is an orthonormal basis in ⃗Rn +t0 and βiR.⃗ci = R.U.⃗ci = αiR.⃗ci gives βi = αi and the +⃗ci are eigenvectors of U, for all i. +I.2 +Linear elasticity: A Classical “tensorial” approach +I.2.1 +Classical approach (“isometric objectivity”), and an issue +With the infinitesimal strain “tensor” (which is not a tensor but a matrix) +ε = F + F T +2 +− I, +(I.17) +the homogeneous isotropic elasticity constitutive law reads (matrix equation for the stress) +(σ(Φ) =) +σ = λTr(ε)I + 2µε, +(I.18) +where λ, µ are the Lamé coefficients and σ is the Cauchy stress “tensor”. +Issue: Recall: Adding F and F T to make ε is functionally a mathematical nonsense since F : ⃗Rn +t0 → +⃗Rn +t and F T : ⃗Rn +t → ⃗Rn +t0 and I is some identity operator: σ is not a tensor. In particular the meaning of +Tr(ε) is questionable (since ε is not an endomorphism and Tr(ε) means Tr([ε]) = Tr([F ])+Tr([F T ]) +2 +− n), as +well as the meaning of ε.⃗n = 1 +2(F.⃗n + F T .⃗n), or the meaning of +σ.⃗n = λTr(ε)⃗n + 2µε.⃗n +(I.19) +since ⃗n has to be defined at (t0, pt0) for F and at (t, pt) for F T . +(Cauchy’s approach: ⃗n is defined +at (t, pt).) +So, despite the eventual claims, neither ε nor σ are tensors (they don’t have a functional meaning): +They only have a questionable matrix meaning (observer dependent) [ε] := [F ]+[F ]T +2 +− [I] and +[σ] = λTr([ε])[I] + 2µ[ε], +and +[σ].[⃗n] = λTr([ε])[⃗n] + 2µ[ε].[⃗n]. +(I.20) +Remark I.6 To justify the name “tensor” applied to ε, you may read: “For small displacements the +Eulerian variable pt and the Lagrangian variable pt0 can be confused”: pt ≃ pt0 (so Ωt0 and Ωt are +“almost equal”, so F(pt0) + F T (pt) can be considered). Which means that you use the zero-th order +Taylor expansions pt = Φt0 +pt0 (t) = pt0 + o(1). But then, you cannot also use the first (or higher) order +Taylor expansion in following calculations, e.g. you cannot use velocities... +I.2.2 +A functional (tensorial) formulation (“isometric objectivity”) +Question: Can the constitutive law (I.18) be modified into a tensorial expression (a functional expression)? +Proposal for a yes: +1. Consider the “right polar decomposition” F = R.U where U ∈ L(⃗Rn +t0; ⃗Rn +t0), cf. (I.3). The Green +Lagrange tensor E = C−I +2 +(endomorphism in ⃗Rn +t0) then reads, with (I.5), +E = U 2−It0 +2 += (U−It0)2 + 2(U − It0) +2 +(I.21) +(the Green–Lagrange tensor is independent of the rotation R), thus, with U − It0 = O(h) (small defor- +mation approximation), we get the modified infinitesimal strain tensor at pt0 ∈ Ωt0 +�ε = U−It0 ∈ L(⃗Rn +t0; ⃗Rn +t0), +(I.22) +endomorphism in ⃗Rn +t0 (to compare with ε which is not a function, cf. the previous §). (Full notation +�εt0 +t,Gg(pt0) = U t0 +t,Gg(pt0)−It0(pt0) in L(⃗Rn +t0; ⃗Rn +t0).) And, for all ⃗W ∈ ⃗Rn +t0 we get +�ε. ⃗W = U. ⃗W − ⃗W = R−1.⃗w − ⃗W +∈ ⃗Rn +t0, +when +⃗w = F. ⃗W (push-forward). +(I.23) +Interpretation: From ⃗w = F. ⃗W = R.U. ⃗W ∈ ⃗Rn +t (the deformed by the motion), remove the “rigid +body rotation” to get R−1.⃗w = U. ⃗W ∈ ⃗Rn +t0, to which you remove the initial ⃗W to obtain �ε. ⃗W ∈ ⃗Rn +t0. +136 + +137 +I.2. +Linear elasticity: A Classical “tensorial” approach +In particular ||�ε. ⃗W||G = ||(U−It0). ⃗W||G measures the relative elongation undergone by ⃗W. And you can +then apply R to get back into ⃗Rn +t at pt: +R.(�ε. ⃗W) = F. ⃗W − R. ⃗W = ⃗w − R. ⃗W ∈ ⃗Rn +t , +when +⃗w = F. ⃗W = (push-forward). +(I.24) +2. Then, at pt0 ∈ Ωt0, consider the stress tensor �Σ(Φ) =noted �Σ ∈ L(⃗Rn +t0; ⃗Rn +t0) (functionally well) +defined by +�Σ = λTr(�ε)It0 + 2µ�ε = λTr(U−It0)It0 + 2µ(U−It0). +(I.25) +(The trace Tr(�ε) is well defined since �ε is an endomorphism.) Then for any ⃗W ∈ ⃗Rn +t0 you get in ⃗Rn +t0, at +pt0 ∈ Ωt0, +�Σ. ⃗W = λTr(�ε) ⃗W + 2µ�ε. ⃗W = λTr(U−It0) ⃗W + 2µ(U. ⃗W− ⃗W) +(I.26) +(functionally well defined in ⃗Rn +t0). Then rotate and shift with R to get into ⃗Rn +t at pt, +R.�Σ. ⃗W = λTr(�ε)R. ⃗W + 2µR.�ε. ⃗W = λTr(U−It0)R. ⃗W + 2µR.(U−It0). ⃗W += λTr(U−It0)R. ⃗W + 2µ(F − R). ⃗W, += λTr(U−It0)R. ⃗W + 2µ(⃗w − R. ⃗W), +where +⃗w = F. ⃗W = R.U. ⃗W. +(I.27) +You have defined the two point tensor (functionally well defined) +R.�Σ = λTr(�ε)R + 2µR.�ε ∈ L(⃗Rn +t0; ⃗Rn +t ). +(I.28) +3. Then you can propose the constitutive law with the stress tensor (the symmetric endomorphism) +in ⃗Rn +t given by +(�σ(Φ) =) +�σ = R ◦ �Σ ◦ R−1 +noted += +R.�Σ.R−1 ∈ L(⃗Rn +t ; ⃗Rn +t ). +(I.29) +(Functionally well defined.) And, for all vector fields ⃗w defined in Ωt, you get the (functionally well +defined) vector field +�σ.⃗w = R.�Σ.R−1.⃗w +∈ ⃗Rn +t . +(I.30) +Interpretation of (I.29)-(I.30): Shift and rigid rotate backward by applying R−1, apply the elastic stress +law with Σ which corresponds to a rotation free motion (Noll’s frame indifference principle), then shift +and rigid rotate forward by applying R. +Detailed expression for (I.29)-(I.30): With Tr(R.�ε.R−1) = Tr(�ε) (see exercise I.8), we get, at (t, pt), +�σ = λTr(�ε) It + 2µR.�ε.R−1 = λTr(U−It0) It + 2µR.(U−It0).R−1 += λTr(U−It0) It + 2µ(F.R−1−It). +(I.31) +And for any ⃗w ∈ ⃗Rn +t , and with ⃗w = R. ⃗W, you get +�σ.⃗w = λTr(�ε) ⃗w + 2µR.�ε. ⃗W = λTr(U−It0) ⃗w + 2µR.(U−It0). ⃗W += λTr(U−It0) ⃗w + 2µ(R.U.R−1.⃗w−⃗w). +(I.32) +To compare with the classical functionally meaningless (I.19). +Remark I.7 Doing so, you avoid the use of the Piola–Kirchhoff tensors. +Exercice I.8 Prove: Tr(R.�ε.R−1) = Tr(�ε) = � +i(αi−1). +(NB: �ε is an endomorphism in ⃗Rn +t0 while +R.�ε.R−1 is an endomorphism in ⃗Rn +t .) +Answer. det|⃗e(R.�ε.R−1 − λIt) = det|⃗e(R.(�ε−λIt0).R−1) = det| ⃗ +E,⃗e(R). det| ⃗ +E(�ε−λI). det|⃗e, ⃗ +E(R−1) = det| ⃗ +E(�ε−λI) +for all Euclidean bases ( ⃗Ei) and (⃗ei) in ⃗Rn +t0 and ⃗Rn +t . +(With L = �ε and components, Tr(R.L.R−1) = +� +i(R.L.R−1)i +i = � +ijk Ri +jLj +k(R−1)k +i = � +jk(R−1.R)k +j Lj +k = � +jk δk +j Lj +k = � +j Lj +j = Tr(L).) +137 + +138 +I.3. +Elasticity with a covariant objective approach? +Exercice I.9 Elongation in R2 along the first axis : origin O, same Euclidean basis ( ⃗E1, ⃗E2) and Eu- +clidean dot product at all time, ξ > 0, t ≥ t0, L, H > 0, P ∈ [0, L] × [0, H], [−−→ +OP]| ⃗E = +� +X0 +Y0 +� +, and +[−−−−−−→ +OΦt0 +t (P)]| ⃗E = +� +X0 + ξ(t−t0)X0 +Y0 +� += +� +X0(κ+1) +Y0 +� += +� +x +y +� += [−→ +Op]| ⃗E, where κ = ξ(t−t0) > 0 for t > t0. +1- Give F, C, U = +√ +C and R = F.U −1. Relation with the classical expression ? +2- Spring −−→ +OP = −−→ +Oct0(s) = X0 ⃗E1+Y0 ⃗E2+s ⃗W, i.e. [−−→ +OP]| ⃗E = [−−→ +Oct0]| ⃗E = +� +X0+sW1 +Y0+sW2 +� +| ⃗E +with s ∈ [0, L] +and ⃗W = W1 ⃗E1 + W2 ⃗E2. Give the deformed spring, i.e. give p = ct(s) = Φt0 +t (ct0(s)), and ⃗ct′, and the +stretch ratio. +Answer. 1- [F] = [dΦ] = +� κ+1 +0 +0 +1 +� +, same Euclidean dot product and basis at all time, thus [F T ] = [F]T = [F], +then [C] = [F T ].[F] = [F]2 = +� (κ+1)2 +0 +0 +1 +� +, thus [U] = [F] = +� κ+1 +0 +0 +1 +� +, thus [R] = [I]. All the matrices are +given relative to the basis ( ⃗Ei), thus F, C, U, R (e.g., C. ⃗E1 = (κ+1)2 ⃗E1 and C. ⃗E2 = ⃗E2). +Since R = I and [ε] = [�ε], (I.31) gives the usual result [σ] = λTr([ε])I + 2µ[ε], cf (I.18) (matrix meaning). +2- −−−−→ +Oct(s) += +−−−−−−−−−→ +OΦt0 +t (ct0(s)) += +� (X0+sW1)(κ+1) +Y0+sW2 +� +| ⃗ +E +, +thus ⃗ct +′(s) += +� W1(κ+1) +W2 +� +| ⃗ +E +, +stretch ration +W 2 +1 (κ+1)2+W 2 +2 +W 2 +1 +W 2 +2 +at (t, pt). +Exercice I.10 Simple shear in R2 : [−−−−−−→ +OΦt0 +t (P)]| ⃗E = +� +X + ξ(t−t0)Y +Y +� +=noted +� +X + κY +Y +� += +� +x +y +� += +[−→ +Op]| ⃗E. Same questions, and moreover give the eigenvalues of C. +Answer. +1- [F] = +� 1 +κ +0 +1 +� +, [C] = +� 1 +0 +κ +1 +� +. +� 1 +κ +0 +1 +� += +� 1 +κ +κ +κ2+1 +� +. +Eigenvalues: det(C − λI) = λ2 − +(2+κ2)λ + 1. Discriminant ∆ = (2+κ2)2 − 4 = κ2(κ2+4). Eigenvalues α± = 1 +2(2+κ2 ± κ +√ +κ2+4). (We check that +α± > 0.) Eigenvectors ⃗v±(main directions of deformations) given by (1−α±)x+κy = 0, i.e., y = 1 +2(κ± +√ +κ2+4)x, +thus, e.g., ⃗v± = +� +2 +κ ± +√ +κ2+4 +� +. (We check that ⃗v+ ⊥ ⃗v−.) With P the transition matrix from ( ⃗E1, ⃗E2) to +( +⃗v+ +||⃗v+||, +⃗v− +||⃗v−||) and D = diag(α+, α−) we get C = P.D.P −1 (with P −1 = P T since here ( +⃗v+ +||⃗v+||, +⃗v− +||⃗v−||) is an +orthonormal basis), thus U = P. +√ +D.P −1 (we check that U T = U and U 2 = C). And R = F.U −1. +2- −−−−→ +Oct(s) = −−−−−−−−−→ +OΦt0 +t (ct0(s)) = +� (X0+sW1) + κ(Y0+sW2) +Y0+sW2 +� +, thus [⃗ct +′(s)] = +� W1 + κW2 +W2 +� +. +Stretch ratio +(W1+κW2)2+W 2 +2 +W 2 +1 +W 2 +2 +at (t, pt). +I.2.3 +Second functional formulation: With the Finger tensor +The above approach uses the push-forward: It uses F, i.e. you arrive with your memory. You may prefer +to use the pull-back, i.e. use F −1 (you remember the past which is Cauchy’s point of view): Then you +use F −1 = R−1.V −1 the right polar decomposition of F −1, and you consider the tensor +��εt = V −1−It ∈ L(⃗Rn +t ; ⃗Rn +t ), +(I.33) +and +(σt(Φ) =) +σt = λTr(��εt)It + 2µ��εt, +and +σt.⃗nt = λTr(��εt)⃗nt + 2µ��εt.⃗nt. +(I.34) +(Quantities functionally well defined: Give a tensorial approach). +I.3 +Elasticity with a covariant objective approach? +In § I.2 you need to start with Euclidean dot products, so from the start the result can’t be covariant +objective. Can you start without Euclidean dot products to set up general laws? Proposal: +Hypothesis: The Cauchy stress ⃗w is a Eulerian vector field. +Then we could use the (covariant objective) Lie derivative which characterizes the rate of stress, +cf. § 9.3 and 9.5: With a particle PObj ∈ Obj, with ⃗v(τ, pτ) = ∂�Φ +∂τ (τ, PObj) its Eulerian velocity at τ at +138 + +139 +J.1. +The displacement vector ⃗U +pτ = �Φ(τ, PObj), the Lie derivative of a Eulerian vector field ⃗w along ⃗v is, at (t, pt), +L⃗v ⃗w(t, pt) = lim +τ→t +⃗w(τ, pτ) − ⃗wt∗(τ, pτ) +τ − t += (∂ ⃗w +∂t + d⃗w.⃗v − d⃗v.⃗w)(t, pt). +(I.35) +Hence +the +proposal, +with +the +virtual +power +principle +to +measure +the +rate +of +stress +(see +https://arxiv.org/abs/2208.10780v1 for a full description). +1- Hypotheses: 1.1- Suppose that n Eulerian vector fields ⃗wj (“force fields”), j = 1, ..., n, enable to +characterize a material. (In fact, for elasticity problems it could be better to replace vector fields ⃗wj with +1-forms αi to characterize the work.) +1.2- With a basis (⃗ei) chosen in ⃗Rn +t , with (ei) its (covariant) dual basis in ⃗Rn∗ +t , assume that the internal +power density at (t, pt) is given by (at first order): +pint(⃗v) = +n +� +j=1 +ej.L⃗v ⃗wj = +n +� +j=1 +ej.(∂ ⃗wj +∂t + d⃗wj.⃗v − d⃗v.⃗wj). +(I.36) +(At second order you can add second order Lie derivatives as L⃗v(L⃗v ⃗wj), similarly for higher orders.) +2- Then, so that this pint satisfies the frame invariance hypothesis, choose a Euclidean dot prod- +uct (·, ·)g in ⃗Rn +t ; Then, first, the internal power has to vanish if d⃗v = 0, thus we are left with +pint(⃗v) = − +n +� +j=1 +ej.d⃗v.⃗wj = −τ 0.. d⃗v, +where +τ = +n +� +j=1 +⃗wj ⊗ ej, +(I.37) +defined at t. (The j-th column of [τ]|⃗e is [⃗wj]|⃗e.) And, second, the internal power vanishes if d⃗v +d⃗vT = 0 +(rotation), thus we are left with +pint(⃗v) = −τ 0.. d⃗v + d⃗vT +2 += −σ 0.. d⃗v +where +σ = τ + τ T +2 +, +(I.38) +this pint(⃗v) = −σ 0.. d⃗v being the usual expression of the internal power at first order. +Example I.11 (I.36) may be applied to orthotropic elasticity, e.g. for a material which fibers at some +time t0 are along ⃗e1, in a 2-D case for simplicity: 1- With an elongation type motion (Φe) given by +[(Fe)(pt0)] = [d(Φe)(pt0)] = +� +1+α11(pt0) +0 +0 +1−α22(pt0) +� +you measure the Young moduli in the direc- +tions ⃗e1 and ⃗e2; 2- And with a shear type motion given by [(Fs)(pt0)] = [d(Φs)(pt0)] = +� +1 +γ12 +0 +1 +� +you +measure the shear modulus. +For more complex material, you may need more vectors ⃗wj to describe the constitutive law, that is, +(I.36) may be considered with �m +i=1ej.L⃗v ⃗wj with m > n. +Remark I.12 The Lie approach is different from the usual classic approach: +1- The classic approach looks for an order two stress tensor [σ] as a function of the deformation +gradient [F], cf. (I.18). +2- The Lie approach begins with the internal power (which measures forces), cf. (I.36), which then +enable to build τ and the σ (the stress tensor), cf. (I.37)-(I.38). +E.g., application to visco-elasticity: With the Lie approach, you automatically use Lie derivative of +vector fields (and/or of differential forms), instead of Lie derivative of order 2 tensor fields (which does +not seem to give good result, see e.g. the Maxwell visco-elastic type laws, as well as footnote1 page 25). +J +Displacement +J.1 +The displacement vector ⃗U +In Rn, let pt = Φt0 +t (pt0). Then the bi-point vector +⃗Ut0 +t (pt0) = Φt0 +t (pt0) − It0(pt0) = pt − pt0 = −−→ +pt0pt +(J.1) +is called the displacement vector at pt0 relative to t0 and t. This defines the map +⃗Ut0 +t +: +� +Ωt0 → ⃗Rn +pt0 → ⃗Ut0 +t (pt0) := pt − pt0 = −−→ +pt0pt +when +pt = Φt0 +t (pt0). +(J.2) +139 + +140 +J.2. +The differential of the displacement vector +Remark J.1 ⃗Ut0 +t (pt0) doesn’t define a vector field (it is not tensorial), because ⃗Ut0 +t (pt0) = pt−pt0 = −−→ +pt0pt +is a bi-point vector which is neither in ⃗Rn +t0 or in ⃗Rn +t since pt0 ∈ Ωt0 and pt ∈ Ωt (it requires time and +space ubiquity gift). In particular, it makes no sense on a non-plane surface (manifold). More at § J.5. +Remark J.2 For elastic solids in Rn, the function ⃗Ut0 is often considered to be the unknown (to be +computed); But the “real” unknown is the motion Φt0. And it is sometimes confused with the extension +of a spring 1-D case; But see figure 4.1 where ||⃗wt0(pt0)|| represents the initial length and ||⃗wt0∗(t, pt)|| +represents the current length of the spring, while the length of the displacement vector ⃗Ut0 +t += pt − pt0 +can be very long for a very small elongation ||⃗wt0∗(t, pt)|| − ||⃗wt0(pt0)|| of the spring. +J.2 +The differential of the displacement vector +The differential of ⃗Ut0 +t +at pt0 is +d⃗Ut0 +t (pt0) = dΦt0 +t (pt0) − It0 = F t0 +t (pt0) − It0, +written +d⃗U = F − I, +(J.3) +thus isn’t defined as a function, because F t0 +t (pt0) : ⃗Rn +t0 → R while It0 : ⃗Rn +t0 → ⃗Rn +t0. +So d⃗Ut0 +t (pt0) as to be understood as a matrix: With +[⃗Ut0 +t (pt0)] = [−−→ +pt0pt] = [−−−−−−→ +OΦt0 +t (pt0)] − [−−→ +Opt0], +(J.4) +relative to an origin O and a unique basis at all time, compute [d⃗Ut0 +t (pt0)] = [dΦt0 +t (pt0)] − I, abusively +written d⃗U = dΦ − I. Then, with ⃗W ∈ ⃗Rn +t0 , +d⃗U. ⃗W = F. ⃗W − ⃗W, +which means += [F t0 +t (pt0)].[ ⃗W] − [ ⃗W]. +(J.5) +Thus we have defined (matrix meaning) +⃗Ut0 : +� +[t0, T] × Ωt0 → ⃗Rn +(t, pt0) → ⃗Ut0(t, pt0) := ⃗Ut0 +t (pt0), +and +⃗Ut0 +pt0 : +� +[t0, T] → ⃗Rn +t → ⃗Ut0 +pt0 (t) := ⃗Ut0 +t (pt0). +(J.6) +J.3 +Deformation “tensor” ε (matrix), bis +(J.3) gives (matrix meaning) +F t0 +t (pt0) = It0 + d⃗Ut0 +t (pt0), +written +F = I + d⃗U. +(J.7) +Therefore, Cauchy–Green deformation tensor C = F T .F reads, in short, (matrix meaning) +C = I + d⃗U + d⃗UT + d⃗UT .d⃗U +(matrix meaning), +(J.8) +i.e. [Ct0 +t (pt0)] = [It0] + [d⃗Ut0 +t (pt0)] + [d⃗Ut0 +t (pt0)]T + [d⃗Ut0 +t (pt0)]T .[d⃗Ut0 +t (pt0)]. +Thus the Green–Lagrange deformation tensor E = C−I +2 , cf. (G.47), reads, in short, (matrix meaning) +E = d⃗U + d⃗UT +2 ++ 1 +2d⃗UT .d⃗U +(matrix meaning). +(J.9) +Thus the deformation tensor ε, cf. (G.54), reads (matrix meaning) +ε = E − 1 +2(d⃗U)T .d⃗U, +(J.10) +with ε the “linear part” of E (small displacements: we only used the first order derivative dΦt0 +t ). +140 + +141 +J.4. +Small displacement hypothesis, bis +J.4 +Small displacement hypothesis, bis +(Usual introduction.) Let pt = Φt0 +t (pt0), ⃗Wi ∈ ⃗ +Rn +t0, ⃗wi(pt) = F t0 +t (pt0). ⃗Wi(pt0) ∈ ⃗Rn +t (the push-forwards), +written ⃗wi = F. ⃗Wi. Then define (matrix meaning) +⃗∆i := ⃗wi − ⃗Wi = dU. ⃗Wi, +and +||⃗∆||∞ = max(||⃗∆1||Rn, ||⃗∆2||Rn). +(J.11) +Then the small displacement hypothesis reads (matrix meaning): +||⃗∆||∞ = o(|| ⃗W||∞). +(J.12) +Thus ⃗wi = ⃗Wi + ⃗∆i (with ⃗∆i “small”) and the hypothesis (·, ·)g = (·, ·)G (same inner dot product at t0 +and t) give +(⃗w1, ⃗w2)G − ( ⃗W1, ⃗W2)G = (⃗∆1, ⃗W2)G + (⃗∆2, ⃗W1)G + (⃗∆1, ⃗∆2)G. +So (J.10) gives 2(E. ⃗W1, ⃗W2)G = 2(ε. ⃗W1, ⃗W2)G + (d⃗UT .d⃗U. ⃗W1, ⃗W2)G, And (J.12) gives +(E. ⃗W1, ⃗W2)G = (ε. ⃗W1, ⃗W2)G + O(||⃗∆||2 +∞), +(J.13) +so Et0 +t +is approximated by εt0 +t , that is, Et0 +t ≃ εt0 +t (matrix meaning). +J.5 +Displacement vector with differential geometry +J.5.1 +The shifter +We give the steps, see Marsden–Hughes [12]. The complexity introduced is due to the small displacement +hypothesis applied to the Green–Lagrange tensor E = F T .F −I +2 +which linearization gives ε = F +F T +2 +− I +(the classical approach “squares the motion” to get E, then “linearizes” E ... to get back to F... with a +spurious F T ). +Let P ∈ Ωt0, ⃗WP ∈ ⃗Rn +t0, pt = Φt0 +t (P) ∈ Ωt, and ⃗wpt = F t0 +t (P). ⃗WP ∈ ⃗Rn +t (push-forward). +• Affine case Rn (continuum mechanics): With pt = Φt0 +t (P), the shifter is: +� +St0 +t : +� Ωt0 × ⃗Rn +t0 → Ωt × ⃗Rn +t +(P, ⃗ZP ) → � +St0 +t (P, ⃗ZP ) = (pt, St0 +t (⃗ZP )) +with +St0 +t (⃗ZP ) = ⃗ZP . +(J.14) +(The vector is unchanged but the time and the application point have changed: A real observer has no +ubiquity gift). So: +St0 +t ∈ L(⃗Rn +t0; ⃗Rn +t ) +and +[St0 +t ]|⃗e = I identity matrix, +(J.15) +the matrix equality being possible after the choice of a unique basis at t0 and at t. And (simplified +notation) � +St0 +t (P, ⃗ZP ) =noted St0 +t (⃗ZP ). Then the deformation tensor ε at P can be defined by +εt0 +t (P).⃗Z(P) = (St0 +t )−1(F t0 +t (P).⃗Z(P)) + F t0 +t (P)T .(St0 +t (P).⃗Z(P)) +2 +− ⃗Z(P), +(J.16) +in short: ε.⃗Z = (St0 +t )−1(F.⃗Z)+F T .(St0 +t .⃗Z) +2 +− ⃗Z). +• In a manifold: Ω is a manifold (like a surface in R3 from which we cannot take off). Let TP Ωt0 +be the tangent space à P (the fiber at P), and TptΩt be the tangent space à pt (the fiber at pt). +In general TP Ωt0 ̸= TptΩt (e.g. on a sphere “the Earth”). +The bundle (the union of fibers) at t0 is +TΩt0 = � +P ∈Ωt0 ({P} × TP Ωt0), and the bundle at t is TΩt = � +pt∈Ωt({pt} × TptΩt). Then the shifter +� +St0 +t : +� +TΩt0 → TΩt +(P, ⃗ZP ) → � +St0 +t (P, ⃗ZP ) = (pt, St0 +t (⃗ZP )), +(J.17) +is defined such that ⃗ZP ∈ TP Ωt0 “as little distorted as possible” along a path. E.g., on a sphere, if the +path is a geodesic, if θt0 is the angle between ⃗ZP and the tangent vector to the geodesic at P, then +θt0 is also the angle between St0 +t (⃗ZP ) and the tangent vector to the geodesic at pt, and St0 +t (⃗ZP ) has +the same length than ⃗ZP (at constant speed in a car you think the geodesic is a straight line, although +St0 +t (⃗ZP ) ̸= ⃗ZP : the Earth is not flat). +141 + +142 +K.1. +Alternating multilinear form +J.5.2 +The displacement vector +(Affine space framework, Ωt0 open set in Rn.) +Let P ∈ Ωt0, ⃗WP ∈ ⃗Rn +t0, pt = Φt0 +t (P) ∈ Ωt, and +dΦt0 +t = F t0 +t +∈ L(⃗Rn +t0; ⃗Rn +t ). Define +δ� +⃗Ut0 +t +: +� +� +� +Ωt0 × ⃗Rn +t0 → Ωt × L(⃗Rn +t0; ⃗Rn +t ) +(P, ⃗ZP ) → δ� +⃗Ut0 +t (P, ⃗ZP ) = (pt, δ ⃗Ut0 +t (⃗ZP )) +with +δ ⃗Ut0 +t (⃗ZP ) = (F t0 +t +− St0 +t ).⃗ZP . +(J.18) +Then δ� +⃗Ut0 +t += F t0 +t +− St0 +t +is a two-point tensor. And +Ct0 +t = (F t0 +t )T .F t0 +t += (δUt0 +t + St0 +t )T .(δUt0 +t + St0 +t ) += I + (St0 +t )T .δUt0 +t + (δUt0 +t )T .St0 +t + (δUt0 +t )T .δUt0 +t , +(J.19) +since (St0 +t )T .St0 +t += I identity in TΩt0: Indeed, ((St0 +t )T .St0 +t . ⃗A, ⃗B)Rn = (St0 +t . ⃗A, St0 +t . ⃗B)Rn = ( ⃗A, ⃗B)Rn, +cf. (J.14), for all ⃗A, ⃗B. Then the Green–Lagrange tensor is defined on Ωt0 by +Et0 +t = 1 +2(Ct0 +t − It0) = (St0 +t )T .δUt0 +t + St0 +t .(δUt0 +t )T +2 ++ 1 +2(δUt0 +t )T .δUt0 +t , +(J.20) +to compare with (G.47). +K +Determinants +K.1 +Alternating multilinear form +Let E be a vector space, and let L(E, ..., E; R) =noted L(En; R) be the set of multilinear forms, i.e. +m ∈ L(En; R) iff +m(..., ⃗x + λ⃗y, ...) = m(..., ⃗x, ...) + λm(..., ⃗y, ...) +(K.1) +for all ⃗x, ⃗y +∈ +E +and all λ +∈ +R and for all “slot”. +In particular, +m(λ1⃗x1, ..., λn⃗xn) += +(� +i=1,...,n λi) m(⃗x1, ..., ⃗xn), for all λ1, ..., λn ∈ R and all ⃗x1, ..., ⃗xn ∈ E. +Definition K.1 If n = 1 then a 1-alternating multilinear function is a linear form, also called a 1-form. +If n ≥ 2 then Aℓ : +� +En → R +(⃗v1, ...,⃗vn) → Aℓ(⃗v1, ...,⃗vn) +� +∈ L(En; R) is a n-alternating multilinear form iff, for +all ⃗u,⃗v ∈ E, +Aℓ(..., ⃗u, ...,⃗v, ...) = −Aℓ(...,⃗v, ..., ⃗u, ...), +(K.2) +the other elements being unchanged. If n = 1, the set of 1-forms is Ω1(E) = E∗. If n ≥ 2, the set of +n-alternating multilinear forms is +Ωn(E) = {m ∈ L(En; R) : m = Aℓ is alternating}. +(K.3) +If Aℓ, Bℓ ∈ Ωn(E) and λ ∈ R then Aℓ + λBℓ ∈ Ωn(E) thanks to the linearity for each variable. Thus +Ωn(E) is a vector space, sub-space in (F(En; R), +, .). +K.2 +Leibniz formula +Particular case dim E=n. Let Aℓ ∈ Ωn(E) (a n-alternating multilinear form). Recall (see e.g. Cartan [5]): +1- A permutation σ : [1, n]N → [1, n]N is a bijective map (i.e. one-to-one and onto); Let Sn be the set +of permutations of [1, n]N. +2- A transposition τ : [1, n]N → [1, n]N is a permutation that exchanges two elements, that is, ∃i, j s.t. +τ(..., i, ..., j, ...) = (..., j, ..., i, ...), the other elements being unchanged. +3- A permutation is a composition of transpositions (theorem left as an exercise, of see Cartan). And +a permutation is even iff the number of transpositions is even, and a permutation is odd iff the number +of transpositions is odd. Based on: The parity (even or odd) of a permutation is an invariant. +4- The signature ε(σ) = ±1 of a permutation σ is +1 if σ is even, and is −1 if σ is odd. +142 + +143 +K.3. +Determinant of vectors +Proposition K.2 (Leibniz formula) Let Aℓ ∈ Ωn(E). Let (⃗ei)i=1,...,n =noted (⃗ei) be a basis in E. For +all vectors ⃗v1, ...,⃗vn ∈ E, with ⃗vj = �n +i=1vi +j⃗ei for all j, +Aℓ(⃗v1, ...,⃗vn) = c +� +σ∈Sn +ε(σ) +n +� +i=1 +vσ(i) +i += c +� +τ∈Sn +ε(τ) +n +� +i=1 +vi +τ(i) +(with c := Aℓ(⃗e1, ...,⃗en)). +(K.4) +Thus if c = Aℓ(⃗e1, ...,⃗en) is known, then Aℓ is known. Thus dim(Ωn(E)) = 1. +(Classic not.: ⃗vj = �n +i=1vij⃗ei, Aℓ(⃗v1, ...,⃗vn) = c � +σ∈Sn ε(σ) �n +i=1 vσ(i),i = c � +τ∈Sn ε(τ) �n +i=1 vi,τ(i).) +Proof. Let F := F([1, n]N; [1, n]N) =noted [1, n][1,n]N +N +be the set of functions i : +� +[1, n]N → [1, n]N +k → ik = i(k) +� +. +Aℓ being multilinear, Aℓ(⃗v1, ...,⃗vn) = �n +j1=1 vj1 +1 Aℓ(⃗ej1,⃗v2, ...,⃗vn) (“the first column” development). By +recurrence we get Aℓ(⃗v1, ...,⃗vn) = �n +j1,...,jn=1 vj1 +1 ...vjn +n Aℓ(⃗ej1, ...,⃗ejn) = � +j∈F +�n +k=1 vj(k) +k +Aℓ(⃗ej(1), ...,⃗ej(n)). +And Aℓ(⃗ei1, ...,⃗ein) ̸= 0 iff i : k ∈ {1, ..., n} → i(k) = ik ∈ {1, ..., n} is one-to-one (thus bijective). Thus +Aℓ(⃗v1, ...,⃗vn) = � +σ∈Sn +�n +i=1 vσ(i) +i +Aℓ(⃗eσ(1), ...,⃗eσ(n)) = � +σ∈Sn ε(σ) �n +i=1 vσ(i) +i +Aℓ(⃗e1, ...,⃗en), which is the +first equality in (K.4). Then � +σ∈Sn ε(σ) �n +i=1 vσ(i) +i += � +σ∈Sn ε(σ) �n +i=1 vσ(σ−1(i)) +σ−1(i) +since σ is bijectif, thus +� +σ∈Sn ε(σ) �n +i=1 vσ(i) +i += � +τ∈Sn ε(τ −1) �n +i=1 vi +τ(i), thus the second equality in (K.4) since ε(τ)−1 = ε(τ). +(See Cartan [5].) +K.3 +Determinant of vectors +Definition K.3 (⃗ei)i=1,...,n being a basis in E, the alternating multilinear form det|⃗e ∈ Ωn(E) defined +by +det +|⃗e (⃗e1, ...,⃗en) = 1 +(K.5) +is called the determinant relative to (⃗ei). And, with prop. K.2 (here c = 1), +det +|⃗e (⃗v1, ...,⃗vn) = +� +σ∈Sn +ε(σ) +n +� +i=1 +vσ(i) +i += +� +τ∈Sn +ε(τ) +n +� +i=1 +vi +τ(i) +(K.6) +is called the determinant of the vectors ⃗vi relative to (⃗ei). And we write +Ωn(E) = Vect{det +|⃗e } +(the 1-D vector space spanned by det|⃗e). +(K.7) +Thus, if Aℓ ∈ Ωn(E) then +Aℓ = Aℓ(⃗e1, ...,⃗en) det +|⃗e , +(K.8) +thus if (⃗bi) is another basis then +∃c ∈ R, +det +|⃗b += c det +|⃗e , +with +c = det +|⃗b +(⃗e1, ...,⃗en). +(K.9) +Exercice K.4 Change of measuring unit: If (⃗ai) is a basis and ⃗bj = λ⃗aj for all j, prove +∀j = 1, ..., n, +⃗bj = λ⃗aj +=⇒ +det +|⃗a = λn det +|⃗b +(K.10) +(relation between volumes relative to a change of measuring unit in the Euclidean case). +Answer. det +|⃗a (⃗b1, ...,⃗bn) = det +|⃗a (λ⃗a1, ..., λ⃗an) +multi += +linear λn det +|⃗a (⃗a1, ...,⃗an) +(K.5) += λn (K.5) += λn det +|⃗b +(⃗b1, ...,⃗bn). +Proposition K.5 det|⃗e(⃗v1, ...,⃗vn) ̸= 0 iff (⃗v1, ...,⃗vn) is a basis, or equivalently, det|⃗e(⃗v1, ...,⃗vn) = 0 iff +⃗v1, ...,⃗vn are linearly dependent. +Proof. If one of the ⃗vi is = ⃗0 then det|⃗e(⃗v1, ...,⃗vn) = 0 (multilinearity), and if �n +i=1ci⃗vi = 0 and one +of the ci ̸= 0 and then a ⃗vi is a linear combination of the others thus det|⃗e(⃗v1, ...,⃗vn) = 0 (since det|⃗e +is alternate). Thus det|⃗e(⃗v1, ...,⃗vn) ̸= 0 ⇒ the ⃗vi are independent. And if the ⃗vi are independent then +(⃗v1, ...,⃗vn) is a basis, thus det|⃗v(⃗v1, ...,⃗vn) = 1 ̸= 0, with det|⃗v = c det|⃗e, thus det|⃗e(⃗v1, ...,⃗vn) ̸= 0. +143 + +144 +K.4. +Determinant of a matrix +Exercice K.6 In R2. Let ⃗v1 = �2 +i=1 vi +1⃗ei and ⃗v2 = �2 +j=1 vj +2⃗ej (duality notations). Prove: +det +|⃗e (⃗v1,⃗v2) = v1 +1v2 +2 − v2 +1v1 +2. +(K.11) +Answer. +Development relative to the first column (linearity used for the first vector ⃗v1 = v1 +1⃗e1 + v2 +1⃗e2): +det|⃗e(⃗v1,⃗v2) = det|⃗e(v1 +1⃗e1 + v2 +1⃗e2,⃗v2) = v1 +1 det|⃗e(⃗e1,⃗v2) + v2 +1 det|⃗e(⃗e2,⃗v2). +Thus (linearity used for the second +vector ⃗v2 = v1 +2⃗e1 + v2 +2⃗e2): det|⃗e(⃗v1,⃗v2) = 0 + v1 +1v2 +2 det(⃗e1,⃗e2) + v2 +1v1 +2 det(⃗e2,⃗e1) + 0 = v1 +1v2 +2 − v2 +1v1 +2. +Exercice K.7 In R3, with ⃗vj = �3 +i=1 vi +j⃗ei, prove: +det(⃗v1,⃗v2,⃗v3) = +3 +� +i,j,k=1 +εijkvi +1vj +2vk +3, +(K.12) +where εijk = 1 +2(j−i)(k−j)(k−i), i.e. εijk = 1 if (i, j, k) = (1, 2, 3), (3, 1, 2) or (2, 3, 1) (even signature), +εijk = −1 if (i, j, k) = (3, 2, 1), (1, 3, 2) and (2, 1, 3) (odd signature), and εijk = 0 otherwise. +Answer. Development relative to the first column (as in exercise K.6). +Result = v1 +1v2 +2v3 +3 + v1 +2v2 +3v3 +1 + v1 +3v2 +1v3 +2 − v3 +1v2 +2v1 +3 − v3 +2v2 +3v1 +1 − v3 +3v2 +1v1 +2. +K.4 +Determinant of a matrix +Let M = [Mij] i=1,...,n +j=1,...,n be a n2 real matrix. Let ⃗Rn = R × ... × R (Cartesian product n-times) with its +canonical basis ( ⃗Ei). Let ⃗vj ∈ ⃗Rn, ⃗vj = �n +i=1Mij ⃗Ei; So M = +� [⃗v1]| ⃗E, ..., [⃗vn]| ⃗E +� +. +Definition K.8 The determinant of the matrix M = +� [⃗v1]| ⃗E, ..., [⃗vn]| ⃗E +� +is +det(M) := det +| ⃗E +(⃗v1, ...,⃗vn). +(K.13) +Proposition K.9 Let M T be the transposed matrix, i.e., (M T )ij = Mji for all i, j. Then +det(M T ) = det(M). +(K.14) +Proof. det[Mij] = det +⃗E +(⃗v1, ...,⃗vn) +(K.6) += +� +σ∈Sn +ε(σ) +n +� +i=1 +vσ(i) +i += +� +τ∈Sn +ε(τ) +n +� +i=1 +vi +τ(i) = det[Mji]. +K.5 +Volume +Definition K.10 Let (⃗ei) be a Euclidean basis. Consider a parallelepiped in Rn which sides are vectors +⃗v1, ...,⃗vn; Its algebraic volume relative to (⃗ei) is +algebraic volume = det +|⃗e (⃗v1, ...,⃗vn). +(K.15) +And its volume relative to (⃗ei) is (non negative) +volume = +��det +|⃗e (⃗v1, ...,⃗vn) +��. +(K.16) +E.g., if n = 1 and ⃗v = v1⃗e1, then det|⃗e(⃗v) = v1 is the algebraic length of ⃗v (relative to the unit +of measurement given by ⃗e1). And | det|⃗e(⃗v)| = |v1| is the length of ⃗v (the norm of ⃗v). (The volume +function (⃗v1, ...,⃗vn) → +��det|⃗e(⃗v1, ...,⃗vn) +�� is not a multilinear form, because the absolute value function is +not linear.) E.g., if n = 2 or 3, see exercises K.6-K.7. +Notation. Let (⃗ei) be a Cartesian basis and (ei) = (dxi) be the dual basis. Then, cf. Cartan [6], +det +|⃗e +noted += +e1 ∧ ... ∧ en = dx1 ∧ ... ∧ dxn. +(K.17) +And, for integration, the volume element (non negative) uses a Euclidean basis (⃗ei) and is +dΩ(⃗x) = | det +|⃗e | = |dx1 ∧ ... ∧ dxn| noted += +dx1...dxn. +(K.18) +Thus +the +volume +of +a +parallelepiped +at +⃗x +which +sides +are +given +by +δx1⃗u1, ..., δxn⃗un +is +dΩ(⃗x)(δx1⃗u1, ..., δxn⃗un) = |δx1...δxn| | det|⃗e(⃗u1, ..., ⃗un)|; Thus the volume of a polygonal domain Ω = +144 + +145 +K.6. +Determinant of an endomorphism +�N +i=1 Pi where Pi is a parallelepiped which sides are given by δxi,1⃗ui,1, ..., δxi,n⃗ui,n is +|Ω| = +N +� +i=1 +| det +|⃗e (⃗ui,1, ..., ⃗ui,n)|δxi,1...δxi,n. +(K.19) +And thus (Riemann approach), the volume of a regular domain Ω is written +|Ω| = +� +Ω +dΩ = +� +⃗x∈Ω +| det +|⃗e (⃗ui,1, ..., ⃗ui,n)| dx1...dxn. +(K.20) +In particular, since any regular volume Ω can be approximated with cubes as small as wished, |Ω| = +�N +i=1 |δxi,1...δxi,n det|⃗e(⃗e1, ...,⃗en)| = �N +i=1 |δxi,1...δxi,n| gives +|Ω| = +� +Ω +dΩ = +� +⃗x∈Ω +dx1...dxn. +(K.21) +Exercice K.11 Let Ψ : ⃗q = (q1, ..., qn) ∈ [a1, b1] × ... × [an, bn] → ⃗x = (x1 = Ψ1(⃗q), ..., xn = Ψn(⃗q)) ∈ Ω +be a parametric description of a domain Ω; Prove +dΩ(⃗x) = |JΨ(⃗q)| dq1...dqn +(= | det +|⃗e (⃗p1(⃗x), ..., ⃗pn(⃗x))| dq1...dqn), +(K.22) +where (⃗pi(x)) = ( ∂Ψ +∂qi (⃗q)) is the parametric basis at ⃗x = Ψ(⃗q) and JΨ(⃗q) = det|⃗e[dΨ(⃗q)]|⃗e is the Jacobian +matrix of Ψ at ⃗q. And thus |Ω| = +� +⃗q |JΨ(⃗q)| dq1...dqn. +Answer. Polar coordinates for illustration purpose (immediate generalization): Consider the disk Ω parametrized +with the polar coordinate system Ψ : ⃗q = (ρ, θ) ∈ [0, R] × [0, 2π] → ⃗x = (x = ρ cos θ, y = ρ sin θ) ∈ R2 +where a Euclidean basis (⃗e1,⃗e2) has been used in R2 (so ⃗x = ρ cos θ⃗e1 + ρ sin θ⃗e2). The associated polar ba- +sis at ⃗x = Ψ(⃗q) is (⃗p1(⃗x) = +∂Ψ +∂ρ (ρ, θ), ⃗p2(⃗x) = +∂Ψ +∂θ (ρ, θ)), so [⃗p1(⃗x)]|⃗e = +� cos θ +sin θ +� +and [⃗p2(⃗x)]|⃗e = +� −ρ sin θ +ρ cos θ +� +. +Thus det|⃗e(⃗p1(⃗x), ⃗p2(⃗x)) = ρ (> 0 here), thus dΩ = |ρ| dρdθ = ρ dρdθ. Thus the volume is |Ω| = +� +⃗x∈Ω dΩ = +� R +ρ=0 +� 2π +θ=0 ρ dρdθ (= πR2). +Exercice K.12 What is the “volume element” on a regular surface Σ in R3, called the “surface element”? +Answer. Let (⃗e1,⃗e2,⃗e3) be a Euclidean basis in R3. We need a regular parametric description Ψ : (u, v) ∈ [a1, b2]× +[a2, b2] → ⃗x = Ψ(u, v) = x1(u, v)⃗e1 + ... + x3(u, v)⃗e3 of the geometric surface Σ = Im(Ψ). Thus ⃗t1(⃗x) = ∂Ψ +∂u (u, v) +and ⃗t2(⃗x) = ∂Ψ +∂v (u, v) are tangent vectors at Σ at ⃗x = Ψ(u, v). Hence a normal unit vector is ⃗n(⃗x) = +⃗t1(⃗x)∧⃗t2(⃗x) +||⃗t1(⃗x)∧⃗t2(⃗x)||, +and thus det|⃗e(⃗t1,⃗t2,⃗n) = ||⃗t1(⃗x) ∧ ⃗t2(⃗x)|| is the area of the parallelogram which sides are given by ⃗t1 and ⃗t2 +(volume with height 1). Thus the surface element at ⃗x = Ψ(u, v) is dΣ(⃗x) = || ∂Ψ +∂u (u, v) ∧ ∂Ψ +∂v (u, v)|| dudv. Thus +|Σ| = +� +⃗x∈Σ dΣ(⃗x) = +� b1 +u=a1 +� b2 +v=a2 || ∂Ψ +∂u (u, v) ∧ ∂Ψ +∂v (u, v)|| dudv. +K.6 +Determinant of an endomorphism +K.6.1 +Definition and basic properties +Definition K.13 The determinant of an endomorphism L ∈ L(E; E) relative to a basis (⃗ei) is +� +det +|⃗e (L) := det +|⃗e (L.⃗e1, ..., L.⃗en). +(K.23) +This define � +det|⃗e : L(E; E) → R. (If the context is not ambiguous, then � +det|⃗e =noted det|⃗e.) +Proposition K.14 Let L ∈ L(E; E). +1- If L = I the identity, then � +det|⃗e(I) = 1 for all basis (⃗ei). +2- For all ⃗v1, ...,⃗vn ∈ E, +det +|⃗e (L.⃗v1, ..., L.⃗vn) = � +det +|⃗e (L) det +|⃗e (⃗v1, ...,⃗vn). +(K.24) +3- If L.⃗ej = �n +i=1Lij⃗ei, i.e. [L]|⃗e = [Lij], then +� +det +|⃗e (L) = det([L]|⃗e) = det([Lij]). +(K.25) +4- For all M ∈ L(E; E), and with M ◦ L =noted M.L (thanks to linearity), +� +det +|⃗e (M.L) = � +det +|⃗e (M) � +det +|⃗e (L) = � +det +|⃗e (L.M). +(K.26) +5- L is invertible iff � +det|⃗e(L) ̸= 0. +145 + +146 +K.6. +Determinant of an endomorphism +6- If L is invertible then +� +det +|⃗e (L−1) = +1 +� +det|⃗e(L) +. +(K.27) +7- If (·, ·)g is an inner dot product in E and LT +g is the (·, ·)g transposed of L (i.e., (LT +g ⃗w, ⃗u)g = (⃗w, L.⃗u)g +for all ⃗u, ⃗w ∈ E) then +� +det +|⃗e (LT +g ) = � +det +|⃗e (L). +(K.28) +8- If (⃗ei) and (⃗bi) are two (·, ·)g-orthonormal bases in ⃗Rn +t (e.g. two Euclidean basis for the same +measuring unit), then det|⃗b = ± det|⃗e. +Proof. 1- � +det|⃗e(I) =(K.23) det|⃗e(I.⃗e1, ..., I.⃗en) =(K.5) det|⃗e(⃗e1, ...,⃗en) = 1, true for all basis. +2- Let m : (⃗v1, ...,⃗vn) → m(⃗v1, ...,⃗vn) := det|⃗e(L.⃗v1, ..., L.⃗vn): It is a multilinear alternated form, since +L is linear; Thus m =(K.8) m(⃗e1, ...,⃗en) det|⃗e; With m(⃗e1, ...,⃗en) =(K.23) � +det|⃗e(L), thus (K.24). +3- Apply (K.13) with M = [L]|⃗e to get (K.25). +4- det|⃗e((M.L).⃗e1, ..., (M.L).⃗en) = det|⃗e(M.(L.⃗e1), ..., M.(L.⃗en)) =(K.24) � +det|⃗e(M) det|⃗e(L.⃗e1, ..., L.⃗en). +5- If L is invertible, then 1 = � +det|⃗e(I) = � +det|⃗e(L.L−1) = � +det|⃗e(L) � +det|⃗e(L−1), thus � +det|⃗e(L) ̸= 0. +If � +det|⃗e(L) ̸= 0 then det|⃗e(L.⃗e1, ..., L.⃗en) ̸= 0, thus (L.⃗e1, ..., L.⃗en) is a basis, thus L is invertible. +6- (K.26) gives 1 = � +det|⃗e(I) = � +det|⃗e(L−1.L) = � +det|⃗e(L). � +det|⃗e(L−1), thus (K.27). +7- +(LT +g ⃗w, ⃗u)g += +(⃗w, L.⃗u)g +gives +[g]|⃗e.[LT +g ]|⃗e += +([L]|⃗e)T .[g]|⃗e, +thus +det([g]|⃗e) det([LT +g ]|⃗e) += +det(([L]|⃗e)T ) det([g]|⃗e), and det([g]|⃗e) ̸= 0 (exercise), thus (K.28). +8- Let P be the change of basis endomorphism from (⃗ei) to (⃗bi), and P be the transition matrix +from (⃗ei) to (⃗bi). +Both basis being (·, ·)g-orthonormal, P T .P = I, thus det(P) = ±1 = � +det|⃗e(P). +And det|⃗e(⃗b1, ...,⃗bn) = det|⃗e(P.⃗e1, ..., P.⃗en) = � +det|⃗e(P) det|⃗e(⃗e1, ...,⃗en) = � +det|⃗e(P) det|⃗b(⃗b1, ...,⃗bn), thus +det|⃗e = � +det|⃗e(P) det|⃗b = ± det|⃗b. +Definition K.15 Two (·, ·)g-orthonormal bases (⃗ei) and (⃗bi) have the same orientation iff det|⃗b = + det|⃗e. +Exercice K.16 Prove � +det|⃗e(λL) = λn � +det|⃗e(L). +Answer. � +det +|⃗e (λL) = det +|⃗e (λL.⃗e1, ..., λL.⃗en) = λn det +|⃗e (L.⃗e1, ..., L.⃗en) = λn � +det +|⃗e (L). +K.6.2 +The determinant of an endomorphism is objective +Proposition K.17 Let (⃗ai) and (⃗bi) be bases in E. The determinant of an endomorphism L ∈ L(E; E) +is objective (observer independent, here basis independent): +(det([L]|⃗a) =) +� +det +|⃗a (L) = � +det +|⃗b +(L) +(= det([L]|⃗b)). +(K.29) +NB: But the determinant of n vectors is not objective, cf. (K.9) (compare the change of basis formula +for vectors [⃗w]|⃗b = P −1.[⃗w]|⃗a with the change of basis formula for endomorphisms [L]|⃗b = P −1.[L]|⃗a.P). +Proof. Let (⃗ai) and (⃗bi) be bases in E, and P be the transition matrix from (⃗ai) to (⃗bi). The change +of basis formula [L]|⃗b = P −1.[L]|⃗a.P and (K.26) give det([L]|⃗b) = det(P −1) det([L]|⃗a) det(P) = det([L]|⃗a), +thus (K.25) gives (K.29). +Exercice K.18 Let (⃗ai) and (⃗bi) be bases in E, and P ∈ L(E; E) be the change of basis endomorphism +from (⃗ai) to (⃗bi) (i.e., P.⃗aj = ⃗bj for all j). Prove +det +|⃗a (⃗b1, ...,⃗bn) = � +det +|⃗a (P), +thus +det +|⃗a = � +det +|⃗a (P) det +|⃗b +, +i.e. +det +|⃗b += +det|⃗a +� +det|⃗a(P) +, +(K.30) +Answer. +det +|⃗a (⃗b1, ...,⃗bn) = det +|⃗a (P.⃗a1, ..., P.⃗an) +(K.24) += +� +det +|⃗a (P) det +|⃗a (⃗a1, ...,⃗an) = +� +det +|⃗a (P) 1 = +� +det +|⃗a (P) det +|⃗b +(⃗b1, ...,⃗bn), +thus (K.30) and det|⃗a = � +det|⃗a(P) det|⃗b and det|⃗a(⃗v1, ...,⃗vn) = � +det|⃗a(P) det|⃗b(⃗v1, ...,⃗vn). +146 + +147 +K.7. +Determinant of a linear map +K.7 +Determinant of a linear map +(Needed for the deformation gradient F t0 +t (P) = dΦt0 +t (P) : ⃗Rn +t0 → ⃗Rn +t .) +Let A and B be vector spaces, dim A = dim B = n, and (⃗ai) and (⃗bi) be bases in A and B. +K.7.1 +Definition and first properties +Definition K.19 The determinant of a linear map L ∈ L(A; B) relative to the bases (⃗ai) and (⃗bi) is +� +det +|⃗a,⃗b +(L) := det +|⃗b +(L.⃗a1, ..., L.⃗an). +(K.31) +(And � +det|⃗a,⃗b(L) =noted det(L) if the bases are implicit.) +Thus, (K.13) gives, with L.⃗aj = �n +i=1Lij⃗bi, i.e. [L]|⃗a,⃗b = [Lij]: +� +det +|⃗a,⃗b +(L) = det([L]|⃗a,⃗b) = det([Lij]). +(K.32) +Proposition K.20 Let ⃗u1, ..., ⃗un ∈ A. Then +det +|⃗b +(L.⃗u1, ..., L.⃗un) = � +det +|⃗a,⃗b +(L) det +|⃗a (⃗u1, ..., ⃗un). +(K.33) +Proof. m : (⃗u1, ..., ⃗un) ∈ An → m(⃗u1, ..., ⃗un) := det|⃗b(L.⃗u1, ..., L.⃗un) ∈ R is a multilinear alternated form, +since L is linear; And m(⃗a1, ...,⃗an) = det|⃗b(L.⃗a1, ..., L.⃗an) =(K.31) � +det|⃗a,⃗b(L) = � +det|⃗a,⃗b(L) det|⃗a(⃗a1, ...,⃗an). +Thus m = � +det|⃗a,⃗b(L) det|⃗a, cf. (K.9), thus (K.33). +Corollary K.21 Let A, B, C be vector spaces such that dim A = dim B = dim C = n. Let (⃗ai), (⃗bi), (⃗ci) +be bases in A, B, C. Let L : A → B and M : B → C be linear. Then, with M ◦ L =noted M.L (thanks +to linearity), +� +det +|⃗a,⃗c(M.L) = � +det +|⃗a,⃗b +(L) � +det +|⃗b,⃗c +(M). +(K.34) +Proof. � +det +|⃗a,⃗c(M.L) = det +|⃗c (M.L.⃗a1), ..., M.L.⃗an)) = � +det +|⃗b,⃗c +(M) det +|⃗b +(L.⃗a1, ..., L.⃗an) = � +det +|⃗b,⃗c +(M) � +det +|⃗a,⃗b +(L). +K.7.2 +Jacobian of a motion, and dilatation +Let �Φ be a motion, let t0, t ∈ R, let Φt0 +t be the associated motion, let F t0 +t (pt0) := dΦt0 +t (pt0) : ⃗Rn +t0 → ⃗Rn +t +the deformation gradient at pt0 ∈ Ωt0 relative to t0 and t, cf. (4.1). Let ( ⃗Ei) be a Euclidean basis in ⃗ +Rn +t0 +and (⃗ei) be a Euclidean basis in ⃗ +Rn +t for all t ≥ t0, and [F t0 +t (pt0)]| ⃗E,⃗e = [Fij(pt0)], i.e., F t0 +t (pt0). ⃗Ej = +�n +i,j=1Fij(pt0)⃗ei for all j. +Definition K.22 The “volume dilatation” at pt0, relative to the Euclidean bases ( ⃗Ei) in +⃗ +Rn +t0 and (⃗ei) +in ⃗ +Rn +t , is +J| ⃗E,⃗e(Φt0 +t )(pt0) := � +det +| ⃗E,⃗e +(F t0 +t (pt0)) +(= det +|⃗e (F t0 +t (pt0). ⃗E1, ..., F t0 +t (pt0). ⃗En) = det([Fij(pt0)])), +(K.35) +usually written J| ⃗E,⃗e := det([F]| ⃗E,⃗e) (or simply J = det(F) when everything is implicit). +So, at t0 at pt0, (pt0, ⃗E1, ..., ⃗En) is a unit parallelepiped which volume is 1 relative to the unit of mea- +surement chosen in ⃗Rn +t0, and, at t at pt = Φt0 +t (pt0), J| ⃗E,⃗e(Φt0 +t )(pt0) = det|⃗e(F t0 +t (pt0). ⃗E1, ..., F t0 +t (pt0). ⃗En) +is the volume of the parallelepiped (pt, F t0 +t (pt0). ⃗E1, ..., F t0 +t (pt0). ⃗En) relative to the unit of measurement +chosen in ⃗Rn +t . +Interpretation: With t2 > t1 ≥ t0, and [⃗ei) is the basis at t1 and t2: +• Dilatation if J| ⃗E,⃗e(Φt0 +t2)(pt0) > J| ⃗E,⃗e(Φt0 +t1)(pt0) (volume increase), +147 + +148 +K.8. +Dilatation rate +• contraction if J| ⃗E,⃗e(Φt0 +t2)(pt0) < J| ⃗E,⃗e(Φt0 +t1)(pt0) (volume decrease), and +• incompressibility if J| ⃗E,⃗e(Φt0 +t2)(pt0) = J| ⃗E,⃗e(Φt0 +t1)(pt0) for all t (volume conservation). +In particular, if (⃗ei) = ( ⃗Ei) then J|⃗e,⃗e(Φt0 +t0)(pt0) = 1, and if t > t0, then +• Dilatation if J|⃗e,⃗e(Φt0 +t )(pt0) > 1 (volume increase), +• contraction if J|⃗e,⃗e(Φt0 +t )(pt0) < 1 (volume decrease), and +• incompressibility if J|⃗e,⃗e(Φt0 +t )(pt0) = 1 for all t (volume conservation). +Exercice K.23 Let ( ⃗Ei) be a Euclidean basis in ⃗Rn +t0, and let (⃗ai) and (⃗bi) be two Euclidean bases in ⃗Rn +t +for the same Euclidean dot product (·, ·)g. Prove: +J| ⃗E,⃗a(Φt0 +t (P)) = ±J| ⃗E,⃗b(Φt0 +t (P)). +(K.36) +Answer. P being the transition matrix from (⃗ai) to (⃗bi), det(P) = ±1 here. And (4.30) gives [F]| ⃗ +E,⃗a = P.[F]| ⃗ +E,⃗b, +thus det([F]| ⃗ +E,⃗a) = ± det([F]| ⃗ +E,⃗b), thus det|⃗a(F. ⃗E1, ..., F. ⃗En) = ± det|⃗b(F. ⃗E1, ..., F. ⃗En). +K.7.3 +Determinant of the transposed +Let (A, (·, ·)g) and (B, (·, ·)h) be finite dimensional Hilbert spaces. +Let L ∈ L(A; B) (a linear map). +Recall: The transposed LT +gh ∈ L(B; A) is defined by, for all ⃗u ∈ A and all ⃗w ∈ B, cf. (A.68) +(LT +gh.⃗w, ⃗u)g := (⃗w, L.⃗u)h. +(K.37) +Let (⃗ai) be a basis in A and (⃗bi) be a basis in B. Then +� +det([LT +gh]|⃗b,⃗a) = det([L]|⃗a,⃗b)det([(·, ·)g]|⃗a) +det([(·, ·)h]|⃗b). +(K.38) +Indeed, (K.37) gives [(·, ·)g]|⃗a.[LT +gh]|⃗b,⃗a = ([L]|⃗a,⃗b)T .[(·, ·)h]|⃗b. +K.8 +Dilatation rate +A unique Euclidean basis (⃗ei) at all time is chosen, and (·, ·)g is the associated inner dot product. +K.8.1 +∂Jt0 +∂t (t, pt0) = Jt0(t, pt0) div⃗v(t, pt) +A regular motion �Φ is considered, cf. (1.5), and the Eulerian velocity is ⃗v(t, pt) = ∂�Φ +∂t (t, PObj) at pt = +�Φ(t, PObj). Let t0 be given; The associated motion Φt0 is given by Φt0(t, pt0) = �Φ(t, PObj) =noted pt when +pt0 = �Φ(t0, PObj), +(3.1), and is supposed to be at least C2; The Lagrangian velocity is ⃗V (t, pt0) = +∂Φt0 +∂t (t, pt0), and the Eulerian velocity satisfies ⃗v(t, pt) = ∂Φt0 +∂t (t, pt0) when pt = Φt0(t, pt0), cf. (3.25). Let +F t0(t, pt0) = dΦt0(t, pt0) = F t0 +t (pt0) = dΦt0 +t (pt0), and consider the Jacobian +Jt0 +t (pt0) = det +|⃗e (F t0 +t (pt0)) = Jt0(t, pt0), +(K.39) +Lemma K.24 +∂Jt0 +∂t (t, pt0) satisfies, with pt = Φt0 +t (pt0), +∂Jt0 +∂t (t, pt0) = Jt0(t, pt0) div⃗v(t, pt) +(K.40) +(value to be considered at t at pt). In particular, �Φ is incompressible iff div⃗v(t, pt) = 0. +Proof. Let O be a origin in Rn. Let −−−→ +OΦt0 = �n +i=1Φi⃗ei, ⃗V t0 = �n +i=1V i⃗ei, ⃗v = �n +i=1vi⃗ei, F t0. ⃗Ej = +dΦt0. ⃗Ej = �n +i=1 +∂Φi +∂Xj ⃗ei. Let [F t0]| ⃗E,⃗e =noted F, Jt0 =noted J and [dΦi]| ⃗E = +� ∂Φi +∂X1 +... +∂Φi +∂Xn +� +=noted dΦi +148 + +149 +K.8. +Dilatation rate +(row matrix). Thus J = det F = det +� +� +dΦ1 +... +dΦn +� +�, thus (a determinant is multilinear) +∂J +∂t = det +� +� +� +� +� +� +∂(dΦ1) +∂t +dΦ2 +... +dΦn +� +� +� +� +� +� ++ . . . + det +� +� +� +� +� +dΦ1 +... +dΦn−1 +∂(dΦn) +∂t +) +� +� +� +� +� . +With Φt0 C2, thus ∂(dΦi) +∂t +(t, pt0) Swhartz += +d(∂Φi +∂t )(t, pt0) = dV i(t, pt0) = dvi(t, pt).F(t, pt0), cf. (3.27). +Thus +det +� +� +� +� +� +� +∂(dΦ1) +∂t +dΦ2 +... +dΦn +� +� +� +� +� +� += det +� +� +� +� +� +� +� +n +� +i=1 +∂v1 +∂xi dΦi +dΦ2 +... +dΦn +� +� +� +� +� +� +� +det is += +alternating det +� +� +� +� +� +� +∂v1 +∂x1 dΦ1 +dΦ2 +... +dΦn +� +� +� +� +� +� += ∂v1 +∂x1 det +� +� +� +� +dΦ1 +dΦ2 +... +dΦn +� +� +� +� = ∂v1 +∂x1 J +Idem for the other terms, thus +∂J +∂t (t, pt0) = ∂v1 +∂x1 (t, pt) J(t, pt0) + . . . + ∂vn +∂xn (t, pt) J(t, pt0) = div⃗v(t, pt) J(t, pt0), +i.e. (K.40). +Definition K.25 div⃗v(t, pt) is the dilatation rate. +K.8.2 +Leibniz formula +Proposition K.26 (Leibniz formula) Under regularity assumptions (e.g. hypotheses of the Lebesgue +theorem to be able to derive under +� +) we have +d +dt +�� +pt∈Ωt +f(t, pt) dΩt +� += +� +pt∈Ωt +�Df +Dt + f div⃗v +� +(t, pt) dΩt += +� +pt∈Ωt +�∂f +∂t + df.⃗v + f div(⃗v) +� +(t, pt) dΩt += +� +pt∈Ωt +�∂f +∂t + div(f⃗v) +� +(t, pt) dΩt. +(K.41) +Proof. Let +Z(t) := +� +p∈Ωt +f(t, p) dΩt = +� +P ∈Ωt0 +f(t, Φt0(t, P)) Jt0(t, P) dΩt0. +(The Jacobian is positive for a regular motion.) Then (derivation under +� +) +Z′(t) = +� +P ∈Ωt0 +Df +Dt (t, pt) Jt0(t, P) + f(t, pt)∂Jt0 +∂t (t, P) dΩt0 += +� +P ∈Ωt0 +(Df +Dt (t, pt) + f(t, pt) div⃗v(t, pt))Jt0(t, P) dΩt0, +thanks to (K.40). And div(f⃗v) = df.⃗v + f div⃗v gives (K.41). +Corollary K.27 With (⃗u, ⃗w)g =noted ⃗u • ⃗w (in the given Euclidean framework), +d +dt +� +Ωt +f(t, pt) dΩt = +� +Ωt +∂f +∂t (t, pt) dΩt + +� +∂Ωt +(f⃗v • ⃗n)(t, pt) dΓt, +(K.42) +sum of the temporal variation within Ωt and the flux through the surface ∂Ωt. +Proof. Apply (K.41)3. +149 + +150 +K.9. +∂J/∂F = J F −T +K.9 +∂J/∂F = J F −T +K.9.1 +Meaning of +∂ det +∂Mij ? +Let Mnn = {M = [Mij] ∈ Rn2} be the set of n ∗ n matrices, and consider the function +Z := det : +� +Mnn → R +M = [Mij] → Z(M) := det(M) = det([Mij]). +(K.43) +Question: What does +∂Z +∂Mij (M) mean? +Answer: It is the “standard meaning” of a directional derivative +∂f +∂xi (⃗x) = df(⃗x).⃗ei... +where here +f = Z, thus ⃗x =noted M is a matrix (a vector in Mnn), and (⃗ei) is the canonical basis (mij) in Mnn +(all the elements of the matrix mij vanish but the element at intersection of line i and column j which +equals 1). So: +∂Z +∂Mij +(M) := dZ(M).mij = lim +h→0 +Z(M + hmij) − Z(M) +h +(∈ R). +(K.44) +K.9.2 +Calculation of +∂ det +∂Mij +Proposition K.28 +∀i, j, +∂Z +∂Mij +(M) = Z(M) (M −T )ij, +written +∂Z +∂M = Z M −T . +(K.45) +Proof. +∂Z +∂Mij (M) := limh→0 +det(M+hmij)−det(M) +h +; The development of the determinant det(M + hmij) +relative to the column j gives +det(M + h[mij]) = det(M) + h cij +(K.46) +where cij is the (i, j)-th cofactor of M; Thus +∂Z +∂Mij (M) = limh→0 +Z(M+hmij)−Z(M) +h += cij; And since +M −1 = +1 +det(M)[cij]T , i.e. [cij] = det(M)M −T , we get +∂Z +∂Mij (M) = det(M)(M −T )ij, i.e. (K.45). +K.9.3 +∂J/∂F = J F −T usually written [ ∂J +∂Fij ] = J F −T +Setting of § K.8: With F := dΦ(pt0) we have F. ⃗Ej = �n +i=1Fij⃗ei where Fij = ∂Φi +∂Xj (pt0), and +JΦ,pt0, ⃗E,⃗e +noted += +J : +� +� +� +� +� +L(⃗Rn +t0; ⃗Rn +t ) → R +F → J(F) := det([Fij]) +(= det([ ∂Φi +∂Xj +(pt0)]), +(K.47) +so, J(F) is the Jacobian � +det| ⃗E,⃗e(dΦ(pt0)) of Φ at pt0 relative to ( ⃗Ei) and (⃗ei). Thus (K.45) gives: +Corollary K.29 +∀i, j, +∂J +∂Fij +(F) = J(F) ([F]−T )ij, +written +∂J +∂F = J F −T . +(K.48) +K.9.4 +Interpretation of +∂J +∂Fij ? +The first derivations into play are along the directions ⃗Ej at t0: The Fij = ∂Φi +∂Xj (pt0) := dΦi(pt0). ⃗Ej. +Question: +∂J +∂Fij is the usual notation for a directional derivative, cf. § K.9.1. So +∂J +∂Fij is the derivative +in which direction? +Answer: 1- “Identify” F ∈ L(⃗Rn +t0; ⃗Rn +t ) with the tensor �F ∈ L(⃗Rn∗ +t , ⃗Rn +t0; R) given by �F(ℓ, ⃗U) = ℓ.(F.⃗U); +So, if F. ⃗Ej = �n +i=1Fij⃗ei then �F = �n +i,j=1Fij⃗ei ⊗ πEj, relative to a basis ( ⃗Ei) and its covariant dual basis +(πEi) in ⃗Rn +t0 and a basis (⃗ei) and ⃗Rn +t . +2- Define the function � +det ⃗E,⃗e = �J : +� +� +� +L(⃗Rn∗ +t , ⃗Rn +t0; R) → R +�F → �J( �F) := J(F) = det +⃗E,⃗e +(F) = det([Fij]) +� +� +�; +150 + +151 +L.1. +Transformed parallelepiped +3- Then it is meaningful to differentiate �J along the direction ⃗ei ⊗ πEj ∈ L(⃗Rn∗ +t , ⃗Rn +t0; R) to get +∂ �J +∂Fij +( �F) := lim +h→0 +�J|⃗e, ⃗E( �F + h⃗ei ⊗ πEj) − �J|⃗e, ⃗E( �F) +h +( noted += +∂J +∂Fij +(F)); +(K.49) +This is a derivation in both directions πEj in ⃗Rn +t0 (past at pt0) and ⃗ei in ⃗Rn +t (present at pt). What does +this derivative mean? (The answer is unknown to the author.) +L +Transport of volumes and areas +Here Rn = R3 the usual affine space. Let t0, t ∈ R, and Φt0 +t : R × Ωt0 → Ωt, see (3.1). Let FP = dΦt0 +t (P). +Let (·, ·)g be a Euclidean dot product in ⃗Rn (English, French...), with ||.||g the associated norm. +L.1 +Transformed parallelepiped +The Jacobian of Φt0 +t at P relative to a (·, ·)g-Euclidean bases is defined in (K.35): With FP = F t0 +t (P), +JP = J(P) := det +|⃗e (F t0 +t (P). ⃗E1, ..., F t0 +t (P). ⃗En)), +and +JP > 0 +(L.1) +the motion being supposed regular. Thus, if (⃗U1P , ..., ⃗UnP ) is a parallelepiped at P at t0, if ⃗uip = FP .⃗UiP , +then (⃗u1p, ..., ⃗unp) is a parallelepiped at p at t which volume is +det +|⃗e (⃗u1p, ..., ⃗unp) = JP det +|⃗e (⃗U1P , ..., ⃗UnP ). +(L.2) +L.2 +Transformed volumes +Riemann integrals and (L.2) give the change of variable formula: For any regular function f : Ωt → R, +� +pt∈Ωt +f(pt) dΩt = +� +P ∈Ωt0 +f(Φt0 +t (P)) |J(P)| dΩt0. +(L.3) +(See (K.18): dΩt is a positive measure: It is not a multilinear form.) In particular, +|Ωt| = +� +pt∈Ωt +dΩt(pt) = +� +P ∈Ωt0 +|J(P)| dΩt0(P). +(L.4) +(With J(P) > 0 for regular motions.) +L.3 +Transformed parallelogram +Consider two independent vectors ⃗U1P , ⃗U2P ∈ ⃗Rn +t0 at t0 at P, and the vectors ⃗u1p = FP .⃗U1P and ⃗u2p = +FP .⃗U2P at t at p = Φt0 +t (P). Since Φt0 +t is a diffeomorphism, ⃗u1p and ⃗u2p are independent. +Then choose a Euclidean dot product (·, ·)g (English, French...) to be able to use the vectorial product, +cf. (E.15), the same at all time t. Then the areas of the parallelograms are +||⃗U1P ∧ ⃗U2P ||g +and +||⃗u1p ∧ ⃗u2p)||g, +(L.5) +and unit normal vectors to the quadrilaterals are +⃗NP = +⃗U1P ∧ ⃗U2P +||⃗U1P ∧ ⃗U2P ||g +∈ ⃗Rn +t0, +and +⃗np = +⃗u1p ∧ ⃗u2p +||⃗u1p ∧ ⃗u2p||g +∈ ⃗Rn +t . +(L.6) +Proposition L.1 If ⃗u1p = FP .⃗U1P and ⃗u2p = FP .⃗U2P , then +⃗u1p ∧ ⃗u2p = JP F −T +P +. +�⃗U1P ∧ ⃗U2P +� +, +and +||⃗u1p ∧ ⃗u2p||g = JP ||F −T +P +.(⃗U1P ∧ ⃗U2P )||g, +(L.7) +since JP > 0 (for regular motions), and +⃗np = +F −T +P +. ⃗NP +||F −T +P +. ⃗NP ||g +(̸= FP . ⃗NP in general). +(L.8) +151 + +152 +L.4. +Transformed surface +Proof. Let ⃗WP ∈ ⃗ +Rn +t0 and ⃗wp = FP . ⃗WP . Then the volume of the parallelepiped (⃗u1p, ⃗u2p, ⃗wp) is +(⃗u1p ∧ ⃗u2p, ⃗wp)g = det(⃗u1p, ⃗u2p, ⃗wp) = det(FP .⃗U1P , FP .⃗U2P , FP . ⃗WP ) = det(FP ) det(⃗U1P , ⃗U2P , ⃗WP ) += JP (⃗U1P ∧ ⃗U2P , ⃗WP )g = JP (⃗U1P ∧ ⃗U2P , F −1 +P .⃗wp)g = JP (F −T +P +.(⃗U1P ∧ ⃗U2P ), ⃗wp)g, +(L.9) +for all ⃗wp, thus (L.7), thus +⃗u1p∧⃗u2p +||⃗u1p∧⃗u2p||g = +JP F −T +P +.(⃗U1P ∧⃗U2P ) +JP ||F −T +P +.(⃗U1P ∧⃗U2P )||g , thus (L.8). +L.4 +Transformed surface +L.4.1 +Deformation of a surface +A parametrized surface Ψt0 in Ωt0 and the associated geometric surface St0 are defined by +Ψt0 : +� +[a, b] × [c, d] → Ωt0 +(u, v) → P = Ψt0(u, v) +� +and +St0 = Im(Ψt0) ⊂ Ωt0. +(L.10) +(It is also represented after a choice of an origin O by the vector valued parametrized surface ⃗rt0 = −−−→ +OΨt0.) +The transformed parametric surface is Ψt := Φt0 +t ◦ Ψt0 and the associated geometric surface is St: +Ψt := Φt0 +t ◦ Ψt0 : +� +[a, b] × [c, d] → Ωt0 +(u, v) → p = Ψt(u, v) = Φt0 +t (Ψt0(u, v)) = Φt0 +t (P) +� +and +St = Φt0 +t (St0). +(L.11) +(After a choice of an origin O, the associated vector valued parametrized surface is ⃗rt = −−→ +OΨt.) +Let ( ⃗E1, ⃗E2) be the canonical basis in the space R × R ⊃ [a, b] × [c, d] = {(u, v)} of parameters. +The surface Ψt0 is supposed to be regular, that is, Ψt0 is C1 and, for all P = Ψt0(u, v) ∈ St0, the +tangents vectors ⃗T1P and ⃗T2P at P are independent, that is, +⃗T1P := dΨt0(u, v). ⃗E1 +noted += +∂Ψt0 +∂u (u, v), +⃗T2P := dΨt0(u, v). ⃗E2 +noted += +∂Ψt0 +∂v (u, v), +� +� +� +� +� +and +⃗T1P ∧ ⃗T2P ̸= ⃗0. +(L.12) +And the tangent vectors at St at p = Φt0 +t (P) at t are +� +� +� +� +� +⃗t1p := dΨt(u, v). ⃗E1 = ∂Ψt +∂u (u, v), +so +⃗t1p = FP .⃗T1P +(= dΦt0 +t (P).∂Ψt0 +∂u (u, v)), +⃗t2p := dΨt(u, v). ⃗E2 = ∂Ψt +∂v (u, v), +so +⃗t2p = FP .⃗T2P +(= dΦt0 +t (P).∂Ψt0 +∂v (u, v)). +(L.13) +These vectors are independent since Φt0 +t is a diffeomorphism and Ψt0 is regular. In facts, we used tangent +vectors to curves and their push-forwards, cf. figure 4.1 and § 6.5.1. +L.4.2 +Euclidean dot product and unit normal vectors +Then choose a Euclidean dot product (·, ·)g (English, French...), to be able to use the vectorial product, +cf. (E.15), the same at all time t. Then the scalar area elements dΣP at P at St0 relative to Ψt0, and dσp +at p at St relative to Ψt, are +� +� +� +� +� +dΣP := ||∂Ψt0 +∂u (u, v) ∧ ∂Ψt0 +∂v (u, v)||g du dv +(= ||⃗T1P ∧ ⃗T2P ||g du dv), +dσp := ||∂Ψt +∂u (u, v) ∧ ∂Ψt +∂v (u, v)||g du dv +(= ||⃗t1p ∧ ⃗t2p||g du dv). +(L.14) +And the areas of St0 and St are +� +� +� +� +� +� +� +� +� +|St0| = +� +P ∈St0 +dΣP := +� b +u=a +� d +v=c +||∂Ψt0 +∂u (u, v) ∧ ∂Ψt0 +∂v (u, v)||g du dv, +|St| = +� +p∈St +dσp := +� b +u=a +� d +v=c +||∂Ψt +∂u (u, v) ∧ ∂Ψt +∂v (u, v)||g du dv. +(L.15) +(See (K.18): dΣP and dσp are positive measures: They are not multilinear forms.) +152 + +153 +L.5. +Piola identity +And the unit normal vectors ⃗NP at St0 at P at t0 and ⃗np at St at p at t are +� +� +� +� +� +� +� +� +� +� +� +⃗NP = +∂Ψt0 +∂u (u, v) ∧ ∂Ψt0 +∂v (u, v) +|| ∂Ψt0 +∂u (u, v) ∧ ∂Ψt0 +∂v (u, v)||g +(= +⃗T1P ∧ ⃗T2P +||⃗T1P ∧ ⃗T2P ||g +) +⃗np = +∂Ψt +∂u (u, v) ∧ ∂Ψt +∂v (u, v) +|| ∂Ψt +∂u (u, v) ∧ ∂Ψt +∂v (u, v)||g +(= +⃗t1p ∧ ⃗t2p +||⃗t1p ∧ ⃗t2p||g += . +(L.16) +Then the vectorial area elements d⃗ΣP at P at St0 = Im(Ψt0) relative to ⃗rt0 and d⃗σp at p at St = Im(Ψt) +relative to Ψt are +� +� +� +� +� +d⃗ΣP := ⃗NP dΣP = ∂Ψt0 +∂u (u, v) ∧ ∂Ψt0 +∂v (u, v) du dv +(= ⃗T1P ∧ ⃗T2P du dv) +d⃗σp := ⃗np dσp = ∂Ψt +∂u (u, v) ∧ ∂Ψt +∂v (u, v) du dv +(= ⃗t1p ∧ ⃗t2p du dv). +(L.17) +(Useful to get the flux through a surface: +� +Γ ⃗f • ⃗n dσ = +� +Γ ⃗f • d⃗σ.) +(NB: d⃗ΣP and d⃗σp are not multilinear since dΣP and dσp are not.) +L.4.3 +Relations between surfaces +⃗t1p ∧ ⃗t2p = JP F −T +P +.(⃗T1P ∧ ⃗T2P ), cf. (L.7), gives +∂Ψt +∂u (u, v) ∧ ∂Ψt +∂v (u, v) = JP F −T +P +.(∂Ψt0 +∂u (u, v) ∧ ∂Ψt0 +∂v (u, v)). +(L.18) +This gives the relation between vectorial and scalar area elements, +⃗n dσp = d⃗σp = JP F −T +P +.d⃗ΣP = JP F −T +P +. ⃗NP dΣP , +and +dσp = JP ||F −T +P +. ⃗NP ||g dΣP . +(L.19) +(Check with (L.8).) +L.5 +Piola identity +Reminder: Let M = [M i +j] be a 3∗3 matrix function. We use the usual divergence in continuum mechanics +(non objective) given by divM := +� +� +� +∂M 1 +1 +∂X1 + ∂M 1 +2 +∂X2 + ∂M 1 +3 +∂X3 +∂M 2 +1 +∂X1 + ∂M 2 +2 +∂X2 + ∂M 2 +3 +∂X3 +∂M 3 +1 +∂X1 + ∂M 3 +2 +∂X2 + ∂M 3 +3 +∂X3 +� +� +� = +� +� +� +� +�n +j=1 +∂M 1 +j +∂Xj +�n +j=1 +∂M 2 +j +∂Xj +�n +j=1 +∂M 3 +j +∂Xj +� +� +� +�, cf. (S.65). And if Cof(M) +is the matrix of cofactors (in R3: Cof(M)i +j = M i+1 +j+1M i+2 +j+2 − M i+1 +j+2M i+2 +j+1), then M −1 = +1 +det M Cof(M)T , +i.e., +(det M)M −1 = Cof(M)T . +(L.20) +The framework being Euclidean, we use a Euclidean basis and the associated matrix, and thus (matrix +meaning) +J(P)F(P)−T = Cof(F(P)) noted += +Cof(F)(P), +written +JF −T = Cof(F). +(L.21) +Proposition L.2 (Piola identity) In R3, we have +div(JF −T )(P) = 0, +i.e. +∀i, +n +� +j=1 +∂Cof(F)i +j +∂Xj +(P) = 0. +(L.22) +Also written �n +j=1 +∂ +∂Xj (J ∂Xi +∂xj ) = 0... +NB: (L.22) is a just a matrix computation since we used the +divergence of a matrix (we used components relative to a given basis). +Proof. We are in R3, thus Cof(F)i +j = F i+1 +j+1F i+2 +j+2 − F i+1 +j+2F i+2 +j+1, and F = [dΦt] = [ ∂ϕi +∂Xj ], that is, F i +j = ∂ϕi +∂Xj . +Thus +∂Cof(F)i +j +∂Xj += +∂2ϕi+1 +∂Xj∂Xj+1 +∂ϕi+2 +∂Xj+2 + ∂ϕi+1 +∂Xj+1 +∂2ϕi+2 +∂Xj∂Xj+2 − +∂2ϕi+1 +∂Xj∂Xj+2 +∂ϕi+2 +∂Xj+1 − ∂ϕi+1 +∂Xj+2 +∂2ϕi+2 +∂Xj∂Xj+1 . +Thus, for all i = 1, 2, 3, we get �n +j=1 +∂Cof(F )i +j +∂Xj += 0 (the terms cancel each other out two by two). +153 + +154 +L.6. +Piola transformation +L.6 +Piola transformation +Let ⃗u be a vector field in Ωt. The goal is to find a vector field ⃗UPiola in Ωt0 s.t. for all open subset ωt ⊂ Ωt +with ωt0 = Φt0 +t +−1(ωt) ⊂ Ωt0, +� +∂ωt0 +⃗UPiola • ⃗N dΣ = +� +∂ωt +⃗u • ⃗n dσ, +(L.23) +or +� +ωt0 +div(⃗UPiola) dΩt0 = +� +ωt +div(⃗u) dΩt, +(L.24) +i.e. +� +P ∈ωt0 +div(⃗UPiola)(P) dΩt0 = +� +P ∈ωt0 +div(⃗u)(Φt0 +t (P)) J(P) dΩt0. +(L.25) +(The motion is supposed to be regular, so J(P) > 0). Thus we want +div⃗UPiola(P) = J(P) div⃗u(p) +when +p = Φt0 +t (P). +(L.26) +Definition L.3 The Piola transform is the map +� +� +� +C∞(Ωt; Rn) → C∞(Ωt0; Rn) +⃗u → ⃗UPiola, +⃗UPiola(P) := J(P)F(P)−1.⃗u(p) +when +p = Φt0 +t (P). +(L.27) +(So ⃗UPiola(P) = J(P)Φ∗(⃗u)(P) where Φ∗(⃗u)(P) = F(P)−1.⃗u(p) = the pull-back with Φ = Φt0 +t .) +Proposition L.4 With p = Φt0 +t (P), ⃗UPiola = �n +i=1U i +Piola⃗ei and ⃗u = �n +i=1ui⃗ei we get +div⃗UPiola(P) = J(P) div⃗u(p), +i.e. +n +� +i=1 +∂U i +Piola +∂Xi +(P) = J(P) +n +� +i=1 +∂ui +∂xi (p). +(L.28) +Proof. div(τ.⃗w) =(S.61) � +div(τ).⃗w + τ 0.. d⃗w gives div((JF −1).(⃗u ◦ Φt0 +t ))(P) = (div(JF −T )(P), ⃗u(p))g + +(J(P)F(P)−1) 0.. (d⃗u(p).F(P))=(L.22)0+J(P)(F(P).F(P)−1) 0.. d⃗u(p) = J(P) I 0.. d⃗u(p) = J(P)div⃗u(p), +which gives (L.28). +M +Work and power +M.1 +Definitions +M.1.1 +Work +(Thermodynamic like approach.) The elementary work is a differential form α, e.g. α = dU (internal +energy density), α = δW = (elementary work). Consider a regular curve c : t ∈ [t0, T] → c(t) ∈ Rn. And +let ⃗v(t, c(t)) := ⃗c ′(t). The work of α along the curve is +� +c +α := +� T +t=t0 +α(t, c(t)).⃗c ′(t) dt noted += +� T +t=t0 +α.d⃗c += +� T +t=t0 +α(t, c(t)).⃗v(t, c(t)) dt noted += +� T +t=t0 +α.⃗v dt. +(M.1) +E.g., W t0 +T (α, c) = +� +c δW = work along c of the differential form α = δW. +Then consider an object Obj and its motion �Φ : (t, PObj) → p(t) = �Φ(t, PObj) = �ΦPObj (t) ∈ Rn, the +curves cPObj = �ΦPObj : t ∈ [t0, T] → p(t) = �ΦPObj (t) ∈ Rn, and the Eulerian velocities ⃗v(t, p(t)) = �ΦPObj +′(t). +The work for Obj and a Eulerian differential form α along �Φ is the sum of work of α of all particles, +formally � +PObj ∈Obj( +� +cPObj αPObj ). So with the associated motion Φt0(t, pt0) = �Φ(t, PObj) = p(t) = Φt0 +pt0 (t) +when pt0 = �Φ(t0, PObj), and with Ωt = �Φ(t, Obj), +W t0 +T (�Φ) := +� +pt0∈Ωt0 +� T +t=t0 +α(t, Φt0 +pt0 (t)).⃗v(t, Φt0 +pt0 (t)) dt dΩt0 = +� T +t=t0 +� +pt∈Ωt +α(t, pt)).⃗v(t, pt) dΩt dt. +(M.2) +(The last equality if Fubini theorem can be applied, e.g. if α is C0 and Φt0 is C1, Obj being bounded.) +154 + +155 +M.2. +Piola–Kirchhoff tensors +Exercice M.1 If α is a stationary and exact differential form, α = dU, then prove that +� +c +dU = U(c(T)) − U(c(t0)) noted += +∆U +(M.3) +only depends on the extremities c(t0) and c(T) of the curve c. +Answer. +� +c dU = +� T +t=t0 dU(c(t)).⃗c ′(t) dt = +� T +t=t0 +d(U◦c) +dt +(t) dt = [U ◦ c]T +t0 = U(c(T)) − U(c(t0)). +Remark (continuum mechanics): An observer chooses a Euclidean dot product (·, ·)g = . •g . = . • . (if +(·, ·)g is imposed and implicit). And if he chooses to represent a linear form αt(pt) with its (·, ·)g-Riesz +representation vector ⃗ft(pt) (observer dependent), cf. (B.8), then +� +c +α = +� T +t=t0 +α(t, c(t)).⃗c ′(t) dt = +� T +t=t0 +⃗f • d⃗c = +� T +t=t0 +⃗f • ⃗v dt. +(M.4) +M.1.2 +And its associated power density +Definition: The power density of a differential form α along �Φ is the Eulerian function +ψ := α.⃗v : +� +� +� +� +� +C = +� +t∈[t0,T ] +({t} × Ωt) → R +(t, p) → ψ(t, p) = α(t, p).⃗v(t, p). +(M.5) +And the power at t is +Pt(�Φ) = P(t, �Φ) := +� +p∈Ωt +ψ(t, p) dΩt = +� +p∈Ωt +αt(p).⃗vt(p) dΩt +noted += +P(t,⃗vt) = Pt(⃗vt). +(M.6) +Remark: With a Euclidean dot product (·, ·)g, then with the (·, ·)g-Riesz representation vector ⃗f of α +(observer dependent) we get +ψ = ⃗f • ⃗v, +i.e. +ψ(t, p) = ⃗f(t, p) •g ⃗v(t, p) +(= (⃗f(t, p),⃗v(t, p))g), +(M.7) +which gives P(t, �Φ) := +� +p∈Ωt ⃗f(t, p) • ⃗v(t, p) dΩt. +M.2 +Piola–Kirchhoff tensors +Consider a regular Eulerian velocity field ⃗v, so d⃗v is an endomorphism (identified with a +�1 +1 +� +tensor). +Then we need another endomorphism τ (identified with a +�1 +1 +� +to get the objective double contraction +τ 0.. d⃗v := Tr(τ.d⃗v), +(M.8) +which means τ(t, p) 0.. d⃗v(t, p) := Tr(τ(t, p).d⃗v(t, p)). +Quantification: With a basis (⃗ei) at t and τ = � +ij τ i +j⃗ei ⊗ ej and d⃗v = � +jk vj +|k⃗ej ⊗ ek (the endomor- +phisms have been written like +�1 +1 +� +tensors for calculation purpose), τ.d⃗v = � +ijk τ i +jvj +|k⃗ei ⊗ ek and +τ 0.. d⃗v = +n +� +i,j=1 +τ i +jvj +|i +(objective value), +(M.9) +see (Q.32). Then choose a Euclidean dot product (·, ·)g, to be able to use the double matrix product +τ : d⃗v := +n +� +i,j=1 +τ i +jvi +|j = [τ]|⃗e +T : [d⃗v]|⃗e +(subjective value). +(M.10) +M.2.1 +Objective internal power for the stress: function of d⃗v +Usual hypothesis for the internal stress in a material: At first order, the power density is of the +type +ψ = τ 0.. d⃗v, +i.e. +ψ(t, p) = τ(t, p) 0.. d⃗v(t, p), ∀(t, p) ∈ C, +(M.11) +thus the power at t is +Pt(⃗vt) = +� +p∈Ωt +ψ(t, p) dΩt = +� +p∈Ωt +τ t(p) 0.. d⃗vt(p) dΩt. +(M.12) +155 + +156 +M.2. +Piola–Kirchhoff tensors +M.2.2 +The first Piola–Kirchhoff tensor +The Piola–Kirchhoff approach consists in transforming Eulerian quantities into Lagrangian quantities to +refer to the initial configuration. (M.12) gives +Pt(⃗vt) = +� +P ∈Ωt0 +ψt(Φt0 +t (P)) |Jt0 +t (P)| dΩt0 = +� +P ∈Ωt0 +τ t(Φt0 +t (P)) 0.. d⃗vt(Φt0 +t (P)) Jt0 +t (P) dΩt0 +(M.13) +(the Jacobian Jt0 +t (P) = det(F t0 +t (P)) of Φt0 +t at P is positive for a regular motion). The Lagrangian velocity +⃗V t0(t, P) = ⃗vt(Φt0 +t (P)) satisfies d⃗V t0 +t (P) = d⃗vt(pt).F t0 +t (P) where pt = Φt0(t, P). Thus +τ t(pt) 0.. d⃗vt(pt) = τ(pt) 0.. (d⃗V t0 +t (P).F t0 +t (P)−1) = (F t0 +t (P)−1.τ(pt)) 0.. d⃗V t0 +t (P). +(M.14) +Quantification: Choose a basis and a Euclidean dot product (·, ·)g, thus +Pt(⃗vt) = +� +P ∈Ωt0 +(Jt0 +t (P)τ(pt)T .F t0 +t (P)−T +� +�� +� +PKt0 +t (P ) +) : d⃗V t0 +t (P) dΩt0. +(M.15) +Definition M.2 The first Piola–Kirchhoff (two point) tensor at P ∈ Ωt0, relative to t0, t and a basis (⃗ei), +is the linear map PKt0 +t (P) ∈ L(⃗Rn +t0; ⃗Rn +t ) defined by +PKt0 +t (P) = Jt0 +t (P) σt(Φt0 +t (P)).F t0 +t (P)−T , +where +σ = τ T , +(M.16) +abusively written +PK = J σ.F −T . +(M.17) +Hence +Pt(⃗vt) = +� +Ωt0 +PKt0 +t (P) : d⃗V t0 +t (P) dΩt0. +(M.18) +Remark M.3 The Piola–Kirchhoff tensor is not that easy to master: Everything is quite simple in +a Eulerian framework (the configuration at t where the laws are expressed to begin with), but then +everything is made more complicated when expressed in an initial configuration (at t0)... So, when the +Piola–Kirchhoff tensor is used to introduce the Lie derivatives (Eulerian type), it makes the Lie derivative +quite a mysterious mathematical object, see footnote page 25. +Remark M.4 Continuation of the remark: With the pull-backs, (M.13) reads (Jt0 +t (P) being positive) +P(t, �Φ) = +� +pt∈Ωt +ψt(pt) dΩt = +� +P ∈Ωt0 +� +(Φt0 +t )∗ψt +� +(P) +� +(Φt0 +t )∗dΩt +� +, +(M.19) +since ((Φt0 +t )∗dΩt) = Jt0 +t (P) dΩt0 and ((Φt0 +t )∗ψt)(P) = ψt(pt) (scalar valued functions). +It gives the +Piola–Kirchhoff tensor (pull-back to the initial configuration) since (Φt0 +t )∗(αt.⃗vt)(pt) = (αt.⃗vt)(Φt0 +t (P)) = +αt(Φt0 +t (P)).⃗vt(Φt0 +t (P)). +M.2.3 +The second Piola–Kirchhoff tensor +The first Piola–Kirchhoff tensor PK may confuse Eulerian and Lagrangian variables, linear maps and +endomorphisms... +And PK(pt0) is not symmetric: It can’t be since PK(pt0) ∈ L(⃗Rn +t0; ⃗Rn +t ) is not an +endomorphism. To get a symmetric tensor, the second Piola–Kirchhoff tensor is defined: +Definition M.5 The second Piola–Kirchhoff tensor is the endomorphism SKt0 +t (P) ∈ L(⃗Rn +t0; ⃗Rn +t0) defined +by, in short, +SK = F −1.PK = JF −1.σ.F −T . +(M.20) +Full notation: SKt0 +t (P) = (F t0 +t (P))−1.PKt0 +t (P) = Jt0 +t (P)(F t0 +t (P))−1.σt(p).(F t0 +t (P))−T . +So if σt(p) ∈ L(⃗Rn +t ; ⃗Rn +t ) is symmetric then SKt0 +t (P) ∈ L(⃗Rn +t0; ⃗Rn +t0) is symmetric. +156 + +157 +M.3. +Classical hyper-elasticity: ∂W/∂F +Thus, with the pull-back of the endomorphism d⃗vt ∈ L(⃗Rn +t , ⃗Rn +t ): +((Φt0 +t )∗d⃗vt)(P) = F t0 +t (P)−1.d⃗vt(pt).F t0 +t (P), +(M.21) +and with d⃗vt(pt) = d⃗V t0 +t (P).F t0 +t (P)−1 and σt(p) symmetric (so SKt0 +t +is symmetric), +Pt(⃗vt) = +� +Ωt0 +PKt0 +t : d⃗V t0 +t +dΩt0 = +� +Ωt0 +(F t0 +t .SKt0 +t ) : d⃗V t0 +t +dΩt0 = +� +Ωt0 +([F t0 +t ].[SKt0 +t ]) : [d⃗V t0 +t ]T dΩt0 += +� +Ωt0 +SKt0 +t : ((d⃗V t0 +t )T .F t0 +t ) dΩt0 = +� +Ωt0 +SKt0 +t : (F t0 +t +T .d⃗V t0 +t ++ d(⃗V t0 +t )T .F t0 +t +2 +) dΩt0. +(M.22) +Remark M.6 It is a “chosen time derivative” of SK(t) = J(t)F(t)−1.σ(t).F(t)−T that leads to some +kind of Lie derivative as explain in books in continuum mechanics, as in footnote page 25. +M.3 +Classical hyper-elasticity: ∂W/∂F +E and F are finite dimensional spaces, dim E = n, dim F = m. +M.3.1 +Definition +Reminder: Consider a function +� +W : +� +L(E; F) → R +L → � +W(L) +(M.23) +Its differential d� +W : +� +L(E; F) → L(L(E; F); R) +L → d� +W(L) +� +is defined at L by, in a direction M, cf. (S.3), +d� +W(L)(M) = lim +h→0 +� +W(L + hM) − � +W(L) +h +noted += +∂� +W +∂L (L)(M). +(M.24) +Also written d� +W(L)(M) = d� +W(L).M since d� +W(L) is linear. +Example M.7 � +W : F ∈ L(⃗Rn +t0; ⃗Rn +t ) → � +W(F) ∈ R (real valued function), with F := F t0 +t (pt0) the +deformation gradient at ∈ Ωt0 at t at pt0. Thus d� +W(F).M = limh→0 +� +W (F +hM)−� +W (F ) +h +=noted ∂� +W +∂F (F).M ∈ +R is the derivative of � +W at F in a direction M ∈ L(⃗Rn +t0; ⃗Rn +t ). +Example M.8 m = n, endomorphisms L ∈ L(⃗Rn; ⃗Rn), and � +W(L) := Tr(L) (the trace). +Here +dTr(L)(M) = limh→0 +Tr(L+hM)−Tr(L) +h += Tr(M) (the trace is linear), thus dTr(L) = Tr for all L. +M.3.2 +Expression with bases (quantification): The ∂W/∂Lij +Let (⃗ai) and (⃗bi) be bases in E and F, with (πai) the (covariant) dual basis of (⃗ai). Let (Lij) i=1,...,m +j=1,...,n = +(⃗bi ⊗ πaj) be the associated basis in L(E; F), i.e. the Lij are defined by Lij.⃗aℓ = δjℓ⃗bi for all i = 1, ..., m +and j, ℓ = 1, ..., n, cf. prop. A.41 (all the elements of the matrix [Lij]|⃗a,⃗b vanish except the element at the +intersection of row i and column j which equals 1). Let L ∈ L(E; F). The derivation of � +W at L in a +direction Lij is +d� +W(L).Lij = lim +h→0 +� +W(L + hLij) − � +W(L) +h +noted += +∂� +W +∂Lij +(L) +(M.25) +(usual notation). +Associated matrix relative to the chosen bases: +[d� +W(L)]|Lij := [ ∂� +W +∂Lij +] i=1,...,m +j=1,...,n +noted += +[d� +W(L)]|⃗a,⃗b +( noted += +[d� +W(L)ij]). +(M.26) +So if M = �m +i=1 +�n +j=1MijLij then d� +W(L)(M) = � +ij Mij d� +W(L)(Lij) since d� +W(L) is linear, so +d� +W(L)(M) = +n +� +i,j=1 +∂� +W +∂Lij +(L)Mij = [d� +W(L)]|⃗a,⃗b : [M]|⃗a,⃗b, +(M.27) +double matrix contraction. Duality notations: d� +W(L)(M) = � +ij +∂� +W +∂Li +j (L)M i +j. +157 + +158 +M.3. +Classical hyper-elasticity: ∂W/∂F +Remark M.9 The notation [M]| ⃗E,⃗e : [d� +W(L)]| ⃗E,⃗e is just a matrix product, since M = L(⃗Rn; ⃗ +Rm) and +d� +W(L) ∈ L(L(⃗Rn; ⃗ +Rm); R) are different kinds of mathematical objects. +Example M.10 Continuing example M.8 with (⃗ei) = ( ⃗Ei): Then � +W(L) = Tr(L) gives d� +W(L).M = +Tr(M) = � +i Mii, thus +∂� +W +∂Lij (L) = δij for all i, j, thus [d� +W(L)]|⃗e = [I] = [ ∂Tr +∂Lij (L)] (identity matrix), and +we recover dTr(L)(M) = [ ∂Tr +∂Lij (L)] : [M] = [I] : [M] = �n +i=1Mii = Tr(M). +Example M.11 Continuing example M.7: The meaning of the derivation +∂� +W +∂Fij is intriguing: It is a +derivation in the direction Lij =noted ⃗ei ⊗ πEj, where (⃗ei) is a basis at p = Φt0 +t (P) in ⃗Rn +t and (πEj) is the +dual basis of a basis ( ⃗Ej) at P in ⃗Rn +t0, i.e. +∂� +W +∂Fij (F) = d� +W(F).Lij =noted d� +W(F).(⃗ei ⊗ πEj) is a derivation +“at the same time” in the directions ⃗ei (at (t, p)) and πEj (at (t0, P)), where F stands for F t0 +t (P). +M.3.3 +Motions and ω-lemma +Generalization of (M.23) to C1 functions, with UE open subset in a affine space which associated vector +space is E, +� +W : +� +UE × L(E; F) → R +(P, L) → � +W(P, L). +(M.28) +At P, let � +WP (L) := � +W(P, L). The differential d� +WP (L) =noted ∂2� +W(P, L) in a direction M ∈ L(E; F) is +∂2� +W(P, L).M := d(� +WP )(L).M = lim +h→0 +� +W(P, L + hM) − � +W(P, L) +h +noted += +∂� +W +∂L (P, L).M. +(M.29) +With a motion Φ := Φt0 +t : Ωt0 → Ωt define +f : +� +C1(Ωt0; Ωt) → C0(Ωt0; R) +Φ → f(Φ) := � +W(., dΦ(.)), +(M.30) +a function of Φ which only depends on its first (covariant) gradient; So, for all P ∈ Ωt0, +f(Φ)(P) = � +W(P, dΦ(P)) ∈ R. +(M.31) +(This kind of relation is generally deduced after application of the frame invariance principle, and the +hypothesis of dependence on only the first order derivative dΦ = F.) +Lemma M.12 (ω-lemma) For all Φ, Ψ ∈ C1(Ωt0; Ωt), +df(Φ)(Ψ) = ∂2� +W(., dΦ)(dΨ) noted += +∂� +W +∂F (., dΦ)(dΨ) , +(M.32) +i.e., for all P ∈ Ωt0, df(Φ)(Ψ)(P) = ∂� +W +∂F (P, dΦ(P))(dΨ(P)). +With bases ( ⃗Ei) and (⃗ei) in ⃗Rn +t0 and ⃗Rn +t and dΨ. ⃗Ej = �n +i=1 +∂Ψi +∂Xj ⃗ei, we get +df(Φ)(Ψ) = +n +� +i,j=1 +∂� +W +∂Fij +(., dΦ) ∂Ψi +∂Xj +(.) noted += +[ ∂� +W +∂Fij +] : [ ∂Ψi +∂Xj +]. +(M.33) +Marsden notations: df(Φ)(Ψ) = �n +i,J=1 +∂� +W +∂F i +J +∂Ψi +∂XJ = [ ∂� +W +∂F i +J ] : [ ∂Ψi +∂XJ ]. +Proof. C1(Ωt0; Ωt) is a vector space, so df(Φ)(Ψ) = limh→0 +f(Φ+hΨ)−f(Φ) +h +∈ C0(Ωt0; Ωt), i.e., for any +P ∈ Ωt0 we have df(Φ)(Ψ)(P) = limh→0 +f(Φ+hΨ)(P )−f(Φ)(P ) +h += limh→0 +� +W (P,dΦ(P )+h dΨ(P ))−� +W (P,dΦ(P )) +h += +d ⃗WP (dΨ(P), i.e. (M.32) +158 + +159 +M.3. +Classical hyper-elasticity: ∂W/∂F +M.3.4 +Application to classical hyper-elasticity: PK = ∂W/∂F +Let (·, ·)g be a unique Euclidean dot product in ⃗Rn +t at all times t, and let ( ⃗Ei) and (⃗ei) be Euclidean +bases at t0 and at t. Let σt(p) = the Cauchy stress tensor at t at p = Φt0 +t (P). +Let PK(P) = J(P) σt(p).F −T (P) = the first Piola–Kirchhoff (two point) tensor at P, cf. (M.16). +Since PK depends on Φ, the full notation is PK = PK(Φ) given by +PK(Φ)(P) = J(P) σt(Φ(P)).dΦ(P)−T . +(M.34) +Definition M.13 If there exists a function � +PK such that PK reads +PK(Φ)(P) = � +PK(P, dΦ(P)) +(M.35) +then � +PK is called a constitutive function. (First order hypothesis: � +PK only depends on dΦ = F the first +order derivative of Φ.) +Definition M.14 The material is hyper-elastic iff there exists a function � +W : +� +Ωt0 × L(⃗Rn +t0; ⃗Rn +t ) → R +(P, L) → � +W(P, L) +� +such that +(PK(Φ) =) +� +PK(., dΦ) = ∂� +W +∂F (., dΦ), +written +� +PK = ∂� +W +∂F , +(M.36) +that is, � +PK(P, F(P)) = ∂� +W +∂F (P, F(P)) for all P ∈ Ωt0, where F = dΦ. +With bases ( ⃗EI) and (⃗ei) in ⃗Rn +t0 and ⃗Rn +t , and (EI) the dual basis of ( ⃗EI), and PK = �n +i,J=1PKi +J⃗ei⊗EJ, +[PK(Φ)]| ⃗E,⃗e = [� +PK(., F)]| ⃗E,⃗e = [∂� +W +∂F (., F)]| ⃗E,⃗e, +i.e. +[PKi +J] = [ ∂� +W +∂F i +J +(., F)]. +(M.37) +Thus, for any (virtual) motion Ψ : Ωt0 → Ωt, with (M.32) and (M.27), +� +PK(dΦ)(dΨ) = ∂� +W +∂F (dΦ)(dΨ) += +� +iJ +∂� +W +∂F i +J +(F) ∂Ψi +∂XJ +noted += +[� +PK] : [dΨ], +(M.38) +that is, � +PK(dΦ)(dΨ)(P) = � +iJ +∂� +W +∂F i +J (P, F t0 +t (P)) ∂Ψi +∂XJ (P) for all P ∈ Ωt0. +Exercice M.15 With a unique Euclidean dot product (·, ·)g both in ⃗Rn +t0 and ⃗Rn +t , with Euclidean bases +( ⃗Ei) ∈ ⃗Rn +t0 and (⃗ei) ∈ ⃗Rn +t , and with (Ei) the dual basis of ( ⃗Ei), with C = F T .F, prove (derivation in the +direction ⃗ei ⊗ EJ): +∂C +∂F i +J +(F) = +� +K +F i +K ⃗EJ ⊗ EK + +� +K +F i +K ⃗EK ⊗ EJ +(= dC(F).(⃗ei ⊗ EJ)). +(M.39) +∂ +√ +C +∂F (F) = 1 +2 +�� +C(F) +�−1.∂C +∂F (F). +(M.40) +∂ +√ +C +∂C += 1 +2( +√ +C)−1. +(M.41) +Answer. Let F = � +iJ F i +J⃗ei ⊗ EJ, so F T = � +Ij(F T )I +j ⃗EI ⊗ ej = � +Ij F j +I ⃗EI ⊗ ej, and C = � +IJ CI +J ⃗Ei ⊗ EJ = +F T .F = � +IJ +� +k(F T )I +kF k +J ⃗EI ⊗ EJ = � +IJ +� +k F k +I F k +J ⃗EI ⊗ EJ = C(F), so CI +J = � +k F k +I F k +J = CI +J(F). And +C(F + h⃗ei ⊗ EJ) = (F + h⃗ei ⊗ EJ)T .(F + h⃗ei ⊗ EJ) = (F T + h ⃗EJ ⊗ ei).(F + h⃗ei ⊗ EJ) += C(F) + h ( ⃗EJ ⊗ ei).F + h F T .(⃗ei ⊗ EJ) + h2 ⃗EJ ⊗ Ei += C(F) + h ( +� +K +F i +K ⃗EJ ⊗ EK + +� +K +(F T )K +i ⃗EK ⊗ EJ) + h2 ⃗EJ ⊗ EJ +(M.42) +Thus (M.39). And C(F + h⃗ei ⊗ EJ) − C(F) = ( +√ +C(F + h⃗ei ⊗ EJ) + +√ +C(F)).( +√ +C(F + h⃗ei ⊗ EJ) − +√ +C(F)) gives +dC(F)(⃗ei ⊗ EJ) = 2 +√ +C(F).d +√ +C(F)(⃗ei ⊗ EJ). Thus ∂ +√ +C +∂F i +J (F) = 1 +2( +� +C(F))−1. ∂C +∂F i +J (F). +(C + h⃗ei ⊗ ej) − C = ( +√ +C + h⃗ei ⊗ ej + +√ +C).( +√ +C + h⃗ei ⊗ ej − +√ +C), divided by h, gives ⃗ei ⊗ ej += +2 +√ +C. limh→0 +√ +C+h⃗ei⊗ej− +√ +C +h += 2 +√ +C.d +√ +C.(⃗ei ⊗ ej), thus L = 2 +√ +C.(d +√ +C.L) for all L (linearity of d +√ +C), thus +d +√ +C.L = 1 +2( +√ +C)−1.L. +159 + +160 +M.3. +Classical hyper-elasticity: ∂W/∂F +M.3.5 +Corollary (hyper-elasticity): SK = ∂W/∂C +With the symmetry of the second Piola–Kirchhoff tensor SK = F −1.PK, we deduce SKt0 +t (Φt0 +t )(P) = +� +SKt0 +t (P, F t0 +t (P)) +(constitutive +function). +And +we +deduce +the +existence +of +a +function +� +W +: +� +Ωt0 × L(⃗Rn +t0; ⃗Rn +t0) → R +(P, L) → � +W(P, L) +� +such that, +� +SKt0 +t (., C) = ∂� +W +∂C (., C). +(M.43) +(See Marsden and Hughes [12] for details and the thermodynamical hypotheses required.) +N +Conservation of mass +Let ρ(t, p) = ρt(p) be the (Eulerian) mass density at t at p ∈ Ωt, supposed to be > 0; The mass m(ωt) of +a subset ωt ⊂ Ωt is +m(ωt) = +� +p∈ωt +ρt(p) dωt. +(N.1) +Conservation of mass principle (no loss nor production of particles): For all ωt0 ⊂ Ωt0 and all t, +m(ωt) = m(ωt0), +i.e. +� +p∈ωt +ρt(p) dωt = +� +P ∈ωt0 +ρt0(P) dωt0. +(N.2) +Proposition N.1 If (N.2) then, with Jt0 +t (P) = det(dΦt0 +t (P)) (positive Jacobian the motion being sup- +posed regular) and p = Φt0 +t (P), +ρt(p) = ρt0(P) +Jt0 +t (P). +(N.3) +Proof. The change of variable formula gives +� +p∈ωt +ρt(p) dωt = +� +P ∈ωt0 +ρt(Φt0 +t (P)) Jt0 +t (P) dωt0, +thus (N.2) gives ρt(Φt0 +t (P))Jt0 +t (P) = ρt0(P). +Proposition N.2 ⃗v = ⃗v(t, pt) being the Eulerian velocity at (t, pt) ∈ R × Ωt, (N.2) gives +Dρ +Dt + ρ div⃗v = 0, +i.e. +∂ρ +∂t + div(ρ⃗v) = 0. +(N.4) +Thus, for all ωt ⊂ Ωt, +� +ωt +∂ρ +∂t dωt = − +� +∂ωt +ρ⃗v.⃗n dσt. +(N.5) +Proof. (N.2) gives d +dt( +� +p(t)∈ωt ρ(t, p(t)) dωt) = 0, and Leibniz formula (K.41) applied for all ωt gives (N.4). +Then the Green formula +� +Ωt div(ρ⃗v) dΩt = +� +∂Ωt ρ⃗v.⃗n dσt gives (N.5). +Exercice N.3 Use (N.3) to prove (N.4). +Answer. J(t, P)ρ(t, Φ(t, P)) = ρt0(P) give, with pt = Φ(t, P), +∂J +∂t (t, P) ρ(t, pt) + J(t, P) +�∂ρ +∂t (t, pt) + dρ(t, pt).dΦ(t, P) +� += 0. +Thus ∂J +∂t (t, P) = J(t, P) div⃗v(t, p), cf. (K.40), gives (N.4). +160 + +161 +O.1. +Framework +O +Balance of momentum +O.1 +Framework +�Φ : [t0, T] × Obj → Rn is a regular motion, Ωt = �Φ(t, Obj), Γt = ∂Ωt (the boundary), ⃗v is the Eulerian +velocity field, ωt is a regular sub domain in Ωt and ∂ωt is its boundary. +An observer chooses a Euclidean basis (⃗ei) (e.g. made with the foot or the metre) and call (·, ·)g the +associated Euclidean dot product. And ⃗n(t, p) = ⃗nt(p) is the outer unit normal at t at p ∈ ∂ωt. +All the functions are assumed to be regular enough to validate the following calculations. +Let ρ : +� +� +� +� +� +� +t∈[t0,T ] +({t} × Ωt) → R +(t, pt) → ρ(t, pt) +� +� +� +� +� +(a mass density), let ⃗f : +� +� +� +� +� +� +t∈[t0,T ] +({t} × Ωt) → ⃗Rn +(t, pt) → ⃗f(t, pt) +� +� +� +� +� +(a +body force density), and let ⃗T : +� +� +� +� +� +� +t∈[t0,T ] +({t} × ∂ωt × ⃗ +Rn +t ) → ⃗Rn +(t, pt,⃗n(pt)) → ⃗T(t, pt,⃗n(pt)) +� +� +� +� +� +(a surface force density) +defined for any regular subset ωt ⊂ Ωt. +O.2 +Master balance law +Definition O.1 The balance of momentum is satisfied by ρ, ⃗f and ⃗T iff, for all regular open subset ωt +in Ωt, +d +dt( +� +ωt +ρ⃗v dΩt) = +� +ωt +⃗f dΩt + +� +∂ωt +⃗T∂ωt dΓt +(master balance law). +(O.1) +(It is in fact a linearity hypothesis, see theorem O.2.) +Thus, with (K.41), +� +ωt +D(ρ⃗v) +Dt ++ ρ⃗v div⃗v dΩt = +� +ωt +⃗f dΩt + +� +∂ωt +⃗T∂ωt dΓt. +(O.2) +And with the conservation of mass hypothesis, cf. (N.4), we get +� +ωt +ρD⃗v +Dt dΩt = +� +ωt +⃗f dΩt + +� +∂ωt +⃗T∂ωt dΓt, +(O.3) +with D⃗v +Dt = ⃗γ = the Eulerian acceleration. +O.3 +Cauchy theorem ⃗T = σ.⃗n (stress tensor σ) +Theorem O.2 (Cauchy first law: Cauchy stress tensor) If the master balance law (O.1) is satis- +fied, then ⃗T is linear in ⃗n, that is, there exists a Eulerian endomorphism σ, identified to a Eulerian tensor +σ ∈ T 1 +1 (Ωt), called the Cauchy stress tensor, s.t. on all ∂ωt, in short +⃗T = σ.⃗n, +(O.4) +where ⃗n is the unit outward normal to ∂ωt (i.e., ⃗T(t, pt) = σ(t, pt).⃗n(t, pt) for all t and pt ∈ ∂ωt). +The proof is based on: +Lemma O.3 Let ϕ : +� +Ω → R +p → ϕ(p) +� +∈ C1(Ω; R) and ψ : +� +Ω × ⃗R3 → R +(p, ⃗w) → ψ(p, ⃗w) +� +∈ C1(Ω, ⃗R3; R). If +∀ω open in Ω, +� +p∈ω +ϕ(p) dΩ = +� +p∈∂ω +ψ(p,⃗n(p)) dΓ +(O.5) +(no dependence on the curvature or on higher derivatives since at any p ∈ ∂ω, ψ only depends on ⃗n(p)), +then +∃⃗k ∈ C1(Ω; ⃗R3) s.t. ψ = (⃗k,⃗n)g, and ϕ = div⃗k, +(O.6) +i.e. ψ depends linearly on ⃗n, and ϕ is a divergence. +161 + +162 +O.3. +Cauchy theorem ⃗T = σ.⃗n (stress tensor σ) +Proof. (Lemma O.3.) (This proof is standard: We recall it.) Let p ∈ Ω ⊂ R3. Consider the tetrahedral +defined by its vertices p, p + (h1, 0, 0), p + (0, h2, 0) and p + (0, 0, h3), with hi > 0 for all i. (On each +face of a tetrahedron, the unit normal vector is uniform.) Let Σ1 the side which outer unit normal is +− ⃗E1: It area is σ1 = 1 +2h2h3 (square triangle). Idem for Σ2 and Σ3. Let Σ be the fourth side: its area +is σ = 1 +2 +� +h2 +2h2 +3 + h2 +3h2 +1 + h2 +1h2 +2 and its outer unit normal is ⃗n = +1 +2σ(h2h3, h3h1, h1h2) (see exercise O.5), +that is ⃗n = (n1, n2, n3) with ni = σi +σ pour i = 1, 2, 3. The volume of the tetrahedral is 1 +6h1h2h3 =noted ℓ3. +Let M := supp∈Ω |ϕ(p)|; We have M < ∞, since ϕ is continuous in Ω. Then (O.6) give +Mℓ3 ≥ | +� +∂ωt +ψ(p,⃗n(p)) dΓ|, +so +� +∂ωt +ψ(p,⃗n(p)) dΓ = O(ℓ3). +(O.7) +And ψ being continuous, the mean value theorem applied on Σi gives: There exists pi ∈ Σi s.t. +� +Σi +ψ(p,⃗n(p)) dΓ = σiψ(pi,⃗ni). +Thus +� +∂ωt +ψ(p,⃗n(p)) dΓ = +� +σ1ψ(p1, − ⃗E1) + σ2ψ(t, p2, − ⃗E2) + σ3ψ(p3, − ⃗E3) + σψ(p4,⃗n) +� +. +Then, Ψ being continous, (O.7) gives +σ1ψ(p1, − ⃗E1) + σ2ψ(p2, − ⃗E2) + σ3ψ(p3, − ⃗E3) + σψ(p4,⃗n) = O(ℓ3). +(O.8) +We flatten the tetrahedron on the yz face by taking h2 = h3 =noted h and h1 = h2; Thus σ1 = 1 +2h2, +σ2 = o(h2), σ3 = o(h2), σ ∼ σ1, ℓ3 = 1 +6h4, with ⃗n ∼ −⃗n1 = ⃗E1 and pi ∼ p; Then +ψ(p, − ⃗E1) + ψ(p, + ⃗E1) = 0. +(O.9) +Idem with xz and xy. And for a fixed tetrahedron with h1, h2, h3 given, consider the smaller tetrahedron +with εh1, εh2, εh3. Then as ε → 0 (O.8) with (O.9) give +ψ(p,⃗n) = − σi +σ ψ(p, − ⃗E1) − σ2 +σ ψ(p, − ⃗E2) − σ3 +σ ψ(p, − ⃗E3) = +3 +� +i=1 +niψ(p, ⃗Ei), +since ni = σi +σ pour i = 1, 2, 3. The same steps can be done for any (inclined) tetrahedron (or apply a +change of variable to get back to the above tetrahedron). Thus ψp is a linear map in ⃗np, that is, there +exists a linear form αp s.t. ψp(⃗np) = αp.⃗np for any p ∈ ∂ω. And the Riesz representation theorem gives: +∃⃗kp s.t. αp.⃗np = (⃗kp,⃗np)g =noted ⃗kp • ⃗np. +Proof. (Theorem.) +Apply Lemma O.3 component by component with ⃗ϕ = ρ D⃗v +Dt − ⃗f = �n +i=1ϕi⃗ei, +cf. (O.3). +Corollary O.4 With divσ := �n +i=1(�n +j=1 +∂σij +∂xj )⃗ei (definition of “the matrix divergence” see (S.65)), +� +� +� +⃗f + divσ = ρD⃗v +Dt +in Ωt, +σ.⃗n = ⃗T +on Γt +(O.10) +(matrix meaning). (With duality notations, divσ := �n +i=1(�n +j=1 +∂σi +j +∂xj )⃗ei.) +Proof. Apply the divergence Formula to (O.3). +Exercice O.5 Consider a triangle T in R3 which vertices are A = (h1, 0, 0), B = (0, h2, 0), C = (0, 0, h3). +Prove that ⃗n = (h2h3, h3h1, h1h2) is orthogonal to T and that σ = 1 +2 +� +h2 +2h2 +3 + h2 +3h2 +1 + h2 +1h2 +2 is its area. +Answer. Consider the parametric surface ⃗r(t, u) = A + t ⃗ +AB + u ⃗ +AC for t, u ∈ [0, 1] describing the triangle. Thus +⃗n = +∂⃗r +∂t ∧ ∂⃗r +∂u = +⃗ +AB ∧ ⃗ +AC = +� +� +−h1 +h2 +0 +� +� ∧ +� +� +−h1 +0 +h3 +� +� = +� +� +h2h3 +h3h1 +h1h2 +� +� is orthonormal. And dσ = || ∂⃗r +∂t ∧ ∂⃗r +∂u||dudt = +� +h2 +2h2 +3 + h2 +3h2 +1 + h2 +1h2 +2dudt. Thus σ = +� 1 +t=0 +� 1 +u=0 dσ = +� +h2 +2h2 +3 + h2 +3h2 +1 + h2 +1h2 +2 is twice the aera of the triangle. +162 + +163 +Q.1. +Tensorial product and multilinear forms +P +Balance of moment of momentum +Definition P.1 The balance of moment of momentum is satisfied by ρ, ⃗f and ⃗T iff for all regular sub-open +set ωt ⊂ Ωt +d +dt +� +ωt +ρ −−→ +OM ∧ ⃗v dΩt = +� +ωt +ρ −−→ +OM ∧ ⃗f dΩt + +� +∂ωt +−−→ +OM ∧ ⃗T dΓt, +(P.1) +equality called the master balance of moment of momentum law. (This excludes e.g. Cosserat continua +materials.) +Theorem P.2 (Cauchy second law.) If the master balance law (so ⃗T = σ.⃗n) and the master balance of +moment of momentum law are satisfied then σ is symmetric. +Proof. (Standard proof.) Let ⃗x = −−→ +OM = � +i xi ⃗Ei, and ⃗T = � +i Ti ⃗Ei = σ.⃗n = � +ij σijnj ⃗Ei. Then +(first component) (⃗x ∧ ⃗T)1 = x2T3 − x3T2 = x2(σ31n1 + σ32n2 + σ33n3) − x3(σ21n1 + σ22n2 + σ23n3) = +(x2σ31 − x3σ21)n1 + (x2σ32 − x3σ22)n2 + (x2σ33 − x3σ23)n3. Thus +� +∂ωt(⃗x ∧ ⃗T)1 dΓt = +� +ωt +∂(x2σ31−x3σ21) +∂x1 ++ +∂(x2σ32−x3σ22) +∂x2 ++ ∂(x2σ33−x3σ23) +∂x3 +dΩt = +� +ωt x2(divσ)3 + x3(divσ)2 + σ32 − σ23 dωt. +(O.10) gives ρ D⃗v +Dt − ⃗f = divσ, thus ⃗x ∧ (ρ⃗γ − ⃗f) = ⃗x ∧ divσ, so the first component of ⃗x ∧ (ρ⃗γ − ⃗f) is +x2(divσ)3−x3(divσ)2, cf. (O.10). Thus (P.1) gives +� +ωt σ32−σ23 dωt = 0. True for all ωt, thus σ32−σ23 = 0. +Idem for the other components: σ is symmetric. +Q +Uniform tensors in Lr +s(E) +Uniform tensors enable to define without ambiguity the “objective contraction rules”. Uniform tensors +are scalar valued multilinear functions acting on both vectors and linear forms. +NB: In classical mechanics courses, what is called a “tensor” generally not a tensor but a matrix. +E.g. you may encounter the expression “Euclidean tensor” which means: The matrix representation of +“something” with respect to a Euclidean basis (based on the foot, metre,...) chosen by some observer. +(An “Euclidean tensor” is a non-sense, e.g. can you define a “Euclidean vector”?) +Q.1 +Tensorial product and multilinear forms +Let A1, ..., An be n finite dimension vector spaces. And A∗ +i = L(Ai; R) the set of linear forms. +Q.1.1 +Tensorial product of functions +Let f1 : A1 → R, ..., fn : An → R be n functions. Their tensorial product is the function f1 ⊗ ... ⊗ fn : +A1 × ... × An → R defined by (separate variable function) +(f1 ⊗ ... ⊗ fn)(⃗x1, ..., ⃗xn) = f1(⃗x1)...fn(⃗xn). +(Q.1) +(E.g., n = 2 and A1 = A2 = R and (cos ⊗ sin)(x, y) = cos(x) sin(y).) +Q.1.2 +Tensorial product of linear forms: multilinear forms +Let L(A1, ..., An; R) be the set of R-multilinear forms on the Cartesian product A1 × ... × An, that is, the +set of the functions M : A1 × ... × An → R s.t., for all i = 1, ..., n, all ⃗xi, ⃗yi ∈ Ai and all λ ∈ R, +M(..., ⃗xi + λ⃗yi, ...) = M(..., ⃗xi, ...) + λ M(..., ⃗yi, ...), +(Q.2) +the other variables being unchanged. +Definition: An elementary tensor is multilinear form M = ℓ1 ⊗ .... ⊗ ℓn, with ℓi ∈ A∗ +i for all i; So +∀(⃗xi)i∈N∗ ∈ +n +� +i=1 +Ai, +(ℓ1 ⊗ ... ⊗ ℓn)(⃗x1, ..., ⃗xn) = (ℓ1.⃗x1)...(ℓn.⃗xn) ∈ R. +(Q.3) +(The dot in ℓi.⃗xi is not an inner dot product: It is the duality “outer product” ℓi.⃗xi := ℓi(⃗xi), cf. (A.45).) +163 + +164 +Q.2. +Uniform tensors in L0 +s(E) +Q.2 +Uniform tensors in L0 +s(E) +Let E be a real vector space, with dim(E) = n ∈ N∗. In this section we consider the first overlay on E +made of multilinear forms M on E, called the uniform tensors of type 0 s or of type +�0 +s +� +. +E.g., M ∈ L0 +1(E) a linear form, M ∈ L0 +2(E) an inner dot product, M ∈ L0 +n(E) a determinant... +Notations for quantification purposes: (⃗ei) is a basis in E, (πei) is its (covariant) dual basis (basis in +E∗ = L(E; R)), (∂i) is its bidual basis (basis in E∗∗ = L(E∗; R)). +Q.2.1 +Definition of type +�0 +s +� +uniform tensors +L0 +0(E) := R, and if s ∈ N∗ then +L0 +s(E) := L(E × ... × E +� +�� +� +s times +; R) +(Q.4) +is called the set of uniform tensors of type +�0 +s +� +on E. +Q.2.2 +Example: Type +�0 +1 +� +uniform tensor = linear forms +A type +�0 +1 +� +uniform tensor is an element of L0 +1(E) = L(E; R) = E∗: It is a linear form ℓ ∈ L0 +1(E) = E∗. +Quantification: With ℓi := ℓ(⃗ei) we have, cf. (A.10), +ℓ = +n +� +i=1 +ℓiπei, +and +[ℓ]|πe = ( ℓ1 +... +ℓn ) noted += +[ℓ]|⃗e +(Q.5) +(row matrix for a linear form). Duality notations: (ei) is the covariant dual basis and ℓ = �n +i=1ℓiei. +Thus, if ⃗v ∈ E, ⃗v = �n +i=1vi⃗ei, then ⃗v is represented by [⃗v]|⃗e = +� +� +v1 +... +vn +� +� (column matrix for a vector), +and the matrix calculation rules give +ℓ(⃗v) = [ℓ]|⃗e.[⃗v]|⃗e = ( ℓ1 +... +ℓn ) . +� +� +v1 +... +vn +� +� = +n +� +i=1 +ℓivi +noted += +ℓ.⃗v. +(Q.6) +Duality notations: ⃗v = �n +i=1vi⃗ei and ℓ(⃗v) = �n +i=1ℓivi, and Einstein’s convention is satisfied. +Q.2.3 +Example: Type +�0 +2 +� +uniform tensor +A type +�0 +2 +� +uniform tensor is an element of L0 +2(E) = L(E, E; R): It is a bilinear form T ∈ L(E, E; R). +Quantification: Let Tij := T(⃗ei,⃗ej). Then, with ⃗v = �n +i=1vi⃗ei and ⃗w = �n +i=1wi⃗ei, +T(⃗v, ⃗w) = +n +� +i,j=1 +Tijviwj = [⃗v]T +|⃗e.[T]|⃗e.[⃗w]|⃗e, +i.e. +T = +n +� +i,j=1 +Tijπei ⊗ πej. +(Q.7) +Duality notations: T(⃗v, ⃗w) = �n +i,j=1Tijviwj, and Einstein’s convention is satisfied. +An elementary uniform tensor in L0 +2(E) is a tensor T = ℓ ⊗ m, where ℓ, m ∈ E∗. And so, for all +⃗v, ⃗w ∈ E, +(ℓ ⊗ m)(⃗v, ⃗w) = (ℓ.⃗v)(m.⃗w). +(Q.8) +Q.2.4 +Example: Determinant +The determinant is a alternating +�0 +n +� +uniform tensor, cf. (K.2). +Q.3 +Uniform tensors in Lr +s(E) +In this section we consider an over-overlay on E: The multilinear forms acting on both vectors (∈ E) and +functions ∈ E∗ (linear forms). +164 + +165 +Q.3. +Uniform tensors in Lr +s(E) +Q.3.1 +Definition of type +�r +s +� +uniform tensors +Let r, s ∈ N s.t. r + s ≥ 1. The set of multilinear forms +Lr +s(E) := L(E∗ × ... × E∗ +� +�� +� +r times +, E × ... × E +� +�� +� +s times +; R) +(Q.9) +is called the set of uniform tensors of type +�r +s +� +on E. +The case r = 0 has been considered at § Q.2. +When r ≥ 1, a tensor T ∈ Lr +s(E) is a functional: Its domain of definition contains a set of functions +(the set E∗ = L(E; R)). +Q.3.2 +Example: Type +�1 +0 +� +uniform tensor: Identified with a vector +A uniform +�1 +0 +� +tensor is a element T ∈ L1 +0(E) = L(E∗; R) = L(L(E; R); R) = E∗∗. With the natural +canonical isomorphism +J : +� +E → E∗∗ = L1 +0(E) +⃗w → J (⃗w) = w, +defined by +w(ℓ) := ℓ(⃗w), +∀ℓ ∈ E∗, +(Q.10) +cf. (T.9) and prop. T.5, +w noted += +⃗w, +so +w.ℓ noted += +⃗w.ℓ +(= ℓ.⃗w). +(Q.11) +So a +�1 +0 +� +type uniform tensor w is identified (natural canonical) to the vector ⃗w = J −1(w). +Interpretation: +E∗∗ is the set of directional derivatives. Indeed, if E is an affine space, if E is the +associated vector space, if p ∈ E, and if f is a differentiable function at p, then w.df(p) =(Q.10) df(p).⃗w is +the directional derivative along ⃗w. +Remark: In differential geometry, w.df is written ⃗w(f), so ⃗w(f)(p) := df(p).⃗w, the definition of a +vector being a directional derivative. +Quantification: For all i, j, +∂i.πej = δij = πej.⃗ei, +thus +∂i = J (⃗ei) noted += +⃗ei. +(Q.12) +Duality notations: ∂i.ej = δj +i = ej.⃗ei. +E.g., if f is a C1 function then df(p) = �n +i=1f|i(p) πei (= +�n +i=1f|i(p) ei) and +∂i(df(p)) = df(p).⃗ei = f|i(p) noted += +∂i(f)(p) noted += +⃗ei(f)(p). +(Q.13) +Q.3.3 +Example: Type +�1 +1 +� +uniform tensor +An elementary uniform tensor in L1 +1(E) is a tensor T = u ⊗ β, where u ∈ E∗∗ and β ∈ E∗. And, with +⃗u = J−1(u) ∈ E, cf. (Q.10), we also write T = ⃗u ⊗ β. Thus, for all ℓ ∈ E∗ and ⃗w ∈ E +(u ⊗ β)(ℓ, ⃗w) = u(ℓ)β(⃗w) = ℓ(⃗u)β(⃗w) noted += +⃗u(ℓ)β(⃗w) noted += +(⃗u ⊗ β)(ℓ, ⃗w). +(Q.14) +Quantification: Let T(πei,⃗ej). So +T = +n +� +i,j=1 +Tij ⃗ei ⊗ πej, +and +[T]|⃗e = [Tij], +(Q.15) +[T]|⃗e = [Tij] being the matrix of T relative to the basis (⃗ei). Duality notations: T(ei,⃗ej) = T ij, [T]|⃗e = +[T ij], T = �n +i,j=1T ij⃗ei ⊗ ej, and Einstein’s convention is satisfied. +Thus with ℓ ∈ E∗, ℓ = �n +i=1ℓiei ∈ E∗, and ⃗w ∈ E, ⃗w = �n +i=1wi⃗ei ∈ E, (Q.15) gives +T(ℓ, ⃗w) = +n +� +i,j=1 +Tij⃗ei(ℓ)πej(⃗w) = +n +� +i,j=1 +Tijℓiwj = [ℓ]|⃗e.[T]|⃗e.[⃗w]|⃗e +(Q.16) +([ℓ]|⃗e is a row matrix). Duality notations: T(ℓ, ⃗w) = �n +i,j=1T ijℓiwj and Einstein convention is satisfied. +165 + +166 +Q.4. +Exterior tensorial products +Q.3.4 +Example: Type +�1 +2 +� +uniform tensor +The same steps are applied to any tensor. +E.g., if T ∈ L1 +2(E), then with duality notations, T ijk = +T(ei,⃗ej,⃗ek) and +T = +n +� +i,j,k=1 +T i +jk⃗ei ⊗ ej ⊗ ek, +and +T(ℓ, ⃗u, ⃗w) = +n +� +i,j,k=1 +T i +jkℓiujwk. +(Q.17) +Q.4 +Exterior tensorial products +Let T1 ∈ Lr1 +s1(E) and T2 ∈ Lr2 +s2(E). Their tensorial product is the tensor T1 ⊗ T2 ∈ Lr1+r2 +s1+s2(E) defined by +(T1 ⊗ T2)(ℓ1,1, ..., ℓ2,1, ..., ⃗u1,1, ..., ⃗u2,1, ...) := T1(ℓ1,1, ..., ⃗u1,1, ...)T2(ℓ2,1, ..., ⃗u2,1, ...). +(Q.18) +Particular case: with λ ∈ L0 +0(E) = R and T ∈ Lr +s(E), +λ ⊗ T = T ⊗ λ := λT ∈ Lr +s(E). +(Q.19) +Example Q.1 let T1, T2 ∈ L1 +1(E). +Quantification: +Let T1 = �n +i,j=1(T1)i +j⃗ei ⊗ ej and let T2 = +�n +k,m=1(T2)k +m⃗ek ⊗ em; Then T1 ⊗ T2 = �n +i,j,k,m=1(T1)i +k(T2)j +m⃗ei ⊗ ⃗ej ⊗ ek ⊗ em ∈ L2 +2(E). +Remark Q.2 Alternative +definition: +T1 �⊗T2 +:= +�n +i,j,k,m=1(T1)i +j(T2)k +m⃗ei ⊗ ej ⊗ ⃗ek ⊗ em +∈ +L(E∗, E, E∗, E; R). +And we get back to the previous definition thanks to the natural canonical +isomorphism �J : L(E∗, E, E∗, E; R) → L(E∗, E∗, E, E; R) = L2 +2(E) defined by �J( �T) = T where +T(ℓ, m,⃗v, ⃗w) = �T(ℓ,⃗v, m, ⃗w). +Q.5 +Contractions +Q.5.1 +Contraction of a linear form with a vector +Let ℓ ∈ L0 +1(E) = E∗ and ⃗w ∈ E. Their contraction is the value +ℓ(⃗w) linearity += +ℓ.⃗w noted += +⃗w.ℓ. +(Q.20) +And with a basis (⃗ei) and its dual basis (πei), ℓ = �n +i=1ℓiπei and ⃗w = �n +i=1wi⃗ei give +ℓ.⃗w = +n +� +i=1 +ℓiwi = [ℓ]|⃗e.[⃗w]|⃗e = +n +� +i=1 +wiℓi = ⃗w.ℓ = Tr(⃗w ⊗ ℓ), +(Q.21) +where Tr is the objective trace operator Tr : L(E; E) ≃ L1 +1(E) → R (defined by Tr(⃗ei ⊗ πej) = δi +j). +Duality notations: ℓ.⃗w = �n +i=1ℓiwi, and Einstein convention is satisfied. +Exercice Q.3 Use the change of coordinate formulas to prove that the computation ℓ.⃗w in (Q.21) gives +a result independent of the basis. +Answer. Let P be the change of basis matrix. So [⃗w]new = P −1.[⃗w]old and [ℓ]new = [ℓ]old.P, cf. (A.28), thus +[ℓ]new.[⃗w]new = ([ℓ]old.P).(P −1.[⃗w]old) = [ℓ]old.(P.P −1).[⃗w]old = [ℓ]old[⃗w]old (= ℓ.⃗w). +Q.5.2 +Contraction of a +�1 +1 +� +tensor and a vector +Let ℓ ∈ E∗ and ⃗u ∈ E. The contraction of the elementary tensor ⃗w ⊗ ℓ ∈ L1 +1(E) with ⃗u is defined by: +(⃗w ⊗ ℓ).⃗u +���� +contraction += (ℓ.⃗u)⃗w. +(Q.22) +Thus, if (⃗ei) is a basis in E and (πei) is the dual basis, and T = �n +i,j=1Tij⃗ei ⊗ πej ∈ L1 +1(E) and +⃗u = �n +j=1uj⃗ej ∈ E, then +T = +n +� +i,j=1 +Tij⃗ei ⊗ ej +=⇒ +T.⃗u = +n +� +i,j=1 +Tijuj +j⃗ei +(Q.23) +because πej(⃗u) = uj. Duality notations: T.⃗u = �n +i,j=1T i +juj⃗ei. +166 + +167 +Q.5. +Contractions +Then, with the natural canonical isomorphism (L1 +1(E) =) L(E, E∗; R) ≃ L(E; E), see (T.7), any +endomorphism L ∈ L(E; E) defined by L.⃗ej = �n +i=1Lij⃗ei can be written, for calculation purpose, +�L = +n +� +i,j=1 +Lij⃗ei ⊗ πej +noted += +L, +which means +L.⃗u +(Q.22) += +n +� +i=1 +Lijuj⃗ei +(Q.24) +when ⃗u = � +i uj⃗ej, since πej(⃗u) = uj. Duality notations: L = �n +i,j=1Lij⃗ei ⊗ ej. +Q.5.3 +Contractions of uniform tensors +More generally, the contraction of two tensors, if meaningful, is defined thanks to (Q.20): Let T1 ∈ Lr1 +s1(E), +T2 ∈ Lr2 +s2(E), ℓ ∈ E∗ and ⃗u ∈ E. +Definition Q.4 The objective contraction of T1 ⊗ ℓ ∈ Lr2 +s2+1(E) and ⃗u ⊗ T2 ∈ Lr2+1 +s2 +(E) is the tensor +(T1 ⊗ ℓ).(⃗u ⊗ T2) ∈ Lr1+r2 +s1+s2 given by +(T1 ⊗ ℓ).(⃗u +���� +contraction +⊗T2) := (ℓ.⃗u) T1 ⊗ T2. +(Q.25) +In particular (T1 ⊗ ℓ).⃗u = (ℓ.⃗u) T1 (as in (Q.22)), and ℓ.(⃗u ⊗ T2) = (ℓ.⃗u) T2. +And the objective contraction of T1 ⊗ ⃗u ∈ Lr2+1 +s2 +(E) and ℓ ⊗ T2 ∈ Lr2 +s2+1(E) is the tensor (T1 ⊗ ⃗u).(ℓ ⊗ +T2) ∈ Lr1+r2 +s1+s2 given by +(T1 ⊗ ⃗u).(ℓ ⊗ T2) = (⃗u.ℓ) T1 ⊗ T2 +(= (ℓ.⃗u) T1 ⊗ T2). +(Q.26) +Quantification with a basis (⃗ei), examples to avoid cumbersome notations: +Example Q.5 Let T ∈ L1 +1(E) = L1 +0+1(E), T = �n +i,j=1T i +j⃗ei ⊗ ej. +With ⃗w ∈ E ∼ E∗∗ = L1 +0(E), +⃗w = �n +j=1wj⃗ej, (Q.25) gives T.⃗w ∈ L1 +0(E) ∼ E and +T.⃗w = +n +� +i,j=1 +T i +jwj⃗ei, +i.e. +[T.⃗w]|⃗e = [T]|⃗e.[⃗w]|⃗e +(column matrix). +(Q.27) +(Einstein’s convention is satisfied.) Indeed, T.⃗w = �n +i,j,k=1T i +jwk(⃗ei ⊗ ej).⃗ek = �n +i,j,k=1T i +jwk⃗ei(ej.⃗ek) = +�n +i,j,k=1T i +jwk⃗ei(δj +k) = �n +i,j=1T i +jwj⃗ei. With ℓ ∈ E∗ = L0 +1(E), ℓ = �n +i=1ℓiei, (Q.25) gives ℓ.T ∈ L0 +1(E) = +E∗ and +ℓ.T = +n +� +i,j=1 +ℓiT i +jej, +i.e. +[ℓ.T]|⃗e = [ℓ]|⃗e.[T]|⃗e +(row matrix). +(Q.28) +(Einstein’s convention is satisfied.) Indeed ℓ.T = (�n +i=1ℓiei).(�n +j,k=1T k +j ⃗ek⊗ej) = �n +i,j,k=1ℓiT k +j (ei.⃗ek)ej = +�n +i,j=1ℓiT i +jej. +Example Q.6 Let S, T ∈ L1 +1(E), S = �n +i,k=1Si +k⃗ei ⊗ ek and T = �n +j,k=1T k +j ⃗ek ⊗ ej. Then +S.T = +n +� +i,j,k=1 +Si +kT k +j ⃗ei ⊗ ej, +i.e. +[S.T]|⃗e = [S]|⃗e.[T]|⃗e +(Q.29) +(Einstein’s convention is satisfied.) +Indeed S.T += +(�n +i,k=1Si +k⃗ei ⊗ ek).(�n +j,m=1 T m +j ⃗em ⊗ ej) += +�n +i,j,k,m=1Si +kT m +j ⃗ei(ek.⃗em) ⊗ ej = �n +i,j,k=1Si +kT k +j ⃗ei ⊗ ej. +Example Q.7 Let T ∈ L1 +2(E), T = �n +i,j,k=1T i +jk⃗ei ⊗ ej ⊗ ek, and ⃗u, ⃗w ∈ E ∼ L1 +0(E), ⃗w = �n +i=1wi⃗ei and +⃗u = �n +i=1ui⃗ei. Then +T.⃗w = +n +� +i,j,k=1 +T i +jkwk⃗ei ⊗ ej ∈ L1 +1(E), +and +(T.⃗w).⃗u = +n +� +i,j,k=1 +T i +jkwkuj⃗ei +noted += +T(⃗u, ⃗w). +(Q.30) +(Einstein’s convention is satisfied.) So [T.⃗w]|⃗e = [�n +k=1T i +jkwk] i=1,...,n +j=1,...,n . And with ℓ ∈ E∗, ℓ = �n +i=1ℓiei, +((T.⃗w).⃗u).ℓ = +n +� +i,j,k=1 +T i +jkwkujℓi = T(ℓ, ⃗u, ⃗w) = ℓ.T(⃗u, ⃗w) = ℓ.(T.⃗w).⃗u. +(Q.31) +167 + +168 +Q.5. +Contractions +Q.5.4 +Objective double contractions of uniform tensors +Definition Q.8 Let S, T ∈ L1 +1(E). And let (⃗ei) be a basis in E, (ei) its dual basis, S = �n +i,j=1Si +j⃗ei ⊗ ej +and T = �n +i,j=1T i +j⃗ei ⊗ ej. The double objective contraction S 0.. T of S and T is defined by +S 0.. T = +n +� +i,j=1 +Si +jT j +i . +(Q.32) +(Einstein convention is satisfied.) +Proposition Q.9 S 0.. T defined in (Q.32) is an invariant: It is the trace Tr(LS ◦ LT ) of the endo- +morphisms LS, LT ∈ L(E; E) naturally canonically associated to S and T (given by ℓ.LS.⃗u := S(ℓ, ⃗u) +and ℓ.LT .⃗u := T(ℓ, ⃗u) for all (⃗u, ℓ) ∈ E × E∗). So the real value �n +i,j=1Si +jT j +i has the same real value +regardless of the chosen basis (⃗ei). (Which is not the case of the term to term matrix multiplication +S : T = �n +i,j=1Si +jT i +j, see next § Q.5.5 and example Q.13.) +Proof. Let (⃗ai) and (⃗bi) be two bases and P = [P i +j] be the transition matrix from (⃗ai) to (⃗bi), +i.e., ⃗bj += �n +i=1P i +j⃗ai for all j. +Let Q = [Qi +j] := P −1. +Then bi += �n +i=1Qi +jai. +Let S += +� +ij(Sa)i +j⃗ai ⊗ aj = � +ij(Sb)i +j⃗bi ⊗ bj. +So [(Sb)i +j] = P −1.[(Sa)i +j].P (change of basis formula for +�1 +1 +� +tensors identified with endomorphisms), i.e. (Sb)i +j = � +km Qi +k(Sa)k +mP m +j +for all i, j. +Idem with T. +Thus � +i,j(Sb)i +j(Tb)j +i = � +i,j,k,m,α,β Qi +k(Sa)k +mP m +j Qj +α(Ta)α +βP β +i += � +i,j,k,m,α,β(Sa)k +m(Ta)α +βP β +i Qi +kP m +j Qj +α = +� +k,m,α,β(Sa)k +m(Ta)α +βδβ +k δm +α = � +k,m(Sa)k +m(Ta)m +k . +Definition Q.10 More generally, the objective double contractions S 0.. T of uniform tensors, is obtained +by applying the objective simple contraction twice consecutively, when applicable. +E.g., T1 ⊗ ℓ1,1 ⊗ ℓ1,2 and ⃗u2,1 ⊗ ⃗u2,2 ⊗ T2 give +(T1 ⊗ ℓ1,1 ⊗ ℓ1,2).(⃗u2,1 +� +�� +� +first +⊗⃗u2,2 ⊗ T2) = (ℓ1,2.⃗u2,1)(T1 ⊗ ℓ1,1) ⊗ (⃗u2,2 +� +�� +� +second +⊗T2) += (ℓ1,2.⃗u2,1)(ℓ1,1.⃗u2,2) T1 ⊗ T2. +(Q.33) +Example Q.11 Let S ∈ L1 +2(E), T ∈ L2 +1(E), S = �n +i,j,k=1Si +jk⃗ei ⊗ej ⊗ej, T = �n +α,β,γ=1 T αβ +γ ⃗eα ⊗⃗eβ ⊗eγ. +Then +S.T = +n +� +i,j,k,β,γ=1 +Si +jkT kβ +γ ⃗ei ⊗ ej ⊗ ⃗eβ ⊗ eγ, +and +S 0.. T = +n +� +i,j,k,γ=1 +Si +jkT kj +γ ⃗ei ⊗ eγ. +(Q.34) +(Einstein’s convention is satisfied.) +Exercice Q.12 If S ∈ L(E, F; R), T ∈ L(F, G; R) and U ∈ L(G, E; R) then prove +S 0.. (T.U) = (S.T) 0.. U = (U.S) 0.. T +(circular permutation). +(Q.35) +Answer. +If S = � Si +j⃗ai ⊗ bj, T = � T i +j⃗bi ⊗ cj and U = � U i +j⃗ci ⊗ aj, then T.U = � T i +kU k +j ⃗bi ⊗ aj, thus +S 0.. (T.U) = � Si +mT m +k U k +i , and S.T = � Si +kT k +j ⃗ai ⊗ cj, so (S.T) 0.. U = � Si +kT k +mU m +i . And the second equality +thanks to the symmetry of 0.. , i.e. (S.T) 0.. U = U 0.. (S.T) = (U.S) 0.. T with the previous calculation. +We define in the same way the triple objective contraction (apply the simple contraction three times +consecutively). E.g., with (Q.34) we get +S 0... T = +n +� +i,j,k=1 +Si +jkT kj +i. +(Q.36) +(Einstein’s convention is satisfied.) +168 + +169 +Q.6. +Kronecker (contraction) tensor, trace +Q.5.5 +Non objective double contraction: Double matrix contraction +The engineers often use the double matrix contraction of second order tensors defined by (term to term +multiplication): If S = [Sij] = [Si +j] and T = [Tij] = [T i +j] then +S : T := +n +� +i,j=1 +SijTij = +n +� +i,j=1 +Si +jT i +j +noted += +Tr(S.T T ). +(Q.37) +Einstein’s convention is not satisfied, and the result is observer dependent for associated endomorphism: +Example Q.13 Let (⃗ei) be a basis, let S ∈ L(E; E) given by [S]⃗e = +� +0 +4 +2 +0 +� +(so S.⃗e1 = 2⃗e2 and +S.⃗e2 = 4⃗e1). Then the double matrix contraction (Q.37) gives +S : S = [S]⃗e : [S]⃗e = 4 ∗ 4 + 2 ∗ 2 = 20. +(Q.38) +Change of basis: let ⃗b1 = ⃗e1 and ⃗b2 = 2⃗e2. The transition matrix from (⃗ei) to (⃗bi) is P = +� +1 +0 +0 +2 +� +. Thus +[S]⃗b = P −1.[S]⃗e.P = +� +1 +0 +0 +1 +2 +� +. +� +0 +8 +2 +0 +� += +� +0 +8 +1 +0 +� +. Thus +S : S = [S]⃗b : [S]⃗b = 8 ∗ 8 + 1 ∗ 1 = 65 ̸= 20. +(Q.39) +To be compared with the double objective contraction: [S]⃗e 0.. [S]⃗e = 4∗2+2∗4 = 16 = [S]⃗b 0.. [S]⃗b = S 0.. S +(observer independent result = objective result). +So it is absurd to use S : S (double matrix contraction) if you need objectivity: Recall that the foot is +the international vertical unit in aviation, and thus the use of the double objective contraction is vital, +while the use of the double matrix contraction can be fatal (really). Also see the Mars climate orbiter +probe crash. +Exercice Q.14 Let S ∈ L0 +2(E) (e.g. a metric), let (⃗ai) be a Euclidean basis in foot, and let (⃗bi) = (λ⃗ai) +be the related euclidean basis in metre (change of unit). Give [S]|⃗a : [S]|⃗a and [S]|⃗b : [S]|⃗b and compare. +(The simple and double objective contractions are impossible here since S and T are not compatible.) +Answer. +Let S = �n +i,j=1Sa,ijai ⊗ aj = �n +i,j=1Sb,ijbi ⊗ bj. +Since (⃗bi) = (λ⃗ai) we have bi = +1 +λai. +Thus +�n +i,j=1Sa,ijai ⊗ aj = �n +i,j=1Sa,ijλ2bi ⊗ bj, thus λ2Sa,ij = Sb,ij. Thus +[S]|⃗b : [S]|⃗b = +n +� +i,j=1 +(Sb,ij)2 = λ4 +n +� +i,j=1 +(Sa,ij)2 = λ4[S]|⃗a : [S]|⃗a, +(Q.40) +with λ4 ≥ 100: Quite a difference isn’t it? +Q.6 +Kronecker (contraction) tensor, trace +Definition Q.15 The Kronecker tensor is the +�1 +1 +� +uniform tensor δ ∈ L1 +1(E) defined by +∀(ℓ, ⃗u) ∈ E∗ × E, +δ(ℓ, ⃗u) := ℓ.⃗u. +(Q.41) +And the Kronecker symbols relative to a basis (⃗ei) are the reals defined by, calling (πei) the dual basis, +δij := δ(πei,⃗ej) = +� +1 if i = j, +0 if i ̸= j, +� +i.e. +δ := +n +� +i=1 +πei ⊗ ei, +[δ] = [δj] = [I] +(Q.42) +(identity matrix whatever the basis). Duality notations: δi +j := δ(ei,⃗ej), δ := �n +i=1 ⃗ei ⊗ ei and [δ] = [δi +j]. +Definition Q.16 The trace of a +�1 +1 +� +uniform tensor T ∈ L1 +1(E) is +� +Tr(T) = δ 0.. T +(= Tr(LT )) +(Q.43) +(with the natural canonical isomorphism T ∈ L1 +1(E) ≃ LT ∈ L(E; E) given by T(ℓ,⃗v) := ℓ.LT .⃗v). +Thus � +Tr(T) = �n +i=1T ii. +In particular � +Tr(δ) = n, and � +Tr(⃗v ⊗ ℓ) = � +i viℓi = ℓ.⃗v when ⃗v = � +i vi⃗ei and ℓ = � +j ℓjej. +169 + +170 +R.1. +Introduction, module, derivation +R +Tensors in T r +s (U) +R.1 +Introduction, module, derivation +Let A and B be any sets, and let F(A; B) be the set of functions A → B. The “plus” inner operation +and the “dot” outer operation are defined by, for all f, g ∈ F(A; B), all λ ∈ R and all p ∈ A, +� (f + g)(p) := f(p) + g(p), +and +(λ.f)(p) := λ f(p), +λ.f noted += +λf. +(R.1) +(F(A; B), +, ., R) is thus a vector space on the field R (see any elementary course) called F(A; B). +But the field R is “too small” to define a tensor which can be seen as “a linear tool that satisfies the +change of coordinate system rules”: +Example R.1 Fundamental counter-example: Derivation. Let U be an open set in Rn. The +derivation d : ⃗w ∈ C1(U; ⃗Rn) → d⃗w ∈ C0(U; L(⃗Rn; ⃗Rn)) is R-linear: In particular d(λ⃗w) = λ(d⃗w) for all +λ ∈ R... +...but d doesn’t satisfy the change of coordinate system rules, see (S.35). +So a derivation it not a tensor (it is a “spray”, see Abraham–Marsden [1]). +In fact, one requirement for T to be a tensor is, e.g. with T = ⃗w a vector field: For all ϕ ∈ C∞(U; R), +and all ⃗w ∈ Γ(U) (C∞-vector field), +T(ϕ⃗w) = ϕ T(⃗w). +(R.2) +While +d(ϕ⃗w) ̸= ϕ d(⃗w), +because +d(ϕ⃗w) = ϕ d⃗w + dϕ.⃗w. +(R.3) +Thus the elementary R-linearity requirement “T.(λ⃗w) = λ(T.⃗w) for all λ ∈ R is not sufficient to charac- +terize a tensor: The R-linearity has to be replaced by the C∞(U; R)-linearity, cf. (R.2). +Thus we will have to replace a real vector space (V, +, ., R) over the field R with the “module” +(V, +, ., C∞(U; R)) over the ring C∞(U; R), which mainly amounts to consider (R.1) for all λ = ϕ ∈ +C∞(U; R). Remark: The use of a module is very similar to the use of a vector space, but for the use of +the inverse: all real λ ̸= 0 has a multiplicative inverse in R (namely 1 +λ), but a function f ∈ C∞(U; R) s.t. +“f ̸= 0 and f vanishes at one point” doesn’t have a multiplicative inverse in C∞(U; R). +R.2 +Field of functions and vector fields +Framework of classical mechanics: U is an open set in an affine space E which associated vector +is E. And the definition of tensors is done at a fixed time t (concerns the space variables). As before, the +approach is first qualitative, then quantitative with a basis (⃗ei(p)) and its dual basis (πei(p)) = (ei(p)), +at any p ∈ E. +R.2.1 +Field of functions +Let f ∈ C∞(U; R) be a function. The associated function field is +�f : +� +U → U × R +p → �f(p) := (p; f(p)), +(R.4) +and p is called the base point. So Im �f = {(p; f(p)) : p ∈ U} is the graph of f. Definition: +T 0 +0 (U) := { �f : f ∈ C∞(U; R)} = {field of functions} = the set of +�0 +0 +� +type tensor on U, +(R.5) +or the set of tensors of order 0 on U. Abusive short notations (to lighten the writings): +�f(p) noted += +f(p), +and +T 0 +0 (U) noted += +C∞(U; R), +(R.6) +but keep the base point in mind (no ubiquity gift). +In T 0 +0 (U), the internal sum is defined by, for all �f, �g ∈ T 0 +0 (U) with �f(p) = (p; f(p)) and �g(p) = (p; g(p)), +( �f + �g)(p) := (p; (f + g)(p)) +(= (p; f(p) + g(p))), +(R.7) +and the external multiplication on the ring C∞(U; R) is defined by, for all ϕ ∈ C∞(U; R), +(ϕ �f)(p) := (p; (ϕf)(p)) +(= (p; ϕ(p)f(p))) +(R.8) +(the base point p remains unchanged). Thus (T 0 +0 (U), +, .) is a module over the ring C∞(U; R). +170 + +171 +R.3. +Differential forms +R.2.2 +Vector fields +Let ⃗w ∈ C∞(U, E) be a vector valued function (at least Lipschitzian, to get integral curves, cf. Cauchy– +Lipschitz theorem). The associated vector field is +�⃗w : +� +U → U × E +p → �⃗w(p) = (p; ⃗w(p)). +(R.9) +So Im�⃗w = {(p; ⃗w(p)) : p ∈ U} is the graph of ⃗w, and the definition of �⃗w tells that the vector ⃗w(p) has to +be drawn at p (the base point). Abusive short notation: +�⃗w(p) noted += +⃗w(p) +instead of �⃗w(p) = (p; ⃗w(p)). +(R.10) +It lightens the notations, but keep the base point in mind. Let +Γ(U) := the set of vector fields on U. +(R.11) +More precisely, we will use the following full definition of vector fields (see e.g. Abraham–Marsden [1]): +A vector field is built from tangent vectors to curves. It makes sense on non planar surfaces, and more +generally on differential manifolds. +R.3 +Differential forms +The basic concept is that of vector fields. A first over-layer is made of differential forms (which “measure +vector fields”): +Definition R.2 Let α +� +U → E∗ +p → α(p) +� +(so α(p) is a linear form at p). The associated differential form +(also called a 1-form) is “the field of linear forms” defined by +�α : +� +U → U × E∗ +p → �α(p) = (p; α(p)) +( = “a pointed linear form at p”). +(R.12) +And p is called the base point, and Im�α = {(p; α(p)) : p ∈ U} is the graph of α. +Thus, if �α ∈ Ω1(U) (differential form) and �⃗w ∈ Γ(U) (vector field), then �α.�⃗w ∈ T 0 +0 (U) (field of scalar +valued functions) satisfies +�α.�⃗w : +� +U → U × R +p → (�α.�⃗w)(p) = (p; (α.⃗w)(p)) = (p; α(p).⃗w(p)) ∈ U × R. +(R.13) +Short notation: +�α(p) noted += +α(p), +instead of �α(p) = (p; α(p)), +(R.14) +but keep the base point in mind. And +Ω1(U) := the set of differential forms U. +(R.15) +R.4 +Tensors +A second over-layer is introduced with the tensors with are “functions defined on vector fields and on +differential forms” (which “measure vector fields and differential forms”). +Let r, s ∈ N, r+s ≥ 1, and let T : +� +U → Lr +s(E) +p → T(p) +� +(so T(p) is a uniform +�r +s +� +tensor for each p, +cf. (Q.3.1)). And consider the associated function +�T : +� +U → U × Lr +s(E) +p → �T(p) = (p; T(p)) +(R.16) +Abusive short notation: +�T(p) noted += +T(p) +instead of �T(p) = (p; T(p)), +(R.17) +but keep the base point in mind. +171 + +172 +R.5. +First Examples +Definition R.3 (Abraham–Marsden [1].) �T is a tensor of type +�r +s +� +iff T is C∞(U; R)-multilinear (not only +R-multilinear), i.e., for all f ∈ C∞(U; R), all z1, z2 vector field or differentiable form where applicable, +and all p ∈ U, +� +T(p)(..., z1(p) + z2(p), ...) = T(p)(..., z1(p), ...) + T(p)(..., z2(p), ...), +and +T(p)(..., f(p)z1(p), ...) = f(p) T(p)(..., z1(p), ...), +(R.18) +written in short +� +T(..., z1 + z2, ...) = T(..., z1, ...) + T(..., z2, ...), +and +T(..., fz1, ...) = f T(..., z1, ...). +(R.19) +And +T r +s (U) := the set of +�r +s +� +type tensors on U. +(R.20) +(Recall: T 0 +0 (U) := C∞(U; R) the set of function fields, cf. (R.4).) +Remark R.4 Definition in differential geometry lessons: A tensor is a section of a certain bundle over +a manifold. For classical mechanics, definition R.3 gives an equivalent definition. +R.5 +First Examples +R.5.1 +Type +�0 +1 +� +tensor = differential forms +If T ∈ T 0 +1 (U) then T(p) ∈ E∗, so T = α ∈ Ω1(U) is a differential form: T 0 +1 (U) ⊂ Ω1(U). +Converse: Does a differential form α ∈ Ω1(U) defines a +�0 +1 +� +type tensor on U? Yes: We have to +check (R.18), which is trivial. So α ∈ T 0 +1 (U), so Ω1(U) ⊂ T 0 +1 (U). +Thus +T 0 +1 (U) = Ω1(U). +(R.21) +R.5.2 +Type +�1 +0 +� +tensor (identified to a vector field) +Let T ∈ T 0 +1 (U), so T(p) ∈ L1 +0(E) = L(E∗; R) = E∗∗ for all p ∈ U. Thus, thanks to the natural canonical +isomorphism E∗∗ ≃ E, T(p) can be identified to a vector, thus T 0 +1 (U) ⊂ Γ(U). +Converse: Does a vector field ⃗w ∈ Γ(U) defines a +�1 +0 +� +type tensor on U? Yes: We have to check (R.18), +which is trivial. So Γ(U) ⊂ T 1 +0 (U). +Thus +T 1 +0 (U) ≃ Γ(U). +(R.22) +R.5.3 +A metric is a +�0 +2 +� +tensor +Let T ∈ T 0 +2 (U), so T(p) ∈ L0 +2(E) for all p ∈ U, and T(⃗u, ⃗w) ∈ T 0 +0 (U) for all ⃗u, ⃗w ∈ Γ(U). +Definition R.5 A metric g on U is a +�0 +2 +� +type tensor on U such that, for all p ∈ E, g(p) =noted gp is an +inner dot product on E. +R.6 +�1 +1 +� +tensor, identification with fields of endomorphisms +Let T ∈ T 1 +1 (U), so T(p) ∈ L1 +1(E) for all p ∈ U, and T(α, ⃗w) ∈ T 0 +0 (U) for all α ∈ Ω1(U) and ⃗w ∈ Γ(U) (so +T(p)(α(p), ⃗w(p)) ∈ R for all p). +The associated field of endomorphisms on U is �LT : +� +U → U × L(E; E) +p → �LT (p) = (p, LT (p)) +� +where LT (p) is +identified with T(p) thanks to the natural canonical isomorphism L(E; E) ≃ L(E∗, E; R) = L1 +1(E) given +by +∀ℓ ∈ E∗, ∀⃗w ∈ E, +ℓ.(LT (p).⃗w) = T(p)(ℓ, ⃗w). +(R.23) +R.7 +Unstationary tensor +Let t ∈ [t1, t2] ⊂ R. Let (Tt)t∈[t1,t2] be a family of +�r +s +� +tensors, cf. (R.16). Then T : t → T(t) := Tt is called +an unstationary tensor. And the set of unstationary tensors is also noted T r +s (U). E.g., a Eulerian velocity +field is a +�1 +0 +� +unstationary vector field. +172 + +173 +S.1. +Differential +S +Differential, its eventual gradients, divergences +S.1 +Differential +The definition of the differential of a function is observer independent: All observers have the same +definition (qualitative: no man made tool required, like a basis or an inner dot product). +S.1.1 +Framework +Classical Framework: E are F affine spaces associated with vector spaces E and F, and ||.||E and ||.||F +are norms in E and F such that (E, ||.||E) and (F, ||.||F ) are complete (we need “limit that stay in the +space as h → 0”, ). U is an open set in E, and Φ : +� +U → F +p → pF = Φ(p) +� +is a function. If applicable, E +and/or F can be replaced by E and/or F. (The definitions can be generalized to manifolds.) Reminder: +Definition S.1 Let p ∈ U. The function Φ is said to be continuous at p iff Φ(q) −→ +q→p Φ(p) relative to the +considered norms, i.e., ||Φ(q) − Φ(p)||F −→||q−p||E→0 0, also written (Landau notation): Near p, +Φ(q) = Φ(p) + o(1), +(S.1) +called “the zero-th order Taylor expansion of Φ near p”. In other words: +∀ε > 0, ∃η > 0 s.t. ∀q ∈ E s.t. ||q − p||E < η we have ||Φ(q) − Φ(p)||F < ε. +And C0(U; F) is the set of functions that are continuous at all p ∈ U. +S.1.2 +Directional derivative and differential (observer independent) +Let p ∈ U, ⃗u ∈ E, and let f : R → F defined by +f(h) := Φ(p + h⃗u) +(S.2) +Definition S.2 The function Φ is differentiable at p in the direction ⃗u iff f is derivable at 0, i.e. iff the +limit f ′(0) = limh→0 +Φ(p+h⃗u)−Φ(p) +h +=noted dΦ(p)(⃗u) exists in F, i.e. iff, near p, +Φ(p + h⃗u) = Φ(p) + h dΦ(p)(⃗u) + o(h), +(S.3) +equation called the first order Taylor expansion of Φ at p in the direction ⃗u (it is the first order Taylor +expansion of f near p). +Then dΦ(p)(⃗u) is called the directional derivative of Φ at p in the direction ⃗u. +And if, for all ⃗u ∈ E, dΦ(p)(⃗u) exists (in F) then Φ is called Gâteaux differentiable at p. +Exercice S.3 Prove: If Φ is Gâteaux differentiable at p then dΦ(p) is homogeneous, i.e., dΦ(p)(λ⃗u) = +λ dΦ(p)(⃗u) for all ⃗u ∈ E and all λ ∈ R. +Answer. limh→0 +Φ(p+h(λ⃗u))−Φ(p) +h += λ limh→0 +Φ(p+λh⃗u)−Φ(p) +λh += λ limk→0 +Φ(p+k⃗u)−Φ(p) +k +. +Definition S.4 If Φ is Gateaux differentiable and if moreover dΦ(p) is linear and continuous at p, then +Φ is said to be differentiable at p (or Fréchet differentiable at p). So +Φ(q) = Φ(p) + h dΦ(p).−→ +pq + o(||−→ +pq||E), +(S.4) +since then dΦ(p)(⃗u) =noted dΦ(p).⃗u for all ⃗u ∈ E (linearity). +And the affine function affp : q → affp(q) := Φ(p) + dΦ(p).−→ +pq is the affine approximation of Φ at p. +(So, the graph of affp is the tangent plane of Φ at p.) +Definition S.5 Φ : U → F is said to be differentiable in U iff Φ is differentiable at all p ∈ U. Then its +differential is the map +dΦ : +� +U → L(E; F) +p → dΦ(p). +(S.5) +And C1(U; F) is the set of differentiable functions ψ such that dΦ ∈ C0(U; L(E; F)). +And C2(U; F) is the set of differentiable functions ψ such that dΦ ∈ C1(U; L(E; F)). +... And Ck(U; F) is the set of differentiable functions ψ such that dΦ ∈ Ck−1(U; L(E; F)).... +173 + +174 +S.2. +A basis and the j-th partial derivative +Proposition S.6 The differentiation (or derivation) operator d : +� +C1(U; F) → C0(U; L(E; F)) +Φ → dΦ +� +is +R-linear (“a derivation is linear”). +Proof. d(Φ + λΨ)(p).⃗u = limh→0 +(Φ+λΨ)(p+h⃗u)−(Φ+λΨ)(p) +h += limh→0 +Φ(p+h⃗u)−Φ(p)+λΨ(p+h⃗u)−λΨ(p) +h += +limh→0 +Φ(p+h⃗u)−Φ(p) +h ++ λ limh→0 +Ψ(p+h⃗u)−Ψ(p) +h += dΦ(p).⃗u + λdΨ(p).⃗u = (dΦ(p) + λdΨ(p)).⃗u for all p +and ⃗u, thus d(Φ + λΨ) = dΦ + λdΨ for all λ ∈ R and Φ, Ψ ∈ C1(U; F). +Exercice S.7 Prove: if f ∈ C1(U; R) (scalar values) and Φ ∈ C1(U; F) then, for all ⃗u ∈ E, +d(fΦ).⃗u = (df.⃗u)Φ + f(dΦ.⃗u) +(S.6) +(and we also write d(fΦ) = Φ ⊗ df + f dΦ for a use with contraction rules). +Answer. +d(fΦ)(p).⃗u = lim +h→0 +f(p+h⃗u)Φ(p+h⃗u) − f(p)Φ(p) +h += lim +h→0 +f(p+h⃗u)Φ(p+h⃗u) − f(p)Φ(p+h⃗u) +h ++ f(p)Φ(p+h⃗u) − f(p)Φ(p) +h += lim +h→0 +f(p+h⃗u) − f(p) +h +(Φ(p) + o(1)) + lim +h→0 f(p)Φ(p+h⃗u) − Φ(p) +h += (df(p).⃗u)Φ(p) + f(p)(dΦ(p).⃗u). +(S.7) +Tensorial writing: d(fΦ).⃗u = (Φ ⊗ df).⃗u + (f dΦ).⃗u, thanks to the contraction rule which gives (Φ ⊗ df).⃗u + +(f dΦ).⃗u = Φ(df.⃗u) + f(dΦ.⃗u). +Remark S.8 In differential geometry, the definition of a tangent map is defined by, with definition S.4: +TΦ : +� +U × E → F × F +(p, ⃗u) → TΦ(p, ⃗u) = (Φ(p), dΦ(p).⃗u). +(S.8) +The two points p (input) and Φ(p) (output) are the base points, and the two vectors ⃗u (input) and +dΦ(p).⃗u (output) are the initial vector and its push-forward by Φ. +S.1.3 +Notation for the second order Differential +Let Φ ∈ C2(U; F); Thus dΦ ∈ C1(U; L(E; F)), thus d(dΦ) ∈ C0(U; L(E; L(E; F))); So, for p ∈ U and ⃗u ∈ +E, we have d(dΦ)(p).⃗u = limh→0 +dΦ(p+h⃗u)−dΦ(p) +h +∈ L(E; F), and, with ⃗v ∈ E we have (d(dΦ)(p).⃗u).⃗v ∈ F. +Definition S.9 The bilinear map d2Φ(p) ∈ L(E, E; F) is defined by +d2Φ(p)(⃗u,⃗v) = (d(dΦ)(p).⃗u).⃗v, +(S.9) +thanks to the natural canonical isomorphism L ∈ L(E; L(E; F)) ↔ TL ∈ L(E, E; F) given by +TL(⃗u1, ⃗u2) := (L.⃗u1).⃗u2 for all ⃗u1, ⃗u2 ∈ E; Thus L =noted TL, thus d(dΦ) =noted d2Φ(p) ∈ L(E, E; F). +This gives the usual second order Taylor expansion of Φ (supposed C2) near p in the direction ⃗u: +Φ(p + h⃗u) = Φ(p) + h dΦ(p).⃗u + h2 +2 d2Φ(p)(⃗u, ⃗u) + o(h2) +(S.10) +(=the second order Taylor expansion of f : h → f(h) = Φ(p + h⃗u) near h = 0, cf. (S.2)). +And Schwarz’s theorem tells that d2Φ(p) is symmetric when Φ is C2, i.e. d2Φ(p)(⃗u,⃗v) = d2Φ(p)(⃗v, ⃗u). +S.2 +A basis and the j-th partial derivative +Definition S.10 Let Φ ∈ C1(U; F), ⃗u ∈ Γ(U) (a vector field), p ∈ U. The derivative of Φ at p along ⃗u +is defined by +∂⃗uΦ(p) := dΦ(p).⃗u(p) +(= lim +h→0 +Φ(p + h⃗u(p)) − Φ(p) +h +∈ F). +(S.11) +This defines the directional derivative operator along ⃗u: +∂⃗u : +� +C1(U; F) → C0(U; F) +Φ → ∂⃗u(Φ) := dΦ.⃗u, +i.e. +∂⃗u(Φ)(p) := dΦ(p).⃗u(p). +(S.12) +(And ∂⃗u(Φ)(p) =noted ⃗u(Φ)(p) in differential geometry thanks to E ≃ E∗∗ which gives ∂⃗u ≃ ⃗u.) +174 + +175 +S.3. +Application 1: Scalar valued functions +In particular, if (⃗ei(p)) is a basis at p, then the j-th partial derivative of Φ at p is ∂⃗ejΦ(p) =noted ∂jΦ(p) +(the derivative along ⃗ej), and the j-th directional derivative operator is +∂j : +� C1(U; F) → C0(U; F) +Φ → ∂jΦ := dΦ.⃗ej , +i.e. +∂j(Φ)(p) := dΦ(p).⃗ej(p). +(S.13) +(In differential geometry ∂jΦ =noted ⃗ej(Φ), so ⃗ej(Φ)(p) := dΦ(p).⃗ej(p).) +S.3 +Application 1: Scalar valued functions +S.3.1 +Differential of a scalar valued function (objective) +Here Φ noted += +f : +� +U → R +p → f(p) +� +is a C1 scalar valued function, so df ∈ Ω1(U)∩C0(U; E∗) (a C0 differential +form). So df(p) ∈ E∗ for all p ∈ U, and df(p).⃗u = limh→0 +f(p+h⃗u)−f(p) +h +∈ R for all ⃗u ∈ E. +Exercice S.11 Prove: If f, g ∈ C1(U; R) then (derivative of a product) +d(fg) = (df)g + f(dg), +(S.14) +i.e., d(fg).⃗w = (df.⃗w)g + f(dg.⃗w) for all ⃗w ∈ Γ(U). +Answer. limh→0 +f(p+h ⃗w)g(p+h ⃗w)−f(p)g(p) +h += limh→0 +f(p+h ⃗w)g(p+h ⃗w)−f(p)g(p+h ⃗w) +h ++ limh→0 +f(p)g(p+h ⃗w)−f(p)g(p) +h += +limh→0 +f(p+h ⃗w)−f(p) +h +(g(p) + o(1)) + limh→0 f(p) g(p+h ⃗w)−g(p) +h +, calculation that only requires the first order (affine) +approximation of f and g: We get the same result as with the affine functions f(x) = a0+a1x and g(x) = b0+b1x, +which give (fg)(x) = a0b0 + (a0b1+a1b0)x + a1b1x2, and then (fg)′(x) = a0b1+a1b0 + 2a1b1x, which is indeed +equal to (f ′g + fg′)(x) = a1(b0+b1x) + (a0+a1x)b1. +S.3.2 +Quantification +Let (⃗ei(p)) be a basis at p. So ∂jf(p) =(S.13) df(p).⃗ej(p) (= limh→0 +f(p+h⃗ej(p))−f(p) +h +), and we write +∂jf(p) noted += +f|j(p). +(S.15) +So, with (πei(p)) the dual basis of the basis (⃗ei(p)), and with f|j(p) := πei(p).df(p) (j-th component of +df(p) in the basis (πei(p))), we have +df = +n +� +j=1 +f|jπej, +and +[df(p)]|⃗e = ( f|1(p) +... +f|n(p) ) +(row matrix). +(S.16) +So df.⃗u = �n +j=1f|juj = [df]|⃗e.[⃗u]|⃗e when ⃗u(p) = � +i ui(p)⃗ei(p). In particular with a Cartesian basis, +(πei(p)) =noted (dxj), and df = �n +j=1 +∂f +∂xj dxj. +Duality notations: πei = ei, ⃗u = �n +j=1uj⃗ej, df = �n +j=1f|j ej, df.⃗u = �n +j=1f|juj, and with a Cartesian +basis, πei = dxi and df = �n +j=1 +∂f +∂xj dxj. +Exercice S.12 Prove: (fg)|j = f|j g + f g|j when f, g : U → R are C1 scalar valued functions. +Answer. Apply (S.7): here d(fg) = g df + f dg, i.e. d(fg).⃗ej = (df.⃗ej) g + f (dg.⃗ej) for all j. +And df(p) ∈ E∗ satisfies the covariant change of basis formula for linear forms, i.e., if (⃗ai(p)) and +(⃗bi(p)) are two bases at p and P(p) is the transition matrix from (⃗ai(p)) to (⃗bi(p)), then [df(p)]|⃗b =(A.28) +[df(p)]|⃗a.P(p), or in short: +[df]|⃗b = [df]|⃗a.P +(covariance formula). +(S.17) +175 + +176 +S.4. +Application 2: Coordinate system basis and Christoffel symbols +S.3.3 +Gradients (subjective) associated with a differential through inner dot products +Let f ∈ C1(U; R) (a C1 scalar valued function). Choose (subjective) an inner dot product (·, ·)g in E. +Definition S.13 The conjugate gradient +⃗ +gradgf(p) of f at p ∈ U relative to (·, ·)g, also called the +(·, ·)g-conjugate gradient of f at p, is the (·, ·)g-Riesz representation vector of the linear form df(p) ∈ E∗: +⃗ +gradgf(p) := ⃗Rg(df(p)). +(S.18) +I.e., the vector +⃗ +gradgf(p) ∈ E is characterized by, cf. (F.2), +∀⃗u ∈ E, +df(p).⃗u = ( ⃗ +gradgf(p), ⃗u)g = +⃗ +gradgf(p) •g ⃗u. +(S.19) +Fundamental: An English observer with his Euclidean dot product (·, ·)a in foot and a French observer +with his Euclidean dot product (·, ·)b in metre have the same differential df (defined independently of +any unit of measurement); But do not have the same gradient: +⃗ +gradbf +(F.12) += +λ2 ⃗ +gradaf +with +λ2 > 10. +(S.20) +Quite different vectors isn’t it? The “gradient vector” strongly depends on the chosen inner dot product. +And to forget this fact leads to accidents like the crash of the Mars Climate Orbiter probe, cf. remark A.14. +Subjective first order Taylor expansion: +If an inner dot product (·, ·)g exists and is used, then the +first order Taylor expansion (S.3) gives +f(p + h⃗u) = f(p) + h ( ⃗ +gradgf(p), ⃗u)g + o(h) +(= f(p) + h +⃗ +gradgf(p) •g ⃗u + o(h)). +(S.21) +Fundamental once again (we insist): +• An inner dot product does not always exist (as a meaningful tool), see § B.3.2 (thermodynamics), +thus, for a C1 function, a gradient does not always exists (contrary to a differential). +• df(p) is a linear form (covariant) while +⃗ +gradgf(p) is a vector (contravariant). In particular the +change of basis formulas differ, cf. (A.28): +[df]|new = [df]|old.P, +while +[ ⃗ +gradg]|new = P −1.[ ⃗ +gradg]|old. +(S.22) +• df cannot be identified +⃗ +gradf (with one?) (Recall; there is no natural canonical isomorphims between +E and E∗.) The differential df is also called the “covariant gradient”, and any of its associated gradient +vectors is also called the “contravariant gradient relative to an inner dot product”. +Isometric Euclidean framework: If one Euclidean dot product can be imposed to all observers (foot? +metre?) then +⃗ +gradgf =noted +⃗ +gradf = ⃗∇f and (S.19) is written df.⃗u = +⃗ +gradf • ⃗u = ⃗∇f • ⃗u (isometric +framework). +Exercice S.14 Cartesian basis (⃗ei) and (·, ·)g given by [g][⃗e = +� +1 +0 +0 +2 +� +. Give [df]|⃗e and [ ⃗ +gradgf]|⃗e. +Answer. [df]|⃗e = ( ∂f +∂x1 +∂f +∂x2 ) (row matrix) and (S.19) gives [ ⃗ +gradgf]|⃗e = +� +∂f +∂x1 +1 +2 +∂f +∂x2 +� +(column matrix ̸= [df]T ). +S.4 +Application 2: Coordinate system basis and Christoffel symbols +(Necessary when dealing with covariance.) +S.4.1 +Coordinate system, and coordinate system basis +Consider a (open) set Upar = {⃗q ∈]a1, b1[×...×]an, bn[}, called the set of parameters, in the Cartesian +space Rn, consider an open set U ⊂ Rn, called the set of geometric positions, and consider a C2- +diffeomorphism Ψ : ⃗q ∈ Upar → p ∈ U, called a coordinate system. +Let (⃗ai) the canonical basis of the parameter space, let ⃗q = � +i qi⃗ai ∈ Upar (the qi are called the +parameters). E.g., see the polar coordinate system at § 6.6.2 where ⃗q = (q1, q2) = (r, θ). +176 + +177 +S.4. +Application 2: Coordinate system basis and Christoffel symbols +Ψ being a diffeomorphism, at any p = Ψ(⃗q) ∈ U the vectors +⃗ai∗(p) := dΨ(⃗q).⃗ai +(S.23) +make a basis in E at p, and (⃗ai∗(p)) is called the coordinate system basis at p. Its dual basis at p is made +of the linear forms dqi(p), so where, for all i, j, +dqi(p).⃗aj∗(p) = δj +i . +(S.24) +Duality notations: dqi(p).⃗aj∗(p) = δi +j for all i, j. +S.4.2 +Parametric expression of the differential of a scalar valued function +With a coordinate system Ψ, a scalar valued function f : +� +U → R +p → f(p) +� +defined in U can be described +with the function g = f ◦ Ψ : +� +Upar → R +⃗q → g(⃗q) := f(p) when p = Ψ(⃗q) +� +defined in Upar, and g is called the +parametric expression of f. Thus +dg(⃗q) = df(p).dΨ(⃗q) +when +p = Ψ(⃗q), +(S.25) +in particular, +∂g +∂qj +(⃗q) := dg(⃗q).⃗aj = df(p).dΨ(⃗q).⃗aj = df(p).⃗aj∗(p) noted += +∂f +∂qj +(p). +(S.26) +Warning, pay attention: f is a function of p, not a function of ⃗q, and the notations +∂f +∂qj (p) means +:= ∂(f◦Ψ) +∂qj +(⃗q) when p = Ψ(⃗q), and nothing else. +Thus with (dqj(p)) the dual basis of the coordinate basis (⃗ai∗(p)) at p, +df(p) +(S.26) += +n +� +j=1 +∂f +∂qj +(p) dqj(p). +(S.27) +Duality notations: df(p) = � +j +∂f +∂qj (p) dqj(p). +Remark S.15 Pay attention to the notations that could contradict themselves: +1- In Upar the dual basis (πai) of the Cartesian basis (⃗ai) is a uniform basis (independent of ⃗q)... and +is (almost) never written (dqi)... +2- Indeed, (dqi(p)) is the name reserved for the dual basis of (⃗ai∗(p)) in the geometric space... Mind +the notations! E.g. for polar coordinates (dq1(p), dq2(p)) = (dr(p), dθ(p)) is the dual basis of the polar +coordinate system basis (⃗a1∗(p),⃗a2∗(p)) at p, cf. (6.6.2). +Exercice S.16 Bases (⃗ai) and (⃗bi) at p. A vector ⃗x is expressed as ⃗x = � +i xa,i⃗ai = � +i xb,i⃗bi. Prove: +⃗bi = λ⃗ai, ∀i +=⇒ +∂f +∂xb,i += λ ∂f +∂xa,i +or +∂f +∂xa,i += +∂f +∂(λxa,i). +(S.28) +(Change of unit formula.) Duality notations: +∂f +∂xj +b = λ ∂f +∂xj +a . +Answer. df(p).⃗bj(p) = λdf(p).⃗aj(p) (linearity of df(p)) reads (S.28). (Or [df]|⃗b = [df]|⃗a.P with P = λI here.) +Exercice S.17 [df]|⃗b = [df]|⃗a.P, cf. (S.17), i.e. +∂f +∂xj +b = �n +i=1 +∂f +∂xia P i +j is also noted +∂f +∂xj +b += +n +� +i=1 +∂f +∂xia +∂xi +a +∂xj +b +. +(S.29) +Why? +177 + +178 +S.4. +Application 2: Coordinate system basis and Christoffel symbols +Answer. Quick answer. [⃗x]|⃗b = P −1.[⃗x]|⃗a, i.e. [⃗x]|⃗a = P.[⃗x]|⃗b, which means [⃗x]|⃗a([⃗x]|⃗b) = P.[⃗x]|⃗b, i.e. +� +� +� +x1 +a(x1 +b, ..., xn +b ) +... +x1 +a(x1 +b, ..., xn +b ) +� +� +� = +� +� +� +�n +j=1P 1 +j xj +b +... +�n +j=1P n +j xj +b +� +� +� , +thus +∂xi +a +∂xj +b +(x1 +b, ..., xn +b ) = P i +j , ∀i, j. +(S.30) +Thus (S.29) means +∂f +∂xj +b +(p) = +n +� +i=1 +∂f +∂xia +(p)∂xi +a +∂xj +b +(x1 +b, ..., xn +b ) +thus += +n +� +i=1 +∂f +∂xia +(p) P i +j , +(S.31) +as given in (S.17). +Detailed answer. Let O be a point (origin) in U. If p ∈ U, let ⃗x = −→ +Op = �n +i=1xi +a⃗ai = �n +i=1xi +b⃗bi. +This define the function [⃗x]|⃗a : [⃗x]|⃗b → [⃗x]|⃗a([⃗x]|⃗b), and we have [⃗x]|⃗a([⃗x]|⃗b) = P.[⃗x]|⃗b (change of basis formula). +Then let fa, fb : Rn → R be defined by fa([⃗x]|⃗a) := f(p) and fb([⃗x]|⃗b) := f(p). (NB: fa and fb don’t have the +same definition domain: They are different). +Thus fa([⃗x]|⃗a) = fb([⃗x]|⃗b) (= f(p)) when [⃗x([⃗x]|⃗b)]|⃗a = P.[⃗x]|⃗b, so (fa ◦ [⃗x]|⃗a)([⃗x]|⃗b) = fb([⃗x]|⃗b). +Thus +∂(fa◦[⃗x]|⃗a) +∂xi +b +([⃗x]|⃗b) = ∂fb +∂xi +b ([⃗x]|⃗b), thus the meaning of (S.29) is +n +� +j=1 +∂fa +∂xj +a +([⃗x]|⃗a)∂xj +a +∂xi +b +([⃗x]|⃗b) = ∂fb +∂xi +b +([⃗x]|⃗b). +(S.32) +Question: Why did we introduce fa and fb (and not just keep f)? +Answer: Because a vector is not just a collection of components (is not just a matrix), and −→ +Op cannot be +reduced to a matrix of components (which one: [⃗x]|⃗a? [⃗x]|⃗b?). Here f is a function acting on a point p (independent +of a referential), while fa and fb are functions acting on matrices (dependent on the choice of a referential): The +domain of definitions are different, so the functions f, fa and fb are different. +S.4.3 +Christoffel symbols +We use duality notations for readability and usage. +Definition S.18 In a coordinate system basis (⃗ei(p)) in E (previously called (⃗ai∗(p)), the Christoffel +symbol γi +jk(p) ∈ R are the components of the vector d⃗ek(p).⃗ej(p), i.e. d⃗ek(p).⃗ej(p) = �n +k=1γi +jk(p)⃗ei(p), so +d⃗ek.⃗ej = +n +� +i=1 +γi +jk⃗ei , +or +d⃗ej.⃗ei = +n +� +k=1 +γk +ij⃗ek. +(S.33) +(So, with (ei(p)) the dual basis of (⃗ei(p)), γi +jk := ei.d⃗ek.⃗ej, and, for calculations with contractions, +d⃗ek = � +ij γi +jk⃗ei ⊗ ej.) +(The Christoffel symbols vanish in a Cartesian framework.) +(Differential geometry in manifolds: ∇⃗ej⃗ek = �n +i=1γi +jk⃗ei, i.e. the γi +jk = ei.∇⃗ej⃗ek are the component +of the connection ∇, the usual connection in a surface in Rn being the Riemannian connection, in which +case ∇⃗ej⃗ek is the orthogonal projection of d⃗ek.⃗ej on the surface relative to a Euclidean dot product.) +E.g. for the polar coordinate system, see remark 6.12, d⃗e2.⃗e2 = −r⃗e1, thus γ1 +22 = −r and γ2 +22 = 0. +Exercice S.19 Prove: If (⃗ei(p)) is the coordinate system basis of a C2 coordinate system, then: +∀i, j, d⃗ei.⃗ej = d⃗ej.⃗ei +(= +∂2Ψ +∂qi∂qj ), +and +∀i, j, k, γk +ji = γk +ij +(symmetry for lower indices). +(S.34) +Answer. ⃗ei(p) = (⃗ei ◦ Ψ)(⃗q) =(S.23) dΨ(⃗q).⃗ai gives d(⃗ei ◦ Ψ)(⃗q).⃗aj = d(dΨ(⃗q).⃗ai).⃗aj, thus d⃗ei(Ψ(⃗q)).dΨ(⃗q).⃗aj = +∂ ∂Ψ +∂qi +∂qj += +∂ ∂Ψ +∂qj +∂qi +(Schwarz theorem since Ψ is C2) = d⃗ej.⃗ei =noted +∂2Ψ +∂qj∂qi , thus �n +k=1γk +ij⃗ek = �n +k=1γk +ji⃗ek. +Exercice S.20 Consider two coordinate system bases (⃗ai(p)) and (⃗bi(p)) at p, P(p) = [P i +j(p)] the tran- +sition matrix from (⃗ai(p)) to (⃗bi(p)), and Q = P −1. Using the generic notation d⃗ek.⃗ej = �n +i=1γi +jk,e⃗ei, +prove the change of basis formula for the Christoffel symbols: +γi +jk,b = +n +� +λ,µ,ν=1 +Qi +λP µ +j P ν +k γλ +µν,a+ +n +� +λ,µ=1 +Qi +λP µ +j (dP λ +k .⃗aµ) +(= +n +� +λ,µ,ν=1 +Qi +λP µ +j P ν +k γλ +µν,a+ +n +� +λ=1 +Qi +λ(dP λ +k .⃗bj)). (S.35) +(Because of the term � +µν Qi +λP µ +j (dP λ +k .⃗aµ), a derivation is not a tensor.) +178 + +179 +S.5. +Application 3: Differential of a vector field +Answer. +⃗bk(p) = � +ν P ν +k (p)⃗aν(p) gives d⃗bk.⃗bj = � +ν(dP ν +k .⃗bj)⃗aν + � +ν P ν +k (d⃗aν.⃗bj) = � +µν P µ +j (dP ν +k .⃗aµ)⃗aν + +� +µν P ν +k P µ +j (d⃗aν.⃗aµ); And bi = � +λ Qi +λaλ, thus +γi +jk,b = bi.d⃗bk.⃗bj = +� +λµν +Qi +λP µ +j (dP ν +k .⃗aµ)aλ.⃗aν + +� +λµν +Qi +λP µ +j P ν +k aλ.(d⃗aν.⃗aµ) = +� +λµ +Qi +λP µ +j (dP λ +k .⃗aµ) + +� +λµν +Qi +λP µ +j P ν +k γλ +µν,a, +thus (S.35). +S.5 +Application 3: Differential of a vector field +Here F = E = ⃗Rn, Φ =noted ⃗w ∈ Γ(U) is a vector field. Thus d⃗w(p) ∈ L(E; E) and d⃗w.⃗u is a vector field +in E for all ⃗u ∈ Γ(U), given by (d⃗w.⃗u)(p) = d⃗w(p).⃗u(p) = limh→0 +⃗w(p+h⃗u(p))− ⃗w(p) +h +∈ E. +Quantification: (⃗ei(p)) is a basis at p in E. +Call wi(p) ∈ R the components of ⃗w(p), i.e. ⃗w(p) = +�n +i=1wi(p)⃗ei(p). And call wi|j(p) the components of d⃗w(p) (endomorphism in E): +⃗w = +n +� +i=1 +wi⃗ei, +d⃗w.⃗ej = +n +� +i=1 +wi|j⃗ei, +[d⃗w]|⃗e = [wi|j] +(Jacobian matrix). +(S.36) +And tensorial notations for calculations with contractions: (πei(p)) being the dual basis, +d⃗w = +n +� +i,j=1 +wi|j⃗ei ⊗ πej. +(S.37) +Duality notations: ⃗w = �n +i=1wi⃗ei, d⃗w.⃗ej = �n +i,j=1wi +|j⃗ei, [d⃗w]|⃗e = [wi +|j], and d⃗w = �n +i,j=1wi +|j⃗ei ⊗ ej. +In a Cartesian basis: Here (⃗ei) is uniform, so ⃗w(p) = �n +i=1wi(p)⃗ei gives d⃗w(p).⃗ej = �n +i=1(dwi(p).⃗ej)⃗ei, +thus (S.36) gives +wi|j = ∂wi +∂xj +(p) noted += +wi,j, +so +[d⃗w]|⃗e = [∂wi +∂xj +]. +(S.38) +Duality notations: wi +|j = ∂wi +∂xj and [d⃗w]|⃗e = [ ∂wi +∂xj ]. +In a coordinate system basis: With the coordinate system described in § S.4 and the duality notations +for readability (and usage). ⃗w(p) = �n +i=1wi(p)⃗ei(p) gives, for all j, +d⃗w.⃗ej = +n +� +i=1 +(dwi.⃗ej)⃗ei + +n +� +i=1 +wi(d⃗ei.⃗ej) +(= +n +� +i=1 +wi +|j⃗ei). +(S.39) +(Tensorial notations to be used with contractions: d⃗w = � +i ⃗ei ⊗ dwi + � +i wi d⃗ei = � +ij wi +|j⃗ei ⊗ ej.) +And � +i wi(d⃗ei.⃗ej) =(S.33) � +ik wiγk +ji⃗ek = � +ik wkγi +jk⃗ei, thus, for all i, j, +wi +|j = ∂wi +∂qj + +n +� +k=1 +wkγi +jk +where +∂wi +∂qj := dwi.⃗ej. +(S.40) +( ∂wi +∂qj := dwi.⃗ej is the derivation along the j-th coordinate line of the scalar valued function wi). +(In particular, if ⃗w = ⃗eℓ = � +i δi +ℓ⃗ei, we recover d⃗eℓ.⃗ej = � +i 0⃗ei + � +ik δk +ℓ γi +jk⃗ei = � +i γi +jℓ⃗ei, cf. (S.33).) +Exercice S.21 With exercise S.20, and ⃗w = �n +i=1ui⃗ai = � +n vi⃗bi, check with calculations (d⃗w is an +endomorphism defined independently of any basis): +[d⃗w]|⃗b = P −1.[d⃗w]|⃗a.P, +i.e. +vi +|j = +n +� +k,ℓ=1 +Qi +kuk +|ℓP ℓ +j . +(S.41) +Answer. ⃗bj = � +ℓ P ℓ +j⃗aℓ for all i, Q = P |1, and [⃗w]|⃗b = Q.[⃗w]|⃗a reads vi = � +k Qi +kuk for all i. +Cartesian basis: dvi.⃗bj = d(� +k Qi +kuk).(� +ℓ P ℓ +j⃗aℓ) = � +kℓ Qi +k(duk.⃗aℓ)P ℓ +j , qed (here the Qi +k are uniform i.e. +independent of p). +179 + +180 +S.6. +Application 4: Differential of a differential form +Coordinate system basis: vi = � +λ Qi +λuλ gives dvi.⃗bj = � +λ(dQi +λ.⃗bj)uλ + � +λ Qi +λ(duλ.⃗bj); Thus +vi +|j +(S.33) += +dvi.⃗bj + +� +k +vkγi +jk,b += +� +λµ +uλP µ +j (dQi +λ.⃗aµ) + +� +λµ +Qi +λP µ +j (duλ.⃗aµ) + +� +kλµν +(Qk +λuλ)Qi +νP µ +j (dP ν +k .⃗aµ) + +� +kωλµν +(Qk +ωuω)Qi +λP µ +j P ν +k γλ +µν,a +And Qk +ωP λ +k = δλ +ω gives (dQk +ω.⃗aµ)P λ +k + Qk +ω(dP λ +k .⃗aµ) = 0, thus the third term reads +� +kλµν +uλQi +νP µ +j Qk +λ(dP ν +k .⃗aµ) = − +� +kλµν +uλQi +νP µ +j P ν +k (dQk +λ.⃗aµ) = − +� +λµ +uλP µ +j (dQi +λ.⃗aµ), +which cancels the first term: Thus vi +|j = � +λµ Qi +λP µ +j (duλ.⃗aµ) + � +λµν uνQi +λP µ +j γλ +µν = � +λµ Qi +λui +|jP µ +j , i.e. (S.41). +S.6 +Application 4: Differential of a differential form +Here F = R, Φ =noted ℓ ∈ Ω1(U) (differential form) supposed C1, p ∈ U, so ℓ(p) ∈ E∗. Its differential at p +in a direction ⃗u is dℓ(p).⃗u = limh→0 +ℓ(p+h⃗u)−ℓ(p) +h +∈ E∗. And (dℓ(p).⃗u).⃗v = limh→0 +ℓ(p+h⃗u).⃗v−ℓ(p).⃗v +h +∈ R +for all ⃗u,⃗v ∈ E. +Quantification: (πei(p) its the dual basis. +Call ℓi(p) ∈ R the components of ℓ(p), i.e. ℓ(p) = �n +i=1ℓi(p)πei(p). And call ℓi|j(p) the components +of dℓ(p) ∈ L(E; E∗): +ℓ = +n +� +i=1 +ℓiπei, +dℓ.⃗ej = +n +� +i=1 +ℓi|jπei, +[dℓ]|⃗e = [ℓi|j]. +(S.42) +Tensorial notations, to be used with contractions: dℓ = �n +i,j=1ℓi|jπei ⊗ πej. +Duality notations: ℓ = � +i ℓiei, dℓ.⃗ej = �n +i=1ℓi|jei, [dℓ]|⃗e = [ℓi|j], and dℓ = �n +i,j=1ℓi|jei ⊗ ej. +In a Cartesian basis: Here (⃗ei) is uniform, so +ℓi|j = ∂ℓi +∂xj +(p) noted += +ℓi,j, +so +[dℓ]|⃗e = [ ∂ℓi +∂xj +]. +(S.43) +Duality notations: ℓi|j = dℓi.⃗ej = ∂ℓi +∂xj and [dℓ]|⃗e = [ ∂ℓi +∂xj ]. +In a coordinate system basis: With duality notations and Christoffel symbols: +dei.⃗ej = − +n +� +k=1 +γi +jkek . +(S.44) +Indeed, ei.⃗ek = δi +k gives (dei.⃗ej).⃗ek + ei.(d⃗ek.⃗ej) = 0, thus (dei.⃗ej).⃗ek = −ei. � +ℓ γℓ +jk⃗eℓ = −γi +jk. Thus +ℓi|j = ∂ℓi +∂qj − +n +� +k=1 +ℓkγk +ji +where +∂ℓi +∂qj (p) := dℓi(p).⃗ei(p). +(S.45) +Indeed, ℓ = � +i ℓiei gives dℓ.⃗ej = � +i(dℓi.⃗ej)ei + � +i ℓi(dei.⃗ej) = � +i(dℓi.⃗ej)ei − � +ik ℓiγi +jkek. +S.7 +Application 5: Differential of a 1 1 tensor +Consider a C1 �1 +1 +� +tensor τ : +� +U → L(E∗, E; R) +p → τ(p) +� +. Its differential dτ : +� +U → L(E; L(E∗, E; R)) +p → dτ(p) +� +is +defined by dτ(p).⃗u = limh→0 +τ(p+h⃗u)−τ(p) +h +∈ L(E∗, E; R), so (dτ(p).⃗u)(ℓ,⃗v) = limh→0 +τ(p+h⃗u)(ℓ,⃗v)−τ(p)(ℓ,⃗v) +h +(∈ R), for all ⃗u,⃗v ∈ E and ℓ ∈ E∗. +Quantification (duality notations): Basis (⃗ei(p)) in E at p, dual basis (ei(p)), call τ i +j(p) the components +of τ(p), call τ i +j|k(p) the components of dτ(p): +τ = +� +ij +τij⃗ei ⊗ ej, +dτ.⃗ek = +n +� +i,j=1 +τ i +j|k⃗ei ⊗ ej . +(S.46) +Tensorial notations, to be used with contractions: dτ = �n +i,j,k=1τ i +j|k⃗ei ⊗ ej ⊗ ek. +(Classical notations: τ = � +ij τij⃗ei ⊗ πej, dτ.⃗ek = � +ij τij|k⃗ei ⊗ πej, and dτ = � +ijk τij|k⃗ei ⊗ πej ⊗ πek.) +180 + +181 +S.8. +Divergence of a vector field: Invariant +Cartesian basis: dτ(p).⃗ek = � +ij(dτ i +j(p).⃗ek)⃗ei ⊗ ej, so +τ i +j|k = ∂τ i +j +∂xk +noted += +τ i +j,k +(:= dτ i +j.⃗ek). +(S.47) +Coordinate system basis: τ(p) = �n +i,j=1τ i +j(p)⃗ei(p) ⊗ ej(p) gives, for all k, +dτ.⃗ek = � +ij(dτ i +j.⃗ek)⃗ei ⊗ ej + � +ij τ i +j(d⃗ei.⃗ek) ⊗ ej + � +ij τ i +j⃗ei ⊗ (dej.⃗ek) += � +ij(dτ i +j.⃗ek)⃗ei ⊗ ej + � +ijℓ τ i +jγℓ +ki⃗eℓ ⊗ ej − � +ijℓ τ i +jγj +kℓ⃗ei ⊗ eℓ += � +ij(dτ i +j.⃗ek)⃗ei ⊗ ej + � +ijℓ τ ℓ +j γi +kℓ⃗ei ⊗ ej − � +ijℓ τ i +ℓγℓ +kj⃗ei ⊗ ej +(S.48) +thus +τ i +j|k = ∂τ i +j +∂qk + +n +� +ℓ=1 +τ ℓ +j γi +kℓ − +n +� +ℓ=1 +τ i +ℓγℓ +kj +where +∂τ i +j +∂qk := dτ i +j.⃗ek. +(S.49) +(We have the + sign from vector fields, cf. (S.40), and the − sign from differential forms, cf. (S.45).) +Exercice S.22 If ⃗u ∈ E, ℓ ∈ E∗ then for the elementary +�1 +1 +� +tensor τ = ⃗u ⊗ ℓ prove: +d(⃗u ⊗ ℓ).⃗ek = (d⃗u.⃗ek) ⊗ ℓ + ⃗u ⊗ (dℓ.⃗ek), +and +(⃗u ⊗ ℓ)i +j|k = ui +|kℓj + uiℓj|k, +(S.50) +when ⃗u = � +i ui⃗ei, ℓ = � +j ℓjej, d⃗u.⃗ek = � +i ui +|k⃗ei, dℓ.⃗ek = � +j ℓj|kej. +Answer. τ = ⃗u ⊗ ℓ = � +ij τ i +j⃗ei ⊗ ej. where τ i +j = uiℓj, and dτ.⃗ek = �n +i,j=1τ i +j|k⃗ei ⊗ ej where τ i +j|k = (uiℓj)|k = +ui +|kℓj + uiℓj|k = (⃗u ⊗ ℓ)i +j|k. Thus (similar to the derivation of a product): +d(⃗u ⊗ ℓ)(p).⃗ek(p) = lim +h→0 +(⃗u ⊗ ℓ)(p+h⃗ek(p)) − (⃗u ⊗ ℓ)(p) +h += lim +h→0 +⃗u(p+h⃗ek(p)) ⊗ ℓ(p+h⃗ek(p)) − ⃗u(p) ⊗ ℓ(p) +h += lim +h→0 +⃗u(p+h⃗ek(p)) ⊗ ℓ(p+h⃗ek(p)) − ⃗u(p+h⃗ek(p)) ⊗ ℓ(p) +h ++ lim +h→0 +⃗u(p+h⃗ek(p)) ⊗ ℓ(p) − ⃗u(p) ⊗ ℓ(p) +h += lim +h→0(⃗u(p+h⃗ek(p)) ⊗ (ℓ(p+h⃗ek(p)) − ℓ(p) +h +) + lim +h→0(⃗u(p+h⃗ek(p)) − ⃗u(p) +h +) ⊗ ℓ(p) += ⃗u(p) ⊗ (dℓ(p).⃗ek(p)) + (d⃗u(p).⃗ek(p)) ⊗ ℓ(p), +thus (S.50)1. Which gives d(⃗u ⊗ ℓ).⃗ek = (� +i ui⃗ei) ⊗ (� +j ℓj|kej) + (� +i ui +|k⃗ei) ⊗ (� +j ℓjej), thus (S.50)2. +S.8 +Divergence of a vector field: Invariant +Γ(U) is the set of C1 vector fields in U, and Tr : L(E; E) → R is the trace operator. +Definition S.23 The divergence operator is +div := Tr ◦ d : +� +Γ(U) → C0(U; R) +⃗w → div⃗w := Tr(d⃗w), +i.e. +div⃗w(p) := Tr(d⃗w(p)). +(S.51) +(So div⃗w(p) = trace of the endomorphism d⃗w(p)). +Tr and d are linear, hence div = Tr ◦ d is R-linear (composed of two R-linear maps). +Proposition S.24 The divergence of a vector field is objective (is an invariant): Same value for all +observers (objective quantity) intrinsic to ⃗w. +Proof. The differential and the trace are objective. (Or computation: ⃗w = � +i ui⃗ai = � +i vi⃗bi gives +vi +|j = � +kℓ Qi +kuk +|ℓP ℓ +j , see (S.41), thus � +i vi +|i = � +ikℓ P ℓ +i Qi +kuk +|ℓ = � +kℓ δℓ +kuk +|ℓ = � +k uk +|k.) +Quantification: ⃗w ∈ Γ(U), (⃗ei) is a basis, ⃗w = �n +i=1wi⃗ei with classical notations, and wi|j(p) are the +components of the vector d⃗w(p).⃗ej(p) in the basis (⃗ei(p)). Thus +div⃗w = +n +� +i=1 +wi|i . +(S.52) +Duality notations: ⃗w = �n +i=1wi⃗ei, d⃗w.⃗ej = �n +i=1wi +|j⃗ei, [d⃗w]|⃗e = [wi +|j], div⃗w = �n +i=1wi +|i. +181 + +182 +S.9. +Objective divergence for 1 1 tensors +Cartesian basis (⃗ei) (classical notations): dwi.⃗ej =noted ∂wi +∂xj and +wi|j = ∂wi +∂xi +, +thus +div⃗w = +n +� +i=1 +∂wi +∂xi +. +(S.53) +Coordinate system basis (⃗ei) (duality notations): With the Christoffel symbols, cf. (S.33), (S.40) gives +wi +|i = ∂wi +∂qi + +n +� +i=1 +wkγi +ik, +thus +div⃗w = +n +� +i=1 +∂wi +∂qi + +n +� +i,k=1 +wkγi +ik. +(S.54) +Exercice S.25 Prove: +div(f ⃗w) = df.⃗w + f div⃗w. +(S.55) +Answer. d(f ⃗w) = ⃗w ⊗ df + f d⃗w, thus Tr(d(f ⃗w)) = Tr(⃗w ⊗ df) + Tr(f d⃗w) = df.⃗w + f Tr(d⃗w). Use a coordinate +system if you prefer. +Remark S.26 If α is a differential form, if (⃗ei) is a basis and (ei) its dual basis, and if α = �n +i=1αiei, +then dα = �n +i=1αi|jei ⊗ ej, with αi|j := ⃗ei.dα.⃗ej. Here it is impossible to define an objective trace of dα +like �n +i=1αi|i: The result depends on the choice of the basis (the Einstein convention is not satisfied, and +e.g. with a Euclidean basis the result depends on the choice of unit of length: Foot? Meter?). Thus the +objective (or intrinsic) divergence of a differential form is a nonsense. +S.9 +Objective divergence for 1 1 tensors +To create an objective divergence for a second order +�1 +1 +� +tensor τ ∈ T 1 +1 (U), in (S.46) we have to contract an +admissible index with the “differential index k”, So, no choice: Contract i and k to get � +divτ := �n +i,j=1τ i +j|iej. +Let us start with: +Definition S.27 Let ⃗u ∈ Γ(U) and ℓ ∈ Ω1(U) be C1. The objective divergence of the elementary +�1 +1 +� +tensor ⃗u ⊗ ℓ ∈ T 1 +1 (U) is the differential form � +div(⃗u ⊗ ℓ) ∈ Ω1(U) defined by +� +div(⃗u ⊗ ℓ) = (div⃗u)ℓ + dℓ.⃗u, +(S.56) +i.e. defined by � +div(⃗u ⊗ ℓ).⃗w = (div⃗u)(ℓ.⃗w) + (dℓ.⃗w)⃗u for all ⃗w ∈ E. And the objective divergence operator +� +div : +� +T 1 +1 (U) → Ω1(U) +τ → � +divτ +� +is the linear map defined on elementary tensors with (S.56). +Quantification: (⃗ei) is a basis, (ei) its dual basis, ⃗u = � +i ui⃗ei, ℓ = � +j ℓjej. Thus ⃗u⊗ℓ = � +ij uiℓj⃗ei⊗ej, +and (S.50)-(S.56) give +� +div(⃗u ⊗ ℓ) = +n +� +i,j=1 +(ui +|iℓj + ℓj|iui)ej. +(S.57) +So for an elementary tensor τ = ⃗u ⊗ ℓ, τ = � +ij τ i +j⃗ei ⊗ ej, and dτ.⃗ek = � +ijk τ i +j|k⃗ei ⊗ ej and � +div(τ) = +� +ij τ i +j|iei, with τ i +j|k = ui +|kℓj + uiℓj|k, here with τ i +j = uiℓj, and τ i +j|k = ui +|kℓj + uiℓj|k, so τ i +j|i = ui +|iℓj + uiℓj|i. +Thus, by linearity of � +div, for all tensors τ ∈ T 1 +1 (U), we have with (S.46): +� +divτ = +n +� +i,j=1 +τ i +j|iej , +i.e. +[ � +divτ]|⃗e = +� � +i τ i +1|i +... � +i τ i +n|i +� +(S.58) +(row matrix since � +divτ is a differential form). I.e., we have contracted i and k in (S.46). +(Classical notations: � +divτ := �n +i,j=1τij|iπej, i.e. [ � +divτ]|⃗e = ( � +i τi1|i +... � +i τin|i ).) +So +Cartesian bases: � +divτ = +n +� +i,j=1 +∂τ i +j +∂xi ej, +Coord. sys. bases: � +divτ = +n +� +i,j=1 +�∂τ i +j +∂qi + +n +� +k=1 +τ k +j γi +ik − +n +� +k=1 +τ k +i γi +kj +� +ej. +(S.59) +Indeed: With τ = � +j(� +i τ i +j⃗ei) ⊗ ej = � +j ⃗wj ⊗ ej where ⃗wj = � +i τ i +j⃗ei, the linearity of � +div gives +� +divτ = � +j � +div(⃗wj ⊗ ej); Thus, with (S.56): 1- Cartesian basis: div⃗wj = � +i +∂τ i +j +∂xi and dej = 0 give � +divτ = +182 + +183 +S.9. +Objective divergence for 1 1 tensors +� +j +� +i +∂τ i +j +∂xi ej = � +ij τ i +j,iej, thus (S.59)1; And 2- Coordinate system basis: div⃗wj=(S.54)� +i +∂τ i +j +∂qi +� +ik τ k +j γi +ik +and dej.⃗wj = � +k τ k +j dej.⃗ek = � +k τ k +j (− � +i γj +kiei), thus � +j dej.⃗wj = − � +ijk τ k +j γj +kiei = − � +ijk τ k +i γi +kjej, +thus (S.59)2. +Exercice S.28 Prove: If f ∈ C1(U; R) and τ = �n +i,j=1τ i +j⃗ei ⊗ ej ∈ T 1 +1 (U) ∩ C1 then +� +div(fτ) = df.τ + f � +divτ. +(S.60) +Answer. fτ = � +ij fτ i +j⃗ei ⊗ ej gives d(fτ) = � +ijk(fτ i +j)|k⃗ei ⊗ ej ⊗ ek = � +ijk(f|kτ i +j + fτ i +j|k)⃗ei ⊗ ej ⊗ ek, thus +� +div(fτ) = � +ij(f|iτ i +j + fτ i +j|i)ej; And df.τ + f � +divτ = � +ij f|iτ i +jej + f � +ij τ i +j|iej. +Exercice S.29 Prove: If τ ∈ T 1 +1 (U) and ⃗w ∈ Γ(U) then +div(τ.⃗w) = � +div(τ).⃗w + τ 0.. d⃗w . +(S.61) +Answer. τ = � +ij τ i +j⃗ei ⊗ ej and ⃗w = � +i wi⃗ei give τ.⃗w = � +ij τ i +jwj⃗ei, thus div(τ.⃗w) = � +ij τ i +j|iwj + τ i +jwj +|i. +Exercice S.30 If τ ∈ T 1 +1 (U) check with component calculations (since � +div(τ) is objective): +[ � +div(τ)]|b = [ � +div(τ)]|a.P +(covariance formula), +(S.62) +where P is the transition matrix from a basis (⃗ai) to a basis (⃗bi). +Answer. Let τ = � +ij σi +j⃗ai ⊗ aj = � +ij τ i +j⃗bi ⊗ bj, so τ i +j = � +λµ Qi +λσλ +µP µ +j . +1- Cartesian bases: � +i τ i +j|i = � +i dτ i +j.⃗bi = � +i d(� +λµ Qi +λσλ +µP µ +j ).(� +ν P ν +i .⃗aν) = � +iλµν Qi +λP µ +j P ν +i (dσλ +µ.⃗aν) = +� +λµν δν +λP µ +j (dσλ +µ.⃗aν) = � +λµ P µ +j (dσλ +µ.⃗aλ) = � +µ(� +λ σλ +µ|λ)P µ +j as desired. +2- Coordinate system bases: � +i τ i +j|i =(S.59) � +i dτ i +j.⃗ei + � +iℓ τ ℓ +j γi +iℓ,b − � +iℓ τ i +ℓγℓ +ij,b (with j fixed); With +� +i +(dτ i +j.⃗bi) = +� +iλµ +Qi +λ (dσλ +µ.⃗bi) P µ +j + +� +iλµ +(dQi +λ.⃗bi) σλ +µ P µ +j + +� +iλµ +Qi +λ σλ +µ (dP µ +j .⃗bi) += +� +iλµν +Qi +λP µ +j P ν +i (dσλ +µ.⃗aν) + +� +iλµν +σλ +µ P µ +j P ν +i (dQi +λ.⃗aν) + +� +iλµν +σλ +µ Qi +λP ν +i (dP µ +j .⃗aν) += +� +λµ +P µ +j (dσλ +µ.⃗aλ) − +� +iλµν +σλ +µ P µ +j Qi +λ(dP ν +i .⃗aν) + +� +λµ +σλ +µ (dP µ +j .⃗aλ) +since P ν +i Qi +λ = δν +λ gives P ν +i (dQi +λ.⃗aν) − Qi +λ(dP ν +i .⃗aν). And, with (S.35), +� +iℓ +τ ℓ +j γi +iℓ,b = +� +iℓ +( +� +λµ +Qℓ +λσλ +µP µ +j )( +� +αβω +Qi +αP β +i P ω +ℓ γα +βω,a + +� +αβ +Qi +αP β +i (dP α +ℓ .⃗aβ)) += +� +λµα +σλ +µP µ +j γα +αλ,a + +� +ℓλµα +σλ +µQℓ +λP µ +j (dP α +ℓ .⃗aα), +(S.63) +and +− +� +iℓ +τ i +ℓγℓ +ij,b = − +� +iℓ +( +� +λµ +Qi +λσλ +µP µ +ℓ )( +� +αβω +P α +i P β +j Qℓ +ωγω +αβ,a + +� +αω +P α +i Qℓ +ω(dP ω +j .⃗aα)) += − +� +λµβ +σλ +µP β +j γµ +λβ,a − +� +λµ +σλ +µ(dP µ +j .⃗aλ). +(S.64) +Thus � +i τ i +j|i = � +λµ P µ +j (dσλ +µ.⃗aλ) + � +λµα σλ +µP µ +j γα +αλ,a − � +λµβ σλ +µP β +j γµ +λβ,a = � +λµ P µ +j σλ +µ|λ as desired. +S.9.1 +Divergence of a 2 0 tensor +Let τ ∈ T 2 +0 (U) and τ = �n +i,j=1τ ij⃗ei⊗⃗ej, thus dτ = �n +i,j,k=1τ ij +|k⃗ei⊗⃗ej⊗ek; Then two objective divergences +may be defined: by contracting k with i, or k with j. (The Einstein convention is then satisfied.) +S.9.2 +Divergence of a 0 2 tensor +Let τ = �n +i,j=1τijei ⊗ ej ∈ T 0 +2 (U). Thus dτ = �n +i,j,k=1τij|kei ⊗ ej ⊗ ek, and there are no indices to +contract to satisfy Einstein convention: There is no objective divergence of 0 2 tensors. +183 + +184 +S.10. +Euclidean framework and “classic divergence” of a tensor (subjective) +S.10 +Euclidean framework and “classic divergence” of a tensor (subjective) +Let σ be a C1 tensor of order 2 of any kind. An observer chooses a (Cartesian) Euclidean basis (⃗ei) +and call (·, ·)g the associated Euclidean dot product. And he calls σij the components of σ, e.g. writes +σ.⃗ej = � +i σij⃗ei. +Definition S.31 (Usual divergence in classical mechanics.) The divergence divσ of σ relative to the +basis (⃗ei), is the column matrix (it is not a vector) +diveσ = +� +� +� +�n +j=1 +∂σ1j +∂xj +... +�n +j=1 +∂σnj +∂xj +� +� +� +noted += +divσ +(a matrix). +(S.65) +(Take the divergences of the “row vectors” of [σ]|e = [σij] to make the “column vector” [diveσ].) +Proposition S.32 The “so called vector” divσ, in (S.67), is not a vector: It does not satisfy the change +of basis formula: If (⃗ai) and (⃗bi) are bases, if P is the transition matrix from (⃗ai) to (⃗bi), if [σ]|⃗a = [Aij] +and [σ]|⃗b = [Bij], with the divergence of σ relative to (⃗ai) and (⃗bi) called divaσ and divbσ, then neither a +contravariant nor a covariant change of basis formula applies in general: +neither +[divbσ]|⃗b ̸= P −1.[divaσ]|⃗a +nor +[divbσ]T +|⃗b = [divaσ]T +|⃗a.P +(S.66) +(compare with (S.62)). So divσ as given in (S.67) is neither a contravariant vector nor a covariant vector +(it is just a matrix which depends on an observer). +Proof. Consider the simple case ⃗bi = λ⃗ai, for all i, λ > 1: Transition matrix P = λI, and P −1 = 1 +λI. +For a +�1 +1 +� +tensor: σ = � +ij(σb)i +j⃗bi ⊗ bj = � +ij(σa)i +j⃗ai ⊗ aj, [σ]|⃗b = P −1.[σ]|⃗a.P = 1 +λ.[σ]|⃗a.λ = [σ]|⃗a, i.e. +(σa)i +j = (σb)i +j for all i, j. Thus (S.67) gives divbσ = � +ij(d(σb)i +j.⃗bj)⃗bi = � +ij(d(σa)i +j.(λ⃗aj))(λ⃗ai) = λ2divaσ. +Thus [divbσ]|⃗b ̸= P −1.[divbσ]|⃗a and [divbσ]T +|⃗b ̸= [divaσ]T +|⃗a.P. +For a +�0 +2 +� +tensor: σ = � +ij σb,ijbi ⊗ bj = � +ij σa,ijai ⊗ aj, and [σ]|⃗b = P T .[σ]|⃗a.P = λ2[σ]|⃗a, i.e. σb,ij = +λ2σa,ij for all i, j. Thus (S.67) gives divbσ = � +ij(dσb,ij.⃗bj)⃗bi = λ2 � +ij(dσa,ij.(λ⃗aj))(λ⃗ai) = λ4divaσ. +Thus [divbσ]|⃗b ̸= P −1.[divbσ]|⃗a and [divbσ]T +|⃗b ̸= [divaσ]T +|⃗a.P. +For a +�2 +0 +� +tensor: σ = � +ij σij +b ⃗bi ⊗ ⃗bj = � +ij σij +a ⃗ai ⊗ ⃗aj, and [σ]|⃗b = P −T .[σ]|⃗a.P −1 = +1 +λ2 [σ]|⃗a, i.e. +σij +b = +1 +λ2 σij +a for all i, j. Thus (S.67) gives divbσ = � +ij(dσij +b .⃗bj)⃗bi = +1 +λ2 +� +ij(dσij +a .(λ⃗aj))(λ⃗ai) = divaσ. +Thus [divbσ]|⃗b ̸= P −1.[divbσ]|⃗a and [divbσ]T +|⃗b ̸= [divaσ]T +|⃗a.P. +Remark: (S.65) can be written +divσ = +� +ij +∂σij +∂xj +⃗Ei +(S.67) +where ( ⃗Ei) is the canonical basis in Mn1 the space of n ∗ 1 column vectors. +T +Natural canonical isomorphisms +T.1 +The adjoint of a linear map +Setting of § A.12: E and F are vector spaces, E∗ = L(E; R) and F ∗ = L(F; R) are their dual spaces, +and the adjoint of a linear map P ∈ L(E; F) is the linear map P∗ ∈ L(F ∗; E∗) canonically defined by +∀ℓ ∈ F ∗, +P∗(ℓ) := ℓ ◦ P, +written +P∗.ℓ = ℓ.P +(T.1) +(dot notations P∗(ℓ) =noted P∗.ℓ and ℓ◦P =noted ℓ.P since ℓ and P∗ are linear), i.e., for all (ℓ, ⃗u) ∈ F ∗×E, +P∗(ℓ)(⃗u) = ℓ(P(⃗u)), +written +(P∗.ℓ).⃗u = ℓ.P.⃗u. +(T.2) +Interpretation: If P is the push-forward of vector fields, then P∗ is the pull-back of differential forms, +see remark 7.5. In particular, it will be interpreted with P ∈ Li(E; F) (linear and invertible = a change +of observer). +184 + +185 +T.2. +An isomorphism E ≃ E∗ is never natural (never objective) +T.2 +An isomorphism E ≃ E∗ is never natural (never objective) +Two observers A and B consider a linear map L ∈ L(E; E∗); Let P ∈ L(E; E) be the change of observer +endomorphism. Willing to work together, A and B (“naturally”) consider the diagram +E +L +−→ E∗ +← considered by observer A +P ↓ +↑ P∗ +E −→ +L +E∗ +← considered by observer B +(T.3) +Definition T.1 (Spivak [17].) A linear map L ∈ L(E; E∗) is natural iff the diagram (T.3) commutes for +all P ∈ L(E; E): +L ∈ L(E; E∗) is natural +⇐⇒ +∀P ∈ L(E; E), P∗ ◦ L ◦ P = L. +(T.4) +(In that case, if A computes L.⃗u with the top line of the diagram, if B computes with the bottom line of +the diagram, then they can easily check their results since here L.⃗u = (P∗ ◦ L ◦ P).⃗u.) +Question: Does there exist an endomorphism L such that the diagram (T.3) commutes for all change +of observers? That is, do we have +∃?L ∈ L(E; E), ∀P ∈ Li(E; E), +P∗ ◦ L ◦ P = L ? +(T.5) +Answer: Always no (if L ̸= 0): +Theorem T.2 A (non-zero) linear map L ∈ L(E; E∗) is not natural: If L ∈ L(E; E∗) − {0}, then +∃P ∈ Li(E; E) +s.t. +L ̸= P∗ ◦ L ◦ P. +(T.6) +Proof. (Spivak [17].) It suffices to prove this proposition for E = ⃗R. Let L ∈ L(⃗R; (⃗R)∗), L ̸= 0. +Let (⃗a1) be a basis in ⃗R (chosen by A). Let (⃗b1) be a basis in ⃗R (chosen by B). +Consider P ∈ Li(⃗R; ⃗R) defined by P(⃗a1) = ⃗b1 (change of observer), and let λ ∈ R s.t. ⃗b1 = λ⃗a1. Then +(T.1) gives P∗(ℓ)(⃗a1) := ℓ(P(⃗a1)) = ℓ(⃗b1) = ℓ(λ⃗a1) = λℓ(⃗a1), thus P∗(ℓ) = λℓ for all ℓ ∈ (⃗R)∗. +Thus P∗(L(P(⃗a1))) = P∗(L(λ⃗a1)) = λP∗(L(⃗a1)) = λ2L(⃗a1) ̸= L(⃗a1) when λ2 ̸= 1. E.g., P = 2I gives +L ̸= P∗ ◦ L ◦ P (= 4L), thus (T.6): A (non-zero) linear map E → E∗ cannot be natural. +Example T.3 Consider E s.t. dim E = 1, and consider the linear map L ∈ L(E; E∗) which sends a basis +(⃗a1) onto its dual basis (πa1), so L is defined by L.⃗a1 := πa1. +Question: If (⃗b1) is another basis, λ ̸= ±1 and ⃗b1 = λ⃗a1 (change of unit of measurement), does +L.⃗b1 = πb1, i.e. does L also sends (⃗b1) onto its dual basis? +Answer: No. Indeed, ⃗b1 = λ⃗a1 gives πb1 = +1 +λπa1, thus L.⃗b1 = λL.⃗a1 = λπa1 = λ2πb1 ̸= πb1 since +λ2 ̸= 1. In words: L is not natural, cf. (T.6). +A different presentation: Let LA and LB be defined by LA.⃗aj = πaj and LB.⃗bj = πbj for all j. And +suppose that ⃗bj = λ⃗aj for all j. Then, LA.⃗bj = λLA.⃗aj = λπaj = λ2πbj = λ2LB.⃗bj ̸= LB.⃗bj when λ2 ̸= 1, +that is, LA ̸= LB when λ2 ̸= 1: An operator that sends a basis onto its dual basis is not natural. +Example T.4 Let (·, ·)g be an inner dot product in E = ⃗Rn. Let ⃗Rg ∈ L(E∗; E) be the Riesz rep- +resentation map, that is, defined by ⃗Rg(ℓ) = ⃗ℓg where ⃗ℓg is defined by (⃗ℓg,⃗v)g = ℓ.⃗v for all ⃗v ∈ ⃗Rn, +cf (F.3). +Question: Is ⃗Rg natural? +Answer: No: Consider the diagram +� +E∗ +⃗Rg +−→ E +P∗ ↓ +↑ P +E∗ −→ +⃗Rg +E +� +with P = λI, λ ̸= ±1. Then P∗ = λI, and +P.⃗Rg.P∗.ℓ = λ2 ⃗Rg.ℓ ̸= ⃗Rg.ℓ gives P.⃗Rg.P∗ ̸= ⃗Rg: So ⃗Rg is not natural, cf. (T.6). (You may prefer to +consider the diagram (T.3) with L = ⃗R−1 +g .) +A different presentation: Consider two distinct Euclidean dot products (·, ·)g and (·, ·)h (e.g., built with +a foot and built with a metre). So (·, ·)h = λ2(·, ·)g with λ2 ̸= 1. Let ⃗Rg, ⃗Rh ∈ L(Rn∗; Rn) be the Riesz +operators relative to (·, ·)g and (·, ·)h, that is ⃗Rg.ℓ = ⃗ℓg and ⃗Rh.ℓ = ⃗ℓh are given by ℓ.⃗v = (⃗ℓg,⃗v)g = (⃗ℓh,⃗v)h +for all ⃗v ∈ ⃗Rn. We have ⃗ℓh = λ2⃗ℓg, cf. (F.11), thus ⃗Rh = λ2 ⃗Rg ̸= ⃗Rg since λ2 ̸= 1: A Riesz representation +operator is not natural (it is observer dependent). +185 + +186 +T.3. +Natural canonical isomorphism E ≃ E∗∗ +T.3 +Natural canonical isomorphism E ≃ E∗∗ +Two observers A and B consider the same linear map L ∈ L(E; E∗∗) (where E∗∗ = (E∗)∗ = L(E∗; R)). +Willing to work together, they (“naturally”) consider the diagram +E +L +−→ E∗∗ +← considered by observer A +P ↓ +↓ P∗∗ +E −→ +L +E∗∗ +← considered by observer B +(T.7) +where P ∈ L(E; E) is a linear diffeomorphism, P∗ ∈ L(E∗; E∗) its adjoint, given by P∗(ℓ) = ℓ ◦ P +cf. (T.1), and P∗∗ ∈ Li(E∗∗; E∗∗) the adjoint of P∗, thus given by P∗∗(u) = u ◦ P∗ for all u ∈ E∗∗ +cf. (T.1), i.e. P∗∗ is given by, for all (ℓ, u) ∈ E∗ × E∗∗, +(P∗∗(u))(ℓ) = u(ℓ ◦ P), +i.e. +(P∗∗.u).ℓ = u.(ℓ.P). +(T.8) +Question: Does there exist a linear map L ∈ L(E; E∗∗) that is natural? +Answer: Yes (particular case of the next proposition): +Proposition T.5 The canonical isomorphism +JE : +� +E → E∗∗ +⃗u → u = JE(⃗u) +defined by +JE(⃗u)(ℓ) := ℓ.⃗u, +∀ℓ ∈ E∗, +(T.9) +is natural, that is, F being another finite dimensional vector space, the diagram +E JE +−→ E∗∗ +P ↓ +↓ P∗∗ +F −→ +JF +F ∗∗ +written +E +J +−→ E∗∗ +P ↓ +↓ P∗∗ +F −→ +J +F ∗∗ +(T.10) +commutes for all P ∈ L(E; F), i.e. +∀P ∈ L(E; F), +P∗∗ ◦ JE = JF ◦ P, +and we write +E ≃ E∗∗. +(T.11) +Thus we can use the unambiguous notation (observer independent) +J (⃗u) noted += +⃗u, +and +J (⃗u).ℓ noted += +⃗u.ℓ +(= ℓ.⃗u). +(T.12) +(And u = J (⃗u) is the derivation operator in the direction ⃗u.) +Proof. (Spivak [17].) +It is trivial that JE is linear and bijective (E is finite dimensional): It is an +isomorphism. Then (P∗∗ ◦ JE(⃗u))(ℓ) +(T.8) += JE(⃗u)(ℓ.P) +(T.9) += (ℓ ◦ P)(⃗u) = ℓ(P(⃗u)) +(T.9) += JF (P(⃗u))(ℓ), for all +ℓ ∈ F ∗ and all ⃗u ∈ E, thus P∗∗ ◦ JE(⃗u) = JF (P(⃗u)), for all ⃗u ∈ E, thus P∗∗ ◦ JE = JF ◦ P. +Proposition T.6 (Characterization of JE.) JE sends any basis (⃗ai) onto its bidual basis. (Expected, +since JE(⃗u) is the directional derivative in the direction ⃗u, whatever ⃗u.) +Proof. Let (⃗ai) be a basis and (πai) be its dual basis (defined by πai.⃗aj = δij for all i, j). Then (T.9) +gives JE(⃗aj).πai = πai.⃗aj = δij for all i, j, thus (JE(⃗aj)) is the dual basis of (πai), i.e., is the bidual basis +of (⃗ai); True for all basis: JE(⃗bj).πbi = πbi.⃗bj = δij for all i, j. +T.4 +Natural canonical isomorphisms L(E; F) ≃ L(F ∗, E; R) ≃ L(E∗; F ∗) +E, F, A, B are finite dimensional vector spaces. Consider the canonical isomorphism +JEF : +� +L(E; F) → L(F ∗, E; R) +L → �L = JEF (L) +where +�L(ℓ, ⃗u) := ℓ.L.⃗u, +∀(ℓ, ⃗u) ∈ F ∗ × E. +(T.13) +186 + +187 +T.5. +Natural canonical isomorphisms L(E; L(E; F)) ≃ L(E, E; F) ≃ L(F ∗, E, E; R) +Let P1 ∈ Li(E; A) and P2 ∈ L(F; B), and consider the diagram +L(E; F) JEF +−→ L(F ∗, E; R) +IP ↓ +↓ � +IP +L(A; B) −→ +JAB +L(B∗, A; R) +(T.14) +where +IP(L) = P2.L.P−1 +1 +and +� +IP(�L)(b,⃗a) = �L(b.P2, P−1 +1 .⃗a) +∀(b,⃗a) ∈ B∗ × A. +(T.15) +(IP and � +IP are the push-forwards for linear maps L ∈ L(E; F) and for bilinear forms �L ∈ L(F ∗, E; R).) +Proposition T.7 The canonical isomorphism JEF is natural, that is, the diagram (T.14) commutes for +all P1 ∈ Li(E, A) and all P2 ∈ L(F, B): +� +IP ◦ JEF = JAB ◦ IP, +and +L(E; F) +natural +≃ +L(F ∗, E; R). +(T.16) +Thus L(E∗; F ∗) +natural +≃ +L(E; F). +Proof. JAB(IP(L))(b,⃗a) +(T.13) += +b.IP(L).⃗a +(T.15) += +b.(P2.L.P−1 +1 ).⃗a = (b.P2).L.(P−1 +1 .⃗a) +(T.13) += +JEF (L)(b.P2, P−1 +1 .⃗a) +(T.15) += +� +IP(JEF (L))(b,⃗a), true for all L ∈ L(E; F), b ∈ B∗, ⃗a ∈ A, thus (T.16). +Thus L(E∗; F ∗) +(T.16) +≃ L((F ∗)∗, E∗; R) +(T.11) +≃ L(F, E∗; R) +(T.16) +≃ L(E∗∗; F) +(T.11) +≃ L(E; F). +Consider the canonical isomorphism (defines the transposed of a bilinear map) +KEF : +� +L(E, F; R) → L(F, E; R) +T → KEF (T) +� +, +KEF (T)(⃗u,⃗v) := T(⃗v, ⃗u), +∀(⃗u,⃗v) ∈ E × F, +(T.17) +and ZAB ∈ L(E, F; R) → L(A, B; R) defined by ZAB(T)(⃗a,⃗b) := T(P−1 +1 .⃗a, P−1 +2 .⃗b) for all (⃗a,⃗b) ∈ A × B. +Proposition T.8 The canonical isomorphism KEF is natural: For all (P1, P2) ∈ Li(E; A)×L(F; B), the +diagram +L(E, F; R) KEF +−→ L(F, E; R) +ZAB ↓ +↓ ZBA +L(A, B; R) −→ +KAB +L(B, A; R) +commutes: L(E, F; R) +natural +≃ +L(F, E; R). +Proof. KEF (ZAB(T))(⃗b,⃗a) = ZAB(T)(⃗a,⃗b) = T(P−1 +2 .⃗b, P−1 +1 .⃗a) and ZBA(KEF (T))(⃗a,⃗b) = KEF (T)(P−1 +1 .⃗a, P−1 +2 .⃗b) = +T(P−1 +2 .⃗b, P−1 +1 .⃗a), thus KAB ◦ ZAB = ZBA ◦ KEF . +T.5 +Natural canonical isomorphisms L(E; L(E; F)) ≃ L(E, E; F) ≃ L(F ∗, E, E; R) +For application to the second order derivative d(d⃗u) ≃ d2⃗u and, with ⃗u ∈ T 1 +0 (U), the notation d⃗u ∈ T 1 +1 (U), +then d2⃗u ∈ T 1 +2 (U), ..., dk⃗u ∈ T 1 +k (U), ... +Consider the canonical isomorphism +J12E : +� +L(E; L(E; F)) → L(E, E; F) +T1 → T2 = J12E(T1) +� +, +J12E(T1)(⃗u1, ⃗u2) := T1(⃗u1).⃗u2 ∈ F, +∀⃗u1, ⃗u2 ∈ E, +(T.18) +and the canonical isomorphism +J23E : +� +L(E, E; F) → L(F ∗, E, E; R) +T2 → J23E(T2) = T3 +� +, +T3(ℓ, ⃗u,⃗v) := ℓ.T2(⃗u1, ⃗u2), +∀⃗u1, ⃗u2 ∈ E, ∀ℓ ∈ F ∗. (T.19) +Proposition T.9 J12 and J23 are natural. Thus J23 ◦ J12 is natural. +187 + +188 +U.1. +Definitions +Proof. 1- We have to prove that the following diagram commutes: +L(E; L(E; F)) J12E +−→ +L(E, E; F) +ZAB ↓ +↓ YAB +L(A; L(A; B)) J12A +−→ +L(A, A; B) +where +ZAB(T1)(⃗a1).⃗a2 := T1(P−1 +1 .⃗a1).(P−1 +1 .⃗a2), +YAB(T2)(⃗a1,⃗a2) = T2(P−1 +1 .⃗a1, P−1 +1 .⃗a2), +(T.20) +(the “push-forwards) for all ⃗a1,⃗a2 ∈ A and LAB ∈ L(A; B). +Let T1 ∈ L(E; L(E; F)). We have +J12A(ZAB(T1))(⃗a1).⃗a2 = ZAB(T1)(⃗a1).⃗a2 = T1(P−1 +1 .⃗a1).(P−1 +1 .⃗a2), and +YAB(J12E(T1))(⃗a1,⃗a2) = J12E(T1)(P−1 +1 .⃗a1, P−1 +1 .⃗a2) = T1(P−1 +1 .⃗a1).(P−1 +1 .⃗a2), +thus J12A ◦ ZAB = YAB ◦ J12E, thus J12 is natural. +2- We have to prove that the following diagram commutes: +L(E, E; F) J23E +−→ +L(F ∗, E, E; R) +ZAB ↓ +↓ YAB +L(A, A; B) J23A +−→ +L(B∗, A, A; R) +where +ℓB.ZAB(T2)(⃗a1,⃗a2) := (ℓB.P2).T2(P−1 +1 .⃗a1, P−1 +1 .⃗a2), +YAB(T3)(ℓB,⃗a1,⃗a2) = T3(ℓB.P2, P−1 +1 .⃗a1, P−1 +1 .⃗a2), +(T.21) +(the “push-forwards) for all ⃗a1,⃗a2 ∈ A and ℓB ∈ B∗. +Let T2 ∈ L(E, E; F). We have +J23A(ℓB, ZAB(T2)(⃗a1,⃗a2)) = ℓB.ZAB(T2)(⃗a1,⃗a2) = (ℓB.P2).T2(P−1 +1 .⃗a1, P−1 +1 .⃗a2), and +YAB(J23A(T2))(ℓB,⃗a1,⃗a2) = J23A(T2)(ℓB.P2, P−1 +1 .⃗a1, P−1 +1 .⃗a2) = ℓB.P2.T2(P−1 +1 .⃗a1, P−1 +1 .⃗a2) +thus J23A ◦ ZAB = YAB ◦ J23E, thus J23 is natural. +U +Distribution in brief: A covariant concept +We refer to the books of Laurent Schwartz for a full description. In continuum mechanics, with Ω an +open set in Rn and for the space of the finite energy functions L2(Ω) and its sub-spaces, a distribution +gives a covariant formulation for the virtual power, as used by Germain. +U.1 +Definitions +Usual notations: Let p ∈ [1, ∞[ (e.g. p = 2 for finite energy functions), and let +Lp(Ω) := {f : Ω → R : +� +Ω +|f(x)|p dΩ < ∞} +and +||f||p = ( +� +Ω +|f(x)|p dΩ) +1 +p , +(U.1) +the space of functions such that |f|p is Lebesgue integrable with ||.||p its usual norm. Then (Lp(Ω), ||.||Lp) +is a Banach space (a complete normed space). And let +L∞(Ω) := {f : Ω → R : sup +x∈Ω +(|f(x)|) < ∞}, +and +||f||∞ = sup +x∈Ω +(|f(x)|), +(U.2) +the space of Lebesgue measurable bounded functions with ||.||∞ its usual norm. Then (L∞(Ω), ||.||L∞) +is a Banach space (a complete normed space). +Definition U.1 If f ∈ F(Ω; R), then its support is the set +supp(f) := {x ∈ Ω : f(x) ̸= 0} = the closure of {x ∈ Ω : f(x) ̸= 0} +(U.3) += the set where it is interesting to study f. +The closure is required: E.g., if Ω =]0, 2π[ and f(x) = sin x for all x ∈ ω, then {x ∈ Ω : f(x) ̸= 0} = +]0, π[∪]π, 2π[; And the point π is a point of interest since sin varies in its vicinity: f ′(π) = cos(π) = −1 ̸= 0; +So it is the closure supp(f) := ]0, π[∪]π, 2π[ = [0, 2π] that is considered. +Definition U.2 (Schwartz notation, D being the letter after C:) Let +D(Ω) := C∞ +c (Ω; R) = {ϕ ∈ C∞(Ω; R) s.t. supp(ϕ) is compact in Ω}. +(U.4) +188 + +189 +U.2. +Derivation of a distribution +E.g., Ω = R, ϕ(x) := e− +1 +1−x2 if x ∈]−1, 1[ and ϕ(x) := 0 elsewhere: ϕ ∈ D(R) with supp(ϕ) = [−1, 1]. +And D(Ω) is a vector space which is dense in (Lp(Ω), ||.||Lp) for any p ∈ [1, ∞[. +Definition U.3 A distribution in Ω is a linear D(Ω)-continuous3 function +T : +� D(Ω) → R +ϕ → T(ϕ) noted += +⟨T, ϕ⟩ +(U.5) +The space of distribution in Ω is named D′(Ω) (the dual of D(Ω)). +The notation ⟨T, ϕ⟩D′(Ω),D(Ω) = ⟨T, ϕ⟩ is the “duality bracket” = the “covariance–contravariance +bracket” between a linear function T ∈ D′(Ω) and a vector ϕ ∈ D(Ω). +Definition U.4 Let f ∈ Lp(Ω). The regular distribution Tf ∈ D′(Ω) associated to f is defined by +Tf(ϕ) := +� +Ω +f(x)ϕ(x) dΩ, +∀ϕ ∈ D(Ω). +(U.6) +So Tf is a measuring instrument with density dmf(x) = f(x) dΩ, i.e. Tf(ϕ) := +� +Ω ϕ(x) dmf(x). +Definition U.5 Let x0 ∈ Rn. The Dirac measure at x0 is the distribution T noted += +δx0 ∈ D′(R) defined +by, for all ϕ ∈ D(R), +δx0(ϕ) = ϕ(x0), +i.e. +⟨δx0, ϕ⟩ = ϕ(x0). +(U.7) +And δx0 is not a regular distribution (δx0 is not a density measure): There is no integrable function f +such that Tf = δx0. Interpretation: δx0 corresponds to an ideal measuring device: The precision is perfect +at x0 (gives the exact value ϕ(x0) at x0). In real life δx0 is the ideal approximation of Tfn where fn is +e.g. given by fn(x) = n1[x0,x0+ 1 +n ] (drawing): For all ϕ ∈ D(Ω), Tfn(ϕ) −→n→∞ δx0(ϕ) = ϕ(x0). +Generalization of the definition: In (U.5) D(Ω) = C∞ +c (Ω; R) is replaced by C∞ +c (Ω; ⃗Rn). So if you +consider a basis (⃗ei) then ⃗ϕ ∈ C∞ +c (Ω; ⃗Rn) reads ⃗ϕ = �n +i=1ϕi⃗ei with ϕi ∈ D(Ω) for all i. +Example U.6 Power: +Let α : Ω → T 0 +1 (Ω) be a differential form. +Then P += Tα defined by +P(⃗v) = +� +Ω α.⃗v dΩ gives the virtual power associated to α relative to the vector field ⃗v (mechanics and +thermodynamics). +U.2 +Derivation of a distribution +Let O be a point in Rn (an origin). If p ∈ Rn and if (⃗ei) is a basis in ⃗Rn, let ⃗x = −→ +Op = �n +i=1xi⃗ei. +Definition U.7 The derivative +∂T +∂xi of a distribution T ∈ D′(Ω) is the distribution ∈ D′(Ω) defined by, +for all ϕ ∈ D(Ω), +∂T +∂xi +(ϕ) := −T( ∂ϕ +∂xi +), +i.e. +⟨ ∂T +∂xi +, ϕ⟩ := −⟨T, ∂ϕ +∂xi +⟩. +(U.8) +( ∂T +∂xi is indeed a distribution: Easy check.) +Example U.8 If T = Tf is a regular distribution with f ∈ C1(Ω), then ∂(Tf ) +∂xi += T( ∂f +∂xi ). Indeed, for all +ϕ ∈ D(Ω), ∂(Tf ) +∂xi (ϕ) = −Tf( ∂ϕ +∂xi ) = − +� +Ω f(x) ∂ϕ +∂xi dΩ = + +� +Ω +∂f +∂xi ϕ(x) dΩ + +� +Γ 0 dΓ, since ϕ vanishes on +Γ = ∂Ω (the support of ϕ is compact in Ω), thus ∂(Tf ) +∂xi (ϕ) = T( ∂f +∂xi )(ϕ) for all ϕ ∈ D(Ω). +Example U.9 Consider the Heaviside function (the unit step function) H0 := 1R+ and the associated +distribution T = TH0. Then ⟨(TH0)′, ϕ⟩ := −⟨TH0, ϕ′⟩ = − +� +Ω H0(x)ϕ′(x) dx = − +� ∞ +0 +ϕ′(x) dx = ϕ(0) = +⟨δ0, ϕ⟩ for any ϕ ∈ D(R), thus (TH0)′ = δ0. Written H0 +′ = δ0 in D′(Ω), which is not in a equality between +functions, because H0 is not derivable at 0 as a function, and δ0 is not a function; It is equality between +distributions: The notation H0 +′ can only be used to compute H0 +′(ϕ) (= ⟨H0 +′, ϕ⟩ := −⟨H0, ϕ′⟩). +3The D(Ω)-continuity of T is defined by: 1- A sequence (ϕn)N∗ in D(Ω) converges in D(Ω) towards a function ϕ ∈ D(Ω) +iff there exists a compact K ⊂ Ω s.t. supp(ϕn) ⊂ K for all n, and || +∂kϕ +∂xi1 ...∂xik − +∂kϕn +∂xi1 ...∂xik ||∞ −→n→∞ 0 for all k ∈ N +and all ij; 2- T is continuous at ϕ ∈ D(Ω) iff T(ϕn) −→ +n→∞ T(ϕ) for any sequence (ϕn)N ∈ D(Ω)N −→ +n→∞ ϕ in D(Ω). +189 + +190 +U.3. +Hilbert space H1(Ω) +U.3 +Hilbert space H1(Ω) +U.3.1 +Motivation +Consider the hat function Λ(x) +� +� +� +� +� += x + 1 if x ∈ [−1, 0], += 1 − x if x ∈ [0, 1], += 0 otherwise +� +� +� +� +� +(drawing). When applying the finite element +method, it is well-known that, if you use integrals (if you use the virtual power principle which makes +you compute average values), then you can consider the derivative of the hat function Λ as if it was the +usual derivative, i.e. at the points where the usual computation of Λ′ is meaningful, that is, +Λ′(x) +� +� +� +� +� += 1 if x ∈] − 1, 0[, += −1 if x ∈]0, 1[, += 0 if x ∈ R − {−1, 0, 1} +(U.9) +(drawing). +Problem: Λ′ is not defined at −1, 0, 1 (the function Λ is not derivable at −1, 0, 1); +Question: So does the “usual” computation I = +� +R Λ′(x)ϕ(x) dx with (U.9) gives the good result? (This +is not a trivial question: E.g., with H0 = 1R+ instead of Λ, we would get the absurd result H′ +0 = 0, +absurd since H′ +0 = δ0.) +Answer: Yes: +1- Consider TΛ the regular distribution associated to Λ, cf. (U.6); +2- Then consider (TΛ)′, cf. (U.8): +We get ⟨(TΛ)′, ϕ⟩ +(U.8) += −⟨TΛ, ϕ′⟩ += +− +� +R +Λ(x)ϕ′(x) dx += +− +� 0 +−1 +Λ(x)ϕ′(x) dx − +� 1 +0 +Λ(x)ϕ′(x) dx = + +� 0 +−1 +1]−1,0[(x)ϕ(x) dx + +� 1 +0 +1]0,1[ϕ(x) dx, for any ϕ ∈ D(R); +3- Thus (TΛ)′ = Tf where f = 1]−1,0[ + 1]0,1[, that is (TΛ)′ is a regular distribution, And its is named +f = Λ′ within the distribution framework, i.e., for computations ⟨Λ′, ϕ⟩ := ⟨(TΛ)′, ϕ⟩ with ϕ ∈ D(R) +(value = +� +R f(x)ϕ(x) dx). +U.3.2 +Definition of H1(Ω) +The space C1(Ω; R) is too small in many applications (e.g., for the Λ function above); We need a larger +space where the functions are “derivable is a weaker sense” which is the distribution sense. Consider a +basis in Rn: +Definition U.10 The Sobolev space H1(Ω) is the subspace of L2(Ω) restricted to functions whose gen- +eralized derivatives are in L2(Ω): +H1(Ω) = {v ∈ L2(Ω) : ∂v +∂xi +∈ L2(Ω), ∀i = 1, ..., n}. +(U.10) +Usual shortened notation: H1(Ω) = {v ∈ L2(Ω) : +⃗ +gradv ∈ L2(Ω) +n}. +So to check that v ∈ H1(Ω), even if ∂v +∂xi does not exists in the classic way (see the above hat function Λ), +you have to: +1- Consider its associated regular distribution Tv, +2- Compute ∂Tv +∂xi in D′(Ω), +3- And if, for all i, there exists fi ∈ L2(Ω) s.t. ∂Tv +∂xi = Tfi, then v ∈ H1(Ω). +4- And then fi is noted +∂v +∂xi when used within the Lebesgue integrals +� +Ω +∂v +∂xi (x)ϕ(x) dx with ϕ ∈ D(Ω). +E.g., Λ ∈ H1(R) since (TΛ)′ = Tf with f = 1]−1,0[ + 1]0,1[ ∈ L2(R); And (TΛ)′ =noted Λ′ (= f) in the +distribution context (integral computations). +Let (·, ·)L2 and ||.||L2 be the usual inner dot product and norm in L2(Ω), i.e. +(u, v)L2 = +� +Ω +u(x)v(x) dΩ, +and +||v||L2 = +� +(v, v)L2 = ( +� +Ω +v(x)2 dΩ) +1 +2 . +(U.11) +(L2(Ω), (·, ·)L2) is a Hilbert space. Then define, for all u, v ∈ H1(Ω), +(u, v)H1 = (u, v)L2 + +n +� +i=1 +( ∂u +∂xi +, ∂v +∂xi +)L2, +and +||v||H1 = (v, v) +1 +2 +H1. +(U.12) +Then (H1(Ω), (·, ·)H1) is a Hilbert space (Riesz–Fisher theorem). +190 + +191 +U.3. +Hilbert space H1(Ω) +With a Euclidean dot product (·, ·)g in ⃗Rn and a (·, ·)g-orthonormal basis, +(u, v)H1 = (u, v)L2 + ( ⃗ +gradu, +⃗ +gradv)L2. +(U.13) +U.3.3 +Subspace H1 +0(Ω) and its dual space H−1(Ω) +The boundary Γ = ∂Ω of Ω is supposed to be regular. Let +H1 +0(Ω) := {v ∈ H1(Ω) : v|Γ = 0}. +(U.14) +Then (H1 +0(Ω), (·, ·)H1) is a Hilbert space. +More generally (without any regularity assumption on Γ), H1 +0(Ω) := D(Ω) +H1 += the closure of D(Ω) +in (H1(Ω), ||.||H1): This closure of D(Ω) in H1(Ω) enables the use of the distribution framework. +Notation : The dual space of H1 +0(Ω) is the space +H−1(Ω) = (H1 +0(Ω))′ = L(H1 +0(Ω); R) +(U.15) +equipped with the (usual) norm ||T||H−1 := +sup +||v||H1 +0 =1 +|T(v)|. And (duality bracket), if v ∈ H1 +0(Ω) and +T ∈ H−1(Ω) then +T(v) noted += +⟨T, v⟩H−1,H1 +0 +noted += +⟨T, v⟩. +(U.16) +Theorem U.11 (Characterization of H−1(Ω) = (H1 +0(Ω))′.) A distribution T is in H−1(Ω) iff +∃(f,⃗g) ∈ L2(Ω) × L2(Ω) +n +s.t. +T = f − div⃗g (∈ D′(Ω)), +(U.17) +that is, for all v ∈ H1 +0(Ω), +⟨T, v⟩H−1,H1 +0 = +� +Ω +fv dΩ + +� +Ω +dv.⃗g dΩ. +(U.18) +And if Ω is bounded then we can choose f = 0. If moreover ⃗g ∈ H1(Ω) +n then +⟨T, v⟩H−1,H1 +0 = +� +Ω +f(x)v(x) dx − +� +Ω +div⃗g(x)v(x) dx. +(U.19) +(In fact we only need ⃗g ∈ Hdiv(Ω) = {⃗g ∈ L2(Ω) +n : div⃗g ∈ L2(Ω)}.) +Proof. E.g., see Brezis [4]. +For boundary value problems with Neumann boundary conditions, we then need (H1(Ω))′ the dual +space of H1(Ω). +Characterization of (H1(Ω))′: We still have (U.18), but we have to replace (U.17) +or (U.19) by, with a Euclidean dot product in ⃗Rn, see Brezis [4], +⟨T, v⟩(H1)′,H1 = +� +Ω +f(x)v(x) dx − +� +Ω +div⃗g(x)v(x) dx + +� +Γ +⃗g(x) • ⃗n(x) v(x) dx. +(U.20) +191 + +192 +REFERENCES +References +[1] Abraham R., Marsden J.E.: Foundation of mechanics, 2nd edition. Addison-Wesley, 1985. +[2] Arnold V.I.: Equations différentielles ordinaires. Editions Mir, 1974. +[3] Arnold V.I.: Mathematical Methods of Classical Mechanics. Second Edition, Springer 1989. +[4] Brézis H.: Functional Analysis, Sobolev spaces and Partial differential equations. Springer, 2011. +[5] Cartan H.: Formes différentielles. Hermann 1967. +[6] Cartan H.: Cours de calcul différentiel. 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Freeman and Cie, 1973 +[15] Petropoulos +M.: +Relativité +Générale. +La +gravitation +en +une +leçon +et +demie. +https://gargantua.polytechnique.fr/siatel-web/linkto/mICYYYTEsNY +[16] Sadd Martin H.: Elasticity. Theory, Applications, and Numerics. Elsevier 2005. +[17] Spivak M.: A comprehensive introduction to differential geometry, Volume 1. Publish or Perish, Inc. +1979. +[18] Strang G.: Introduction to linear algebra. 5th edition. Wellesley - Cambridge Press 2016. +[19] Truesdell C., Noll W.: The Non-Linear Field Theories of Mechanics. Springer, 2004. +[20] http://planet-terre.ens-lyon.fr/article/force-de-coriolis.xml, 2018. +192 + diff --git a/JdAzT4oBgHgl3EQfH_vd/content/tmp_files/load_file.txt b/JdAzT4oBgHgl3EQfH_vd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fbb36977797061dfb9a0d7e441f3d63e1065abf3 --- /dev/null +++ b/JdAzT4oBgHgl3EQfH_vd/content/tmp_files/load_file.txt @@ -0,0 +1,27313 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf,len=27312 +page_content='Objectivity in continuum mechanics, an introduction Motions, Eulerian and Lagrangian variables and functions, deformation gradient, Lie derivatives, velocity-addition formula, Coriolis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Gilles Leborgne, www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='isima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='fr/leborgne January 4, 2023 In classical mechanics, there are two objectivities: 1- The covariant objectivity concerns the universal laws of physics required to be observer independent (true in any reference frame);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This is a main topic in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- The isometric objectivity concerns the constitutive laws of materials once expressed in a reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Covariant objectivity in continuum mechanics follows Maxwell’s requirements, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [13] page 1: “2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') The formula at which we arrive must be such that a person of any nation, by substituting for the different symbols the numerical value of the quantities as measured by his own national units, would arrive at a true result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') The introduction of coordinate axes into geometry by Des Cartes was one of the greatest steps in mathematical progress, for it reduced the methods of geometry to calculations performed on numerical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The position of a point is made to depend on the length of three lines which are always drawn in determinate directions (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') But for many purposes in physical reasoning, as distinguished from calculation, it is desirable to avoid explicitly introducing the Cartesian coordinates, and to fix the mind at once on a point of space instead of its three coordinates, and on the magnitude and direction of a force instead of its three components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This mode of contemplating geometrical and physical quantities is more primitive and more natural than the other,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..” And see the (short) historical note given in the introduction of Abraham and Marsden book “Foun- dations of Mechanics” [1], about qualitative versus quantitative theory: “Mechanics begins with a long tradition of qualitative investigation culminating with Kepler and Galileo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Following this is the period of quantitative theory (1687-1889) characterized by concomitant developments in mechanics, mathemat- ics, and the philosophy of science that are epitomized by the works of Newton, Euler, Lagrange, Laplace, Hamilton, and Jacobi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') For celestial mechanics (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') resolution we owe to the genius of Poincaré, who resurrected the qualitative point of view (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') One advantage (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') is that by suppressing unnecessary coordinates the full generality of the theory becomes evident.” After having defined motions, Eulerian and Lagrangian variables and functions, we give the definition of the deformation gradient as a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We then obtain a simple understanding of the Lie derivatives of vector fields which meet the needs of engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then we get the velocity addition formula and verify that the Lie derivatives are objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Note that Cauchy would certainly have used the Lie derivatives if they had existed during his lifetime: To get a stress, Cauchy had to compare two vectors, whereas one vector is enough when using the derivatives of Lie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We systematically start with qualitive definitions (observer independent), before quantifying with bases and/or Euclidean dot products (observer dependent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A fairly long appendix tries to give in one manuscript the definitions, properties and interpretations, usually scattered across several books (and not always that easy to find).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='01056v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='class-ph] 3 Jan 2023 2 CONTENTS Contents I Motions, Eulerian and Lagrangian descriptions, flows 11 1 Motions 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Referential .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Quantification in a basis: df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u is written (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ grad)f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Representation relative to a Euclidean dot product: ⃗ gradf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Lagrangian function associated with a Eulerian function .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Lagrangian velocity versus Eulerian velocity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Relation between differentials .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Computation of d⃗v called L = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −1 wih Lagrangian variables .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Lagrangian acceleration .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 28 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Where does this unfortunate notation come from?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Interpretation: Vector approach .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 The ambiguous notation d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 52 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Lie derivative, first definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 52 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 A more general definition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 52 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Equivalent definition (differential geometry) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 53 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Lie derivative of a scalar function .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Lie Derivative of a vector field along itself .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 57 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Incompatibility with Riesz representation vectors .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 66 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 The velocity-addition formula .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 66 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Coriolis acceleration, and the acceleration-addition formula .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 69 11 Objectivities 71 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 “Isometric objectivity” and “Frame Invariance Principle” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 72 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Covariant objectivity of differential forms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 72 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Covariant objectivity of tensors .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Eulerian velocity ⃗v : not covariant (and not isometric) objective .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 73 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Linear forms = “Covariant vectors” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 79 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Covariant dual basis (= the functions that give the components of a vector) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 80 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Example: aeronautical units .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 81 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Matrix representation of a linear form .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 90 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition (requires an inner dot product: Not objective) .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 90 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Quantification with bases .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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(·, ·)g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 95 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 Tensorial representation of a linear map .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 96 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Notations for transitions matrices for bilinear forms and linear maps .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 96 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Change of coordinate system for bilinear forms ∈ L(A, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 97 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Change of coordinate system for bilinear forms ∈ L(A∗, B∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 105 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 106 E.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 106 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Reminder .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 106 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Definition of the vector product (cross product) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 107 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Calculation of the vector product .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 107 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Antisymmetric endomorphism represented by a vector .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 108 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Curl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 109 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Pseudo-cross product, and pseudo-vector .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 109 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Antisymmetric matrix represented by a pseudo-vector .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 110 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Rectilinear motion .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 110 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Circular motion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Motion of a planet (centripetal acceleration) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 112 F Riesz representation theorem 115 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The Riesz representation theorem .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 116 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Quantification with a basis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 116 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Change of Riesz representation vector, and Euclidean case .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 A basis and its many associated “dual vectorial basis” .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 119 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Components of ⃗ejg in the basis (⃗ei) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 120 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Multiple admissible notations for the components of ⃗ejg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 121 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 (Huge) differences between “the (covariant) dual basis” and “a dual vectorial basis” 121 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 About the notation gij = shorthand notation for (g♯)ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 121 G Cauchy–Green deformation tensor C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F 122 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='0 Goal .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 122 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Transposed F T : Inner dot products required .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 133 I Polar decomposition, elasticity and objectivity 133 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Polar decompositions of F (“isometric objectivity”) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 134 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 F = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U (shifted right polar decomposition for covariant objectivity) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 134 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 F = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R (left polar decomposition) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Classical approach (“isometric objectivity”), and an issue .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 136 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 A functional (tensorial) formulation (“isometric objectivity”) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 139 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The differential of the displacement vector .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 140 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Deformation “tensor” ε (matrix), bis .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 140 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Small displacement hypothesis, bis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 141 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Displacement vector with differential geometry .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 141 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The shifter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 142 K Determinants 142 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Alternating multilinear form .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 150 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Calculation of ∂ det ∂Mij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 150 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 ∂J/∂F = J F −T usually written [ ∂J ∂Fij ] = J F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 150 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Interpretation of ∂J ∂Fij ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 150 L Transport of volumes and areas 151 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Transformed parallelepiped .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Example: Type �0 1 � uniform tensor = linear forms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 164 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition of type �r s � uniform tensors .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 165 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Example: Type �1 0 � uniform tensor: Identified with a vector .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 166 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Contractions of uniform tensors .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 167 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Objective double contractions of uniform tensors .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 168 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Non objective double contraction: Double matrix contraction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 169 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Kronecker (contraction) tensor, trace .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Type �0 1 � tensor = differential forms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Type �1 0 � tensor (identified to a vector field) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 172 9 10 CONTENTS R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 A metric is a �0 2 � tensor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 173 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Directional derivative and differential (observer independent) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 174 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 A basis and the j-th partial derivative .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 183 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 Euclidean framework and “classic divergence” of a tensor (subjective) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 184 T Natural canonical isomorphisms 184 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The adjoint of a linear map .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 184 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 An isomorphism E ≃ E∗ is never natural (never objective) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 185 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Natural canonical isomorphism E ≃ E∗∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 186 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Natural canonical isomorphisms L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) ≃ L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) ≃ L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F ∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 186 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Natural canonical isomorphisms L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F)) ≃ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) ≃ L(F ∗, E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 191 10 11 A quantity f being given then: g defined by « g equals f » is noted g := f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Part I Motions, Eulerian and Lagrangian descriptions, flows 1 Motions The framework is classical mechanics, time being decoupled from space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R3 is the classical geometric affine space (the space we live in), and ( ⃗R3, +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') = { ⃗pq : p, q ∈ R3} =noted ⃗R3 is the associated vector space of bipoint vectors equipped with its usual rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We also consider R and R2 as subspaces of R3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' we consider Rn and ⃗Rn, n = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Referential Origin: An observer chooses an origin O ∈ Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus a point p ∈ Rn can be located by the observer thanks to the bipoint vector −→ Op = ⃗x ∈ ⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence p = O + ⃗x, and ⃗x = −→ Op =noted p − O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Another observer chooses an origin � O ∈ Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the point p can also be located by this observer with the bipoint vector −→ � Op = �⃗x ∈ ⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So p = O + ⃗x = � O + �⃗x, and �⃗x = −−→ O � O + ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Cartesian coordinate system: A Cartesian coordinate system in the affine space Rn is a set RCart = (O, (⃗ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n), where O is an origin and (⃗ei) := (⃗ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n is a basis in ⃗Rn chosen by the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the location of a point p ∈ Rn can quantified by the observer ∃⃗x ∈ ⃗Rn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' p = O + ⃗x with ⃗x = n � i=1 xi⃗ei, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [−→ Op]|⃗e = [⃗x]|⃗e = � � x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' xn � � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) [⃗x]|⃗e = [−→ Op]|⃗e being the column matrix containing the components xi ∈ R of −→ Op = ⃗x in the basis (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Another observer with his origin Ob and his Cartesian basis (⃗bi)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n make the Cartesian coordinate system RCart,b = (Ob, (⃗bi)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n), and gets for the same position p in Rn, p = Ob + ⃗y with ⃗y = n � i=1 �yi�⃗bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [−−→ Obp]|⃗b = [⃗y]|⃗b = � � � y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' yn � � � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) [⃗y]|⃗b = [−−→ Obp]|⃗b being the column matrix containing the components yi ∈ R of −−→ Obp = ⃗y in the basis (⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And −−→ Obp = −−→ ObO + −→ Op, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗y = −−→ � OO + ⃗x, gives the relation between ⃗x and ⃗y (drawing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Chronology: A chronology (or temporal coordinate system) is a set Rtime = (t0, (∆t)) chosen by an observer, where t0 ∈ R is the time origin, and (∆t) is the time unit (a basis in ⃗R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Referentiel: A referential R is the set R = (Rtime, RCart) = (t0, (∆t), O, (⃗ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n) = (“chronologie”,“Cartesian coordinate system”), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) made of a chronology and a Cartesian coordinate system, chosen by an observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In the following, to simplify the writings, the same implicit chronology is used by all observers, and a referential R = (Rtime, RCart) will simply be noted as the reference frame R = (O, (⃗ei)) (so := RCart).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Einstein’s convention (duality notation) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Einstein’s convention (duality notation) Starting point: The classical notation xi for the components of a vector ⃗x relative to a basis, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then the duality notion is introduced: xi =noted xi (enables to see the difference between a vector and a function when using components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So ⃗x = n � i=1 xi⃗ei � �� � classic not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = n � i=1 xi⃗ei � �� � duality not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' , and [⃗x]|⃗e clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = � � x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' xn � � dual = � � x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' xn � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) The duality notation is part of the Einstein’s convention;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Moreover Einstein’s convention uses the notation �n i=1xi⃗ei =noted xi⃗ei, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the sum sign �n i=1 can be omitted when an index (i here) is used twice, once up and once down, details at § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' However this omission of the sum sign � will not be made in this manuscript (to avoid ambiguities): The TEX-LATEX program makes it easy to print �n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The height of a child is represented on a wall by a vertical bipoint vector ⃗x starting from the ground up to a pencil line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Question: What is the size of the child ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: It depends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' on the observer (quantitative value = subjective result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', an English observer chooses a vertical basis vector ⃗a1 which length is one English foot (ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So he writes ⃗x = x1⃗a1, and for him the size of the child (size of ⃗x) is x1 in foot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' x1 = 4 means the child is 4 ft tall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A French observer chooses a vertical basis vector ⃗b1 which length is one metre (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So he writes ⃗x = y1⃗b1, and for him the size of the child (size of ⃗x) is y1 metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', if x1 = 4 then y1 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22, since 1 ft := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3048 m: The child is both 4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22 tall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' in foot or metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This quantification is written ⃗x = 4 ft = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22 m, where ft means ⃗a1 and m means ⃗b1 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: The qualitative vector ⃗x is the same vector for all observers, not the quantitative values 4 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22 (depends on a choice of a unit of measurement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With duality notation: ⃗x = x1⃗a1 = y1⃗b1, so if x1 = 4 then y1 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This manuscript insists on covariant objectivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus an English engineer (and his foot) and a French engineer (and his metre) will be able to work together .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' and be able to avoid crashes like that of the Mars Climate Orbiter probe, see remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And they will be able to use the results of Galileo, Descartes, Newton, Euler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' who used their own unit of length, and knew nothing about the metre defined in 1793 and adopted in 1799 in France (after 6 years of measurements), and considered by the scientific community at the end of the ninetieth century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' and couldn’t explicitly use the “Euclidean dot products” either (which seems to have been defined mathematically by Grassmann around 1844).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Motion of an object Let Obj be a “real object”, or “material object”, made of particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', the Moon: Exists independently of an observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let t1, t2 ∈ R, t1 < t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The motion of Obj in Rn is the map �Φ : � � � � � [t1, t2] × Obj → Rn (t, PObj) � �� � particle → p = �Φ(t, PObj) � �� � its position at t in the Universe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) And t is the time variable, p is the space variable, and (t, p) ∈ R × Rn is the time-space variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And �Φ is supposed to be C2 in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With an origin O (observer dependent), the motion can be described with the bi-point vector ⃗x = −−−−−−−→ O�Φ(t, PObj) = −→ Op noted = �⃗ϕ(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) But then, two observers with different origins O and Ob have different description of the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' There- fore, in the following we won’t use �⃗ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (quantification) with a Cartesian basis (⃗ei) to make a referential R, we get (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Virtual and real motion Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 A virtual (or possible) motion of Obj is a function �Φ “regular enough for the calculations to be meaningful”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Among all the virtual motions, the observed motion is called the real motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 12 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hypotheses (Newton and Einstein) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Hypotheses (Newton and Einstein) Hypotheses of Newtonian mechanics (Galileo relativity) and general relativity (Einstein): 1- You can describe a phenomenon only at the actual time t and from the location p you are at (you have no gift of ubiquity in time or space);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- You don’t know the future;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3- You can use your memory, so use some past time t0 and some past position pt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4- You can use someone else memory (results of measurements) if you can communicate objectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Configurations Fix t ∈ [t1, t2], and define �Φt : � Obj → Rn PObj �→ p = �Φt(PObj) := �Φ(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 The “configuration at t” of Obj is the range (or image) of �Φt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' is the subset of Rn (affine space) defined by Ωt := {p ∈ Rn : ∃PObj ∈ Obj s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' p = �Φt(PObj)} noted = �Φt(Obj) noted = Im(�Φt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) If t is the actual time then Ωt is the actual (or current or Eulerian) configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If t0 is a time in the past then Ωt0 is the past (or initial or Lagrangian) configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hypothesis: At any time t, Ωt is supposed to be a “smooth domain” in Rn, and the map �Φt is assumed to be one-to-one (= injective): Obj does not crash onto itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Definition of the Eulerian and Lagrangian variables If t is the actual time, then pt = �Φt(PObj) ∈ Ωt is called the Eulerian variable relative to PObj and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If t0 is a time in the past, then pt0 = �Φt0(PObj) ∈ Ωt0 is called the Lagrangian variable relative to PObj and t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A Lagrangian variable is a “past Eulerian variable”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Two observers with two different origin of time t0 and t0′ get two different Lagrangian variable while they have the same Eulerian variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Trajectories Let �Φ be a motion of Obj, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5), and PObj ∈ Obj (a particle in Obj = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the Moon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 The (parametric) trajectory of PObj is the function �ΦPObj : � [t1, t2] → Rn, t �→ p(t) = �ΦPObj (t) := �Φ(t, PObj) (position of PObj at t in the Universe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) Its geometric trajectory is the range (image) of �ΦPObj , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' geometric trajectory of PObj := {q ∈ Rn : ∃t ∈ [t1, t2] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' q = �ΦPObj (t)} = Im(�ΦPObj ) = �ΦPObj ([t1, t2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 Pointed vector, tangent space, fiber, vector field, bundle (See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Abraham–Marsden [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') To deal with surfaces S in R3, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' with S = a sphere (and more generally with manifolds in Rn), a vector cannot simply be a “bi-point vector connecting two points of S” (would get “through the surface”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A vector is defined to be tangent to S: Consider a “regular” curve c : s ∈] − ε, ε[→ c(s) ∈ S where S is a surface in an affine space, and the vector tangent to S at c(0) is ⃗w(c(0)) = limh→0 c(h)−c(0) h (it is defined with a parametrization of c in a general manifold);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Considering all the possible curves, we get “all possible vectors on S”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Notation: TpS := {tangent vectors ⃗wp at S at p} = The tangent space at p ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', if S is a sphere in R3 and p ∈ S, then TpS is its usual tangent plane at p at S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', particular case: If S = Ω is an open set in Rn, then TpS = TpΩ = ⃗Rn is independent of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 13 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The set of configurations Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 The fiber at p := {p} × TpS = { (p, ⃗wp) � �� � pointed vector ∈ {p} × TpS}, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', the fiber at p is the set of “pointed vectors at p”, a pointed vector being the couple (p, ⃗wp) made of the “base point” p and the vector ⃗wp defined at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Drawing: A vector in ⃗Rn can be drawn anywhere in Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' While a “pointed vectors at p” has to be drawn at the point p in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If the context is clear, a pointed vector is simply noted �⃗w(p) =noted ⃗w(p) (lighten the writing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Particular case: If S = Ω is an open set in Rn, then the fiber at p is TpΩ = {p} × ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 The tangent bundle TS := � p∈S ({p} × TpS), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) that is, is the union of the fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 A vector field �⃗w in S is a C∞ function (or at least C2 in the following) �⃗w : � S → TS p → �⃗w(p) = (p, ⃗w(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) If the context is clear, a vector field is simply noted �⃗w =noted ⃗w (lighten the writing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2 Eulerian description (spatial description at actual time t) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The set of configurations Let �Φ be a motion of Obj, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5), and Ωt = �Φt(Obj) ⊂ Rn be the configuration at t, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The set of configurations is the subset C ⊂ R × Rn (the “time-space”) defined by C := � t∈[t1,t2] ({t} × Ωt) (= set in which you find particles in “time-space”) = {(t, p) ∈ R × Rn : ∃(t, PObj) ∈ [t1, t2] × Obj, p = �Φ(t, PObj)}, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) Question: Why don’t we simply use � t∈[t1,t2] Ωt instead of C = � t∈[t1,t2]({t} × Ωt)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: C gives the film of the life of Obj = the succession of the photos Ωt taken at each t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And Ωt is obtained from C thanks to the pause feature at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Whereas � t∈[t1,t2] Ωt ⊂ Rn is the superposition of all the photos on the image � t∈[t1,t2] Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' and we don’t distinguish the past from the present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Eulerian variables and functions Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 In short: A Eulerian function relative to Obj is a function, with m ∈ N∗, Eul : � C → ⃗ Rm (or more generally a suitable set of tensors) (t, p) → Eul(t, p), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) the spatial variable p being the Eulerian variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In details: A function Eul being given as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2), the associated Eulerian function � Eul is the function � Eul : � C → C × ⃗ Rm (or C× some suitable set of tensors) (t, p) → � Eul(t, p) = ((t, p), Eul(t, p)) = (time-space position , value), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) and is called “a field of functions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So � Eul(t, p) is the “pointed Eul(t, p)” at (t, p) (in time-space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, the range Im( � Eul) = � Eul(C) of an Eulerian function � Eul is the graph of Eul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Recall: The graph of a function f : x ∈ A → f(x) ∈ B is the subset {(x, f(x)) ∈ A × B} ⊂ A × B: gives the “drawing of f”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If there is no ambiguity, � Eul =noted Eul for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 14 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Eulerian velocity (spatial velocity) and speed At t, the Eulerian vector field at t is � Eult : � Ωt → Ωt × ⃗Rn p → � Eult(p) := (p, Eult(p)) = (position , value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Eul(t, p) = θ(t, p) ∈ R = temperature of the particle PObj which is at t at p = �Φ(t, PObj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Eul(t, p) = ⃗u(t, p) ∈ ⃗Rn = force applied on the particle PObj which is at t at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Eul(t, p) = d⃗u(t, p) ∈ L(⃗Rn : ⃗Rn) = the differential at t at p of a Eulerian function ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Question: Why introduce � Eul?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Isn’t Eul sufficient?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: The “pointed value” � Eul(t, p) = ((t, p), Eul((t, p))) is drawn on the graph of Eul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', at t at p the velocity vector ⃗v(t, p) ∈ ⃗R3 can be drawn anywhere, while the “pointed vector” �⃗v(t, p) = ((t, p);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v(t, p)) is ⃗v(t, p) drawn at t at p (and �⃗v is called the velocity field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Moreover (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) emphasizes the difference between a Eulerian vector field and a Lagrangian vector function, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', the initial framework of Cauchy for his description of forces is Eulerian: The Cauchy stress vector ⃗t = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n is considered at the actual time t at a point p ∈ Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (It is not Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Eulerian velocity (spatial velocity) and speed Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 In short: Consider a particle PObj and its (regular) trajectory �ΦPObj : t → p(t) = �ΦPObj (t), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its Eulerian velocity at t at p(t) = �ΦPObj (t) is ⃗v(t, p(t)) := �ΦPObj ′(t) noted = ∂�Φ ∂t (t, PObj), when p(t) = �ΦPObj (t), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗v(t, p(t)) is the tangent vector at t at p(t) = �ΦPObj (t) to the trajectory �ΦPObj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This defines the vector field (in short) ⃗v : � C → ⃗Rn (t, pt) → ⃗v(t, pt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In details: cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3), the Eulerian velocity is the function �⃗v : � C → C × ⃗ Rm (t, p) → �⃗v(t, p) = ((t, p),⃗v(t, p)) � (pointed vector) where ⃗v(t, p) is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 d�ΦPObj dt (t) = ⃗v(t, �ΦPObj (t)), with p(t) = �ΦPObj (t), is often written dp dt (t) = ⃗v(t, p(t)), or d⃗x dt (t) = ⃗v(t, ⃗x(t)), or d⃗x dt = ⃗v(t, ⃗x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) the two last notations when an origin O is chosen and ⃗x(t) = −−−→ Op(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Such an equation is the pro- totype of an ODE (ordinary differential equation) solved with the Cauchy–Lipschitz theorem, see § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A Lagrangian velocity does not produce an ODE, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 If an observer chooses a Euclidean dot product (·, ·)g (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' foot or metre built), the associated norm being ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||g, then the length ||⃗v(t, p)||g is the speed (or scalar velocity) of PObj (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' in ft/s or in m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the context must remove the ambiguities: the “velocity” is either the vector velocity ⃗v(t, p) = �ΦPObj ′(t) or the speed (the scalar velocity) ||⃗v(t, p)||g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 Euclidean dot product (·, ·)g, ⃗x(t) = −−−→ Op(t), ⃗T(t) = ⃗x ′(t) ||⃗x ′(t)||g , and f(t) = ||⃗x ′(t)||g (speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove : df dt(t) = (⃗x ′′(t), ⃗T(t))g =noted ⃗x ′′(t) • ⃗T(t) (= tangential acceleration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2-D and Euclidean basis: ⃗x(t) = � x(t) y(t) � gives f(t) = (x′(t)2 + y′(t)2) 1 2 , thus f ′(t) = x′(t)x′′(t)+y′(t)y′′(t) f(t) = ⃗r ′(t) • ⃗r ′′(t) ||⃗r ′(t)|| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Idem in n-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Spatial derivative of the Eulerian velocity t ∈ [t1, t2] is fixed, Eul is a given Eulerian function, and Eult : � Ωt → ⃗ Rm p → Eult(p) := Eul(t, p) � is C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 15 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Spatial derivative of the Eulerian velocity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition Recall: If Ω is an open set in Rn and if f : Ω → R is differentiable at p, then its differential at p is the linear form df(p) ∈ L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) (linear map with real values) defined by, for all ⃗u ∈ ⃗Rn (vector at p), df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = lim h→0 f(p+h⃗u) − f(p) h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) This expression is the same for all observers (English, French.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=': There is no inner dot product here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 The space derivative of Eul at (t, p) is the differential dEult at p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all t ∈ [t1, t2], all p ∈ Ωt and all ⃗wp ∈ ⃗Rn t (vector at p), (dEult(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wp =) dEul(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wp = lim h→0 Eul(t, p+h⃗wp) − Eul(t, p) h noted = ∂Eul ∂p (t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) In Ωt (the photo at t), dEul(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wp gives the rate of variations of Eult at p in the direction ⃗wp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', at t, the space derivative d⃗v of the Eulerian velocity field is defined by d⃗v(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wp = lim h→0 ⃗v(t, p+h⃗wp) − ⃗v(t, p) h (= d⃗vt(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 In differential geometry, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) is also written ⃗u(f)(p) = d dhf(p+h⃗u)|h=0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Don’t use this notation if you are not at ease with differential geometry (where a vector is defined to be a derivation, so ⃗u[f] is the derivation of f by ⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The convective derivative dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 If ⃗v is the Eulerian velocity field, then dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v is called the convective derivative of Eul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Quantification in a basis: df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u is written (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ grad)f Quantification: Let f : p ∈ Rn → f(p) ∈ R be C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗ei) be a basis in ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (usual definition) ∂f ∂xi (p) := df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei and [df(p)]|⃗e = ( ∂f ∂x1 (p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂f ∂xn (p) ) (line matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) (Recall: The matrix which represents a linear form is a line matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') And [df(p)]|⃗e is the Jacobian matrix of f at p relative to (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, with ⃗u = �n i=1ui⃗ei a vector at p, and with the usual matrix multiplication rule, we have df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = [df(p)]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u]|⃗e = n � i=1 ∂f ∂xi (p)ui = n � i=1 ui ∂f ∂xi (p) noted = (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ grad)|ef(p), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) where (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ grad)|e : C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) → C0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is the differential operator defined relative to a basis (⃗ei) by (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ grad)|e(f) = n � i=1 ui ∂f ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) If the basis (⃗ei) is unambiguously imposed, then (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ grad)|e =noted ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ grad For vector valued functions ⃗f : Ω → ⃗ Rm, the above steps apply to the components of ⃗f in a basis (⃗bi) in ⃗ Rm: If ⃗f = �m i=1fi⃗bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗f(p) = �m i=1fi(p)⃗bi, then (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ grad)|e(⃗f) = m � i=1 (dfi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u)⃗bi = m � i=1 ((⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ grad)|efi)⃗bi = m � i=1 n � j=1 (uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂fi ∂xj )⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) 16 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Streamline (current line) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Representation relative to a Euclidean dot product: ⃗ gradf An observer chooses a distance unit (foot, metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') and uses the associated Euclidean dot product (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Ω be an open set in Rn, f ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) (scalar valued function), and p ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then the (·, ·)g-Riesz representation vector of the differential form df(p) is called the gradient of f at p relative to (·, ·)g, and named ⃗ gradgf(p) ∈ ⃗Rn: It is defined by ∀⃗u ∈ ⃗Rn, ( ⃗ gradgf(p), ⃗u)g = df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u, written ⃗ gradf • ⃗u = df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) the last notation iff a Euclidean dot product (·, ·)g is imposed to all observer (quite subjective: foot, metre ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (The first order Taylor expansion f(p+h⃗u) = f(p) + h df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u + o(h) can therefore, after a choice of an Euclidean dot product, be written f(p+h⃗u) = f(p) + h ⃗ gradgf(p) •g ⃗u + o(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Quantification: Let (⃗ei) be a Cartesian basis in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) gives [df].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u] = [ ⃗ gradf]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u], for all ⃗u ∈ ⃗Rn t (more precisely [df]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u]|⃗e = [ ⃗ gradgf]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u]|⃗e), thus (since [g]|⃗e is symmetric) [ ⃗ gradf] = [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [df]T (column matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', if ⃗ gradf = �n i=1ai⃗ei then ai = �n j=1gij ∂f ∂xj for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular, if (⃗ei) is a (·, ·)g-orthonormal basis then [ ⃗ gradf] = [df]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With duality notations, ⃗ gradf = �n i=1ai⃗ei and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) gives ai = �n j=1gij ∂f ∂xj : The Einstein convention is not satisfied (the index j is twice bottom), which is expected since the definition of ⃗ gradgf depends on a subjective choice (unit of length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In comparison, df = �n i=1 ∂f ∂xi dxi satisfies the Einstein convention (a differential is objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Mind the notations: The gradient ⃗ gradgf =noted ⃗ gradf depends on (·, ·)g, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14), while (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ grad)f does not (only depends on a basis), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) (historical notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Vector valued functions For vector valued functions ⃗f : Ω → ⃗ Rm, the above steps apply to the components fi of ⃗f relative to a basis (⃗bi) in ⃗ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But, depending on the book you read: 1- Ambiguous: d⃗f, the differential of ⃗f, is unfortunately also sometimes called the “gradient matrix” (although no Euclidean dot product is required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- Ambiguous: It could mean the differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' or the Jacobian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' or its transposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' because an orthonormal basis relative to an imposed Euclidean dot product is chosen (which one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') and then [ ⃗ gradfi] = [dfi]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And calculations confuses [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='] and [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3- Non ambiguous: In the objective framework of this manuscript, we will use the differential d⃗f (objective) to begin with;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And only after an explicit choice of bases (⃗ei) for quantitative purposes, the Jacobian matrix, which is [df]|⃗e, will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 A Euclidean framework being chosen, prove: (⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ grad)⃗v = 1 2 ⃗ grad(||⃗v||2) + ⃗ rot⃗v ∧ ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Euclidean basis ( ⃗Ei), Euclidean dot product (·, ·)g =noted (·, ·), associated norm ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||g =noted ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ⃗v = �n i=1vi ⃗Ei gives ||⃗v||2 = � i v2 i , thus ∂||⃗v||2 ∂xk = � i 2vi ∂vi ∂xk , for any k = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And, the first component of ⃗ rot⃗v is ( ⃗ rot⃗v)1 = ∂v3 ∂x2 − ∂v2 ∂x3 , idem for ( ⃗ rot⃗v)2 and ( ⃗ rot⃗v)3 (circular permutation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (first component) ( ⃗ rot⃗v∧⃗v)1 = ( ∂v1 ∂x3 − ∂v3 ∂x1 )v3−( ∂v2 ∂x1 − ∂v1 ∂x2 )v2, idem for ( ⃗ rot⃗v∧⃗v)2 and ( ⃗ rot⃗v∧⃗v)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ( 1 2 ⃗ grad(||⃗v||2)+ ⃗ rot⃗v∧⃗v)1 = v1 ∂v1 ∂x1 + v2 ∂v2 ∂x1 + v3 ∂v3 ∂x1 + ∂v1 ∂x3 v3 − ∂v3 ∂x1 v3 − ∂v2 ∂x1 v2 + ∂v1 ∂x2 v2 = v1 ∂v1 ∂x1 + v2 ∂v1 ∂x2 + v3 ∂v1 ∂x3 = (⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ grad)v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Idem for the other components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Streamline (current line) Fix t ∈ R, and consider the photo Ωt = �Φt(Obj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let pt ∈ Ωt, ε > 0, and consider the spatial curve in Ωt at pt defined by: cpt : � ] − ε, ε[ → Ωt s → q = cpt(s) � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' cpt(0) = pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) 17 18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Material time derivative (dérivées particulaires) So s is a curvilinear spatial coordinate (dimension of a length), and the graph of cpt is drawn in the photo Ωt at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 ⃗v : (t, p) → ⃗v(t, p) being the Eulerian velocity field of Obj, a streamline through a point pt ∈ Ωt is a (parametric) spatial curve cpt solution of the differential equation dcpt ds (s) = ⃗vt(cpt(s)) with cpt(0) = pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) And Im(cpt) is the geometric associated streamline (⊂ Ωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) cannot be confused with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5): In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) the variable is the time variable t, while in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) the variable is the space variable s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Usual notation: If an origin O is chosen at t by an observer and ⃗x(s) := −−−−−→ Ocpt(s) , then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) is written d⃗x ds (s) = ⃗vt(⃗x(s)) with ⃗x(0) = −−→ Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) Moreover, with a Cartesian basis (⃗ei)) chosen at t by the observer, with ⃗x(s) = �n i=1xi(s)⃗ei we get d⃗x ds (s) = �n i=1 dxi ds (s)⃗ei, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) reads as the differential system of n equations in ⃗Rn ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, dxi ds (s) = vi(t, x1(s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', xn(s)) with xi(0) = (−−→ Opt)i (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) (the n functions xi : s → xi(s) are the unknown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Also written dx1 v1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = dxn vn = ds, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) which means: It is the differential system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) of n equations and n unknowns which must be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (With duality notations, dxi ds (s) = vi(t, x1(s), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', xn(s)) and xi(0) = (−−→ Opt)i for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Material time derivative (dérivées particulaires) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Usual definition Goal: To compute the variations of a Eulerian function Eul along the trajectory �ΦPObj of a particle PObj (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the temperature of a particle along its trajectory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So consider the function gPObj giving the values of Eul relative to a PObj along its trajectory: gPObj (t) := Eul(t, p(t)) when p(t) := �ΦPObj (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15 The Material time derivative of Eul at (t, p(t)) is gPObj ′(t) =noted DEul Dt (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So: DEul Dt (t, p(t)) := gPObj ′(t) (= lim h→0 Eul(t+h, p(t+h)) − Eul(t, p(t)) h ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) Since gPObj (t) := Eul(t, �ΦPObj (t)) we get gPObj ′(t) = ∂Eul ∂t (t, �ΦPObj (t))+dEul(t, �ΦPObj (t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�Φ′ PObj (t), thus, having �Φ′ PObj (t) = ⃗v(t, p(t)) (Eulerian velocity), DEul Dt (t, p(t)) = ∂Eul ∂t (t, p(t)) + dEul(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v(t, p(t)): DEul Dt := ∂Eul ∂t + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) Exercice 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16 Prove, if Eul is C2: D2Eul Dt2 = ∂2Eul ∂t2 + 2d∂Eul ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗v ∂t + d2Eul(⃗v,⃗v) + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' D2Eul Dt2 = D DEul Dt Dt = g′′ PObj (t) = ∂( ∂Eul ∂t + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) ∂t + d(∂Eul ∂t + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v = ∂2Eul ∂t2 + ∂(dEul) ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗v ∂t + d∂Eul ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + d2Eul(⃗v,⃗v) + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v, and Eul C2 gives ∂ ∂t ◦ d = d ◦ ∂ ∂t (Schwarz theorem), hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 18 19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Material time derivative (dérivées particulaires) Exercice 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17 Prove, if Eul is C2, for any vector field ⃗w, D(dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) Dt = d∂Eul ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂ ⃗w ∂t + d2Eul(⃗v, ⃗w) + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' D(dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) Dt = ∂(dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) ∂t + d(dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v = ∂dEul ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂ ⃗w ∂t + (d(dEul).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the Schwarz theorem gives ∂(dEul) ∂t = d( ∂Eul ∂t ) since Eul ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Remark: About notations The notation d dt (lowercase letters) concerns a function of one variable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dgPObj dt (t) := gPObj ′(t) := limh→0 gPObj (t+h))−gPObj (t) h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The notation ∂ ∂t concerns a function with more than one variable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂Eul ∂t (t, p) = limh→0 Eul(t+h,p)−Eul(t,p) h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The notation D Dt (capital letters) concerns a Eulerian function differentiated along a motion, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Other notations, often practical but might be ambiguous if composed functions are considered: dEul(t, p(t)) dt := DEul Dt (t, p(t)), and dEul(t, p(t)) dt |t=t0 := DEul Dt (t0, p(t0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Definition bis: Time-space definition Consider a C1 time-space function f : (t, p) ∈ R × Rn → f(t, p) where t = time and p = space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18 The differential of f : (t, p) ∈ Rn+1 → f(t, p) considered as a function on the Cartesian (time×space) product R × Rn is called the “total differential”, or “total derivative”, and is written Df (here time and space are of a different nature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So if p+ = (t, p) ∈ R×Rn and ⃗w+ = (w0, ⃗w) ∈ ⃗R×⃗Rn (time×space) then, by definition of a differential, Df(p+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w+ := lim h→0 f(p+ + h⃗w+) − f(p+) h , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Df(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (w0, ⃗w) := lim h→0 f(t+hw0, p+h⃗w) − f(t, p) h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) Thus Df(t, p) = ∂f ∂t (t, p) dt + df(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) Along a trajectory �ΦPObj : t → p(t) = �ΦPObj (t) with f = Eul a Eulerian function: Consider the time-space trajectory �ΨPObj : � [t1, t2] → R × Rn t → �ΨPObj (t) := (t, �ΦPObj (t)) (= (t, p(t))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) (So Im(�ΨPObj ) = graph(�ΦPObj ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') The tangent vector to this curve at t is �ΨPObj ′(t) = (1, �ΦPObj ′(t)) = (1,⃗v(t, p(t)) ∈ ⃗R × ⃗Rn (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) where ⃗v(t, p(t)) the Eulerian velocity at p+ = (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) reads gPObj (t) = (Eul ◦ �ΨPObj )(t) = Eul(�ΨPObj (t)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) thus g′ PObj (t) = DEul(�Ψ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�ΨPObj ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) And we recover (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22): g′ PObj (t) =(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) ∂Eul ∂t (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1+dEul(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v(t, p(t)) =noted DEul Dt (t, p(t)) : The ma- terial time derivative is the “total derivative” DEul along the time-space trajectory �ΨPObj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 19 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Eulerian acceleration 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 The material time derivative is a derivation Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19 All the functions are Eulerian and supposed C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Linearity: D(Eul1 + λEul2) Dt = DEul1 Dt + λDEul2 Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) Product rules: If Eul1, Eul2 are scalar valued functions then D(Eul1Eul2) Dt = DEul1 Dt Eul2 + Eul1 DEul2 Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) And if ⃗w is a vector field and T a compatible tensor (so T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w is meaningful) then D(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) Dt = DT Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D ⃗w Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let i = 1, 2, and gi defined by gi(t) := Euli(t, p(t)) where p(t) = �ΦPObj (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (g1 + λg2)′ = g′ 1 + λg′ 2 gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' On the one hand D(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) Dt = ∂(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) ∂t + d(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v = ∂T ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂ ⃗w ∂t + (dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v), and on the other hand DT Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' D ⃗w Dt = ( ∂T ∂t + dT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='( ∂ ⃗w ∂t + d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Commutativity issue The Schwarz theorem that, when Eul is C2, the derivatives ∂Eul ∂t and dEul commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20 The material time derivative D Dt does not commute with the temporal derivation ∂ ∂t or with the spatial derivation d: We have ∂( DEul Dt ) ∂t ̸= D( ∂Eul ∂t ) Dt and d( DEul Dt ) ̸= D(dEul) Dt in general (because the variables t and p are not independent along a trajectory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In facts: ∂( DEul Dt ) ∂t = D( ∂Eul ∂t ) Dt + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗v ∂t = ∂2Eul ∂t2 + d∂Eul ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗v ∂t � � � � � � � , and � � � � � d(DEul Dt ) = D(dEul) Dt + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v = ∂(dEul) ∂t + d2Eul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂ DEul Dt ∂t = ∂( ∂Eul ∂t + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) ∂t = ∂( ∂Eul ∂t ) ∂t + ∂dEul ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗v ∂t Schwarz = ∂( ∂Eul ∂t ) ∂t + d∂Eul ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗v ∂t , thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dDEul Dt = d(∂Eul ∂t + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) = ∂(dEul) ∂t + d(dEul).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v, thus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So ∂( DEul Dt ) ∂t ̸= D( ∂Eul ∂t ) Dt and d(DEul Dt ) ̸= D(dEul) Dt in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21 Prove (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) with components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗ei) is a Cartesian basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂ DEul Dt ∂t = ∂( ∂Eul ∂t +� i ∂Eul ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='vi) ∂t = ∂2Eul ∂t2 + � i ∂2Eul ∂t∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='vi + � i ∂Eul ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂vi ∂t = ∂2Eul ∂t2 + � i ∂2Eul ∂t∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='vi + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂⃗v ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And D( ∂Eul ∂t ) Dt = ∂2Eul ∂t2 + � i ∂ ∂Eul ∂t ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='vi = ∂2Eul ∂t2 + � i ∂2Eul ∂t∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And d( DEul Dt ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = � j ∂ DEul Dt ∂xj wj = � j ∂( ∂Eul ∂t +� i ∂Eul ∂xi vi) ∂xj wj = � j ∂2Eul ∂t∂xj wj +� ij ∂2Eul ∂xi∂xj viwj +� ij ∂Eul ∂xi ∂vi ∂xj wj = � j ∂2Eul ∂t∂xj wj + d2Eul(⃗v, ⃗w) + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And D(dEul) Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = ( ∂(dEul) ∂t + d(dEul).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = ∂(dEul) ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + d2Eul(⃗v, ⃗w) = � i ∂2Eul ∂xi∂t wi + d2Eul(⃗v, ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus d( DEul Dt ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = D(dEul) Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w for all ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Eulerian acceleration Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22 In short: If �ΦPObj is C2, then the Eulerian acceleration of the particle PObj which is at t at pt = �Φ(t, PObj) is ⃗γ(t, pt) := �ΦPObj ′′(t) noted = ∂2�Φ ∂t2 (t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) In details: as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3), the Eulerian acceleration (vector) field �⃗γ is defined with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) by �⃗γ(t, pt) = ((t, pt),⃗γ(t, pt)) ∈ C × ⃗Rn t (pointed vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38) 20 21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Time Taylor expansion of �Φ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23 ⃗γ = D⃗v Dt = ∂⃗v ∂t + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39) And if ⃗v is C2 then d⃗γ = ∂(d⃗v) ∂t + d2⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v = D(d⃗v) Dt + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With g(t) = ⃗v(t, p(t)) = �ΦPObj ′(t) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) we get ⃗γ(t, p(t)) = g′(t) = D⃗v Dt (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ⃗v being C2, the Schwarz theorem gives d ∂⃗v ∂t = ∂(d⃗v) ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24 If an observer chooses a Euclidean dot product (·, ·)g (based on a foot, a metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='), the associated norm being ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||g, then the length ||⃗γ(t, pt)||g is the (scalar) acceleration of PObj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Time Taylor expansion of �Φ Let PObj ∈ Obj and t ∈]t1, t2[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Suppose �ΦPObj ∈ C2(]t1, t2[;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its second-order (time) Taylor expansion of �ΦPObj is, in the vicinity of a t ∈]t1, t2[, �ΦPObj (τ) = �ΦPObj (t) + (τ−t)�Φ′ PObj (t) + (τ−t)2 2 �Φ′′ PObj (t) + o((τ−t)2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' p(τ) = p(t) + (τ−t)⃗v(t, p(t)) + (τ−t)2 2 ⃗γ(t, p(t)) + o((τ−t)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='42) 3 Motion from an initial configuration: Lagrangian description Instead of working on Obj, an observer may prefer to work with an initial configuration Ωt0 = �Φ(t0, Obj) of Obj (essential for elasticity): This is the “Lagrangian approach”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This Lagrangian approach is not objective: Two observers may choose two different initial (times and) configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Initial configuration and Lagrangian “motion” 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition Obj is a material object, �Φ : [t1, t2[×Obj → Rn is its motion, Ωτ = �Φτ(Obj) is its configuration at τ, t0 ∈]t1, t2[ is an “initial time”, and Ωt0 is the initial configuration for the observer who chose t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The motion of Obj relative to the initial configuration Ωt0 = �Φ(t0, Obj) is the function Φt0 : � [t1, t2] × Ωt0 → Rn (t, pt0) �→ pt = Φt0(t, pt0) := �Φ(t, PObj) when pt0 = �Φ(t0, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) So, pt = Φt0(t, pt0) := �Φ(t, PObj) is the position at t of the particle PObj which was at pt0 at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular pt0 = Φt0(t0, pt0) := �Φ(t0, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Marsden and Hughes notations: Once an initial time t0 has been chosen by an observer, then Φt0 =noted Φ, then pt0 =noted P (capital letter for positions at t0) and pt =noted p (lowercase letter for positions at t), so p = Φ(t, P) ∈ Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) (When objectivity is under concern, we need to switch back to the notations Φt0, pt0 and pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') NB: • Talking about the motion of a position pt0 is absurd: A position in Rn does not move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus Φt0 has no existence without the definition, at first, of the motion �Φ of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The domain of definition of Φt0 depends on t0 through Ωt0: The superscript t0 recalls it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And a late observer with initial time t0′ > t0 defines Φt0 ′ which domain of definition is [t1, t2]×Ωt0′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And Φt0 ′ ̸= Φt0 in general because Ωt0′ ̸= Ωt0 in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 21 22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Initial configuration and Lagrangian “motion” The following notation is also used: Φt0(t, pt0) = Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) (The couple (t0, pt0) is “the initial condition”, or t0 and pt0 are the initial conditions, see the § on flows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If a origin O ∈ Rn is chosen by the observer, we may also use, with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6), ⃗xt0 = −−→ Opt0 = ⃗ϕ t0(t0, ⃗xt0) = ⃗X = −−→ OP and ⃗xt = −−→ Opt = ⃗ϕ t0(t, ⃗xt0) = ⃗x = −→ Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Diffeomorphism between configurations With (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1), define Φt0 t : � Ωt0 → Ωt pt0 → pt = Φt0 t (pt0) := Φt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) Hypothesis: For all t0, t ∈]t1, t2[, the map Φt0 t : Ωt0 → Ωt is a Ck diffeomorphism (a Ck invertible function whose inverse is Ck), where k ∈ N∗ depends on the required regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) gives �Φt(PObj) = Φt0 t (�Φt0(PObj)), true for all PObj ∈ Obj, thus Φt0 t ◦ �Φt0 = �Φt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Φt0 t := �Φt ◦ (�Φt0)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) Thus, Φt0 t0 = I and Φt t0 ◦ Φt0 t = (�Φt ◦ (�Φt0)−1) ◦ (�Φt0 ◦ (�Φt)−1) = I give Φt t0 = (Φt0 t )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Trajectories Let (t0, pt0) ∈ [t1, t2] × Ωt0 (initial conditions) and with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) define Φt0 pt0 : � [t1, t2] → Rn t �→ p(t) = Φt0 pt0 (t) := �ΦPObj (t) = Φt0(t, pt0) when pt0 = �ΦPObj (t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Φt0 pt0 is called the (parametric) “trajectory of pt0”, which means: Φt0 pt0 is the trajectory of the particle PObj that is located at pt0 = �Φ(t, PObj) at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the geometric “trajectory of pt0” is Im(Φt0 pt0 ) = Φt0 pt0 ([t1, t2]) = � t∈[t1,t2] {Φt0 pt0 (t)} (= Im(�ΦPObj )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) NB: The terminology “trajectory of pt0” is awkward, since a position pt0 does not move: It is indeed the trajectory �ΦPObj of a particle PObj which is at pt0 at t0 that must be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Streaklines (lignes d’émission) Take a film between t0 and T (start and end).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Let Q be a fixed point in Rn (you see the point Q on each photo that make up the film).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The streakline through Q is the set Et0,T (Q) = {p ∈ Ω : ∃τ ∈ [t0, T] : p = Φτ T (Q) = (ΦT τ )−1(Q)} = {p ∈ Ω : ∃u ∈ [0, T−t0] : p = ΦT −u T (Q) = (ΦT T −u)−1(Q)} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) = the set at T of the positions (a line in Rn) of all the particles which were at Q at a τ ∈ [t0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Smoke comes out of a chimney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Fix a camera nearby, choose a point Q at the top of the chimney where the particles are colored, and make a film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' At T stop filming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (at time T) superimpose the photos in the film: The colored curve we see is the streakline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In other words = � τ∈[t0,T ]{Φτ Q(T)} = � u∈[0,T −t0]{ΦT −u Q (T)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 22 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Lagrangian variables and functions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Lagrangian variables and functions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition Consider a motion �Φ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' An observer chose (subjective) a t0 ∈ [t1, t2] (“in the past”);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So Ωt0 = �Φ(t0, Obj) is his initial configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let m ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 In short: A Lagrangian function, relative to Obj, �Φ and t0, is a function Lagt0 : � [t1, t2] × Ωt0 → ⃗ Rm (or, more generally, some adequat set) (t, pt0) → Lagt0(t, pt0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) and pt0 is called the Lagrangian variable relative to the (subjective) choice t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (To compare with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2): A Eulerian function does not depend on any t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Scalar values: Lagt0(t, pt0) = Θt0(t, pt0) = temperature at t at pt = Φt0 t (pt0) = �Φ(t, PObj) of the particle PObj that was at pt0 at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (So, continuing example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2, Θt0(t, pt0) = θ(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Vectorial values: Lagt0(t, pt0) = ⃗U t0(t, pt0) = force at t at pt = Φt0 t (pt0) = �Φ(t, PObj) acting on the particle PObj that was at pt0 at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (So, continuing example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3, ⃗U t0(t, pt0) = ⃗u(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') If t is fixed or if pt0 ∈ Ωt0 is fixed, then we define Lagt0 t : � Ωt0 → ⃗ Rm (or, more generally, some adequat set) pt0 → Lagt0 t (pt0) := Lagt0(t, pt0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) Lagt0 pt0 : � [t1, t2] → ⃗ Rm (or, more generally, some adequat set) t → Lagt0 pt0 (t) := Lagt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 The position pt0 is also sometimes called a “material point”, which is counter intuitive: PObj (objective) is the material point, and pt0 is just its spatial position at t0 (subjective);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And a Eulerian variable pt is not called a “material point” at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' By the way, the variable pt is also called the “updated Lagrangian variable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 A Lagrangian function is a two point tensor Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 In details: Lagt0 being defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11), a Lagrangian function is a function � Lag t0 : � [t1, t2] × Ωt0 → C × ⃗ Rm (t, pt0) → � Lag t0(t, pt0) = ((t, pt), Lagt0(t, pt0)) when pt = Φt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � Lag t0(t, pt0) = ((t, Φt0 t (pt0)), Lagt0(t, pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (And ⃗ Rm can be replaced by some set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 (Marsden and Hughes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') A Lagrangian function is a “two point vector field” (or more generally a “two point tensor”) in reference to the points pt0 ∈ Ωt0 (departure set) and pt ∈ Ωt (arrival set) where the value Lagt0(t, pt0) is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Interpretation: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) tells that Lagt0(t, pt0) is not represented at (t, pt0), but at (t, pt): That is, having graph(Lagt0) = {((t, pt0), Lagt0(t, pt0)) and Im(� Lag t0) = {((t, pt), Lagt0(t, pt0))}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) we have Im(� Lag t0) ̸= graph(Lagt0) : (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) So a Lagrangian function does not define a tensor in the usual sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' To compare with the Eulerian function Eul which defines a tensor (in particular Im( � Eul) = graph(Eul)), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 23 24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Lagrangian function associated with a Eulerian function 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Lagrangian function associated with a Eulerian function 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition Let �Φ be a motion, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Eul be a Eulerian function, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let t0 ∈ [t1, t2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 The Lagrangian function Lagt0 associated with the Eulerian function Eul is defined by, for all (t, PObj) ∈ [t1, t2] × Obj, Lagt0(t, �Φ(t0, PObj)) := Eul(t, �Φ(t, PObj)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all (t, pt0) ∈ [t1, t2] × Ωt0, Lagt0(t, pt0) := Eul(t, pt), when pt = �Φ(t, PObj) = Φt0 t (pt) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', Lagt0(t, pt0) := Eul(t, pt) when pt0 = (Φt0 t )−1(pt) for all (t, pt) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In other words: Lagt0 t := Eult ◦ Φt0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Remarks If you have a Lagrangian function, then you can associate the function Eult0 t := Lagt0 t ◦ (Φt0 t )−1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) which thus a priori depends on t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But, a Eulerian function is independent of any initial time t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' For one measurement, there is only one Eulerian function Eul, while there are as many associated Lagrangian function Lagt0 as they are t0 (as many as observers): The Lagrangian function Lagt0 ′ of a late observer who chooses t0′ > t0 is different from Lagt0 since the domains of definition Ωt0 and Ωt0′ are different (in general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Lagrangian velocity 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 In short: The Lagrangian velocity at t at pt = �Φ(t, PObj) of the particle PObj is the function ⃗V t0 : � R × Ωt0 → ⃗Rn (t, pt0) → ⃗V t0(t, pt0) := �ΦPObj ′(t) when pt0 = �Φ(t0, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) In details: With (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21), the Lagrangian velocity is the two point vector field given by � ⃗V t0(t, pt0) : � � � R × Ωt0 → C × ⃗Rn (t, pt0) → � ⃗V t0(t, pt0) := ((t, pt), ⃗V t0(t, pt0)), when pt = Φt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) Thus ⃗V t0(t, pt0) = �ΦPObj ′(t) = ⃗v(t, pt) is the velocity at t at pt = �Φ(t, PObj) of the particle PObj which was at pt0 = �Φ(pt0, PObj) at t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ⃗V t0(t, pt0) is not tangent to graph(⃗V t0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16): It is tangent to graph(⃗v) at (t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If t is fixed, or if pt0 ∈ Ωt0 is fixed, then we define ⃗V t0 t (pt0) := ⃗V t0(t, pt0), or ⃗V t0 pt0 (t) := ⃗V t0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) Remark: A usual definition is given without explicit reference to a particle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It is, instead of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21), ⃗V t0(t, pt0) := ∂Φt0 ∂t (t, pt0), ∀(t, pt0) ∈ R × Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Lagrangian velocity versus Eulerian velocity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) give (alternative definition), with pτ = �Φ(τ, PObj), ⃗V t0(t, pt0) = ⃗v(t, pt) (= ∂Φt0 ∂t (t, pt0) = �ΦPObj ′(t) = velocity at t at pt of PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) In other words, ⃗V t0 t = ⃗vt ◦ Φt0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) 24 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Lagrangian acceleration 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Relation between differentials For C2 motions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) gives d⃗V t0 t (pt0) = d⃗vt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΦt0 t (pt0) when pt = Φt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', with F t0 t = dΦt0 t noted = the deformation gradient relative to t0 and t, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) d⃗V t0 t (pt0) = d⃗vt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (pt0) when pt = Φt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) Abusively written (dangerous notation: At what points, relative to what times?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') d⃗V = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Computation of d⃗v called L = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −1 wih Lagrangian variables The Lagrangian approach can be introduced before the Eulerian approach: ⃗V t0 being given, define ⃗vt0(t, pt) := ⃗V t0(t, pt0), when pt = Φt0 t (pt0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗vt0(t, pt) := ⃗V t0(t, Φt0 t −1(pt))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So ⃗vt0(t, Φt0 t (pt0)) = ∂Φt0 ∂t (t, pt0), thus d⃗vt0(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΦt0(t, pt0) = d(∂Φt0 ∂t )(t, pt0) = ∂(dΦt0) ∂t (t, pt0) = ∂F t0 ∂t (t, pt0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) when Φt0 is C2 and F t0 := dΦt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus d⃗vt0(t, pt) = ∂F t0 ∂t (t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0(t, pt0)−1, written in short d⃗v = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −1 (points?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' times?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) And d⃗vt0 t can be written Lt0 t in classical mechanics books, so you can find Lt0 t (pt) := F t0 pt0 (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (pt0)−1, written in short L = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −1 (at what points, what times?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) Here it is not obvious that Lt0 t (pt) does not depend on t0, which is indeed the case, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29): Lt0 t (pt) = d⃗vt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) Reminder: if possible, use Eulerian quantities as long as possible1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Lagrangian acceleration Let PObj ∈ Obj, t0, t ∈ R, pt0 = �ΦPObj (t0) and pt = �ΦPObj (t) (positions of PObj at t0 and t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 In short, the Lagrangian acceleration at t at pt of the particle PObj is ⃗Γ t0(t, pt0) := �ΦPObj ′′(t) when pt0 = �ΦPObj (t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) In other words ⃗Γ t0(t, pt0) := ⃗γ(t, pt) when pt = Φt0(t, pt0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) where ⃗γ(t, pt) = �ΦPObj ′′(t) is the Eulerian acceleration at t at pt = �Φ(t, PObj), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In details, the Lagrangian acceleration is the “two point vector field” defined on R × Ωt0 by � ⃗Γ t0(t, pt0) = ((t, pt), �ΦPObj ′′(t)), when pt = Φt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38) (To compare with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') In particular ⃗Γ t0(t, pt0) is not drawn on the graph of ⃗Γ t0 at (t, pt0), but on the graph of ⃗γ at (t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1To get Eulerian results from Lagrangian computations can make the understanding of a Lie derivative quite difficult: To introduce the “so-called” Lie derivatives in classical mechanics you can find the following steps: 1- At t consider the Cauchy stress vector ⃗t (Eulerian), 2- then with a unit normal vector ⃗n, define the associated Cauchy stress tensor σ (satisfying ⃗t = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3- then use the virtual power and the change of variables in integrals to be back into Ωt0 to be able to work with Lagrangian variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4- then introduce the first Piola–Kirchhoff (two point) tensor PK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 5- then introduce the second Piola–Kirchhoff tensor SK (endomorphism in Ωt0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 6- then differentiate SK in Ωt0 (in the Lagrangian variables although the initials variables are the Eulerian variables in Ωt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 7- then back in Ωt to get back to Eulerian functions (change of variables in integrals),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 8- then you get some Jaumann or Truesdell or other so called Lie derivatives type terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the appropriate choice among all these derivatives being quite obscure because the covariant objectivity has been forgotten en route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' While, with simple Eulerian considerations, it requires a few lines to understand the (real) Lie derivative (Eulerian concept) and its simplicity, see § 9, and deduce second order covariant objective results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 25 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Time Taylor expansion of Φt0 If t is fixed, or if pt0 ∈ Ωt0 is fixed, then define ⃗Γ t0 t (pt0) := ⃗Γ t0(t, pt0), and ⃗Γt0 pt0 (t) := ⃗Γ t0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39) Thus ⃗Γ t0 t = ⃗γt ◦ Φt0 t , and d⃗Γ t0 t (pt0) = d⃗γt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (pt0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) when pt = Φt0 t (pt0) and F t0 t := dΦt0 t (the deformation gradient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Risky notation: d⃗Γ = d⃗γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F (points?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' times?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Time Taylor expansion of Φt0 Let pt0 ∈ Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, at second order, Φt0 pt0 (τ) = Φt0 pt0 (t) + (τ−t)Φt0 pt0 ′(t) + (τ−t)2 2 Φt0 pt0 ′′(t) + o((τ−t)2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) that is, with p(τ) = �ΦPObj (τ) = Φt0 τ (pt0), p(τ) = p(t) + (τ−t)⃗V t0(t, pt0) + (τ−t)2 2 ⃗Γ t0(t, pt0) + o((τ−t)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='42) NB: There are three times involved: t0 (observer dependent), t and τ (for the Taylor expansion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' To compare with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='42): p(τ) = p(t)+(τ−t)⃗v(t, p(t))+ (τ−t)2 2 ⃗γ(t, p(t))+o((τ−t)2), independent of t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 A vector field that let itself be deformed by a motion Consider a C0 Eulerian vector field ⃗w : � C → ⃗Rn (t, pt) → ⃗w(t, pt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let t0 ∈ [t1, t2[ and let ⃗wt0 : � Ωt0 → ⃗Rn pt0 → ⃗wt0(pt0) := ⃗w(t0, pt0) � (vector field in Ωt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then define the (virtual) vector field ⃗wt0∗ : � C → ⃗Rn (t, pt) → ⃗wt0∗(t, pt) := dΦt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0(pt0), when p(t) = Φt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43) (The push-forward = result of the deformation of ⃗wt0 by the motion, see figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 For C2 motions, we have (time variation rate along a virtual trajectory) D ⃗wt0∗ Dt = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0∗, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='44) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L⃗v ⃗wt0∗ = ⃗0, where L⃗v⃗u := D⃗u Dt −d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u (= ∂⃗u ∂t +d⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v −d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u) is the Lie derivative of a (unsteady) vector field ⃗u : C → ⃗Rn along ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Interpretation: We will see that L⃗v ⃗w(t0, pt0) = limt→t0 ⃗w(t,p(t))− ⃗wt0∗(t,p(t)) h measures the “re- sistance of ⃗w to a motion”, see § 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the result L⃗v ⃗wt0∗(t0, pt0) = ⃗0 is “obvious” (= limt→t0 ⃗wt0∗(t,p(t))− ⃗wt0∗(t,p(t)) h ): If ⃗w = ⃗wt0∗ then the vector (“force”) field ⃗w does not oppose any re- sistance to the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' pt0 being fixed, with dΦt0(t, pt0) =noted F(t) we have ⃗wt0∗(t, p(t)) =(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43) F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0(pt0), thus D ⃗wt0∗ Dt (t, p(t)) = F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0(pt0) = F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0∗(t, p(t)) =(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0∗(t, p(t)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4 Deformation gradient F := dΦ Consider a motion �Φ : � R × Obj → Rn (t, PObj) → pt = �Φ(t, PObj) � , Ωt := �Φ(t, Obj) the configuration of Obj at any t, fix t0, t in R, and let Φt0 t : � Ωt0 → Ωt pt0 = �Φ(t0, PObj) → pt = Φt0 t (pt0) := �Φ(t, PObj) � , supposed to be a C1 diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Notations for calculations (quantification), to comply with practices: 26 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definitions 1- Classical (unambiguous) notations as in Arnold, Germain: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', (⃗ai) and (⃗bi) are bases resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' in ⃗Rn t0 and ⃗Rn t , ⃗wt0(pt0) = � i wt0,i(pt0)⃗ai ∈ ⃗Rn t0, ⃗wt,i(pt) = � i wt,i(pt)⃗bi ∈ ⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And 2- Marsden–Hughes duality notations: Capital letters at t0, lower case letters at t, duality notation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ( ⃗EI) and (⃗ei) are bases resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' in ⃗Rn t0 and ⃗Rn t , ⃗W(P) = � I W I(P) ⃗EI ∈ ⃗Rn t0, ⃗w(p) = � i wi(p)⃗ei ∈ ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition of the deformation gradient F Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The differential dΦt0 t =noted F t0 t : � Ωt0 → L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) pt0 → F t0 t (pt0) := dΦt0 t (pt0) � is called “the covari- ant deformation gradient between t0 and t”, or simply “the deformation gradient”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And “the covariant deformation gradient at pt0 between t0 and t”, or in short “the deformation gradient at pt0” is the linear map F t0 t (pt0) ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ), so defined by, for all ⃗wt0(pt0) ∈ ⃗Rn t0 (vector at pt0), F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0(pt0) := lim h→0 Φt0 t (pt0+h⃗wt0(pt0)) − Φt0 t (pt0) h noted = (Φt0 t )∗(⃗wt0)(pt) noted = ⃗wt0∗(t, pt), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) with pt = Φt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Marsden–Hughes notations: Φ := Φt0 t , F := dΦ, P := pt0, ⃗W(P) := ⃗wt0(pt0), p = Φ(P), thus F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W(P) := lim h→0 Φ(P+h ⃗W(P)) − Φ(P) h noted = Φ∗ ⃗W(p) noted = ⃗w∗(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1: ⃗w is a Eulerian vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' At t0 define vector field ⃗wt0 in Ωt0 by ⃗wt0(pt0) := ⃗w(t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The (spatial) curve ct0 : s → pt0 = ct0(s) in Ωt0 is an integral curve of ⃗wt0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' satisfies ct0 ′(s) = ⃗wt0(ct0(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It is transformed by Φt0 t into the (spatial) curve ct = Φt0 t ◦ ct0 : s → pt = ct(s)=Φt0 t (ct0(s)) in Ωt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence ct′(s) = dΦt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ct0 ′(s) = dΦt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0(pt0) =noted ⃗wt0∗(t, pt) is the tangent vector at ct at pt (the push-forward of ⃗wt0 by Φt0 t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ⃗w(t, p(t)) (actual value) is also drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: The “deformation gradient” F t0 t = dΦt0 t is not a “gradient” (its definition does not need a Euclidean dot product);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This lead to confusions when covariance-contravariance and objectivity are at stake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It would be simpler to stick to the name “F t0 t = the differential of Φt0 t ”, but it is not the standard usage, except in thermodynamics: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', the differential dU of the internal energy U is not called “the gradient of U” (there is no meaningful inner dot product): It is just called “the differential of U”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Push-forward (values of F) Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Let ⃗wt0 : � Ωt0 → ⃗Rn t0 pt0 → ⃗wt0(pt0) � be a vector field in Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its push-forward by Φt0 t is the vector field (Φt0 t )∗(⃗wt0) in Ωt defined by (Φt0 t )∗ ⃗wt0(pt) = F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0(pt0) noted = ⃗wt0∗(t, pt) when pt = Φt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) See figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Marsden notation: Φ∗ ⃗W(p) = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W(P) =noted ⃗w∗(p) when p = Φt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 27 042 w(p Cto Ct28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definitions In other words (Φt0 t )∗ ⃗wt0 := (F t0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0) ◦ (Φt0 t )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) Marsden notation: Φ∗ ⃗W = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W) ◦ Φ−1 = ⃗w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 F is a two point tensors With (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1), “the tangent map” is � F t0 t : � Ωt0 → Ωt × L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) pt0 → � F t0 t (pt0) = (pt, F t0 t (pt0)) when pt = Φt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 (Marsden–Hughes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') The function � F t0 t is called a two point tensor, referring to the points pt0 ∈ Ωt0 (departure set) and pt = Φt0 t (pt0) ∈ Ωt (arrival set where ⃗wt0∗(t, pt) = F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0(pt0) is drawn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And in short � F t0 t =noted F t0 t is said to be a two point tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 The name “two point tensor” is a shortcut than can create confusions and errors when dealing with the transposed: F t0 t is not immediately a “tensor”: A tensor is a multilinear form, so gives scalar results (∈ R), while F(P) := F t0 t (P) =noted FP ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) gives vector results (in ⃗Rn t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' However FP can be naturally and canonically associated with the bilinear form �FP ∈ L(⃗Rn∗ t , ⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) defined by, for all ⃗uP ∈ ⃗Rn t0 and ℓp ∈ ⃗Rn∗ t , with p = Φt0 t (P), �FP (ℓp, ⃗uP ) := ℓp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗uP (∈ R), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) see § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13, and it is �FP which defines the so-called “two point tensor”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But don’t forget that the transposed of a linear form (FP here) is not deduced from the transposed of the associated bilinear form ( �FP here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So be careful with the word “transposed” and its two distinct definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Indeed, the transposed of a bilinear form b(·, ·) is intrinsic to b(·, ·) (is objective), given by bT (⃗u, ⃗w) = b(⃗w, ⃗u), while the transposed of a linear function L is not intrinsic to L (is subjective), given by (LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u, ⃗w)g = (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w, ⃗u)h where (·, ·)g and (·, ·)h are inner dot products chosen by human beings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (details in § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 and § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 More generally for manifolds, the differential of Φ := Φt0 t at P ∈ Ωt0 is F(P) := dΦ(P) : � TP Ωt0 → TpΩt ⃗W(P) → ⃗w∗(p) := dΦ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W(P) � with p = Φt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the tangent map is TΦ : � TΩt0 → TΩt (P, ⃗W(P)) → TΦ(P, ⃗W(P)) := (p, dΦ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W(P)) = (p, ⃗w∗(p)), where p = Φt0 t (P), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) called the associated two point tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Evolution: Toward the Lie derivative (in continuum mechanics) Consider a Eulerian vector field ⃗w : � � � C = � t ({t} × Ωt) → ⃗Rn (t, p) → ⃗w(t, p) � � �, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' a “force field”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, at t0 consider ⃗wt0 : � Ωt0 → ⃗Rn t0 pt0 → ⃗wt0(pt0) := ⃗w(t0, pt0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The push-forward of ⃗wt0 by Φt0 t is, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2), ⃗wt0∗(t, p(t)) = F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0(pt0), where p(t) = Φt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) See figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, without any ubiquity gift, at t at p(t) we can compare ⃗w(t, p(t)) (real value of ⃗w at t at p(t)) with ⃗wt0∗(t, p(t)) (transported memory along the trajectory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the rate ⃗w(t, p(t)) − ⃗wt0∗(t, p(t)) t − t0 = actual(t, p(t)) − memory(t, p(t)) t − t0 is meaningful at (t, p(t)) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) (no ubiquity gift required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This rate gives, as h → 0, the Lie derivative L⃗v ⃗w (the rate of stress), and we will see at § 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 that L⃗v ⃗w = D ⃗w Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w (the d⃗v term tells that a “non-uniform flow” acts on the stress).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 28 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification with bases 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Pull-back Formally the pull-back is the push-forward with (Φt0 t )−1: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 The pull-back (Φt0 t )∗ ⃗wt of a vector field ⃗wt defined on Ωt is the vector field defined on Ωt0 by, with pt0 = (Φt0 t )−1(pt), ⃗w∗ t (t0, pt0) = (Φt0 t )∗ ⃗wt(pt0) := (F t0 t )−1(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt(pt), written ⃗W ∗(P) = F −1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Quantification with bases (Simple Cartesian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') (⃗ai) is a Cartesian basis in ⃗ Rn t0, (⃗bi) is a Cartesian basis in ⃗ Rn t , ot is an origin in Rn at t, Φt0 t =noted Φ supposed C1, ϕi : Ωt0 → R is its components in the referential (ot, (⃗bi)): Φ(pt0) = ot + n � i=1 ϕi(pt0)⃗bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' −−−−−→ otΦ(pt0) = n � i=1 ϕi(pt0)⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) Thus, with the classic notation dϕi(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj =noted ∂ϕi ∂Xj (pt0) since (⃗ai) is a Cartesian basis, and (⃗bi) being a Cartesian basis, dΦ(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = n � i=1 (dϕi(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj)⃗bi = n � i=1 ∂ϕi ∂Xj (pt0)⃗bi, thus [dΦ(pt0)][⃗a,⃗b] = [ ∂ϕi ∂Xj (pt0)] = [F(pt0)][⃗a,⃗b], [dΦ(pt0)][⃗a,⃗b] = [F(pt0)][⃗a,⃗b] being the Jacobian matrix of Φ at pt0 relative to the chosen bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In short: dΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = n � i=1 ∂ϕi ∂Xj ⃗bi, thus [dΦ][⃗a,⃗b] = [ ∂ϕi ∂Xj ] = [F][⃗a,⃗b] = [Fij], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) Thus, if ⃗W ∈ ⃗Rn t0 is a vector at pt0 and ⃗W = �n j=1Wj⃗aj then, by linearity of differentials, dΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = n � i=1 FijWj⃗bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W]|⃗b = [F]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗W]|⃗a (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) (more precisely: F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W(pt0) = �n i=1Fij(pt0)Wj(pt0)⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Similarly, for the second order derivative d2Φ = dF (when Φ is C2): With ⃗U = �n j=1Uj⃗aj and ⃗W = �n k=1Wk⃗ak, and with (⃗ai) and (⃗bi) Cartesian bases, we get dF(⃗U, ⃗W) = d2Φ(⃗U, ⃗W) = n � i=1 d2ϕi(⃗U, ⃗W)⃗bi = n � i,j,k=1 ∂2ϕi ∂Xj∂Xk UjWk⃗bi = n � i=1 � [⃗U]T |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[d2ϕi]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗W]|⃗a � ⃗bi, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) [d2ϕi(pt0)]|⃗a = [ ∂2ϕi ∂Xj∂Xk (pt0)] j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n being the Hessian matrix of ϕi at pt0 relative to the basis (⃗ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With Marsden duality notations: p = Φ(P) = ot + n � i=1 ϕi(P)⃗ei, F i J(P) = ∂ϕi ∂XJ (P) (= dϕi(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗EJ), F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = n � i,J=1 F i J(P) W J⃗ei, [F] = [F i J] = [dΦ], dF(⃗U, ⃗W) = d2Φ(⃗U, ⃗W) = n � i,J,K=1 ∂2ϕi ∂XJ∂XK U JW K⃗ei = n � i=1 � [⃗U]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d2ϕi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗W] � ⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 J, j are dummy variables when used in a summation: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = �n j=1 ∂f ∂Xj W j = �n J=1 ∂f ∂XJ W J = �n α=1 ∂f ∂Xα W α = ∂f ∂X1 W 1 + ∂f ∂X2 W 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (there is no uppercase for 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And Marsden–Hughes notations (capital letters for the past) are not at all compulsory, classical notations being just as good and even preferable if you hesitate (because they are not misleading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 29 30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The unfortunate notation d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗X 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 The unfortunate notation d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗X 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Issue (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗w∗(p) := F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W(P), is sometimes written d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗X : “a very unfortunate and misleading notation” (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) which amounts to “confuse a length and a speed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And you also the phrase “(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) is still true if ||d ⃗X|| = 1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' while d ⃗X is supposed to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Where does this unfortunate notation come from?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The notation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) comes from the first order Taylor expansion Φ(Q) = Φ(P) + dΦt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q−P) + o(||Q−P||), where P, Q ∈ Ωt0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', with p = Φt0 t (P) and q = Φt0 t (Q) and h = ||Q−P||, q − p = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q−P) + o(h), written δ⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='δ ⃗X + o(δ ⃗X), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) or −→ pq = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−−→ PQ + o(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So as Q → P we get 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quite useless, isn’t it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' While q − p h = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Q − P h + o(1) is useful: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) As Q → P we get ⃗w∗ = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W which relates tangent vectors, see figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Details: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Interpretation: Vector approach Consider a spatial curve ct0 : � [s1, s2] → Ωt0 s → P := ct0(s) � in Ωt0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It is deformed by Φt0 t to become the spatial curve defined by ct := Φt0 t ◦ ct0 : � [s1, s2] → Ωt s → p := ct(s) = Φt0 t (ct0(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' in Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence, relation between tangent vectors: dct ds (s) = dΦt0 t (ct0(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dct0 ds (s), , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗w∗(p) = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W(P) written d⃗x ds = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗X ds , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) But you can’t simplify by ds to get d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗X: It is absurd to confuse “a slope d ⃗ X ds (s)” and “a length δ ⃗X”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: || dct ds (s)|| = || d ⃗ X ds (s)|| = 1 is meaningful in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19): It means that the parametrization of the curve ct0 in Ωt0 uses a spatial parameter s such that ||ct0 ′(s)|| = 1 for all s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' || ⃗WP || = 1 in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' You cannot simplify by ds: ||d ⃗X|| = 1 is absurd together with d ⃗X “small”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Interpretation: Differential approach (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) is a relation between differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' if you adopt the correct notations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let us do it: With (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11), ⃗x = −→ otp = −−−−−−→ otΦt0 t (P) = n � i=1 ϕi(P)⃗bi noted = n � i=1 xi(P)⃗bi, where ϕi noted = xi (function of P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) Thus, with (πai) = (dXi) the (covariant) dual basis of (⃗ai) we get the system of n equations (functions): dΦ = F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � � � � � � � � � dϕ1(P) = �n j=1 ∂ϕ1 ∂Xj (P) dXj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dϕn(P) = �n j=1 ∂ϕn ∂Xj (P) dXj � � � � � � � � � , which is noted d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗X, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) this last notation being often misunderstood2: It is nothing more than dΦ = F (coordinate free notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2Spivak [17] chapter 4: Classical differential geometers (and classical analysts) did not hesitate to talk about “infinitely small” changes dxi of the coordinates xi, just as Leibnitz had.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' No one wanted to admit that this was nonsense, because true results were obtained when these infinitely small quantities were divided into each other (provided one did it in the right way).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Eventually it was realized that the closest one can come to describing an infinitely small change is to describe a direction in which this change is supposed to occur, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', a tangent vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Since df is supposed to be the infinitesimal change of f under an infinitesimal change of the point, df must be a function of this change, which means that df should be a function on tangent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The dXi themselves then metamorphosed into functions, and it became clear that they must be distinguished from the tangent vectors ∂/∂Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Once this realization came, it was only a matter of making new definitions, which preserved the old notation, and waiting for everybody to catch up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 30 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Tensorial notations, warnings, remarks 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 The ambiguous notation d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗X The bad notation d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗X gives the unfortunate and misunderstood notations d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗X, and then d⃗x = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗x where L = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) Question: What is the meaning (and legitimate notation) of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: d⃗x = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗x means D ⃗wt0∗ Dt = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0∗ = evolution rate of tangent vectors along a trajectory (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) see figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Indeed, ⃗wt0∗(t, p(t)) =(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) F t0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0(pt0) gives D ⃗wt0∗ Dt (t, p(t)) = ∂F t0 ∂t (t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0(pt0) = ∂F t0 ∂t (t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F t0 t (pt0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0∗(t, p(t))), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' D ⃗wt0∗ Dt (t, p(t)) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0∗(t, p(t)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular D ⃗wt0∗ Dt (t0, pt0) = d⃗v(t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0(pt0) is the evolution rate of tangent vectors at t0 at pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Tensorial notations, warnings, remarks As already noted, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6), the linear map F := dΦt0 t (pt0) ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) is naturally canonically associated with the bipoint tensor �F ∈ L(⃗Rn∗ t , ⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) defined by, for all (ℓ, ⃗W) ∈ ⃗Rn∗ t × ⃗Rn t0, �F(ℓ, ⃗W) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) Quantification of �F: basis (⃗ai) with dual basis (πai) in ⃗Rn t0, basis (⃗bi) in ⃗Rn t : if F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = n � i=1 ∂ϕi ∂Xj ⃗bj then �F = n � i,j=1 ∂ϕi ∂Xj ⃗bi ⊗ πaj = n � i=1 ⃗bi ⊗ dϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) And similarly d �F = n � i,j,k=1 ∂2ϕi ∂Xj∂Xk ⃗bi ⊗ (πaj ⊗ πak) = n � i=1 ⃗bi ⊗ d2ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) Warning: The tensorial notation can be misleading, in particular if you use the transposed, see re- mark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, you should always use the standard notation for the linear form F ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) to begin with, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' use F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = �n j=1Fij⃗bi or F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗EJ = �n i,j=1F i J⃗ei (Marsden notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And only use the tensorial notations for calculations purposes at the end (after application of the proper definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 In some manuscripts you find the notation F = dΦ =noted Φ ⊗ ∇X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It does not help to understand what F is (it is the differential dΦ), and should not be used as far as objectivity is concerned: A differentiation is not a tensorial operation, see example R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1, so why use the tensor product notation Φ ⊗ ∇X, when the standard notation dΦ ≃ �F = �n i=1⃗ei ⊗ dϕi is legitimate, explicit, objective and easy to manipulate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And it could be misinterpreted, since, in mechanics, ∇f is often understood to be the vector � i ∂f ∂xi⃗ei (contravariant) which needs a Euclidean dot product to be defined (which one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ), while the differential df is covariant (a differential is unmissable in thermodynamics because you can’t use gradients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It gives the confusing notation Φ ⊗ ∇X ⊗ ∇X, instead of the legitimate d2Φ = �n i=1⃗bi ⊗ d2ϕi which is explicit, objective and easy to manipulate: d2Φ(⃗U, ⃗W) = �n i=1d2ϕi(⃗U, ⃗W)⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 Use Marsden duality notations for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Cartesian bases, with (dXi) the (covariant) dual basis of ( ⃗Ei): with F i J = ∂ϕi ∂XJ , we get dΦ =noted �F = �n i=1⃗ei ⊗ dϕi = �n i,J=1F i J ⃗ei ⊗ dXJ, and d2Φ = �n i=1⃗ei ⊗ d2ϕi = �n i,J,K=1 ∂2ϕi ∂XJ ∂XK ⃗ei ⊗ (dXJ ⊗ dXK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 31 32 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Change of coordinate system at t for F 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Change of coordinate system at t for F Let pt0 ∈ Ωt0, pt = Φt0 t (pt0) ∈ Ωt, ⃗W(pt0) ∈ ⃗Rn t0, ⃗w(pt) = F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W(pt0) ∈ ⃗Rn t (its push-forward), written ⃗w = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The observer at t0 used a basis (⃗ai) in ⃗Rn t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' At t, in ⃗Rn t , a first observer chooses a Cartesian basis (⃗bold,i), and a second observer chooses a Cartesian basis (⃗bnew,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let P = [Pij] be the transition matrix from (⃗bold,i) to (⃗bnew,i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗bnew,j = �n i=1Pij⃗bold,i for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The change of basis formula for vectors from (⃗bold,i) to (⃗bnew,i) (in ⃗Rn t ) gives [⃗w]|⃗bnew = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗bold, thus [F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W]|⃗bnew = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W]|⃗bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) Thus [F]|⃗a,⃗bnew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗W]|⃗a = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[F]|⃗a,⃗bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗W]|⃗a, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) true for all ⃗W, thus [F]|⃗a,⃗bnew = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F]|⃗a,⃗bold .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) NB: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) is not the change of basis formula [L]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P for endomorphisms, which would be nonsense since F := F t0 t (pt0) : ⃗Rn t0 → ⃗Rn t is not an endomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) is just the usual change of basis formula for vectors ⃗w in ⃗Rn t , cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Spatial Taylor expansion of Φ and F Φt0 t =noted Φ is supposed to be C2 for all t0, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let P ∈ Ωt0, dΦ = F, and ⃗W ∈ ⃗Rn t0 vector at P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, in Ωt, Φ(P+h ⃗W) = Φ(P) + h F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W + h2 2 dF(P)( ⃗W, ⃗W) + o(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Time Taylor expansion of F The motion �Φ is supposed to be C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0 ∈ R, Φt0 be the associated motion, p(t) = �Φ(t, PObj) and pt0 = �Φ(t0, PObj), with ⃗v(t, pt) = ∂�Φ ∂t (t, PObj) the Eulerian velocity and ⃗V t0(t, pt0) := ∂Φt0 ∂t (t, pt0) = ⃗v(t, pt) the Lagrangian velocity, and F t0 pt0 (t) = F t0(t, pt0) = dΦt0(t, pt0) =noted F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have ∂F t0 ∂t (t, pt0) = ∂(dΦt0) ∂t (t, pt0) = d(∂Φt0 ∂t )(t, pt0) = d⃗V t0(t, pt0) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t), in short F = d⃗V = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) Then ∂2Φt0 ∂t2 (t, pt0) = ⃗At0(t, pt0) = ⃗γ(t, p(t)) (Lagrangian and Eulerian accelerations), hence ∂2F t0 ∂t2 (t, pt0) = d ⃗At0(t, pt0) = d⃗γ(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t), in short •• F = d ⃗A = d⃗γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) Thus, the second order time Taylor expansion of F t0 pt0 =noted F is, in the vicinity of t, F(t+h) = F(t) + h d⃗V (t) + h2 2 d ⃗A(t) + o(h2) = � I + h d⃗v + h2 2 d⃗γ � (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t) + o(h2) when p(t) = Φt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) NB: They are three times are involved: t and t+h as usual, and t0 through F := F t0 pt0 , ⃗V := ⃗V t0 pt0 and ⃗A := ⃗At0 pt0 (observer dependent), as for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular F(pt0) := F t0 t0 (pt0) = I gives, in the vicinity of t0, F(t0+h) = � I + h d⃗v + h2 2 d⃗γ � (t0, pt0) + o(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 γ = ∂⃗v ∂t + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v is not linear in ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Idem, d⃗γ = d(D⃗v Dt ) = d(∂⃗v ∂t + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) = d∂⃗v ∂t + d2⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v (= D(d⃗v) Dt + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) is non linear in ⃗v, and gives F t0 pt0 ′′(t) = (d ∂⃗v ∂t + d2⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v)(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 pt0 (t), non linear in ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 32 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Homogeneous and isotropic material Exercice 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 Directly check that F ′ = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F gives F ′′ = d⃗γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F ′(t) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t) gives F ′′(t) = D(d⃗v) Dt (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t) + d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F ′(t) with D(d⃗v) Dt = d⃗γ − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36), thus F ′′(t) = (d⃗γ − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v)(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t) + d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t) = d⃗γ(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Homogeneous and isotropic material Let P ∈ Ωt0, let F t0 t (P) := dΦt0 t (P);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Suppose that the “Cauchy stress vector” ⃗ft(pt) à t at pt = Φt0 t (P) only depends on P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' there exists a function ⃗ fun such that ⃗ft(pt) = ⃗ fun(P, F t0 t (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 A material is homogeneous iff ⃗ fun doesn’t depend on the first variable P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', iff, for all P ∈ Ωt0, ⃗ fun(P, F t0 t (P)) = ⃗ fun(F t0 t (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38) (Same mechanical property at any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 (Isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Consider a Euclidean dot product, the same at all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A material is isotropic at P ∈ Ωt0 iff ⃗ fun is independent of the direction you consider, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', iff, for any rotation Rt0(P) in ⃗Rn t0, ⃗ fun(P, F t0 t (P) = ⃗ fun(P, F t0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Rt0(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39) (Mechanical property unchanged when rotating the material first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 A material is isotropic homogeneous iff it is isotropic and homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 The inverse of the deformation gradient ((Φt0 t )−1 ◦ Φt0 t )(P) = P gives, with p = Φt0 t (P), d(Φt0 t )−1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΦt0 t (P) = It0, thus d(Φt0 t )−1(p) = dΦt0 t (P)−1 = F t0 t (P)−1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) where F t0 t = dΦt0 t is the deformation gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have thus define the two point tensor Ht0 t := (F t0 t )−1 : � � � Ωt → L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) p → Ht0 t (p) = (F t0 t )−1(p) := (F t0 t (P))−1 when p = Φt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) So Ht0 t (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(p) = (F t0 t )−1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(p) := F t0 t (P)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(p) ∈ ⃗Rn t0, in short H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='42) for all ⃗w(p) ∈ ⃗Rn t vector at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This defines, with pt = Φt0(t, P), Ht0 : � � � C = � t ({t} × Ωt) → L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) (t, pt) → Ht0(t, pt) := Ht0 t (pt) = (F t0(t, P))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43) NB: Ht0 looks like a Eulerian map, but isn’t: Ht0 depends on a initial time t0 and is a two point tensor (starts in ⃗Rn t , arrives in ⃗Rn t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We will however use the material time derivative D Dt notation in this case, that is, we define, along a trajectory t → p(t) = Φt0(t, P), DHt0 Dt (t, p(t)) := ∂Ht0 ∂t (t, p(t)) + dHt0(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v(t, p(t)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' DHt0 Dt = ∂Ht0 ∂t + dHt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='44) which is the time derivative g′(t) of the function g : t → g(t) = Ht0(t, Φt0(t, P)) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' g(t) = Ht0(t, p(t))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence, with p(t) = Φt0(t, P) and Ht0(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0(t, P) = It0, written H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F = I, we get DH Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂F ∂t = 0, thus DH Dt = −H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45) since ∂F ∂t (t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −1(t, p(t)) = d⃗v(t, p(t)) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 33 34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Introduction: Motion versus flow Exercice 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15 With ⃗wt0∗(t, p(t)) = F t0(t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W(P), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ht0(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0∗(t, p(t)) = ⃗W(P), when p(t) = Φt0(t, P), prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' D ⃗wt0∗ Dt (t, p(t)) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0∗(t, p(t)), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (Ht0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0∗)(t, p(t)) = ⃗W(P) gives DHt0 Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0∗ + Ht0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' D ⃗wt0∗ Dt = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus DHt0 Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0∗ + Ht0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0∗ = 0, thus DH Dt = −H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16 Prove: Ht0 t = Ht0 t1 ◦ Ht1 t and DHt0 Dt (t, p(t)) = Ht0 t1 (pt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' DHt1 Dt (t, p(t)) for all t0, t1 with pt1 = Φt0 t1(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have Φt0 t (pt0) = Φt1 t (Φt0 t1(pt0)), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18), hence F t0 t (pt0) = F t1 t (pt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t1 (pt0), thus F t0 t (pt0)−1 = F t0 t1 (pt0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t1 t (pt1)−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ht0 t (pt) = Ht0 t1 (pt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Ht1 t (p(t)), thus, Ht0(t, p(t)) = Ht0 t1 (pt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Ht1(t, p(t)), thus DHt0 Dt (t, p(t)) = Ht0 t1 (pt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' DHt1 Dt (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 5 Flow 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Introduction: Motion versus flow Motion: A motion �Φ : (t, PObj) → pt = �Φ(t, PObj) locates at t a particle PObj in the affine space Rn, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' From which the Eulerian velocity field ⃗v is deduced: ⃗v(t, pt) := d�ΦPObj dt (t, PObj), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Flow: A flow starts with a Eulerian velocity field ⃗v, from which we deduce a motion by solving the ODE (ordinary differential equation) dΦ dt (t) = ⃗v(t, Φ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Definition Let ⃗v : � R × Rn → ⃗Rn (t, p) → ⃗v(t, p) � be a unstationary vector field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', a Eulerian velocity field which definition domain is C = � t∈[t1,t2]({t} × Ωt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We look for maps Φ : � R → Rn t → p = Φ(t) � which are locally (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' in the vicinity of some t0) solutions of the ODE (ordinary differential equation) dΦ dt (t) = ⃗v(t, Φ(t)), also written dp dt (t) = ⃗v(t, p(t)), or d⃗x dt (t) = ⃗v(t, ⃗x(t)) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) where ⃗x(t) = −−−→ Op(t) after a choice of an origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Also written dp dt = ⃗v(t, p) or d⃗x dt = ⃗v(t, ⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 A solution Φ of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) is a flow of ⃗v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Also called an integral curve of ⃗v since (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) also reads Φ(t) = � t τ=t1 ⃗v(τ, Φ(τ)) dτ + Φ(t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Improper notation for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1): dp dt (t) noted = dp(t) dt (= ⃗v(t, p(t))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) Question: If the notation dp(t) dt is used, then what is the meaning of dp(f(t)) dt ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: It means, either dp dt (f(t)), or d(p◦f) dt (t) = dp dt (f(t)) df dt(t): Ambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So it is better to use dp dt (t), and to avoid dp(t) dt , unless the context is clear (no composite functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Cauchy–Lipschitz theorem Let (t0, pt0) be in the definition domain of ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We look for Φ solution of “the ODE with initial condition (t0, pt0)”, in some vicinity of t0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' such that dΦ dt (t) = ⃗v(t, Φ(t)) and Φ(t0) = pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) (The couple (t0, pt0) is the initial condition, and the values t0 and pt0 are the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Let t1, t2 ∈ R, t1 < t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Ω be an open set in Rn and Ω its closure supposed to be a regular domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='|| be a norm in ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A continuous map ⃗v : [t1, t2] × Ω → ⃗Rn is Lipschitzian iff it is “space Lipschitzian, uniformly in time”, that is, iff ∃k > 0, ∀t ∈ [t1, t2], ∀p, q ∈ Ω, ||⃗v(t, q) − ⃗v(t, p)|| ≤ k||q − p||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) So, ||⃗vt(q)−⃗vt(p)|| ||q−p|| ≤ k, for all t and all p ̸= q (the variations of ⃗v are bounded in space, uniformly in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 34 35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Cauchy–Lipschitz theorem Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 (and definifion) (Cauchy–Lipschitz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If ⃗v : [t1, t2] × Ω → ⃗Rn is Lipschitzian and (t0, pt0) ∈]t1, t2[×Ω, then there exists ε = εt0,pt0 > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) has a unique solution Φ :]t0−ε, t0+ε[→ Rn, noted Φt0 pt0 : dΦt0 pt0 dt (t) = ⃗v(t, Φt0 pt0 (t)) and Φt0 pt0 (t0) = pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) Moreover, if ⃗v is Ck then Φ is Ck+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Arnold [2], or any ODE course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular ||⃗v||∞ := sup t∈]t0−ε,t0+ε[, p∈Ω ||⃗v(t, p)||Rn (maximum speed) exists since ⃗v ∈ C0 on the compact [t1, t2]×Ω), see definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3, hence we can choose ε = min(t0−t1, t2−t0, d(pt0,∂Ω) ||⃗v||∞ ) (the time needed to reach the border ∂Ω from pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have thus defined the function, also called “a flow”, Φ : � ]t1, t2[×]t1, t2[× Ωt0 → Ω (t, t0, pt0) → p = Φ(t, t0, pt0) := Φt0 pt0 (t) noted = Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) And (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) reads ∂Φ ∂t (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, pt0) = ⃗v(t, Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, pt0)), with Φ(t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, pt0) = pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) We have thus defined the function, also called “a flow”, Φt0 : � [t0−ε, t0+ε] × Ωt0 → Rn (t, pt0) → p = Φt0(t, pt0) := Φt0 pt0 (t) : (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) And (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) reads ∂Φt0 ∂t (t, pt0) = ⃗v(t, Φt0(t, pt0)), and Φt0(t0, pt0) = pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) Other definition and notation (can be ambiguous): Φt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0 = Φt0 t : Ωt0 → Rn, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) is written dΦt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0(pt0) dt = ⃗v(t, Φt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0(pt0)), and Φt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0(pt0) = pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Let ⃗v be Lipschitzian, let t0 ∈]t1, t2[, and let Ωt0 be an open set s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ωt0 ⊂⊂ Ω (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' there exists a compact set K ∈ Rn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ωt0 ⊂ K ⊂ Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then there exists ε > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' a flow Φt0 exists on ]t0−ε, t0+ε[×Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let d = d(K, Rn−Ω) (la distance of K to the border of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ||⃗v||∞ := sup t∈[t1,t2],p∈Ω ||⃗v(t, p)||Rn (exists since ⃗v ∈ C0 on the compact [t1, t2] × Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ε = min(t0−t1, t2−t0, d ||⃗v||∞ ) (less that the minimum time to reach the border from K at maximum speed ||v||∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let pt0 ∈ K and t ∈]t0−ε, t0+ε[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then Φt0 pt0 exists, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4, and ||Φt0 pt0 (t) − Φt0 pt0 (t0)||Rn ≤ [t−t0| supτ∈]t0−ε,t0+ε[(||(Φt0 pt0 )′(τ)||Rn) (mean value theorem since, ⃗v being C0, Φ is C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ||Φt0 pt0 (t)− Φt0 pt0 (t0)||Rn ≤ [t − t0| ||v||∞, thus Φt0 pt0 (t) ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus Φt0 pt0 exists on ]t0−ε, t0+ε[, for all pt0 ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 The definition of a flow starts with a Eulerian velocity (independent of any initial time), and then, due to the introduction of initial conditions, leads to the Lagrangian functions Φt0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Once again, Lagrangian functions are the result of Eulerian functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 35 36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Examples 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Examples Example 1 R2 with an origin O, a Euclidean basis (⃗e1,⃗e2) and Ω = [0, 2]×[0, 1] (observation window).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let p ∈ R2, −→ Op =noted ⃗x = x⃗e1 + y⃗e2 =noted (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let t1 = −1, t2 = 1, t0 ∈]t1, t2[, a, b ∈ R, a ̸= 0, and ⃗v(t, p) = � v1(t, x, y) = ay, v2(t, x, y) = b sin(t−t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) (b = 0 corresponds to the stationary case = shear flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') ⃗x(t0) = � x0 y0 � , ⃗x(t) = � x(t) y(t) � = −−−−−−→ OΦt0 pt0 (t) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) give � � � � � dx dt (t) = v1(t, x(t), y(t)) = ay(t), dy dt (t) = v2(t, x(t), y(t)) = b sin(t−t0), with � x(t0) = x0, y(t0) = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) Thus ⃗x(t) = −−−→ Op(t) = −−−−−−→ OΦt0 pt0 (t) = � x(t) = x0 + a(y0 + b)(t−t0) − ab sin(t−t0) y(t) = y0 + b − b cos(t−t0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) Example 2 Similar framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ω > 0 and consider (spin vector field) ⃗v(t, x, y) = � −ωy ωx � = ω � 0 −1 1 0 � � x y � noted = ⃗v(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) With −−→ Opt0 = ⃗xt0 = � xt0 yt0 � , rt0 = � x2 t0 + y2 t0, and θ0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗xt0 = � xt0 = rt0 cos(ωt0) yt0 = rt0 sin(ωt0) � , the solution Φt0 pt0 of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) is ⃗x(t) = −−−→ Op(t) = −−−−−−→ OΦt0 pt0 (t) = � x(t) = rt0 cos(ωt) y(t) = rt0 sin(ωt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) Indeed, � ∂x ∂t (t, ⃗x0) ∂y ∂t (t, ⃗x0) � = � v1(t, x(t, ⃗x0), y(t, ⃗x0)) v2(t, x(t, ⃗x0), y(t, ⃗x0)) � = � −ωy(t, ⃗x0) ωx(t, ⃗x0) � , thus ∂x ∂t (t, ⃗x0) = −ωy(t, ⃗x0) and ∂y ∂t (t, ⃗x0) = ωx(t, ⃗x0), thus ∂2y ∂t2 (t, ⃗x0) = −ω2y(t, ⃗x0), hence y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Idem for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Here d⃗v(t, x, y) = ω � 0 −1 1 0 � = ω � cos π 2 − sin π 2 sin π 2 cos π 2 � is the π/2-rotation composed with the homothety with ratio ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Composition of flows Let ⃗v be a vector field on R × Ω and Φt0 pt0 solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We use the notations pt = Φt0 t (pt0) = Φt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0(pt0) := Φt0 pt0 (t) = Φt0(t, pt0) = Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, pt0) = Φt0,pt0 (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Law of composition of flows (determinism) Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 For all t0, t1, t2 ∈ R, we have (determinism) Φt1 t2 ◦ Φt0 t1 = Φt0 t2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Φt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t1 ◦ Φt1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0 = Φt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) (“The composition of the photos gives the film”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, pt2 = Φt1 t2(pt1) = Φt0 t2(pt0) when pt1 = Φt0 t1(pt0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', pt2 = Φt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t1(pt1) = Φt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0(pt0) when pt1 = Φt1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) Thus dΦt1 t2(pt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΦt0 t1(pt0) = dΦt0 t2(pt0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dΦt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t1(pt1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΦt1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0(pt0) = dΦt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) Summary with commutative diagrams: pt1 Φt1 t2 � pt0 Φt0 t1 � Φt0 t2 � pt2 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' pt1 Φt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t1 � pt0 Φt1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0 � Φt2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0 � pt2 36 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Composition of flows Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let pt1 = Φt0 pt0 (t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) gives � � � � � � � dΦt0 pt0 dt (t) = ⃗v(t, Φt0 pt0 (t)), dΦt1 pt1 dt (t) = ⃗v(t, Φt1 pt1 (t)), � � � � � � � with pt1 = Φt0 pt0 (t1) = Φt1 pt1 (t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus Φt0 pt0 and Φt1 pt1 satisfy the same ODE with the same value at t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus they are equal (uniqueness thanks to Cauchy–Lipschitz theorem), thus Φt1 pt1 (t) = Φt0 pt0 (t) when pt1 = Φt0 t1(pt0), that is, Φt1 t (pt1) = Φt0 t (pt0) when pt1 = Φt0 t1(pt0), which is (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) for any t = t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 A flow is compatible with the motion �Φ of an object Obj: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) gives Φt1 t2 ◦ Φt0 t1 = (�Φt2 ◦ (�Φt1)−1) ◦ (�Φt1 ◦ (�Φt0)−1) = �Φt2 ◦ (�Φt0)−1 = Φt0 t2, that is (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Stationnary case Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 ⃗v is a stationary vector field iff ∂⃗v ∂t = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And then ⃗v(t, p) =noted ⃗v(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the associated flow Φt0 which satisfies ∂Φt0 ∂t (t, pt0) = ⃗v(pt) when pt = Φt0(t, pt0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) is said to be stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 If ⃗v is a stationary vector field then, for all t0, t1, h, when meaningful (h small enough and t1 close enough to t0), Φt1 t1+h = Φt0 t0+h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Φt1+h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t1 = Φt0+h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Φt1 t1+h(q) = Φt0 t0+h(q), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Φ(t1+h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t1, q) = Φ(t0+h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, q) for all q ∈ Ωt0 (see theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In other words, Φt0+h t1+h = Φt0 t1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Φt1+h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0+h = Φt1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Φt0+h t1+h(q) = Φt0 t1(q), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Φ(t1+h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0+h, q) = Φ(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, q) for all q ∈ Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let q ∈ Ωt0, α(h) = Φt0 t0+h(q) = Φt0 q (t0+h) and β(h) = Φt1 t1+h(q) = Φt1 q (t1+h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus α′(h) = dΦt0 q dt (t0+h) = ⃗v(t0+h, Φt0 q (t0+h)) = ⃗v(Φt0 q (t0+h)) = ⃗v(α(h)) (stationary flow), and β′(h) = dΦt1q dt (t1+h) = ⃗v(t1+h, Φt1 q (t1+h)) = ⃗v(Φt1 q (t1+h)) = ⃗v(β(h)) (stationary flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus α and β satisfy the same ODE with the same initial condition α(0) = β(0) = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus α = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, with h = t1−t0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' with t1 = t0+h and t0+h = t1, we get (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 If ⃗v is a stationary vector field, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21), then dΦt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v(pt0) = ⃗v(pt) when pt = Φt0 t (pt0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) that is, if ⃗v is stationary, then ⃗v is transported (push-forwarded by Φt0 t ) along itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18), t2 = t1+s and t1 = t0+s give Φt0+s t1+s(Φt0 t0+s(pt0)) = Φt0 t1+s(pt0), and ⃗v is stationary, thus Φt0 t1(Φt0 t0+s(pt0)) = Φt0 t1+s(pt0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Φ(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, Φt0,pt0 (t0+s)) = Φt0,pt0 (t1+s), thus (s derivative) dΦ(t1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, Φ(t0+s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Φt0,pt0 ′(t0+s) = Φt0,pt0 ′(t1+s), thus dΦt0 t1(Φ(t0+s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v(t0+s, Φt0,pt0 (t0+s)) = ⃗v(t1+s, Φt0,pt0 (t1+s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus with s = 0, and ⃗v being stationary, dΦt0 t1(Φ(t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v(Φt0,pt0 (t0)) = ⃗v(Φt0,pt0 (t1)), thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 37 38 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Velocity on the trajectory traveled in the opposite direction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Velocity on the trajectory traveled in the opposite direction Let t0, t1 ∈ R, t1 > t0, and pt0 ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider the trajectory Φt0 pt0 : � [t0, t1] → Rn t → p(t) = Φt0 pt0 (t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So pt0 is the beginning of the trajectory, pt1 = Φt0 t1(pt0) at the end, ⃗v(t, p(t)) = dΦt0 pt0 dt (t) being the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Define the trajectory traveled in the opposite direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' define Ψt1 pt1 : � [t0, t1] → Rn u → q(u) = Ψt1 pt1 (u) := Φt0 pt0 (t0+t1−u) = Φt0 pt0 (t) = p(t) when t = t0+t1−u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) In particular q(t0) = Ψt1 pt1 (t0) = Φt0 pt0 (t1) = p(t1) and q(t1) = Ψt1 pt1 (t1) = Φt0 pt0 (t0) = p(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 The velocity on the trajectory traveled in the opposite direction is the opposite of the velocity on the initial trajectory: dΨt1 pt1 du (u) = q′(u) = −p′(t) = −⃗v(t, p(t)) when t = t0+t1−u, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ψt1 pt1 (u) = Φt0 pt0 (t0+t1−u) gives dΨt1 pt1 du (u) = − dΦt0 pt0 dt (t0+t1−u) = −⃗v(t0+t1−u, Φt0 pt0 (t0+t1−u)) = −⃗v(t, Φt0 pt0 (t)) when t = t0+t1−u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Variation of the flow as a function of the initial time 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Ambiguous and non ambiguous notations Let Φ : (t, u, p) ∈ R × R × Rn → Φ(t, u, p) ∈ Rn be a C1 function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The partial derivatives are ∂1Φ(t, u, p) := lim h→0 Φ(t+h, u, p) − Φ(t, u, p) h , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) ∂2Φ(t, u, p) := lim h→0 Φ(t, u+h, p) − Φ(t, u, p) h , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) and ∂3Φ(t, u, p), defined for all ⃗w ∈ ⃗Rn (a vector at p) by, ∂3Φ(t, u, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w := lim h→0 Φ(t, u, p+h⃗w) − Φ(t, u, p) h noted = dΦ(t, u, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) When the name of the first variable is systematically noted t, then ∂1Φ(t, u, p) noted = ∂Φ ∂t (t, u, p) noted = ∂Φ(t, u, p) ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) NB: This notation can be ambiguous: What is the meaning of ∂Φ ∂t (t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t, p)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In ambiguous situations, use the notation ∂1Φ, or (if no composed functions inside) use ∂Φ(t,u,p) ∂t |u=t (so t is the derivation variable, and after the calculation you take u = t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' When the name of the second variable is systematically noted u, then ∂2Φ(t, u, p) noted = ∂Φ ∂u (t, u, p) noted = ∂Φ(t, u, p) ∂u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) NB: Idem this notation can be ambiguous: What is the meaning of ∂Φ ∂u (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' u, p)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In ambiguous situations, use the notation ∂2Φ, or use ∂Φ(t,u,p) ∂u |t=u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' When the name of the third variable is systematically a space variable noted p, then ∂3Φ(t, u, p) noted = dΦ(t, u, p) noted = ∂Φ ∂p (t, u, p) noted = ∂Φ(t, u, p) ∂p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) 38 39 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Variation of the flow as a function of the initial time 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Variation of the flow as a function of the initial time The law of composition of the flows gives (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) gives Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' u, Φ(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, p0)) = Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the derivative in u gives ∂2Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' u, Φ(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, p0)) + dΦ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' u, Φ(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, p0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂1Φ(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, p0) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂2Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' u, p(u)) = −dΦ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' u, p(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v(u, p(u)) when p(u) = Φ(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) In particular u = t0 gives, for all (t, t0, p0) ∈ R2 × Ωt0, (∂Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, p0) ∂t0 =) ∂2Φ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, p0) = −dΦ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v(t0, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) In particular (dΦ(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, p0) dt0 |t=t0 =) ∂2Φ(t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, p0) = −⃗v(t0, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) 39 40 Part II Push-forward 6 Push-forward The general tool to describe “transport” is “push-forward by a motion” (the “take with you” operator), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 and figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The push-forward also gives the tool needed to understand the velocity addition formula: In that case, the push-forward is the translator between observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The push-forward can also be used to write coordinate systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' As usual, we start with qualitative results (observer independent results);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, quantitative results are deduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition E and F are affine spaces, E and F are the associated vector spaces equipped with norms ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||E and ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||F with dim E = dim F = n, UE and UF are open sets in the affine space E and F, or possibly the vector spaces E and F, and Ψ : � UE → UF pE → pF = Ψ(pE) is a diffeomorphism (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) (a C1 invertible map which inverse is C1), called the push-forward, and Ψ−1 is the pull-back (push-forward with Ψ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1: cE : s → pE = cE(s) is a curve in UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Push-forwarded by Ψ it becomes the curve cE∗ := Ψ ◦ cE in UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The tangent vector at pE = cE(s) is ⃗wE(pE) = cE ′(s), and the tangent vector at pF = cF(s) = Ψ(cE(s)) is ⃗wE∗(pF) = cF ′(s) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Other illustation: See figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example: Ψ = Φt0 t : Ωt0 → Ωt, the motion that transforms Ωt0 into Ωt, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example: Ψ : UE → UF a coordinate system, see example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example: Ψ = Θt : RB → RA, a change of referential at t (change of observer), see § 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Push-forward and pull-back of points Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 If pE ∈ UE (a point in UE) then its push-forward by Ψ is the point pF = Ψ∗pE := Ψ(pE) = pE∗ ∈ UF, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) see figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1, the last notation if Ψ is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And if pF ∈ UF then its pull-back by Ψ is the point pE = Ψ∗pF := Ψ−1(pF) = pF ∗ ∈ UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) We immediately have Ψ∗ ◦ Ψ∗ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The notations ∗ for push-forward and ∗ for pull-back have been proposed by Spivak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Also see Abraham and Marsden [1] (second edition) who adopt this notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 40 Us 亚 we(pe p =C(s P = 亚(p) Im(c* Im(C41 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Push-forward and pull-back of curves 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Push-forward and pull-back of curves We push-forward (and pull-back) the points on a curve: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Let cE : � ] − ε, ε[ → UE s → pE = cE(s) � be a curve in UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its push-forward by Ψ is the curve Ψ∗cE := Ψ ◦ cE : � ] − ε, ε[ → UF s → pF = Ψ∗cE(s) := Ψ(cE(s)) noted = cE∗(s) (= Ψ(pE)), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) see figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Ψ∗cE =noted cE∗ when Ψ is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') This defines Ψ∗ : � F(] − ε, ε[;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' UE) → F(] − ε, ε[;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' UF) cE → Ψ∗(cE) := Ψ ◦ cE noted = Ψ∗cE = cE∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Let cF : � ] − ε, ε[ → UF s → pF = cF(s) � is a curve in UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its pull-back by Ψ is Ψ∗cF := Ψ−1 ◦ cE � ] − ε, ε[ → UE s → pE = Ψ∗cF(s) := Ψ−1(cF(s)) noted = cF ∗(s) (= Ψ−1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) We have thus defined Ψ∗ : � F(C1(] − ε, ε[;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' UF) → F(C1(] − ε, ε[;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' UE) cF → Ψ∗(cF) := Ψ−1 ◦ cF noted = Ψ∗cF = cF ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Push-forward and pull-back of scalar functions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definitions Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Let fE : � UE → R pE → fE(pE) � (scalar valued function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its push-forward by Ψ is the (scalar valued) function Ψ∗fE := fE ◦ Ψ−1 : � UF → R pF → Ψ∗fE(pF) := fE(pE) noted = fE∗(pF) when pE = Ψ−1(pF), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) (noted fE∗ when Ψ is implicit), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ψ∗fE(Ψ∗pE) := fE(pE), or fE∗(pE∗) := fE(pE) when pE∗ = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have thus defined Ψ∗ : � F(UE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) → F(UF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) fE → fF := Ψ∗(fE) = fE ◦ Ψ−1 noted = Ψ∗fE, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) the notation Ψ∗(fE) = Ψ∗fE since Ψ∗ is linear: ((fE + λgE) ◦ Ψ−1)(pF) = (fE + λgE)(pE) = fE(pE) + λgE(pE) = (fE ◦ Ψ−1)(pF) + λ(gE ◦ Ψ−1)(pF) gives Ψ∗(fE + λgE) = Ψ∗(fE) + λΨ∗(gE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Let fF : � UF → R pF → fF(pF) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its pull-back by Ψ is the push-forward by Ψ−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' is Ψ∗fF := fF ◦ Ψ : � UE → R pE → Ψ∗fF(pE) := fF(pF) noted = fF ∗(pE) when pF = Ψ(pE), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ψ∗fF(Ψ∗pF) := fF(pF), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' fF ∗(pF ∗) := fF(pF) when pF = Ψ∗(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have thus defined Ψ∗ : � F(UF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) → F(UE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) fF → Ψ∗(fF) = fF ∗ := fF ◦ Ψ noted = Ψ∗fF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) We immediately have Ψ∗ ◦ Ψ∗ = I and Ψ∗ ◦ Ψ∗ = I (the first I is the identity in F(UE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), the second I is the identity in F(UF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: We used the same notations Ψ∗ and Ψ∗ than for the push-forward and pull-backs of points: The context removes ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 41 42 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Push-forward and pull-back of vector fields 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Interpretation: Why is it useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' : Let �Φ : R × Obj → Rn be a motion of an object Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' An observer records the temperature θ at all t ∈ [t0, T] and all p = �Φ(t, Obj): He gets θ : � � � C = � t ({t} × Ωt) → R (t, p) → θ(t, p) � � � a Eulerian scalar valued function, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then he chooses an initial time t0 and considers the associated motion Φt0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1), and considers θt0 : � Ωt0 → R pt0 → θt0(pt0) := θ(t0, pt0) � (snapshot of the temperatures at t0 in Ωt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The push-forward of θt0 by Φt0 t is (Φt0 t )∗θt0 := θt0 ◦ (Φt0 t )−1 defines the “memory function” (Φt0 t )∗θt0 : � Ωt → R pt → (Φt0 t )∗θt0(pt) := θt0(pt0) when pt = Φt0 t (pt0), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) And he writes (Φt0 t )∗θt0(pt) =noted θt0∗(t, pt), so the memory transported is at t at pt (along a trajectory) by θt0∗(t, p(t)) = θt0(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) Question: Why do we introduce θt0∗ since we have θt0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: An observer does not have the gift of temporal and/or spatial ubiquity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' He has to do with values at the actual time t and position pt where he is (Newton and Einstein’s point of view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, when he was at t0 at pt0 the observer wrote the value θt0(pt0) on a piece of paper (for memory), puts the piece of paper is his pocket, then once at t at p(t) = Φt0(t, pt0), he takes the paper out of his pocket, and renames the value he reads as θt0∗(t, pt) because he is now at t at pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And, now at t at pt, he can compare the past and present value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular the rate θ(t, p(t)) − θt0∗(t, p(t)) t − t0 = actual(t, p(t)) − memory∗(t, p(t)) t − t0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) is physically meaningful for one observer at t at pt (no ubiquity gift required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' For scalar value functions, we get the usual rate θ(t,p(t))−θ(t0,p(t0)) t−t0 , but it isn’t that simple for vector valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the limit t → t0 in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) defines the Lie derivative for scalar valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Push-forward and pull-back of vector fields This is one of the most important concept for mechanical engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 A definition by approximation Elementary introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let pE and qE be points in UE, and let pF = pE∗ = Ψ(pE) and qF = qE∗ = Ψ(qE) in UF be the push-forwards by Ψ cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The first order Taylor expansion gives (Ψ(qE) − Ψ(pE) =) qF − pF = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (qE − pE) + o(||qE − pE||E), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) thus, −−→ pFqF ||−−→ pEqE||E = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' −−→ pEqE ||−−→ pEqE||E + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) And the definition of the push-forward is obtained by “neglecting” the o(1) (limit as qE → pE): Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 If ⃗wE(pE) ∈ E is a vector at pE ∈ U then its push-forward by Ψ is the vector ⃗wF(pF) =noted ⃗wE∗(pF) =noted Ψ∗ ⃗wE(pF) ∈ F defined at pF = pE∗ = Ψ(pE) ∈ UF by ⃗wF(pF) = ⃗wE∗(pF) := dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE) noted = Ψ∗ ⃗wE(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The definition of the push-forward of a vector field To fully grasp the definition, and to avoid making interpretation errors as in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 (the unfortunate notation d⃗x = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗X), we use the following definition of “a vector”: It is a “tangent vector to a curve” (needed for surfaces and manifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Details: 42 43 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Push-forward and pull-back of vector fields Let cE : � ] − ε, ε[ → UE s → pE = cE(s) � be a C1 curve in UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its tangent vector at pE = cE(s) is ⃗wE(pE) := cE ′(s) (= lim h→0 cE(s + h) − cE(s) h ), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) see figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This defines the function ⃗wE : � Im(cE) → E pE → ⃗wE(pE) � called a vector field along Im(cE)⊂UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The push-forward of cE by Ψ being the image curve cE∗ = Ψ ◦ cE (the curve transformed by Ψ) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4), its tangent vector at pF = cE∗(s) is ⃗wE∗(pF) := cE∗ ′(s) thus = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='cE ′(s) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) Thus we have defined the vector field ⃗wE∗ along Im(cE∗) called the push-forward of ⃗wE by Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With all the integral curves of a vector field defined in UE, we get: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 The push-forward by Ψ of a C0 vector field ⃗wE : � UE → E pE → ⃗wE(pE) � is the vector field Ψ∗ ⃗wE = ⃗wE∗ : � � � UF → F pF → Ψ∗ ⃗wE(pF) := dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE) noted = ⃗wE∗(pF) when pF = Ψ(pE), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) see figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Ψ∗ ⃗wE =noted ⃗wE∗ if Ψ is implicit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In other words, Ψ∗ ⃗wE := (dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE) ◦ Ψ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) This defines the map Ψ∗ : � C∞(UE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) → C∞(UF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) ⃗wE → Ψ∗(⃗wE) := Ψ∗ ⃗wE = ⃗wE∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (We use the same notation Ψ∗ as in definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 for scalar valued functions: The context removes ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Unlike scalar functions, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2: At t0 at pt0 you cannot just draw a vector ⃗wt0(pt0) on a piece of paper, put the paper in your pocket, then let yourself be carried by the flow Ψ = Φt0 t (push-forward), then, once arrived at t at pt, take the paper out of your pocket and read it to get the push-forward: The direction and length of the vector ⃗wt0∗(t, pt) are modified by the flow (a vector is not just a collection of scalar components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 Prove: ⃗cE ′′(s) = d⃗wE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) and d⃗wE∗(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗wE(pE) + d2Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) and cE∗ ′′(s) = d⃗wE∗(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE∗(pF) (= dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗cE ′′(s) + d2Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗cE ′(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗cE ′(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗cE ′(s) = ⃗wE(cE(s)) gives ⃗cE ′′(s) = d⃗wE(cE(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗cE ′(s), hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗wE∗(Ψ(pE)) = dΦ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE) by definition of ⃗wE∗, hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' cF(s) = Ψ(cE(s)) gives ⃗cF ′(s) = dΨ(cE(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗cE ′(s) = dΨ(cE(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(cE(s)) = ⃗wE∗(cF(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ⃗cF ′′(s) = (d2Ψ(cE(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗cE ′(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗cE ′(s) + dΨ(cE(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗cE ′′(s) = d⃗wE∗(cF(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗cF ′(s), hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Pull-back of a vector field Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 If ⃗wF : � UF → F pF → ⃗wF(pF) � is a vector field on UF, then its pull-back by Ψ is the push-forward by Ψ−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' is the vector field on UE defined by Ψ∗ ⃗wF : � � � UE → E pE → Ψ∗ ⃗wF(pE) := dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF(pF) noted = ⃗wF ∗(pE), when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) In other words, Ψ∗ ⃗wF := (dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF) ◦ Ψ noted = ⃗wF ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) 43 44 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification with bases And we get Ψ∗ ◦ Ψ∗ = I and Ψ∗ ◦ Ψ∗ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) Indeed, Ψ∗(Ψ∗ ⃗wE)(pE) = dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Ψ∗ ⃗wE(pF) = dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE) = ⃗wE(pE), for all pE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Idem for the second equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Quantification with bases 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Usual result (⃗ai) is a Cartesian basis in E, OF and (⃗bi) are an origin in F and a Cartesian basis in F, pE ∈ UE, pF = Ψ(pE) = OF + n � i=1 ψi(pE)⃗bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [−−−→ OFpF]|⃗b = � � � ψ1(pE) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ψn(pE) � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) Then, if ⃗wE is a vector field in UE and ⃗wE = � i wj⃗ai, we get Ψ∗ ⃗wE(pF) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE) = �n i=1(dψi(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE))⃗bi = �n i,j=1wj(pE)(dψi(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj)⃗bi = �n i,j=1 ∂ψi ∂xj (pE)wj(pE)⃗bi, so [Ψ∗ ⃗wE(pF)]|⃗b = [dΨ(pE)]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗wE(pE)]|⃗a, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) where [dΨ(pE)]|⃗a,⃗b = [dψi(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj] =noted [ ∂ψi ∂xj (pE)] is the Jacobian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Example: Polar coordinate system Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 Change of coordinate system interpreted as a push-forward: Paradigmatic example of the polar coordinate system (model generalized for the parametrization of any manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Parametric Cartesian vector space R × R =noted ⃗R2 p = {⃗q = (r, θ)}, with its canonical basis (⃗a1,⃗a2), and ⃗q = r⃗a1 + θ⃗a2 =noted (r, θ), so [⃗q]|⃗a = � r θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Geometric affine space R2 (of positions), p ∈ R2, associated vector space ⃗R2, O ∈ R2 (origin), ⃗x = −→ Op, and a Euclidean basis (⃗b1,⃗b2) in ⃗R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The “polar coordinate system” is the associated map Ψ : � ⃗R∗ + × R ⊂ ⃗R2 p → ⃗R2 ⃗q = (r, θ) → ⃗x = Ψ(⃗q) = Ψ(r, θ), � defined by ⃗x = Ψ(⃗q) := r cos θ⃗b1 + r sin θ⃗b2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗b = � x = r cos θ y = r sin θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) The i-th coordinate line at ⃗q in ⃗R2 p (parametric space) is the straight line ⃗c⃗q,i : � R → ⃗R2 p s → ⃗c⃗q,i(s) = ⃗q + s⃗ai � , and its tangent vector at ⃗c⃗q,i(s) is ⃗c⃗q,i′(s) = ⃗ai for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This line is transformed by Ψ into the curve Ψ∗(cq,i) = Ψ ◦ ⃗c⃗q,i =noted c⃗x,i : � R → R2 s → c⃗x,i(s) = Ψ(⃗q + s⃗ai) � (in particular c⃗x,i(0) = ⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So [−−−−−→ Oc⃗x,1(s)]|⃗b = � (r+s) cos θ (r+s) sin θ � (straight line), and [−−−−−→ Oc⃗x,2(s)]|⃗b = � r cos(θ+s) r sin(θ+s) � (circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) And the tangent vector at c⃗x,i(s) is c⃗x,i′(s) =noted ⃗ai∗(⃗x) (push-forward by Ψ), so ⃗a1∗(⃗x) := Ψ∗⃗a1(⃗x) = dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1 = lim h→0 Ψ(⃗q+h⃗a1) − Ψ(⃗q) h = lim h→0 Ψ(r+h, θ) − Ψ(r, θ) h = ∂Ψ ∂r (⃗q), ⃗a2∗(⃗x) := Ψ∗⃗a2(⃗x) = dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2 = lim h→0 Ψ(⃗q+h⃗a2) − Ψ(⃗q) h = lim h→0 Ψ(r, θ+h) − Ψ(r, θ) h = ∂Ψ ∂θ (⃗q), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) Thus ⃗a1∗(⃗x) = cos θ⃗b1 + sin θ⃗b2 and ⃗a2∗(⃗x) = −r sin θ⃗b1 + r cos θ⃗b2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗a1∗(⃗x)]|⃗b = � cos θ sin θ � and [⃗a2∗(⃗x)]|⃗b = � −r sin θ r cos θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) The basis (⃗a1∗(⃗x),⃗a2∗(⃗x)) is called the basis of the polar coordinate system at ⃗x (it is orthogonal but not orthonormal since ||⃗a2∗(⃗x)|| = r ̸= 1 in general);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And [dΨ(⃗q)]|⃗a,⃗b = � [ ∂Ψ ∂r (⃗q)]|⃗b [ ∂Ψ ∂θ (⃗q)]|⃗b � = � [⃗a1∗(⃗x)]|⃗b [⃗a2∗(⃗x)]|⃗b � = � cos θ −r sin θ sin θ r cos θ � = [ ∂Ψi ∂qj (⃗q)] is the Jacobian matrix of Ψ at ⃗q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 44 45 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification with bases And the dual basis of the polar system basis (⃗a1∗(⃗x),⃗a2∗(⃗x)) is called (dq1(⃗x), dq2(⃗x)) (defined by dqi(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj∗(⃗x) = δij), so dq1(⃗x) = cos θ dx1 + sin θ dx2 and dq2(⃗x) = −1 r sin θ dx1 + 1 r cos θ dx2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dq1(⃗x)]|⃗b = ( cos θ sin θ ) and [dq2(⃗x)]|⃗b = − 1 r ( sin θ cos θ ) (row matrices) when ⃗x = Ψ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 The components γk ij(⃗x) of the vector d⃗aj∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai∗(⃗x) ∈ ⃗R2 in the basis (⃗ai∗(⃗x)) are the Christoffel symbols of the polar coordinate system (with duality notations as it is usually presented): d⃗aj∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai∗(⃗x) = n � k=1 γk ij(⃗x)⃗ak∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) At ⃗x = Ψ(⃗q), with ⃗aj∗(⃗x) = dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗aj∗ ◦ Ψ)(⃗q) = ∂Ψ ∂qj , we get d⃗aj∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai∗(⃗x) = ∂2Ψ ∂qi∂qj (⃗q) = d⃗ai∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj∗(⃗x), so γk ij = γk ji (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) for all i, j (symmetry of the bottom indices as soon as Ψ is C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Here for the polar coordinates, ∂Ψ ∂r (⃗q) = cos θ⃗b1 + sin θ⃗b2 gives ∂2Ψ ∂r2 (⃗q) = ⃗0, thus γ1 11 = γ2 11 = 0, and ∂2Ψ ∂θ∂r(⃗q) = − sin θ⃗b1 + cos θ⃗b2 = 1 r⃗a2∗(⃗x), thus γ1 12 = 0 = γ1 21 and γ2 12 = 1 r = γ2 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ∂Ψ ∂θ (⃗q) = −r sin θ⃗b1 + r cos θ⃗b2 gives ∂2Ψ ∂θ2 (⃗q) = −r cos θ⃗b1 − r sin θ⃗b2 = −r⃗a1∗(⃗x), thus γ1 22 = −r and γ2 22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 The (widely used) normalized polar coordinate basis (⃗n1(⃗x),⃗n2(⃗x)) = (⃗a1∗(⃗x), 1 r⃗a2∗(⃗x)) is not holonomic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' is not the basis of a coordinate system (and its use makes higher deriva- tion formulas complicated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Indeed ⃗n2(⃗x) = 1 r⃗a2∗(⃗x) gives d⃗n2(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n1(⃗x) = (d( 1 r)(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n1(⃗x))⃗a2∗(⃗x) + 1 rd⃗a2∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n1(⃗x), and ⃗n1(⃗x) = ⃗a1∗(⃗x) gives d⃗n1(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n2(⃗x) = d⃗a1∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ( 1 r⃗a2∗), thus d⃗n2(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n1(⃗x) − d⃗n1(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n2(⃗x) = (d( 1 r)(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n1(⃗x))⃗a2∗(⃗x) ̸= ⃗0, since 1 r = (x2 + y2)− 1 2 gives d( 1 r)(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n1(⃗x) = ( −x(x2 + y2)− 3 2 −y(x2 + y2)− 3 2 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � cos θ sin θ � = 1 r3 (−r cos2 θ − r sin2 θ) = −1 r2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 (Pay attention to the notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Let f : ⃗q ∈ ⃗R2 p → f(⃗q) ∈ R be C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Call g its push- forward by Ψ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' g : ⃗x ∈ R2 → g(⃗x) = f(⃗q) ∈ R when ⃗x = Ψ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So f(⃗q) = (g ◦ Ψ)(⃗q)and df(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = dg(Ψ(⃗q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = dg(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38) With df(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj =noted ∂f ∂qj (⃗q) and dg(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj =noted ∂g ∂xj (⃗x) and ⃗aj∗(⃗x) = dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = � i ∂Ψi ∂qj (⃗q)⃗aj, we get ∂f ∂qj (⃗q) = � i ∂g ∂xi (⃗x)∂Ψi ∂qj (⃗q) noted = ∂g ∂qj (⃗x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39) Mind this notation!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' g is a function of ⃗x, not of ⃗q, so ∂g ∂qi (⃗x) means = ∂f ∂qi (⃗q), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂g ∂qi (⃗x) means = ∂(g ◦ Ψ) ∂qi (⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' which is [df(⃗q)] = [dg(⃗x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (with f and Ψ C2) ∂ ∂g ∂qi ∂qj (⃗x) means = ∂ ∂(g◦Ψ) ∂qi ∂qj (⃗q) = d(dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai∗)(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = d(dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai∗)(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj∗(⃗x) = d((dg(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj∗(⃗x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai∗(⃗x) + dg(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗ai∗(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj(⃗x)) noted = ∂2g ∂qi∂qj (⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) So ∂2g ∂qi∂qj (⃗x) means = d2g(⃗x)(⃗ai∗(⃗x),⃗aj∗(⃗x)) + n � k=1 ∂g ∂xk (⃗x)γk ij(⃗x)⃗ak(⃗x), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) and ∂2g ∂qi∂qj (⃗x) is not reduced to d2g(⃗x)(⃗ai∗(⃗x),⃗aj∗(⃗x)) (the Christoffel symbols have appeared): First order derivatives ∂g ∂xk are still alive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Contrary to ∂2g ∂xi∂xj (⃗x) = d2g(⃗x)(⃗bi,⃗bj) with a Cartesian basis (⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') NB: The independent variables r and θ don’t have the same dimension (a length and an angle): There is no physical meaningful inner dot product in the parameter space ⃗R2 p = R×R = {(r, θ)}, but this space is very useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (As in thermodynamics: No meaningful inner dot product in the (T, P) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 45 46 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 7 Push-forward and pull-back of differential forms 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition Setting of § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider a differential form αE : � UE → E∗ = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) pE → αE(pE) � on UE (a field of linear forms), and a vector field ⃗wE : � UE → E pE → ⃗wE(pE) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence fE = αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE : � UE → R pE → fE(pE) = (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE)(pE) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE) is a scalar valued function (value of ⃗wE given by αE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) gives (push-forward fE = αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE by Ψ) Ψ∗(αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE)(pF) = (αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE)(pE) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) With ⃗wE∗(pF) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) (push-forward of ⃗wE), we get Ψ∗(αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE)(pF) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE)−1 � �� � =noted αE∗(pF) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF(pF) when pF = Ψ(pE) : (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The push-forward of a differential form αE ∈ Ω1(UE) is the differential form ∈ Ω1(UF) given by Ψ∗αE : � � � UF → F ∗ = L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) pF → Ψ∗αE(pF) := αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE)−1 noted = αE∗(pF) when pF = Ψ(pE), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) the last notation when Ψ is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In other words, Ψ∗αE(pF) = αE(Ψ−1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1(pF), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ψ∗αE := (αE ◦ Ψ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) (Once again, we used the same notation Ψ∗ than for the push-forward of vector fields and functions: The context removes any ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 We cannot always see a vector field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' we can’t see an internal force field): To know it we need to measure it with a well defined tool, the tool being here a differential form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 is a compatbility definition so that we can recover the push-forward of the vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 The pull-forward of a a differential form αF ∈ Ω1(UF) is the differential form Ψ∗αF : � UE → L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) pE → Ψ∗αF(pE) := αF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE) noted = αF ∗(pE) when pF = Ψ(pE), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) In other words, Ψ∗αF := (αF ◦ Ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) (For an alternative definition, see remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 For all αE ∈ Ω1(UE) and αF ∈ Ω1(UF) (differential forms), and ⃗wE ∈ Γ(UE) and ⃗wF ∈ Γ(UF) (vector fields), we have (objectivity result) (Ψ∗αE)(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF(pF) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Ψ∗ ⃗wF)(pE) when pF = Ψ(pE), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' αE∗(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF(pF) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF ∗(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular with αE = df (exact differential form) where f ∈ C1(UE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), d(Ψ∗f) = Ψ∗(df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) (This commutativity result is very particular to the case α = df: In general d(Ψ∗T) ̸= Ψ∗(dT) for a tensor of order ≥ 2, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 46 47 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Incompatibility: Riesz representation and push-forward Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' αE∗(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF(pF) = (αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF(pF) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF(pF)) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w∗ F(pE), for all pF = Ψ(pE) ∈ UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And Ψ∗f(pF) := f(pE) = f(Ψ−1(pF)), thus d(Ψ∗f)(pF) = df(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1(pF) = Ψ∗(df)(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And we have Ψ∗ ◦ Ψ∗ = I and Ψ∗ ◦ Ψ∗ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) Indeed Ψ∗(Ψ∗αE)(pE) = Ψ∗αE(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Idem for Ψ∗ ◦ Ψ∗ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 The pull-back αF ∗ can also be defined thanks to the natural canonical isomorphism � L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) → L(F ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗) L → L∗ � given by L∗(ℓF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗uE = ℓF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗uE) for all (⃗uE, ℓF ) ∈ E×F ∗, and L∗(ℓF ) = ℓF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L is called the pull-back of ℓF by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular with ℓF = αF(pF) and L = dΨ(pE) we get dΨ(pE)∗(αF(pF)) = αF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Incompatibility: Riesz representation and push-forward A push-forward is independent of any inner dot product: It is objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But here we introduce inner dot products (·, ·)g in E and (·, ·)h in F, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Euclidean dot products in ⃗Rn t0 and ⃗Rn t (observer dependent therefore subjective), because some mechanical engineers can’t begin with their beloved Euclidean dot products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let αE ∈ Ω1(UE) and call βF := Ψ∗αE its push-forward by Ψ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' βF(pF) := αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE)−1 when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) Then call ⃗ag(pE) ∈ E and ⃗bh(pF) ∈ F the (·, ·)g and (·, ·)h-Riesz representation vectors of αE and βF, so, for all ⃗uE ∈ Γ(UE) and all ⃗wF ∈ Γ(UF), in short, αE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗uE = (⃗ag, ⃗uE)g, and βF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF = (⃗bh, ⃗wF)h, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) which means αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗uE(pE) = (⃗ag(pE), ⃗uE(pE))g and βF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF(pF) = (⃗bh(pF), ⃗wF(pF))h, for all pE ∈ UE and pF ∈ UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This defines the vector fields ⃗ag ∈ Γ(UE) and ⃗bh ∈ Γ(UF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 ⃗bh ̸= Ψ∗⃗ag in general (although βF = Ψ∗αE), because ⃗bh(pF) = dΨ(pE)−T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ag(pE) ̸= dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ag(pE) in general (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) (unless dΨ(pE)−T = dΨ(pE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dΨ(pE)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE)−1 = I, as a rigid body motion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So the Riesz representation vector of the push-forwarded linear form is not the push-forwarded rep- resentation vector of the linear form push-forwarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This is not a surprise: A push-forward is independent of any inner dot product, while a Riesz repre- sentation vector depends on a chosen inner dot product (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Euclidean foot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' metre?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, as long as possible (not before you need to quantify), you should avoid using a Riesz representation vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' you should use the original (the qualitative differential form) as long as possible, and delay the use of a representative (quantification with which dot product?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') as late as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Recall: The transposed relative to (·, ·)g and (·, ·)h of the linear map dΨ(pE) ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) is the linear map dΨ(pE)T gh =noted dΨ(pE)T ∈ L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) defined by, for all ⃗uE ∈ E and ⃗wF ∈ F vectors at pE and pF, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='68), (dΨ(pE)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF, ⃗uE)g = (⃗wF, dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗uE)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) gives, with pF = Ψ(pE), (⃗ag(pE), ⃗uE)g = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗uE = � βF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗uE = βF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗uE � = (⃗bh(pF), dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗uE)h = (dΨ(pE)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bh(pF), ⃗uE)g, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) true for all ⃗uE, thus ⃗ag(pE) = dΨ(pE)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bh(pF), thus (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 47 48 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Push-forward and pull-back of order 1 tensors 8 Push-forward and pull-back of tensors To lighten the presentation, we only deal with order 1 and 2 tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Similar approach for any tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Push-forward and pull-back of order 1 tensors Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 If T is either a vector field or a differential form, then its push-forward satisfies, for all ξ vector field or differential form (when required) in UF, in short: (Ψ∗T)(ξ) = T(Ψ∗ξ), written Ψ∗T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') = T(Ψ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Ψ∗T)(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ξ(pF) = T(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Ψ∗ξ(pE) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Similarly: in short: (Ψ∗T)(ξ) = T(Ψ∗ξ), written Ψ∗T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') = T(Ψ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Ψ∗T)(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ξ(pE) = T(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Ψ∗ξ(pF) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' • Case T = αE ∈ Ω1(UE) (differential form = a �0 1 � tensor), then here ξ = ⃗wF ∈ Γ(UF) and we have to check: (Ψ∗αE)(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF(pF) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Ψ∗ ⃗wF(pE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1(pE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF(pF) = αE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (dΨ−1(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wF(pF)): True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Case T = ⃗wE ∈ Γ(UE) (vector field ≃ a �1 0 � tensor), then here ξ = αF ∈ Ω1(UF) we have to check: (Ψ∗ ⃗wE)(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='αF(pF) = ⃗wE(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Ψ∗(αF)(pE), where we implicitly use to the natural canonical isomorphism J : � E → E∗∗ ⃗w → w noted = ⃗w � defined by w(ℓ) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w for all ℓ ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So we have to check: αF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Ψ∗ ⃗wE)(pF) = Ψ∗(αF)(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' αF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE(pE)) = (αF(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wE)(pE) : True.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' For (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2), use Ψ−1 instead of Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Push-forward and pull-back of order 2 tensors Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Let T be an order 2 tensor in UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its push-forward by Ψ is the order 2 tensor Ψ∗T in UF defined by, for all ξ1, ξ2 vector field or differential form (when required) in UF, in short: Ψ∗T(ξ1, ξ2) := T(Ψ∗ξ1, Ψ∗ξ2) written Ψ∗T(·, ·) := T(Ψ∗·, Ψ∗·), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ψ∗T(pF)(ξ1(pF), ξ2(pF)) := T(pE)(Ψ∗ξ1(pE), Ψ∗ξ2(pE)) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let T be an order 2 tensor in UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its pull-back by Ψ is the order 2 tensor Ψ∗T in UE defined by, for all ξ1, ξ2 vector field or differential form (when required) in UE, in short: Ψ∗T(ξ1, ξ2) := T(Ψ∗ξ1, Ψ∗ξ2) written Ψ∗T(·, ·) := T(Ψ∗·, Ψ∗·), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', Ψ∗T(pE)(ξ1(pE), ξ2(pE)) := T(pF)(Ψ∗ξ1(pF), Ψ∗ξ2(pF)) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 If T ∈ T 0 2 (UE) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', a metric) then, for all vector fields ⃗w1, ⃗w2 in UF, T∗(⃗w1, ⃗w2) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) = T(⃗w1 ∗, ⃗w2 ∗) = T(dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', T∗(pF)(⃗w1(pF), ⃗w2(pF)) = T(pE)(dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1(pF), dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2(pF)) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Expression with bases (⃗ai) in E and (⃗bi) in F: In short we have (T∗)ij = T∗(⃗bi,⃗bj) = T(⃗bi∗,⃗bj∗) = [⃗b∗ i ]T |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[T]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗b∗ j]|⃗a = ([⃗bi]T |⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dΨ]−T |⃗a,⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[T]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ([dΨ]−1 |⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗bj]|⃗b) = ([dΨ]−T |⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[T]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dΨ]−1 |⃗a,⃗b)ij, thus [T∗]|⃗b = [dΨ]−T |⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[T]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dΨ]−1 |⃗a,⃗b, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) which means [(Ψ∗T)(pF)]|⃗b = ([dΨ(pE)]|⃗a,⃗b)−T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [T(pE)]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ([dΨ(pE)]|⃗a,⃗b)−1 when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Particular case of an elementary tensor T = α1 ⊗ α2 ∈ T 0 2 (UE), where α1, α2 ∈ Ω1(UE), so T(⃗u1, ⃗u2) = (α1 ⊗ α2)(⃗u1, ⃗u2) = (α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u1)(α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2): For all ⃗w1, ⃗w2 ∈ Γ(UF), (α1 ⊗ α2)∗(⃗w1, ⃗w2) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) = (α1 ⊗ α2)(⃗w∗ 1, ⃗w∗ 2) = (α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w∗ 1)(α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w∗ 2) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) = (α1∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1)(α2∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) thus (α1 ⊗ α2)∗ = α1∗ ⊗ α2∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) (And any tensor is a finite sum of elementary tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 48 49 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Push-forward and pull-back of endomorphisms And for the pull-back: For all vector fields ⃗u1, ⃗u2 in UE, T ∗(⃗u1, ⃗u2) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) = T(⃗u1∗, ⃗u2∗) = T(dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u1, dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 If T ∈ T 1 1 (UE) then for all vector fields ⃗w ∈ Γ(UF) and differential forms β ∈ Ω1(UF), T∗(β, ⃗w) = T(β∗, ⃗w∗) = T(β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ, dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', T∗(pF)(β(pF), ⃗w(pF)) = T(pE)(β(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE), dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(pF)) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' For the elementary tensor T = ⃗u ⊗ α ∈ T 1 1 (UE), made of the vector field ⃗u ∈ Γ(UE) and of the differential form α ∈ Ω1(UE): For all β, ⃗w ∈ Ω1(UF) × Γ(UF), in short, (⃗u ⊗ α)∗(β, ⃗w) (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) = (⃗u ⊗ α)(β∗, ⃗w∗) = (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='β∗)(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w∗) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) = (⃗u∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='β)(α∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) = (⃗u∗ ⊗ α∗)(β, ⃗w), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) thus (⃗u ⊗ α)∗ = ⃗u∗ ⊗ α∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) Expression with bases (⃗ai) in E and (⃗bi) in F: In short we have (T∗)ij = T∗(bi,⃗bj) = T(Ψ∗(bi), Ψ∗(⃗bj)) = [Ψ∗(bi)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [Ψ∗(⃗bj)] = [bi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[dΨ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[dΨ−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗bj] = ([dΨ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dΨ−1])ij, thus [T∗]|⃗b = [dΨ]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[T]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dΨ]−1 |⃗a,⃗b, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) which means [(Ψ∗T)(pF)]|⃗b = [dΨ(pE)]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[T(pE)]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dΨ(pE)]−1 |⃗a,⃗b when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Push-forward and pull-back of endomorphisms We have the natural canonical isomorphism J2 : � L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) → L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) L → TL = J2(L) where TL(α, ⃗u) := α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u, ∀(α, ⃗u) ∈ E∗ × E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) Thus Ψ∗TL(m, ⃗w) = TL(Ψ∗m, Ψ∗ ⃗w) = (Ψ∗m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Ψ∗ ⃗w) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w, thus: Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 The push-forward by Ψ of a field of endomorphisms L on UE is the field of endomorphisms Ψ∗L = L∗ on UF defined by in short: Ψ∗L = L∗ = dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1 , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L∗(pF) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1(pF) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus with bases we get [L∗]|⃗b = [dΨ]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[L]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dΨ]−1 |⃗a,⃗b, “as in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Elementary field of endomorphisms L = (J2)−1(⃗u ⊗ α), where ⃗u ∈ Γ(E) and α ∈ Ω1(E): So TL = ⃗u ⊗ α and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2 = (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2)⃗u for all ⃗u2 ∈ Γ(UE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2 = dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2 = dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2∗ = (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2∗)dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = (α∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2)⃗u∗ for all ⃗w2 ∈ Γ(E), thus (TL)∗ = ⃗u∗ ⊗ α∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Let L be a field of endomorphisms on UF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its pull-back by Ψ is the field of endomorphisms Ψ∗L = L∗ on UE defined by in short: Ψ∗L = L∗ = dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L∗(pE) = dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE) when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Application to derivatives of vector fields ⃗u ∈ Γ(UE) is a C1 vector field in UE), pE ∈ UE, so d⃗u : UE → L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) (given by d⃗u(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(pE) = limh→0 ⃗u(pE+h⃗w(pE))−⃗u(pE) h for all ⃗w ∈ Γ(UE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus its push-forward: ((d⃗u)∗ =) Ψ∗(d⃗u) = dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗u)∗(pF) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗u(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(pE)−1 when pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 49 50 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ψ∗(d⃗u) versus d(Ψ∗⃗u): No commutativity 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Ψ∗(d⃗u) versus d(Ψ∗⃗u): No commutativity Here Ψ is C2, ⃗u ∈ Γ(UE), pE ∈ UE, pF = Ψ(pE), so Ψ∗⃗u(pF) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u(pE) = (dΨ(Ψ−1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗u(Ψ−1(pF)), and, for all ⃗w ∈ Γ(UF), d(Ψ∗⃗u)(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(pF) = (d2Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(pF))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u(pE) + dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗u(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(pF), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) with Ψ∗(d⃗u)(pF) = dΨ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗u(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1(pF), thus, in short, d(Ψ∗⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = Ψ∗(d⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + d2Ψ(Ψ∗ ⃗w, ⃗u) ̸= Ψ∗(d⃗u) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) So the differentiation d and the push-forward ∗ do not commute (d(Ψ∗⃗u) = Ψ∗(d⃗u) iff Ψ is affine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Application to derivative of differential forms Let α ∈ Ω1(UE) (a differential form on UE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its derivative dα : UE → L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗) is given by dα(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u(pE) = limh→0 α(pE+h⃗u(pE))−α(pE) h ∈ E∗, for all ⃗u ∈ Γ(UE), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all ⃗u1, ⃗u2 ∈ Γ(UE), (dα(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u1(pE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2(pE) = lim h→0 α(pE + h⃗u1(pE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2(pE) − (α(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u1(pE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2(pE) h ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) With the natural canonical isomorphism L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗) ≃ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) with E∗∗ ≃ E, we can write dα(pE)(⃗u1(pE)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2(pE) = dα(pE)(⃗u1(pE), ⃗u2(pE)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dα(⃗u1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2 = dα(⃗u1, ⃗u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) Thus the push-forward Ψ∗(dα) =noted (dα)∗ of dα, is given by, for all ⃗w1, ⃗w2 ∈ Γ(UF), in short, (dα)∗(⃗w1, ⃗w2) = dα(⃗w∗ 1, ⃗w∗ 2), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', with pF = Ψ(pE), (dα)∗(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2(pF) = (dα(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular, (d2f)∗(⃗w1, ⃗w2) = d2f(dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2) (= d2f(⃗w∗ 1, ⃗w∗ 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Ψ∗(dα) versus d(Ψ∗α): No commutativity Here Ψ is C2, ⃗u ∈ Γ(UE), pE ∈ UE and pF = Ψ(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have Ψ∗α(pF) = α(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1(pF) = α(Ψ−1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1(pF), thus, for all ⃗w1 ∈ Γ(UF), d(ψ∗α)(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1(pF) = (dα(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1(pF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ−1(pF) + α(pE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d2Ψ−1(pF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1(pF) ∈ F ∗, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) thus, for all ⃗w1, ⃗w2 ∈ Γ(UF), in short d(ψ∗α)(⃗w1, ⃗w2) = dα(dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, dΨ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2) + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d2Ψ−1(⃗w1, ⃗w2) ̸= dα(⃗w∗ 1, ⃗w∗ 2) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) So the differentiation d and the push-forward ∗ do not commute (d(Ψ∗α) = Ψ∗(dα) iff Ψ is affine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 50 51 Part III Lie derivative 9 Lie derivative 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='0 Purpose and first results 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Purpose?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Cauchy’s approach may be insufficient, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' - Cauchy’s approach aims to compare two vectors deformed by a motion, thanks to a Euclidean dot product and the deformation gradient F, with the deformation tensor C defined by (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1) • ⃗W2 := (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1) • (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It is a quantitative approach (needs a chosen Euclidean dot product: foot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' metre?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Cauchy’s approach is a first order method (dedicated to linear material): Only the first order Taylor expansion of the motion is used: Only dΦ = F is used (the “slope”), not d2Φ = dF (the “curvature”) or higher derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' - Lie’s approach aims to build qualitative “covariant objective constitutive laws” (some will be discred- ited afterward, because of invariance or thermodynamical requirements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Lie’s approach “naturally” applies to non-linear materials thanks to second order Lie derivatives which uses the second order Taylor expansion of the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In a non planar surface S, you need the Lie derivative if you want to derive along a trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In a Galilean Euclidean framework (quantification), the first order Lie derivatives approach give the same results than Cauchy’s approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Cauchy died in 1857, and Lie was born in 1842: Unfortunately Cauchy could not use the Lie derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Basic results The Eulerian velocity of the motion is ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With the material derivative is DEul Dt := ∂Eul ∂t + dEul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Lie derivative L⃗vf of a Eulerian scalar valued function f is the material derivative L⃗vf = Df Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Lie derivative L⃗v ⃗w of a (Eulerian) vector field ⃗w is more than just the material derivative D ⃗w Dt : L⃗v ⃗w = D ⃗w Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) L⃗v ⃗w gives the rate of stress on ⃗w due to a flow, and in particular the −d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w term in L⃗v ⃗w tells that the spatial variations of ⃗v (variations of the flow) act on the evolution of the stress (anticipated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1)-(9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) enable to define the Lie derivatives of tensors of any order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Issue (ubiquity gift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' �Φ is supposed to be regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗v(t, p(t)) = ∂�Φ ∂t (t, PObj) is the Eulerian velocity at t at p(t) = �Φ(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Recall: If Eul is a Eulerian function then its material time derivative is DEul Dt (t, p(t)) = lim h→0 Eul(t+h, p(t+h)) − Eul(t, p(t)) h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) Issue: The rate Eul(t+h,p(t+h))−Eul(t,p(t)) h raises questions: 1- The difference Eul(t+h, p(t+h)) − Eul(t, p(t)) requires the time and space ubiquity gift to be cal- culated by an observer, since it mixes two distinct times, t and t+h, and two distinct locations, p(t) and p(t+h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- The difference Eul(t+h, p(t+h)) − Eul(t, p(t)) can be impossible: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' if Eul = ⃗w is a vector field in a “non planar surface considered on its own” (manifold) then Eul(t+h, p(t+h)) and Eul(t, p(t)) don’t belong to the same (tangent) vector space (so the difference ⃗w(t+h, p(t+h)) − ⃗w(t, p(t)) is meaningless).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 51 52 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Toward a solution (without ubiquity gift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' To compare Eul(t+h, p(t+h)) and Eul(t, p(t)) (to get the evolution of Eul along a trajectory), you need the duration h to get from t to t+h and to move from p(t) to p(t+h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, you must: take the value Eul(t, pt)) with you (for memory), move along the considered trajectory, and doing so, the value Eul(t, pt) has possibly changed to, with τ = t+h, ((Φt τ)∗Eult)(pτ) noted = Eult∗(τ, pτ) (push-forward);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) And now, at (τ, pτ) where you are, you can compare the actual value Eul(τ, pτ) with the value Eult∗(τ, pτ) you arrived with (the transported memory), thus the difference Eul(τ, pτ) − Eult∗(τ, pτ) (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) is meaningful for a human being since it is computed at a unique time τ and at a unique point pτ (no gift of ubiquity required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1: To compute (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) with Eul = ⃗w a (Eulerian) vector field: At t define the vector field ⃗wt in Ωt by ⃗wt(pt) := ⃗w(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The (spatial) curve ct : s → pt = ct(s) in Ωt is an integral curve of ⃗wt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' satisfies ct′(s) = ⃗wt(ct(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ct is transformed by Φt τ into the (spatial) curve cτ = Φt τ ◦ct : s → pτ = cτ(s)=Φt τ(ct(s)) in Ωτ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence cτ ′(s) = dΦt τ(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='c′(s) = dΦt τ(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt(pt) =noted ⃗wt∗(τ, pτ) is the tangent vector at cτ at pτ (push-forward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the difference ⃗w(τ, pτ)− ⃗wt∗(τ, pτ) can be computed by a human being, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' without ubiquity gift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Lie derivative, first definition Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The Lie derivative L⃗vEul along ⃗v of an Eulerian function Eul is the Eulerian function L⃗vEul defined by, at t at pt = �Φ(t, PObj), L⃗vEul(t, pt) := lim h→0 Eul(t+h, p(t+h)) − (Φt t+h)∗Eult(p(t+h)) h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) Interpretation: L⃗vEul measures the rate of change of Eul along a trajectory: Eul(t+h, p(t+h)) is the value of Eul at t+h at p(t+h), see figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Eult∗(t+h, p(t+h)) = ((Φt t+h)∗Eult)(t+h, p(t+h)) is exclusively strain related: It is the memory transported along a flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the value Eul(t, pt) distorted by the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, with g defined by g(τ) = ((Φt τ)∗Eult)(pτ) (in particular g(t) = Eult(pt)): L⃗vEul(t, pt) := g′(t) = lim τ→t g(τ) − g(t) τ − t also written = d((Φt t+h)∗Eult)(p(t+h)) dt |τ=t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 A more general definition The rate in (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) has to be slightly modified to be adequate in all situations: Eul(t+h, p(t+h)) − Eul∗(t+h, p(t+h)) is computed at (t+h, p(t+h)) which moves as h → 0, and on a “non-planar mani- fold” this is problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The “natural” definition is to arrive with the memory: 52 ex(E, Pe) 2 2 pz= 中(p)= Ex (b) Pt=C (p)53 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Lie derivative of a scalar function Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The Lie derivative L⃗vEul along ⃗v of an Eulerian function Eul is the Eulerian function L⃗vEul defined by, at t at pt = �ΦPObj (t), L⃗vEul(t, pt) := lim h→0 Eul(t, pt) − (Φt−h t )∗Eult−h(pt) h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' with �g defined by �g(τ) = ((Φτ t )∗Eulτ)(pt) (in particular �g(t) = Eul(t, pt)): L⃗vEul(t, pt) := �g′(t) = lim τ→t �g(t) − �g(τ) t − τ = lim τ→t �g(τ) − �g(t) τ − t also written = d((Φτ t )∗Eulτ)(pt) dτ |τ=t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) Here the observer must: At t−h at p(t−h) = �ΦPObj (t−h), take the value Eul(t−h, p(t−h)), travel along the trajectory �ΦPObj , once at t at pt = �ΦPObj (t), this value has become ((Φt t−h)∗Eult−h)(pt) (transported memory), and then the comparison with Eul(t, pt) can be done in Ωt (no ubiquity gift required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Prove: (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With (Φt t+h)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Φt t+h)∗ = I, (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) gives L⃗vEul(t, pt) = limh→0 (Φt t+h)∗Eul(t,pt)−Eult(t,pt) h = limh→0 (Φt t−h)∗Eult−h)(pt)−Eult(pt) −h = limh→0 Eult(pt)−((Φt t−h)∗Eult−h)(pt) h , and (Φt t−h)∗ = (Φt−h t )∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Equivalent definition (differential geometry) Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 The Lie derivative of a Eulerian function Eul along a flow of Eulerian velocity ⃗v is the Eulerian function L⃗vEul defined at (t, pt) by L⃗vEul(t, pt) := lim h→0 ((Φt t+h)∗Eult+h)(pt) − Eul(t, pt) h , rate in Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) In other words, if ˆg is defined by ˆg(τ) = ((Φt τ)∗Eulτ)(pt) (in particular ˆg(t) = Eul(t, pt)), then L⃗vEul(t, pt) := ˆg′(t) = lim τ→t ˆg(τ) − ˆg(t) τ − t also written = d((Φt τ)∗Eulτ)(pt) dτ |τ=t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Prove: (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) also reads L⃗vEul(t, pt) = limh→0 ((Φt t−h)∗Eult−h)(pt)−Eult(pt) −h , and (Φt t−h)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Φt−h t )∗ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 More precise definition, as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3): E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' with (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10), the Lie derivative �L⃗vEul of a Eulerian function � Eul along a flow of Eulerian velocity ⃗v is the Eulerian function defined by, at t at pt = �Φ(t, PObj), �L⃗vEul(t, pt) := ((t, pt), L⃗vEul(t, pt) (pointed function at (t, pt)), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) And, to lighten the notation, �L⃗vEul(t, pt) =noted L⃗vEul(t, pt) (second component of �L⃗vEul(t, pt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Lie derivative of a scalar function Let f be a C1 Eulerian scalar valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With (Φt−h t )∗ft−h(pt) = ft−h(p(t−h)), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10), we get L⃗vf(t, pt) (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) = lim h→0 f(t, pt) − f(t−h, p(t−h)) h , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L⃗vf = Df Dt = ∂f ∂t + df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) So, for scalar functions, the Lie derivative is the material derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Interpretation: L⃗vf measures the rate of change of f along a trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 L⃗vf = 0 iff f is constant along any trajectory (the real value is the memory value): L⃗vf = 0 ⇐⇒ ∀t, τ ∈ [t0, T], (Φt τ ∗)ft(pτ) = f(t, p(t)) when pτ = Φt τ(pt), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' iff f(t, p(t)) = f(t0, pt0) when p(t) = Φt0(t, pt0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' iff f let itself be carried by the flow (unchanged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 53 54 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Lie derivative of a vector field Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let p(t) = �Φ(t, PObj) = pt for all t, so p(τ) = �Φ(τ, PObj) = pτ = Φt t+h(pt) = Φt(τ, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⇐: If fτ = (Φt t+h)∗ft, then fτ(pτ) = ft(pt), thus limτ→t f(τ,p(τ))−f(t,p(t)) τ−t = 0, that is, Df Dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⇒: If Df Dt = 0 then f(t, p(t)) is a constant function on the trajectory t → �Φ(t, PObj), for any parti- cle PObj, so f(τ, p(τ)) = f(t, pt) when p(τ) = Φt t+h(pt), that is, f(τ, pτ) = (Φt t+h)∗ft(pτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Prove: L⃗v(L⃗vf) = D2f Dt2 = ∂2f ∂t2 + 2d( ∂f ∂t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + d2f(⃗v,⃗v) + df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ( ∂⃗v ∂t + d⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Lie derivative of a vector field 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Formula Let ⃗w be a C1 (Eulerian) vector field (interpreted as an “internal force field” in the following).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 L⃗v ⃗w = D ⃗w Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = ∂ ⃗w ∂t + d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) So the Lie derivative is not reduced to the material derivative D ⃗w Dt (unless d⃗v = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' unless ⃗v is uniform): The spatial variations d⃗v of ⃗v influences the rate of stress: ⃗v tries to bend ⃗w (which is expected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗g : τ → ⃗g(τ) = (Φt∗ τ ⃗w)(t, p(t)) = dΦt τ(pt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(τ, p(τ)) when p(τ) = Φt(τ, pt), so (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) reads L⃗v ⃗w(t, pt) = ⃗g ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With ⃗z(τ) := ⃗w(τ, p(τ)) = dΦt(τ, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗g(τ), ⃗z ′(τ) = D ⃗w Dτ (τ, p(τ)) = ∂(dΦt) ∂τ (τ, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗g(τ) + dΦt(τ, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗g ′(τ) = (d⃗v(τ, p(τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t(τ, pt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F t(τ, pt)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(τ, p(τ))) + F t τ(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗g ′(τ) = d⃗v(τ, p(τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(τ, p(τ)) + F t τ(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗g ′(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) Thus D ⃗w Dt (t, pt) = d⃗v(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, pt) + I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗g ′(t), thus ⃗g ′(t) = D ⃗w Dt (t, pt) − d⃗v(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification: Basis (⃗ei), ⃗v = � i vi⃗ei, ⃗w = � i wi⃗ei, d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = � ij vi|j⃗ei, d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = � ij wi|j⃗ei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then L⃗v ⃗w = n � i=1 ∂wi ∂t ⃗ei + n � i,j=1 wi|jvj⃗ei − n � i,j=1 vi|jwj⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) So (column matrix), with [·] := [·]|⃗e, [L⃗v ⃗w] = [D ⃗w Dt ] − [d⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w] (= [∂ ⃗w ∂t ] + [d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v] − [d⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) (And [d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v] = [d⃗w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Duality notations: L⃗v ⃗w = � i ∂wi ∂t ⃗ei + � ij wi |jvj⃗ei − � ij vi |jwj⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Interpretation: Flow resistance measurement Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 Φt0 is supposed to be a C2 motion and a C1 diffeomorphism in space, and ⃗w is a vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L⃗v ⃗w = 0 ⇐⇒ ∀t ∈ [t0, T], ⃗wt = (Φt0 t )∗ ⃗wt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', D ⃗w Dt = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w ⇔ the actual vector ⃗w(t, p(t)) is equal to F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0(pt0) = ⃗wt0∗(t, p(t)) the deformed vector by the flow, see figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So: The Lie derivative L⃗v ⃗w vanishes iff ⃗w does not resist the flow (let itself be deformed by the flow), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' iff ⃗w(t, pt) = ⃗wt0∗(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have L⃗v ⃗w = D ⃗w Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w and ∂F t0 ∂t (t, pt0) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (pt0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⇐ (derivation): Suppose ⃗w(t, p(t)) = F t0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t0, pt0) when p(t) = Φt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then D ⃗w Dt (t, p(t)) = ∂F t0 ∂t (t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t0, pt0) = (d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F t0 t (pt0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, p(t))) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, p(t)), thus D ⃗w Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (See proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') ⇒ (integration): Suppose D ⃗w Dt = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗f(t) = (F t0 t (pt0))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, p(t)) (= pull-back (Φt0 t )∗ ⃗w(t0, pt0)) when p(t) = Φt0(t, pt0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So ⃗w(t, p(t)) = F t0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗f(t) and D ⃗w Dt (t, p(t)) = ∂F t0 ∂t (t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗f(t) + F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗f ′(t) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗f(t) + F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗f ′(t) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, p(t)) + F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗f ′(t) =hyp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' D ⃗w Dt (t, p(t))+F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗f ′(t) for all t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗f ′(t) = ⃗0, thus ⃗f ′(t) = ⃗0 (because Φt0 t is a diffeomorphism), thus ⃗f(t) = ⃗f(t0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗wt = (Φt0 t )∗ ⃗wt0, for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 54 55 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Examples 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Autonomous Lie derivative and Lie bracket The Lie bracket of two vector fields ⃗v and ⃗w is [⃗v, ⃗w] := d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w noted = L0 ⃗v ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) And L0 ⃗v ⃗w = [⃗v, ⃗w] is called the autonomous Lie derivative of ⃗w along ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus L⃗v ⃗w = ∂ ⃗w ∂t + [⃗v, ⃗w] = ∂ ⃗w ∂t + L0 ⃗v ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) NB: L0 ⃗v ⃗w is used when ⃗v et ⃗w are stationary vector fields, thus does not concern objectivity: A stationary vector field in a referential is not necessary stationary in another (moving) referential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Examples 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Lie Derivative of a vector field along itself (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) with ⃗w = ⃗v gives L⃗v⃗v = ∂⃗v ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular, if ⃗v is a stationary vector field then L⃗v⃗v = ⃗0 (= [⃗v,⃗v]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Lie derivative along a uniform flow Here d⃗v = 0, thus L⃗v ⃗w = D ⃗w Dt = ∂ ⃗w ∂t + d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v (when d⃗v = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) Here the flow is rectilinear (d⃗v = 0): there is no curvature (of the flow) to influence the stress on ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Moreover, if ⃗w is stationary, that is ∂ ⃗w ∂t = 0, then L⃗v ⃗w = d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v = the directional derivative ∂ ⃗w ∂⃗v of the vector field ⃗w in the direction ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Lie derivative of a uniform vector field Here d⃗w(t, p) = 0, thus L⃗v ⃗w = ∂ ⃗w ∂t − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w (when d⃗w = 0), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) thus the stress on ⃗w is due to the space variations of ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Moreover, is ⃗w is stationary then L⃗v ⃗w = −d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Uniaxial stretch of an elastic material Strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With [−−→ OP]|⃗e = [ ⃗X]|⃗e = � X Y � , with ξ > 0, t ≥ t0, p(t) = Φt0(t, P) and [⃗x]|⃗e = [−−−→ Op(t)]|⃗e: [⃗x]|⃗e = � x y � = � X Y � + ξ(t−t0) � X 0 � = � X(1 + ξ(t−t0)) Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) Eulerian velocity ⃗v(t, p) = � ξX 0 � = � ξ 1+ξ(t−t0)x 0 � , d⃗v(t, p) = � ξ 1+ξ(t−t0) 0 0 0 � (independent of p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Deformation gradient (independent of P), with κt = ξ(t−t0): Ft = dΦt0 t (P) = � 1 + κt 0 0 1 � = I + κt � 1 0 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) Infinitesimal strain tensor, with F T t = Ft here: εt0 t (P) = Ft − I = κt � 1 0 0 0 � = εt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) Stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Constitutive law = Linear isotropic elasticity: σt(pt) = λTr(εt)I + 2µεt = κt � λ+2µ 0 0 λ � = σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) Cauchy stress vector ⃗T on a surface at p with normal ⃗nt(p) = � n1 n2 � = ⃗n: ⃗Tt(pt) = σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n = κt � (λ+2µ)n1 λn2 � = ξ(t−t0) � (λ+2µ)n1 λn2 � = ⃗Tt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) Push-forwards: ⃗Tt0(pt0) = 0, thus F t0 t0+h(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Tt0(pt0) = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 55 56 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Examples Lie derivative: L⃗v ⃗T(t0, pt0) = lim t→t0 ⃗Tt(pt) − F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Tt0(pt0) t − t0 = ξ � (λ+2µ)n1 λn2 � (rate of stress at (t0, pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) Generic computation with L⃗v ⃗T = ∂ ⃗T ∂t + d⃗T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗T: (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) gives ∂ ⃗T ∂t = ξ � (λ+2µ) n1 λ n2 � and d⃗T = 0 and d⃗vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Tt = � ξ 1+ξ(t−t0) 0 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ξ(t−t0) � (λ+2µ) n1 λ n2 � = ξ2(t−t0) 1+ξ(t−t0) � (λ+2µ) n1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particu- lar, d⃗v(t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗T(t0, pt0) = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus L⃗v ⃗T(t0, pt0) = ξ � (λ+2µ) n1 λ n2 � = rate of stress at the initial (t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Simple shear of an elastic material Euclidean basis (⃗e1,⃗e2) in R2, the same basis at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Initial configuration Ωt0 = [0, L1] ⊗ [0, L2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Initial position [−−→ OP]⃗e = [−−→ Opt0]⃗e = [ ⃗X]⃗e = � X Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ξ ∈ R∗, pt = Φt0 t (pt0), [⃗x]|⃗e = [−−−→ Op(t)]|⃗e, and [⃗x]⃗e = � x = ϕ1(t, X, Y ) = X y = ϕ2(t, X, Y ) � = � X + ξ(t−t0)Y Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) Eulerian velocity ⃗vt(pt) = � ξY 0 � = � ξy 0 � , thus d⃗vt(pt) = � 0 ξ 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With κt = ξ(t−t0), deformation gradient (independent of P): dΦt0 t (P) = � 1 κt 0 1 � = F t0 t , thus F t0 t − I = κt � 0 1 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) Infinitesimal strain tensor: εt0 t (P) = F t0 t (P)−I + (F t0 t (P)−I)T 2 = κt 2 � 0 1 1 0 � = εt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) Stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Constitutive law, usual linear isotropic elasticity (requires a Euclidean dot product): σ(t, pt) = λTr(εt)I + 2µεt = µκt � 0 1 1 0 � = σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) Cauchy stress vector ⃗T(t, pt) (at t at pt) on a surface at p with normal ⃗nt(p) = � n1 n2 � = ⃗n: ⃗Tt = σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n = µκt � n2 n1 � = µξ(t−t0) � n2 n1 � = ⃗T(t) (stress independent of pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) Lie derivative, with ⃗Tt0 = ⃗0: L⃗v ⃗T(t0, pt0) = lim t→t0 ⃗Tt(pt) − F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Tt0(pt0) t − t0 = µξ � n2 n1 � (rate of stress at (t0, pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) Generic computation: L⃗v ⃗T = ∂ ⃗T ∂t + d⃗T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) gives ∂ ⃗T ∂t (t, p) = µξ � n2 n1 � and d⃗T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With d⃗vt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Tt0 = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus L⃗v ⃗T(t0, pt0) = µξ � n2 n1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Shear flow Stationary shear field, see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) with α = 0 and t0 = 0: ⃗v(x, y) = � v1(x, y) = λy, v2(x, y) = 0, d⃗v(x, y) = � 0 λ 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) Let ⃗w(t, p) = � 0 b � = ⃗w(t0, pt0) (constant in time and uniform in space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then L⃗v ⃗w = −d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = � −λb 0 � measures “the resistance to deformation due to the flow”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2, the virtual vector ⃗w∗(t, p) = dΦ(t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t0, pt0) being the vector that would have let itself be carried by the flow (the push-forward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 56 57 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Lie derivative of a differential form Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2: Shear flow, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36), with ⃗w constant and uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L⃗v ⃗w measures the resistance to the deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Spin Rotating flow: Continuing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14): ⃗v(x, y) = ω � 0 −1 1 0 � � x y � , d⃗v(x, y) = ω � 0 −1 1 0 � = ω Rot(π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) In particular d2⃗v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With ⃗w = ⃗w0 constant and uniform we get L⃗v ⃗w0 = −d⃗v(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w0 = −ω Rot(π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w0 (⊥ � a b � = ⃗w0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38) gives “the force at which ⃗w refuses to turn with the flow”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Second order Lie derivative Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 Let ⃗v, ⃗w be C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove: L⃗v(L⃗v ⃗w) = D2 ⃗w Dt2 − 2d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D ⃗w Dt − D(d⃗v) Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w, = ∂2 ⃗w ∂t2 + 2d∂ ⃗w ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v − 2d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂ ⃗w ∂t + d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗v ∂t − d∂⃗v ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + (d2 ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v − 2d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v − (d2⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L⃗v(L⃗v ⃗w) = D(L⃗v ⃗w) Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L⃗v ⃗w) = D( D ⃗w Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (D ⃗w Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) = D2 ⃗w Dt2 − D(d⃗v) Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D ⃗w Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D ⃗w Dt + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w, thus (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39)1, thus (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Lie derivative of a differential form When the Lie derivative of a vector field ⃗w cannot be obtained by direct measurements, you need to use a “measuring device” (Germain: To know the weight of a suitcase you have to lift it: You use work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Here we consider a measuring device which is a differential form α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, if ⃗w is a vector field then f = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v is a scalar function, and (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) gives L⃗v(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) = D(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) Dt = Dα Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' D ⃗w Dt , thus L⃗v(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) = Dα Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w � �� � →(L⃗vα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D ⃗w Dt − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w � �� � =α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L⃗v ⃗w : (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 Let α be a differential form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Lie derivative of α along ⃗v is the differential form L⃗vα := Dα Dt + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v = ∂α ∂t + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) (An equivalent definition is given at (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all vector field ⃗w, L⃗vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w := Dα Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w (= ∂α ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='42) 57 (B ) A q=c (t 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='% w(t) w(t,/p) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(t,p) od T V(B) (t)58 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Lie derivative of a differential form The definition of L⃗vα, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41), immediately gives the “derivation property” L⃗v(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) = (L⃗vα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L⃗v ⃗w) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L⃗v is a derivation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43) Quantification: Relative to a basis (⃗ei) and with [·] := [·]|⃗e, [L⃗vα] = [Dα Dt ] + [α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗v] (row matrix) = [∂α ∂t ] + [dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v] + [α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='44) Thus [L⃗vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w] = [L⃗vα].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w] = [∂α ∂t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w] + [dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w] + [α].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[d⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45) Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 Prove (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='44) with components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And prove [dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v] = [⃗v]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dα]T (row matrix), thus [dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w] = [⃗v]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dα]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w] = [⃗w]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dα].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Basis (⃗ei), dual basis (πei), thus (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) gives [L⃗vα] = [ Dα Dt ] + [α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let α = � i αiπei, ⃗v = � i vi⃗ei, d⃗v = � ij vi|j⃗ei ⊗ πej (tensorial writing convenient for calculations), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗v]|⃗e = [vi|j], thus α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v = � ij αivi|jπej, thus [α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v]|πe = [α]|πe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗v]|⃗e (row matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And dα = � ij αi|jπei ⊗ πej, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dα]|πe = [αi|j], gives dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v = � ij αi|jvjπei = � ij viαj|iπej, and [dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v]|πe is a row matrix (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v is a differential form), thus [dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v]|πe = [⃗v]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dα]T |πe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Or compute (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = � ij αi|jvjwi = [⃗w]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[dα]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗v]|⃗e = [⃗v]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [dα]T |πe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[⃗w]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 Let α be a differential form, and let αt(p) := α(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove, when Φt0 t is a diffeomorphism, L⃗vα = 0 ⇐⇒ ∀t ∈ [t0, T], αt = (Φt0 t )∗αt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='46) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' : Dα Dt = −α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v ⇐⇒ αt(pt) = αt0(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (pt0)−1 for all t, when pt = Φt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⇐: If αt(p(t)) = αt0(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (pt0)−1, then α(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0(t, pt0) = αt0(pt0), thus Dα Dt (t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (pt0) + αt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂F t0 ∂t (t, pt0) = 0, thus Dα Dt (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (pt0) + αt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (pt0) = 0, thus L⃗vα = 0, since Φt0 t is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⇒: If β(t) := (Φt0 t )∗αt0(pt0) = αt(p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (pt0) (pull-back at (t0, pt0)), then β(t) = α(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0(t, pt0), thus β′(t) = Dα Dt (t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (pt0) + α(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (pt0) = 0 (hypothesis L⃗vα = 0), thus β(t) = β(t0) = αt0(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15 A definition equivalent to (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) is, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10), L⃗vα(t, pt) := lim τ→t (Φt τ)∗ατ(pt) − αt(pt) τ − t (= lim τ→t ατ(pτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΦt τ(pt) − αt(pt) τ − t ) noted = D(Φt∗ τ ατ(pt)) Dτ |τ=t noted = D(α∗ τ(pt)) Dτ |τ=t (= D(ατ(pτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΦt τ(pt)) Dτ |τ=t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='47) Indeed, if β(τ) = (Φt τ)∗ατ(pt) = ατ(pτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΦt τ(pt), then β′(τ) and then τ = t give (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16 ⃗v and α being C2, prove: L⃗v(L⃗vα) = ∂2α ∂t2 + 2d∂α ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + 2∂α ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗v ∂t + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂d⃗v ∂t + (d2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) + 2(dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d2⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) + (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='48) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) gives L⃗v(L⃗vα) = L⃗v(∂α ∂t ) + L⃗v(dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) + L⃗v(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v) = ∂2α ∂t2 + d∂α ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + ∂α ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + ∂(dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) ∂t + d(dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + ∂(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v) ∂t + d(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v = ∂2α ∂t2 + d∂α ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + ∂α ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + ∂dα ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗v ∂t + (d2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) + (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + ∂α ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂d⃗v ∂t + (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d2⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v = ∂2α ∂t2 + 2d∂α ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + 2∂α ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗v ∂t + (d2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) + 2(dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂d⃗v ∂t + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d2⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) + (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 58 59 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Incompatibility with Riesz representation vectors 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Incompatibility with Riesz representation vectors The Lie derivative has nothing to do with any inner dot product (the Lie derivative does not compare two vectors, contrary to a Cauchy type approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Here we introduce a Euclidean dot product (·, ·)g and show that the Lie derivative of a linear form α is not trivially deduced from the Lie derivative of a Riesz representation vector of α (which one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Same issue as at § 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Let α be a Eulerian differential form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then let ⃗ag(t, p) ∈ ⃗Rn be the (·, ·)g-Riesz representation vector of the linear form α(t, p) ∈ Rn∗: So, for all Eulerian vector field ⃗w, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = (⃗ag, ⃗w)g (= ⃗ag •g ⃗w), (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='49) which means α(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, p) = (⃗ag(t, p), ⃗w(t, p))g at all admissible (t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This defines the Eulerian vector field ⃗ag (not intrinsic to α: ⃗ag depends on the choice of (·, ·)g, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17 For all ⃗v, ⃗w ∈ ⃗Rn, ∂α ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = (∂⃗ag ∂t , ⃗w)g, (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = (d⃗ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v, ⃗w)g, Dα Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = (D⃗ag Dt , ⃗w)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='50) Thus L⃗vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = (L⃗v⃗ag, ⃗w)g + (⃗ag, (d⃗v+d⃗vT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w)g, and L⃗vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w ̸= (L⃗v⃗ag, ⃗w)g in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='51) So L⃗v⃗ag is not the Riesz representation vector of L⃗vα (but for solid body motions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Expected: A Lie derivative is covariant objective, see § 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4, and the use of an inner dot product ruins this objectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A Euclidean dot product g(·, ·) is bilinear constant and uniform, thus: α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = (⃗ag, ⃗w)g gives ∂α ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂ ⃗w ∂t = ( ∂⃗ag ∂t , ⃗w)g + (⃗ag, ∂ ⃗w ∂t )g, with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂ ⃗w ∂t = (⃗ag, ∂ ⃗w ∂t )g, thus we are left with ∂α ∂t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = ( ∂⃗ag ∂t , ⃗w)g, for all ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = (⃗ag, ⃗w)g gives d(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v = d(⃗ag, ⃗w)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v for all ⃗v, ⃗w, thus (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) = (d⃗ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v, ⃗w)g + (⃗ag, d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v)g, with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) = (⃗ag, d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v)g, thus we are left with (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = (d⃗ag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v, ⃗w)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus Dα Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = ( D⃗ag Dt , ⃗w)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (L⃗vα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = Dα Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = ( D⃗ag Dt , ⃗w)g + (⃗ag, d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w)g = (L⃗v⃗ag + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ag, ⃗w)g + (d⃗vT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ag, ⃗w)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18 Chorus: a “differential form” (measuring instrument, covariant) should not be confused with a “vector field” (object to be measured, contravariant);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, the use of a dot product (which one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') and the Riesz representation theorem should be restricted for computational purposes, after an objective equation has been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See also remark F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Lie derivative of a tensor The Lie derivative of any tensor of order ≥ 2 is defined thanks to L⃗v(T ⊗ S) = (L⃗vT) ⊗ S + T ⊗ (L⃗vS) (derivation formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='52) (Or direct definition: L⃗vT(t0, pt0) = D((Φt0 t )∗Tt)(pt0) Dt |t=t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Lie derivative of a mixed tensor Let Tm ∈ T 1 1 (Ω), and Tm is called a mixed tensor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its Lie derivative, called the Jaumann derivative, is given by L⃗vTm = DTm Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Tm + Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v = ∂Tm ∂t + dTm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Tm + Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='53) Can be checked with an elementary tensor T = ⃗w ⊗α: we have d(⃗w ⊗α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v = (d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v)⊗α+ ⃗w ⊗(dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) and (d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w)⊗α = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗w⊗α), and ⃗w⊗(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v) = (⃗w⊗α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v , thus (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='52) gives L⃗v(⃗w⊗α) = (L⃗v ⃗w)⊗α+ ⃗w⊗(L⃗vα) = ∂ ⃗w ∂t ⊗ α + (d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) ⊗ α − (d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) ⊗ α + ⃗w ⊗ ∂α ∂t + ⃗w ⊗ (dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v) + ⃗w ⊗ (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v) = ∂ ⃗w⊗α ∂t + d(⃗w ⊗ α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗w ⊗ α) + (⃗w ⊗ α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 59 60 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Lie derivative of a tensor Quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Relative to a basis (⃗ei): [L⃗vTm] = [DTm Dt ] − [d⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [Tm] + [Tm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗v] (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54) (the signs ∓ are mixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' “Mixed” also refers to positions of indices (up and down with duality notations): Tm = �n i,j=1T ij⃗ei ⊗ ej with the dual basis (ei), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [Tm]|⃗e = [T ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19 With components, prove (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂Tm ∂t = � ij ∂T ij ∂t ⃗ei ⊗ ej, dTm = � ijk T i j|k⃗ei ⊗ ej ⊗ ek, ⃗v = � i vi⃗ei, d⃗v = � ij vi |j⃗ei ⊗ ej, thus dTm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v = � ijk T i j|kvk⃗ei ⊗ ej, d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Tm = � ijk vi |kT k j⃗ei ⊗ ej, Tm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v = � ijk T i kvk |j⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Lie derivative of a up-tensor Recall: If L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) (a linear map) then its adjoint L∗ ∈ L(F ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗) is defined by, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12, ∀m ∈ F ∗, L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m := m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ∀m, ⃗u ∈ (F ∗ × E), (L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='55) (There is no inner dot product involved here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') In particular, d⃗v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m := m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v for all m ∈ ⃗Rn∗ t , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u) for all m ∈ ⃗Rn∗ t and all ⃗u ∈ ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Tu ∈ T 2 0 (Ω), and Tu is called a up tensor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its Lie derivative is called the upper-convected (Maxwell) derivative or the Oldroyd derivative and is given by L⃗vTu = DTu Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Tu − Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v∗ = ∂Tu ∂t + dTu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Tu − Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='56) Can be checked with an elementary tensor T = ⃗u ⊗ ⃗w and L⃗v(⃗u ⊗ ⃗w) = (L⃗v⃗u) ⊗ ⃗w + ⃗u ⊗ (L⃗v ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Relative to a basis (⃗ei): [L⃗vTu] = [DTu Dt ] − [d⃗v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [Tu] − [Tu].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗v]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='57) “up” also refers to positions of indices (with duality notations): Tu = �n i,j=1T ij⃗ei ⊗ ⃗ej with the dual basis (ei), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [Tu]|⃗e = [T ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20 With components, prove (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂Tu ∂t = � ij ∂T ij ∂t ⃗ei⊗⃗ej, dTu = � ijk T ij |k⃗ei⊗⃗ej⊗ek, ⃗v = � i vi⃗ei, d⃗v = � ij vi |j⃗ei⊗ej, d⃗v∗ = � ij vj |iei⊗⃗ej, thus dTu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v = � ijk T ij |k vk⃗ei ⊗ ej, d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Tu = � ijk vi |kT kj⃗ei ⊗ ⃗ej, Tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v∗ = � ijk T ikvj |kei ⊗ ⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Lie derivative of a down-tensor Let Td ∈ T 0 2 (Ω), and Td is called a down tensor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Lie derivative is called the lower-convected Maxwell derivative and is given by L⃗vTd = DTd Dt + Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + d⃗v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Td = ∂Td ∂t + dTd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + d⃗v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='58) Can be checked with an elementary tensor T = ℓ ⊗ m and L⃗v(ℓ ⊗ m) = (L⃗vℓ) ⊗ m + ℓ ⊗ (L⃗vm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Relative to a basis (⃗ei): [L⃗vTd] = [DTd Dt ] + [Td].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗v] + [d⃗v]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [Td].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='59) “down” also refers to positions of indices (with duality notations): Td = �n i,j=1Tijei ⊗ ej with the dual basis (ei), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [Td]|⃗e = [Tij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21 With components, prove (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂Td ∂t = � ij ∂Tij ∂t ei⊗ej, dTd = � ijk Tij|kei⊗ej⊗ek, ⃗v = � i vi⃗ei, d⃗v = � ij vi |j⃗ei⊗ej, d⃗v∗ = � ij vj |iei⊗⃗ej, thus dTd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v = � ijk Tij|kvkei ⊗ ej, Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v = � ijk Tikvk |jei ⊗ ⃗ej, d⃗v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Td = � ijk vk |iTkjei ⊗ ⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22 Let g = (·, ·)g ∈ T 0 2 (Ω) be a constant and uniform metric (a unique inner dot product for all t, p, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', a Euclidean dot product at all t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then Dg Dt = 0, thus L⃗vg = 0 + g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + d⃗v∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g, thus [L⃗vg] = [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗v] + [d⃗v]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 60 61 Part IV Velocity-addition formula 10 Change of referential and velocity-addition formula 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='0 Issue and result (summary) The velocity-addition formula is (in classical mechanics) ⃗vA = ⃗vB + ⃗vD, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) where ⃗vA, ⃗vB and ⃗vD are the absolute, relative and drive velocity, ⃗vA and ⃗vD being velocities described by an observer A with his referential RA = (OA, ( ⃗Ai)) and ⃗vB being a velocity described by an observer B with his referential RB = (OB, ( ⃗Bi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) is problematic (inconsistent): The velocities ⃗vA and ⃗vD are quantified in RA, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' expressed in foot/s by the absolute observer, The velocity ⃗vB is a quantified in RB, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' expressed in metre/s by the relative observer, Thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) with ⃗vB + ⃗vD tells that you add metre/s and foot/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So: Question: What are we missing (and what does (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) really mean)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: We miss a functional link: The translator between A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Summary: Call �Φ the motion of a observed object Obj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' �Φ is quantified by A in his referential RA = (OA, ( ⃗Ai)) as the “motion” ⃗ϕA = [ −−→ OA�Φ]| ⃗A, and is quantified by B in his referential RB = (OB, ( ⃗Bi)) as the “motion” ⃗ϕB = [ −−→ OB �Φ]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' At t, the translator Θ connects these numerical values: ⃗ϕA(t, PObj) = Θ(t, ⃗ϕB(t, PObj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ∂ ⃗ϕA ∂t (t, PObj) = ∂Θ ∂t (t, ⃗xBt) + dΘ(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂ ⃗ϕB ∂t (t, PObj), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗vA(t, ⃗xAt) = ∂Θ ∂t (t, ⃗xBt) + dΘ(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vB(t, ⃗xBt) where ⃗xAt = ⃗ϕA(t, PObj) and ⃗xBt = ⃗ϕB(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then call dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt) = ⃗vBt∗(⃗xAt) = “the translated relative velocity at t from B to A”, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) thus, with ∂Θ ∂t (t, ⃗xBt) = ⃗vD(t, ⃗xAt) the drive velocity, which gives ⃗vA(t, ⃗xAt) = ⃗vB∗(t, ⃗xAt) + ⃗vD(t, ⃗xAt): so ⃗vA = ⃗vB∗ + ⃗vD = the velocity addition formula in RA, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' : (Absolute velocity) = (Translated relative velocity) + (Drive velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In other words, with ⃗v the velocity of Obj and with ⃗vRB the velocity of RB in RA: For all pt = �Φ(t, PObj), [⃗vt(pt)]| ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗vt(pt)] ⃗B + [⃗vRBt(pt)]| ⃗A, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) relation between the numerical values of the velocities stored by A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Translation motion of RB in RA, so [⃗vRBt(pt)]| ⃗A = [⃗vRBt]| ⃗A is independent of pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' with ( ⃗Bit) = λ( ⃗Ait) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ai in foot and ⃗Bi in meter give λ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28), dΘt = λI, hence [⃗vt(pt)]| ⃗A = λ[⃗vt(pt)] ⃗B + [⃗vRBt]| ⃗A, which is the expected relation (“sum of the velocities with the good units”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Motion of the Earth around the Sun: See § 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Referentials and “matrix motions” 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Absolute and relative referentials Classical mechanics framework: Time and space are decoupled, all the observers share the same time unit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the second) and live in “our” Universe modeled as R3 (affine space) with its usual associated vector space ⃗R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In the following, the affine space is Rn associated to the vector space ⃗Rn, n ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' An observer A, which we will call the absolute observer, chooses a (rigid body) object ObjRA in the Universe, chooses one particle in ObjRA, calls OAt its position at t, and chooses three more particles in ObjRA, calls PAti their positions at t (in the Universe), such that the bi-point vectors ⃗Ait := −−−−−→ OAtPAti make a basis in ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' He has thus built his (Cartesian) referential RAt = (OAt, ( ⃗Ait)), called the absolute referential, and written RA = (OA, ( ⃗Ai)) when used by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ObjRA is the “Sun extended to infinity”, and at t, OAt is the position of the center of the Sun in the Universe, ( ⃗Ait) is a Euclidean basis in foot fixed relative to stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 61 62 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Referentials and “matrix motions” An observer B, which we will call the relative observer, proceeds similarly: He chooses a (rigid body) object ObjRB in the Universe, builds his Cartesian referential RBt = (OBt, ( ⃗Bit)), called the relative referential, written RB = (OB, ( ⃗Bi)) when used by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ObjRB is the “Earth extended to infinity”, and at t, OBt is the position of the center of the Earth and ( ⃗Bit) is a Euclidean basis in metre fixed relative to the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Mn1 is the vectorial space of n ∗ 1 matrices (column matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A and B call Mn1(A) and Mn1(B) the affine spaces of n ∗ 1 matrices made of the “matrix positions” [−−−→ OAtpt]| ⃗A and [−−−→ OBtpt]| ⃗B where pt is the position at t of a particle in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If a function ϕ is given as ϕ(t, x), then ϕt(x) := ϕ(t, x), and conversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Motion of a material object Obj An object Obj is considered by all observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its motion in the Universe is �Φ : � [t1, t2] × Obj → Rn (t, PObj) → pt = �Φ(t, PObj) = position of the particle PObj at t in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) At t at pt = �Φ(t, PObj), the Eulerian velocities and accelerations of PObj are ⃗v(t, pt) = ∂�Φ ∂t (t, PObj) and ⃗γ(t, pt) = ∂2�Φ ∂2t (t, PObj) (∈ ⃗Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Quantification: Absolute and relative “motion” of Obj At t, the position pt = �Φ(t, PObj) of a particle PObj ∈ Obj is spotted by A, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B, with the bi-point vectors −−−→ OAtpt, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' −−−→ OBtpt in ⃗Rn, which components is stored by A, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B, in his referentials: With −−−→ OAtpt = n � i=1 xAti ⃗Ait and −−−→ OBtpt = n � i=1 xBti ⃗Bit, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) and with ( ⃗Ei) the canonical basis in Mn1, the n ∗ 1 matrices ⃗xAt := [−−−→ OApt]| ⃗A = � � xAt1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' xAtn � � = n � i=1 xAti ⃗Ei, and ⃗xBt := [−−−→ OBpt]| ⃗B = � � xBt1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' xBtn � � = n � i=1 xBti ⃗Ei, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) are stored by A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Initial notation: ⃗xAt := [−−−→ OAtpt]|( ⃗Ait), but here OAt and ( ⃗Ait) are fixed in RA, idem for B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Mind the notations: pt is a point, −−−→ OAtpt is a vector, ⃗xAt is a column matrix (components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This defines the “absolute motion” ⃗ϕA and “relative motion” ⃗ϕB of Obj (matrix valued): ⃗ϕA : � � � � � [t1, t2]×Obj → Mn1(A) (t, PObj) → ⃗ϕA(t, PObj) := [ −−−−−−−−→ OA�Φ(t, PObj)]| ⃗A = n � i=1 xAi(t) ⃗Ei noted = ⃗xA(t) = [−−−−→ OAp(t)]| ⃗A, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) ⃗ϕB : � � � � � [t1, t2]×Obj → Mn1(B) (t, PObj) → ⃗ϕB(t, PObj) := [ −−−−−−−−→ OB �Φ(t, PObj)]| ⃗B = n � i=1 xBi(t) ⃗Ei noted = ⃗xB(t) = [−−−−→ OBp(t)]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) And the “absolute” and “relative” velocities and accelerations of PObj are (matrix valued in Mn1): ⃗vA(t, ⃗xAt) := [⃗v(t, pt)]| ⃗A and ⃗γA(t, ⃗xAt) := [⃗γ(t, pt)]| ⃗A, when ⃗xAt := [−−−→ OApt]| ⃗A, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) ⃗vB(t, ⃗xBt) := [⃗v(t, pt)]| ⃗B and ⃗γB(t, ⃗xBt) := [⃗γ(t, pt)]| ⃗B, when ⃗xBt := [−−−→ OBpt]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) 62 63 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Referentials and “matrix motions” Exercice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Prove: ⃗vA(t, ⃗xAt) = ∂ ⃗ϕA ∂t (t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' −−−−−−−→ OAt�ΦPObj (t) = � i xAi(t) ⃗Ai(t) gives ⃗v(t, pt) = � i xAi ′(t) ⃗Ai(t) + xAi(t) ⃗Ai ′(t), thus [⃗v(t, pt)]| ⃗ A = � i xAi ′(t)[ ⃗Ai(t)]| ⃗ A + xAi(t)[ ⃗Ai ′(t)]| ⃗ A (since Mn1 is a vector space) = � i xAi ′(t) ⃗Ei + [⃗0] (the ⃗Ai(t) are static in RA: [ ⃗Ai ′(t)]| ⃗ A = [limh→0 ⃗ Ai(t+h)− ⃗ Ai(t) h ]| ⃗ A = limh→0 [ ⃗ Ai(t+h)]| ⃗ A−[ ⃗ Ai(t)]| ⃗ A h = limh→0 ⃗ Ei− ⃗ Ei h = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ⃗ϕAPObj (t) = [ −−−−−−−→ OA�ΦPObj (t)]| ⃗ A = � i xAi(t)[ ⃗Ai(t)]| ⃗ A = � i xAi(t) ⃗Ei, thus ⃗ϕAPObj ′(t) = � i xAi ′(t) ⃗Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 ⃗u is a C1 vector field, p is a point, ⃗xA := [−−→ OAp]| ⃗A and ⃗uA(⃗xA) := [⃗u(p)]| ⃗A (matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove: d⃗uA(⃗xA) = [d⃗u(p)]| ⃗A (endomorphism in Mn1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' d⃗uA(⃗xA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]| ⃗A = [d⃗u(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w]| ⃗A for all ⃗w ∈ ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The point p+h⃗w ∈ Rn is referenced by A as [−−→ OAp + h⃗w]| ⃗ A = [−−→ OAp]| ⃗ A + h[⃗w]| ⃗ A = ⃗xA + h[⃗w]| ⃗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus d⃗uA(⃗xA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]| ⃗ A = limh→0 ⃗uA(⃗xA+h[ ⃗w]| ⃗ A)−⃗uA(⃗xA) h = limh→0 [⃗u(p+h ⃗w)]| ⃗ A−[⃗u(p)]| ⃗ A h = limh→0 [⃗u(p+h ⃗w)− ⃗w(p)]| ⃗ A h = [limh→0 ⃗u(p+h ⃗w)− ⃗w(p) h ]| ⃗ A = [d⃗u(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w]| ⃗ A = [d⃗u(p)]| ⃗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]| ⃗ A, true for all ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Call Qt the transition matrix from ( ⃗Ait) to ( ⃗Bit) at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove ⃗xAt = [−−−−→ OAOBt]| ⃗A + Qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗xBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗xAt = [−−−→ OApt]| ⃗ A = [−−−−→ OAOBt + −−−→ OBtpt]| ⃗ A = [−−−−→ OAOBt]| ⃗ A + [−−−→ OBtpt]| ⃗ A, and the change of basis formula gives [−−−→ OBtpt]| ⃗ B = Q−1 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [−−−→ OBtpt]| ⃗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Motion of RB Particular case Obj = ObjRB: Its motion in the Universe, also called the motion of RB, is noted �ΦRB : � [t1, t2] × ObjRB → Rn (t, QRB) → qt = �ΦRB(t, QRB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) At t at qt = �ΦRB(t, QRB), the Eulerian velocities and accelerations of QRB are ⃗vRB(t, qt) = ∂�ΦRB ∂t (t, QRB) and ⃗γRB(t, qt) = ∂2�ΦRB ∂2t (t, QRB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Quantification: Drive and static “motion” of RB The “drive motion” ⃗ϕD, also called the motion of RB in RA, and the “static motion” ⃗ϕS is the quantification of �ΦRB by A and by B: ⃗ϕD : � � � [t1, t2] × ObjRB → Mn1(A) (t, QRB) → ⃗ϕD(t, QRB) := [ −−−−−−−−−−→ OA�ΦRB(t, QRB)]| ⃗A noted = ⃗yD(t) = [−−−−→ OAq(t)]| ⃗A, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) ⃗ϕS : � � � ObjRB → Mn1(B) QRB → ⃗ϕS(QRB) := [ −−−−−−−−−−→ OB �ΦRB(t, QRB)]| ⃗B noted = ⃗yS = [−−−−→ OBq(t)]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) (⃗ϕS is independent of t since ObjRB is fixed in RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') The drive velocity, also called the velocity of RB in RA, and static velocity of QRB are ⃗vD(t, ⃗yDt) := [⃗vRB(t, qt)]| ⃗A when ⃗yDt := [−−→ OAqt]| ⃗A, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) ⃗vS(t, ⃗yS) := [⃗vRB(t, qt)]| ⃗B = [⃗0] noted = ⃗0 (null matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) And the drive and static accelerations are ⃗γD(t, ⃗yDt) = [⃗γRB(t, qt)]| ⃗A and ⃗γS(t, ⃗yS) = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Why introduce ⃗ϕS (static)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' You can’t confuse a particle QRB with its stored positions ⃗yS or ⃗yDt at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And see (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 63 64 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The translator Θt 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The translator Θt 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition of Θt Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 At t, the translator Θt : Mn1(B) → Mn1(A) is defined with (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15)-(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) by: � � � [−−−−→ OBq(t)]| ⃗B = ⃗ϕS(QRB) = ⃗yS position of QRB in RB (static), and [−−−−→ OAq(t)]| ⃗A = ⃗ϕDt(QRB) = ⃗yDt position of QRB at t in RA (moving) � � � =⇒ ⃗yDt = Θt(⃗yS), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Θt is the “inter-referential function at t” which translates the “matrix position” ⃗yS = ⃗ϕS(QRB) = [ −−−−−−−−−−→ OB �ΦRB(t, QRB)]| ⃗B ∈ Mn1(B) (position of QRB as stored by B) to the “matrix position” ⃗yDt = ⃗ϕD(t, QRB) = [ −−−−−−−−−−→ OA�ΦRB(t, QRB)]| ⃗A ∈ Mn1(A) (position of QRB as stored by A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So Θt is defined by ⃗ϕDt = Θt ◦ ⃗ϕS : � ObjRB → Mn1(A) QRB → ⃗ϕDt(QRB) := Θt(⃗ϕS(QRB)), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' defined by Θt := ⃗ϕDt ◦ ⃗ϕ −1 S : � Mn1(B) → Mn1(A) ⃗yS → ⃗yDt = Θt(⃗yS) := ⃗ϕDt(⃗ϕ −1 S (⃗yS)) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) (stored position by B to stored position by A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for QOB the particle in ObjRB at t at OBt (chosen by B to locate its origin), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) gives, with ⃗0 the null matrix in Mn1, [−−−−→ OAOBt]| ⃗A = Θt(⃗0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) So, Θt is defined such that the following diagram commutes: ⃗yS = ⃗ϕS(QRB) = localization of QRB by B Θt � QRB ∈ ObjRB ⃗ϕS � ⃗ϕDt � ⃗yDt = ⃗ϕDt(QRB) = Θt(⃗yS) = localization at t of QRB by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Translation at t for the motion �Φ t is fixed, the position pt = �Φ(t, PObj) of a particle PObj ∈ Obj is also the position qt = �ΦRB(t, QRB) of a particle QRB ∈ ObjRB, so ⃗ϕAt(PObj) = [−−−→ OApt]| ⃗A = [−−→ OAqt]| ⃗A = ⃗ϕDt(QRB), and ⃗ϕBt(PObj) = [−−−→ OBpt]| ⃗B = [−−−→ OBqt]| ⃗B = ⃗ϕS(QRB), thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) gives ⃗ϕAt = Θt ◦ ⃗ϕBt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 dΘt 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Push-forward If ⃗yS ∈ Mn1(B), ⃗wS ∈ Mn1 and ⃗yDt = Θt(⃗yS) , then ⃗wSt∗(⃗yDt) := dΘt(⃗yS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wS (= lim h→0 Θt(⃗yS + h⃗wB) − Θt(⃗yS) h ) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) is the push-forward of the matrix ⃗wS ∈ Mn1 by Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, ⃗wSt∗([−−→ OAqt]| ⃗A) = dΘt([−−−→ OBqt]| ⃗B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗wt(qt)]| ⃗B for all qt ∈ Rn and all ⃗wt : Rn → ⃗Rn (vector field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 64 65 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Translated velocities 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Θt is affine in classical mechanics Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 In R3 (in classical mechanics), Θt is affine: For all QB0, QB1 ∈ ObjRB and all t, u ∈ R, with qti = �ΦRB(t, QBi) ∈ Rn (positions at t in our Universe), Θt([−−−→ OBqt0]| ⃗B + u [−−−→ qt0qt1]| ⃗B) = [−−−→ OAqt0]| ⃗A + u [−−−→ qt0qt1]| ⃗A, and [−−−→ qt0qt1]| ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [−−−→ qt0qt1]| ⃗B (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) the differential dΘt(⃗yS0) =noted dΘt being independent of ⃗yS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular [ ⃗Bit] ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗Bi]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In other words, for all ⃗yS0, ⃗yS1 ∈ Mn1(B) and all t, u ∈ R, Θt((1−u)⃗yS0 + u ⃗yS1) = (1−u)Θt(⃗yS0) + u Θt(⃗yS1), and Θt(⃗yS1) = Θt(⃗yS0) + dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(⃗yS1−⃗yS0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider the straight line (possible in classical mechanics in R3) qt : u → qt(u) = qt0 +u −−−→ qt0qt1 ∈ Rn (fixed in RB), in particular, qt(0) = qt0 and qt(1) = qt1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗yS(u) = [−−−−−→ OBqt(u)]| ⃗B (positions stored by B), so ⃗yS(u) = [−−−→ OBqt0 + u −−−→ qt0qt1]| ⃗B = [(1−u)−−−→ OBqt0 + u −−−→ OBqt1]| ⃗B = (1−u)[−−−→ OBqt0]| ⃗B + u [−−−→ OBqt1]| ⃗B = (1−u)⃗yS0 + u ⃗yS1, where ⃗yS0 = [−−−→ OBqt0]| ⃗B = ⃗yS(0) and ⃗yS1 = [−−−→ OBqt1]| ⃗B = ⃗yS(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Idem for A: ⃗yDt(u) = [−−−−−→ OAqt(u)]| ⃗A = (1−u)⃗yDt0 + u⃗yDt1 (positions stored by A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (1−u)Θt(⃗yS0) + uΘt(⃗yS1) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) = (1−u)⃗yDt0 + u⃗yDt1 = ⃗yDt(u) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) = Θt(⃗yS(u)) = Θt((1−u)⃗yS0 + u⃗yS1), thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27)1, thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence (derivation in u): −Θt(⃗yS0) + Θt(⃗yS1) = dΘt(⃗yS(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (−⃗yS0 + ⃗yS1), true for all u, thus dΘt(⃗yS(u)) is independent of u, dΘt(⃗yS(u)) = dΘt(⃗yS0), true for all ⃗yS0, so dΘt(⃗yS0) =noted dΘt, thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27)2, thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus [−−−−−→ OBtPBti]| ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [−−−−−→ OBtPBti]| ⃗B where PBti is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Bit = −−−−−→ OBtPBti, thus [ ⃗Bit] ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗Bi]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 Call Qt = [Qt,ij] the transition matrix from ( ⃗Ait) to ( ⃗Bit) in ⃗Rn, and ( ⃗Ei) the canonical basis in Mn1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove [dΘt]| ⃗E = Qt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ej = n � i=1 Qt,ij ⃗Ei, ∀j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Bjt = �n i=1Qt,ij ⃗Ait gives [ ⃗Bjt]| ⃗ A = �n i=1Qt,ij ⃗Ei, and [ ⃗Bjt]| ⃗ A =(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗Bjt]| ⃗ B = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Translated velocities t is fixed, ⃗vt(pt) = ∂�Φ ∂t (t, PObj) is the velocity of a particle PObj ∈ Obj at t at pt, ⃗xAt := [−−−→ OAtpt]| ⃗A, ⃗xBt := [−−−→ OBtpt]| ⃗B, and Θt affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 The translated relative velocity and acceleration from B to A at t at pt are the matrices ⃗vBt∗(⃗xAt) := dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt) and ⃗γBt∗(⃗xAt) = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗γBt(⃗xBt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗vBt∗(⃗xAt) = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗vt(pt)] ⃗B and ⃗γBt∗(⃗xAt) = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗γt(pt)] ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Interpretation: Let qt0 and qt1 be particles in ObjRB s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗vBt(⃗xBt) = [−−−→ qt0qt1]| ⃗B where ⃗xBt = [−−−→ OBqt0]| ⃗B (here −−−→ qt0qt1 is a tangent vector at qt0 to the curve qt : u → qt(u) = qt0 +u −−−→ qt0qt1 in the proof of prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then [−−−→ qt0qt1]| ⃗A =(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [−−−→ qt0qt1]| ⃗B gives [−−−→ qt0qt1]| ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt) = ⃗vBt∗(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Similarly for ⃗γBt∗(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 ( ⃗Ai) and ( ⃗Bi) are Euclidean basis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' in foot and metre), (·, ·)A and (·, ·)B are the associated Euclidean dot products, λ = || ⃗Bi||A (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28), ( ⃗Ei) is the canonical basis in Mn1, and (·, ·)M is the canonical inner dot product in Mn1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Call ⃗Eit∗ := dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ei and prove: ∀i, j, ( ⃗Eit∗, ⃗Ejt∗)M = λ2δij, and dΘt T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt = λ2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ( ⃗Bit) is a Euclidean basis for B, thus is a Euclidean orthogonal basis for all observers, in particular for A, with || ⃗Bit||A = λ for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ⃗Eit∗ = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗Bjt]| ⃗ B =(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) [ ⃗Bit]| ⃗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ( ⃗Eit∗, ⃗Ejt∗)M = [ ⃗Eit∗]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗Ejt∗] = [ ⃗Bit]T | ⃗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗Bjt]| ⃗ A = ( ⃗Bit, ⃗Bjt)A = λ2( ⃗Bit, ⃗Bjt)B = λ2δij, thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30)1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then λ2δij = (dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ei, dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ej)M = (dΘt T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ei, ⃗Ej)M, true for all i, j, thus dΘt T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt = λ2I, thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 65 66 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition of Θ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Definition of Θ Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 The translator from B to A is the function Θ defined with (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) by Θ : � � � � � � t∈[t1,t2] ({t} × Mn1(B)) → Mn1(A) (t, ⃗yS) → Θ(t, ⃗yS) := Θt(⃗yS) , (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all QRB ∈ ObjRB and all t, Θ(t, [ −−−−−−−−−−→ OB �ΦRB(t, QRB)]| ⃗B) = [ −−−−−−−−−−→ OA�ΦRB(t, QRB)]| ⃗A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Θ(t, ⃗ϕS(QRB)) = ⃗ϕD(t, QRB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) gives Θ(t,⃗0) = [−−−−−→ OAOB(t)]| ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 The translator Θ looks like a motion, but is not: A “usual” motion is defined by one observer and connects one particle to its position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' While Θ connects two “matrix positions” of one particle relative to two referentials: Θ is an “inter-referential” function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 The “Θ-velocity” is the drive velocity Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 The “Θ-velocity” and “Θ-acceleration” are defined by (Eulerian type definition) with ⃗yDt = Θ(t, ⃗yS), � � � � � ⃗vΘ(t, ⃗yDt) := ∂Θ ∂t (t, ⃗yS) (∈ Mn1), ⃗γΘ(t, ⃗yDt) = ∂2Θ ∂t2 (t, ⃗yS) (∈ Mn1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15 ⃗vΘ = ⃗vD and ⃗γΘ = ⃗γD , (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗vΘ(t, ⃗y) = ⃗vD(t, ⃗y) and ⃗γΘ(t, ⃗y) = ⃗γD(t, ⃗y) in Mn1, for all t and all ⃗y ∈ Mn1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Θ(t, ⃗ϕS(QRB)) = ⃗ϕD(t, QRB) gives ∂Θ ∂t (t, ⃗ϕS(QRB)) = ∂⃗ϕD ∂t (t, QRB), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗vΘ(t, Θ(t, ⃗ϕS(QRB))) = ⃗vD(t, ⃗ϕD(t, QRB)), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) thus ⃗vΘ(t, ⃗ϕD(t, QRB)) = ⃗vD(t, ⃗ϕD(t, QRB)), thus ⃗vΘ(t, ⃗y) = ⃗vD(t, ⃗y) for all ⃗y ∈ Mn1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Idem with ∂2 ∂t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 The velocity-addition formula (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) gives ⃗ϕA(t, PObj) = Θ(t, ⃗ϕB(t, PObj)), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) thus ∂⃗ϕA ∂t (t, PObj) = ∂Θ ∂t (t, ⃗ϕB(t, PObj)) + dΘ(t, ⃗ϕB(t, PObj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗ϕB ∂t (t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) Thus ⃗vA(t, ⃗xAt) = ⃗vΘ(t, ⃗xAt) + dΘ(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vB(t, ⃗xBt), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38) where ⃗xBt = ⃗ϕB(t, PObj) and ⃗xAt = ⃗ϕA(t, PObj) = Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, with ⃗vΘ =(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) ⃗vD, ⃗vAt = ⃗vBt∗ + ⃗vDt where ⃗vBt∗(⃗xAt) := dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39) which is the velocity-addition formula in RA: ⃗vAt the absolute velocity = ⃗vBt∗ the translated relative velocity from B to A + ⃗vDt the drive velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) In other words (relation between the numerical values of the velocities stored by A and B), [⃗vt(pt)]| ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗vt(pt)] ⃗B + [⃗vRBt(pt)]| ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) 66 67 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Coriolis acceleration, and the acceleration-addition formula 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Coriolis acceleration, and the acceleration-addition formula (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) gives ∂2⃗ϕA ∂t2 (t, PObj) = ∂2Θ ∂t2 (t, ⃗xBt) + d∂Θ ∂t (t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗ϕB ∂t (t, PObj) + �∂(dΘ) ∂t (t, ⃗xBt) + d2Θ(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗ϕB ∂t (t, PObj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗ϕB ∂t (t, PObj) + dΘ(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂2⃗ϕB ∂t2 (t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='42) And d2Θt = 0 in our classical framework (Θt is affine);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ∂Θ ∂t (t, ⃗yS) = ⃗vΘt(Θt(⃗yS)) gives ∂(dΘ) ∂t (t, ⃗yS) = d( ∂Θ ∂t )(t, ⃗yS) = d⃗vΘt(Θt(⃗yS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗yS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ∂ ⃗ϕB ∂t (t, PObj) = ⃗vBt(⃗xBt) where ⃗xBt = ⃗ϕB(t, PObj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ⃗γAt(⃗xAt) = ⃗γΘt(⃗xAt) + 2d⃗vΘt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt) + dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗γBt(⃗xBt) = ⃗γDt(⃗xAt) + 2d⃗vDt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt∗(⃗xAt) + ⃗γBt∗(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43) Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16 At t, the Coriolis acceleration ⃗γCt at ⃗xAt is ⃗γCt(⃗xAt) = 2d⃗vDt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt∗(⃗xAt), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗γCt = 2d⃗vDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='44) And the Coriolis acceleration ⃗γC at t at ⃗xAt is ⃗γC(t, ⃗xAt) := ⃗γCt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43) gives the acceleration-addition formula in RA: ⃗γAt = ⃗γBt∗ + ⃗γDt + ⃗γCt , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45) ⃗γAt the absolute acceleration = ⃗γBt∗ the translated relative acceleration from B to A + ⃗γDt the drive acceleration + ⃗γCt the Coriolis acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='46) In other words (relation between the numerical values of the acceletations stored by A and B), [⃗γt(pt)]| ⃗A = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗γt(pt)] ⃗B + [⃗γRBt(pt)]| ⃗A + 2[d⃗vRBt]| ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗vt(pt)]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='47) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 With an initial time Let t0, t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider the Lagrangian associated function Φt0 t with the motion �Φ of Obj: Φt0 t : � Ωt0 → Ωt pt0=�Φ(t0, PObj) → pt = Φt0 t (pt0) := �Φ(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='48) And, with ⃗xAt = ⃗ϕA(t, PObj) = [−−−→ OApt]| ⃗A and ⃗xBt = ⃗ϕB(t, PObj) = [−−−→ OBpt]| ⃗B, define the “matrix motions” ⃗ϕt0 At : Mn1(A) → Mn1(A) and ⃗ϕt0 Bt : Mn1(B) → Mn1(B) by � � � ⃗ϕt0 At(⃗xAt0) := ⃗xAt (= [ −−−−−−−−→ OA�Φ(t, PObj)]| ⃗A = [−−−−−−−→ OAΦt0 t (pt0)]| ⃗A = ⃗ϕAt(PObj)), ⃗ϕt0 Bt(⃗xBt0) := ⃗xBt (= [ −−−−−−−−→ OB �Φ(t, PObj)]| ⃗B = [−−−−−−−→ OBΦt0 t (pt0)]| ⃗B = ⃗ϕBt(PObj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='49) And Θt(⃗xBt) = ⃗xAt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Θt(⃗ϕt0 Bt(⃗xBt0)) = ⃗ϕt0 At(⃗xAt0) with ⃗xAt0 = Θt0(⃗xBt0), thus Θt ◦ ⃗ϕt0 Bt = ⃗ϕt0 At ◦ Θt0 : Mn1(B) → Mn1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='50) In other words, the following diagram commutes: ⃗xBt0 = ⃗ϕB(t0, PObj) Θt0 � ⃗ϕt0 Bt � ⃗xBt = ⃗ϕt0 Bt(⃗xBt0) Θt � PObj ∈ Obj ⃗ϕBt0 � ⃗ϕAt0 � ⃗xAt0 = ⃗ϕA(t0, PObj) = Θt0(⃗xBt0) ⃗ϕt0 At � ⃗xAt = ⃗ϕt0 At(⃗xAt0) = Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='51) Thus, for any vector field ⃗uBt0 in RB, dΘt(⃗xBt) � �� � (translation at t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' d⃗ϕt0 Bt(⃗xBt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗uBt0(⃗xBt0) � �� � (deformation from t0 to t) = d⃗ϕt0 At(⃗xAt0) � �� � (deformation from t0 to t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dΘt0(⃗xBt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗uBt0(⃗xBt0) � �� � (translation at t0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='52) 67 68 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Drive and Coriolis forces Exercice 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17 Redo the above steps with ObjRB instead of Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider the Lagrangian associated function Φt0 RBt with the motion �ΦRB of ObjRB: Φt0 RBt : � ΩRBt0 = Rn → ΩRBt = Rn qt0 = �ΦRB(t0, QRB) → qt = Φt0 RBt(qt0) := �ΦRB(t, QRB), � (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='53) then define the “matrix motions” ⃗ϕt0 Dt : Mn1(A) → Mn1(A) and ⃗ϕt0 St : Mn1(B) → Mn1(B) by � � � ⃗ϕt0 Dt(⃗yDt0) := ⃗yDt (= [ −−−−−−−−−−→ OA�ΦRB(t, QRB)]| ⃗ A = [−−−−−−−−→ OAΦt0 RBt(pt0)]| ⃗ A = ⃗ϕDt(QRB)), ⃗ϕt0 St(⃗yS) := ⃗yS (= [ −−−−−−−−−−→ OB �ΦRB(t, QRB)]| ⃗ B = [−−−−−−−−→ OBΦt0 RBt(qt0)]| ⃗ B = ⃗ϕS(QRB)), (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54) Thus ⃗ϕS is a time-shift, which is also abusively noted ⃗ϕt0 St = I (algebraic identity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So with Θt(⃗yS) = ⃗yDt we get Θt(⃗ϕt0 Dt(⃗yS)) = ⃗ϕt0 Dt(⃗yDt0), with ⃗yDt0 = Θt0(⃗yS), thus Θt ◦ ⃗ϕt0 St = ⃗ϕt0 Dt ◦ Θt0 : Mn1(B) → Mn1(A) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='55) (also abusively written Θt = ⃗ϕt0 Dt ◦ Θt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In other words, the following diagram commutes: ⃗yS = ⃗ϕS(QRB) Θt0 � ⃗ϕt0 St = time shift � ⃗yS = ⃗ϕS(QRB) Θt � QRB ∈ ObjRB ⃗ϕS � ⃗ϕt0 D � ⃗yDt0 = ⃗ϕDt0(QRB) = Θt0(⃗yS) ⃗ϕt0 Dt � ⃗yDt = ⃗ϕDt(QRB) = ⃗ϕt0 Dt(⃗yDt0) = Θt(⃗yS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='56) And (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='55) gives, for any ⃗yS = ⃗ϕS(QRB) and all vector field ⃗uS (static in RB), with ⃗yDt0 = Θt0(⃗yS), dΘt(⃗yS) � �� � (translation at t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' d⃗ϕt0 St(⃗yS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗uS(⃗yS) � �� � (time shift from t0 to t) = d⃗ϕt0 Dt(⃗yDt0) � �� � (Drive motion from t0 to t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dΘt0(⃗yS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗uS(⃗yS) � �� � (translation at t0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='57) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 Drive and Coriolis forces 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Fundamental principal: requires a Galilean referential Second Newton’s law of motion (fundamental principle of dynamics): In a Galilean referential, the sum of the external forces ⃗f on an object is equal to its mass multiplied by its acceleration: � external⃗f = m⃗γ (in a Galilean referential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='58) Question: And in a Non Galilean referential?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: Then you have to add “observer dependent forces”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' you have to add “apparent forces” due to the motion of the non Galilean observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Indeed, the motion of an object in our Universe does not care about the observer motion (his accelerations and velocities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='v=_36MiCUS1ro for a carousel (a merry-go-round), See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='v=aeY9tY9vKgs for tornadoes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Drive + Coriolis forces = the inertial force Consider ⃗f(t, pt) = the sum of the external forces acting on PObj at t at pt = �Φ(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In a Galilean referential RA, Newton laws (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='58) means [⃗ft(pt)]| ⃗A = m [⃗γt(pt)]| ⃗A, written ⃗fAt(⃗xAt) = m⃗γAt(⃗xAt) ∈ Mn1, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='59) with ⃗xAt := [−−−→ OApt]| ⃗A, ⃗fAt(⃗xAt) := [⃗ft(pt)]| ⃗A and ⃗γAt(⃗xAt) = [⃗γt(pt)]| ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With ⃗xAt = Θt(⃗xBt), the accelera- tion addition formula gives ⃗fAt(⃗xAt) = m(dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗γB(⃗xBt) + ⃗γDt(⃗xAt) + ⃗γCt(⃗xAt)) ∈ RA, thus, in RB, dΘt −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗fAt(⃗xAt) � �� � ⃗fAt∗(⃗xBt)= ⃗fBt(⃗xBt) = m⃗γB(⃗xBt) + m dΘt −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗γDt(⃗xAt) � �� � m ⃗γDt∗(⃗xBt) + m dΘt −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗γCt(⃗xAt) � �� � m ⃗γCt∗(⃗xBt) , (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='60) and dΘt −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ft(pt)]| ⃗A = dΘt −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗fAt(⃗xAt) =(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) [⃗ft(pt)]| ⃗B =noted ⃗fBt(⃗xBt) is the external forces as quanti- fied by B at t, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) (with Θt supposed to be affine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And with the pull-back notation, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26): 68 69 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Summary for “Sun and Earth” Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18 At t on pt, define The drive force ⃗fBDt(⃗xBt) := −m dΘt −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗γDt(⃗xAt) (= −m⃗γDt ∗(⃗xBt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Coriolis force ⃗fBCt(⃗xBt) := −m dΘt −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗γCt(⃗xAt) (= −m⃗γCt ∗(⃗xBt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (The inertial, or fictitious, force := ⃗fBDt(⃗xBt) + ⃗fBCt(⃗xBt) = −m dΘt −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗γDt + ⃗γCt)(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='61) Then (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='60) gives the fundamental principle quantified in RB (non Galilean referential): ⃗fBt(⃗xBt) + ⃗fBDt(⃗xBt) + ⃗fBCt(⃗xBt) = m⃗γB(⃗xBt) , (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='62) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', at t, in RB: The external force + the Drive and Coriolis forces = m times the acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 Summary for “Sun and Earth” Illustation with a simplified (circular) motion of the Earth around the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Coriolis forces on the Earth 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Referentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Relative referential RB = (OB, ( ⃗B1, ⃗B2, ⃗B3)) chosen by the observer B fixed on the Earth, where OBt = �ΦRB(t, QOB) is the position of the particle QOB at the center of the Earth, written OB by B (fixed for B), and ( ⃗B1t, ⃗B2t, ⃗B3t) is a Euclidean basis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' built with the metre) fixed in the Earth, written ( ⃗B1, ⃗B2, ⃗B3) by B (fixed for B), with ⃗B3 chosen to be along the rotation axis of the Earth and oriented from the south pole to the north pole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (·, ·)B is the associated Euclidean dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, a fixed particle QRB in the Earth at longitude θQRB ∈] − π, π] and latitude ϕQRB ∈ [− π 2 , π 2 ] is referenced by observer B as the matrix ⃗yS = ⃗ϕS(QRB) = [ −−−−−−−−−−→ OB �ΦRB(t, QRB)]| ⃗B = RB � � cos(θQRB ) cos(ϕQRB ) sin(θQRB ) cos(ϕQRB ) sin(ϕQRB ) � � where RB = || −−−−−−−−−−→ OB �ΦRB(t, QRB)||B is the distance between QOB and QRB (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' if QRB is on the surface of the Earth then RB ≃ 6371 km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Initial Galilean referential RA0 = (OA0, ( ⃗A1, ⃗A2, ⃗A3)): OA0 is at the center of the Sun and ( ⃗A1, ⃗A2, ⃗A3) is a Euclidean basis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' built with the foot) fixed relative to the stars, such that ⃗A3 = µ ⃗B3 with µ > 0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3048 and λ = 1 µ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (·, ·)A is the associated Euclidean dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Deduced absolute Galilean referential RA = (OAt, ( ⃗A1, ⃗A2, ⃗A3)) chosen by observer A fixed on Earth, where OAt = OBt, written OA by A (fixed for A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Since it takes more that 365 days for QOB to complete a rotation around the Sun, the motion of QOB will be considered to be rectilinear at constant velocity “in a short interval of time” sufficient for the computation of the Coriolis acceleration with “sufficient accuracy” (simplifies the calculations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (If A prefers to work with the initial Galilean referential RA0, then the absolute matrix motion ⃗ϕA(t, PObj) = [ −−−−−−−−→ OA�Φ(t, PObj)]| ⃗A has to be replaced by ⃗ϕA(t, PObj) = [−−−−−−→ OA0OB(t)]| ⃗A + [ −−−−−−−−−−→ OB(t)�Φ(t, PObj)]| ⃗A, idem for the drive motion ⃗ϕD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Drive motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The motion t → qt = �ΦRB(t, QRB) of a particle QRB in the Earth is stored by A as the drive motion ⃗ϕD given by (matrix valued), with ω the angular velocity of the Earth in RA, ⃗yD(t) = ⃗ϕD(t, QRB) = RA(QRB) � � cos(ωt) cos ϕQRB sin(ωt) cos ϕQRB sin ϕQRB � � = [−−−−→ OAq(t)]| ⃗A = � � yD1(t) yD2(t) yD3 � � , (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='63) where RA(QRB) = ||−−−−−→ QOBQRB||| ⃗A is the distance between QOB and QRB for A (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' RA ≃ 20902231 foot if QRB is on the surface of the Earth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (And (ωt) by replaced by (α0+ω(t−t0)) to be more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Drive velocity: With ⃗ωD := ω ⃗A3, ⃗vD(t, ⃗yD(t)) = ⃗yD ′(t) = ωRA � � − sin(ωt) cos ϕQRB cos(ωt) cos ϕQRB 0 � � = ω � � −y2(t) y1(t) 0 � � = ω � � 0 −1 0 1 0 0 0 0 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗yD(t) = ⃗ωD∧⃗yD(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='64) 69 70 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Summary for “Sun and Earth” 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Drive acceleration: ⃗γD(t, ⃗yDt) = ⃗yD ′′(t) = ⃗ωD ∧ ⃗yD ′(t) = ⃗ωD ∧ ⃗vD(t, ⃗yDt) = ⃗ωD ∧ (⃗ωD ∧ ⃗yD(t)) = −ω2 � � yD1(t) yD2(t) 0 � � (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='65) = the usual centrifugal acceleration (in a plane parallel to the equatorial plane, drawing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Differential of the drive velocity (time and space independent here): (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='64) gives d⃗vD(t, ⃗yDt) = d⃗vD = � � 0 −ω 0 ω 0 0 0 0 0 � � = ⃗ωD ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='66) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Translator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Here OAt = OBt, thus Θt(⃗0) = ⃗0 (with [⃗0] =noted ⃗0 = the null matrix), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Calculation of dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With Θt affine, dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗Bit]| ⃗B = [ ⃗Bit]| ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ⃗B3 = λ ⃗A3 (hypothesis) and dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗B3]| ⃗B = [ ⃗B3t]| ⃗A give dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E3 = λ ⃗E3 where ( ⃗Ei) is the canonical basis in Mn1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then let QBi ∈ ObjRB be the Earth particle which position qti = �ΦRB(t, QBi) makes ⃗Bit := −−−−→ OBtqti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, ⃗B1 and ⃗B2 being in the equatorial plane, (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='63) gives dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E1 = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗B1]| ⃗B = [ ⃗B1]| ⃗A = [−−−→ OAqt1]| ⃗A = λ � � cos(ωt) sin(ωt) 0 � �, and dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E2 = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗B2]| ⃗B = [ ⃗B2]| ⃗A = [−−−→ OAqt2]| ⃗A = λ � � − sin(ωt) cos(ωt) 0 � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus [dΘt]| ⃗E = λ � � cos(ωt) − sin(ωt) 0 sin(ωt) cos(ωt) 0 0 0 1 � � = the expected rotation matrix expanded by λ (change of unit of measurement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Calculation of Θt (affine): Θt(⃗yS) = Θt(⃗0) + dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗yS, so, with OAt = OBt here, ⃗yDt := Θt(⃗yS) = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗yS (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='67) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Motions of Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B quantifies the motion �Φ of Obj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' he stores the relative motion ⃗ϕB of Obj, and the relative velocities and accelerations ⃗vBt and ⃗γB (matrices), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10)-(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Translations for A: With ⃗xAt = Θt(⃗xBt), ⃗vBt∗(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt) and ⃗γBt∗(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗γBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='68) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Drive force (apparent force in RB due to the motion of B): ⃗fBDt(⃗xBt) = −m dΘt −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗γDt(⃗xAt) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='65) = λmω2dΘt −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � � xA1(t) xA2(t) 0 � � (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='67) = λmω2 � � xB1(t) xB2(t) 0 � � , (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='69) centrifugal force (in a “parallel plane” at latitude of PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Coriolis acceleration (apparent acceleration due to the motion of B): ⃗γCt(⃗xAt) = 2 d⃗vDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt)) = 2 dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗vDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt) (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='70) because dΘt commutes with d⃗vDt (composition of “rotations along the same south-north axis” which reads as eiωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ei π 2 = ei π 2 eiωt = ei( π 2 +ωt) in the equatorial plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Coriolis force (apparent force due to the motion of B): ⃗fBCt(⃗xBt) = −m dΘt −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗γCt(⃗xAt) = −2m d⃗vDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt) = −2m⃗ω ∧ ⃗vBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='71) 70 71 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' “Isometric objectivity” and “Frame Invariance Principle” 11 Objectivities Goal: To give an objective expression of the laws of mechanics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' As Maxwell [13] said: “The formula at which we arrive must be such that a person of any nation, by substituting for the different symbols the numerical value of the quantities as measured by his own national units, would arrive at a true result”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Generic notation: if a function z is given as z(t, x), then zt(x) := z(t, x), and conversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 “Isometric objectivity” and “Frame Invariance Principle” This manuscript is not intended to describe “isometric objectivity”: “Isometric objectivity” is the framework in which the “principle of material frame-indifference” (“frame invariance principle”) is settled, principle which states that “Rigid body motions should not affect the stress constitutive law of a material”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', Truesdell–Noll [19] p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 41: « Constitutive equations must be invariant under changes of frame of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' » Or Germain [9] : « Axiom of power of internal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The virtual power of the "internal forces" acting on a system S for a given virtual motion is an objective quantity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', it has the same value whatever be the frame in which the motion is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' » NB: Both of these affirmations are limited to “isometric changes of frame” (the same metric for all), as Truesdell–Noll [19] page 42-43 explain: The “isometric objectivity” concern one observer who defines his Euclidean dot product and consider only orthonormal change of bases to validate a constitutive law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If you want to interpret “isometric objectivity” in the “covariant objectivity” framework, then “isometric objectivity” corresponds to a dictatorial management: One observer with his Euclidean referential (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' based on the English foot), imposes his unit of length to all other users (isometry hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Note: The metre was not adopted by the scientific community until after 1875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Moreover, isometric objectivity leads to despise the difference between covariance and contravariance, due to the uncontrolled use of the Riesz representation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Marsden and Hughes [12] p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 8 use this isometric framework to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But, pages 22 and 163, they write that a “good modelization” has to be “covariant objective” (observer independent) to begin with;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And they propose a covariant modelization for elasticity at § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Definition and characterization of the covariant objectivity 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Framework of classical mechanics Framework of classical mechanics to simplify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider two observers A and B and their referentials RA = (OA, ( ⃗Ai)) and RB = (OB, ( ⃗Bi)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ( ⃗Ai) and ( ⃗Bi) are Euclidean bases in foot and metre, (·, ·)A and (·, ·)B is their associated Euclidean dot products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And Θ is the translator, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider a regular motion �Φ of an object Obj, pt = �Φ(t, PObj) ∈ Rn the position at t of a particle in our Universe, Ωt = �Φ(t, Obj) the configuration at t, and C = � t∈[a,b]({t} × Ωt) the set of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ⃗xAt := [−−−→ OApt]| ⃗A ∈ Mn1(A) and ⃗xBt := [−−−→ OBpt]| ⃗B ∈ Mn1(B) are the stored components of pt relative to the chosen referentials, Mn1(A) and Mn1(B) being the spaces of n ∗ 1 matrices as referred to by A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Covariant objectivity of a scalar function Let f : � C → R (t, pt) → f(t, pt) � be a Eulerian scalar function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', a temperature field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' f is quantified by A and B as the functions fA : � R×Mn1(A) → R (t, ⃗xAt) → fA(t, ⃗xAt) := f(t, pt) � and fB : � R×Mn1(B) → R (t, ⃗xBt) → fB(t, ⃗xBt) := f(t, pt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 f is objective covariant iff, for all referentials RA and RB and for all t, fAt(⃗xAt) = fBt(⃗xBt) when ⃗xAt = Θt(⃗xBt), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' fAt = fBt∗ is the push-forward of fBt by Θt cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 71 72 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition and characterization of the covariant objectivity 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Covariant objectivity of a vector field Let ⃗w : � C → ⃗Rn (t, pt) → ⃗w(t, pt) � be a Eulerian vector field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', a force field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗w is quan- tified by A and B as the functions ⃗wA : � R×Mn1(A) → Mn1(A) (t, ⃗xAt) → ⃗wA(t, ⃗xAt) := [⃗w(t, pt)] ⃗A � and ⃗wB : � R×Mn1(B) → Mn1(B) (t, ⃗xBt) → ⃗wB(t, ⃗xBt) := [⃗w(t, pt)] ⃗B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So ⃗wA(t, ⃗xAt) and ⃗wB(t, ⃗xBt) are the column matrices of the components of ⃗w(t, pt) in RA and RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 ⃗w is objective covariant iff, for all referentials RA and RB and for all t, ⃗wAt(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt(⃗xBt) when ⃗xAt = Θt(⃗xBt), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗wAt = ⃗wBt∗ is the push-forward of ⃗wBt by Θt cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Fundamental counter-example: A Eulerian velocity field is not objective, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39), because of the drive velocity ⃗vD ̸= ⃗0 in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Neither is a Eulerian acceleration field, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 The field of gravitational forces (external forces) is objective covariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Covariant objectivity of differential forms Let α : � C → Rn∗ (t, pt) → α(t, pt) � be a Eulerian differential form (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' a measuring device used to get the inter- nal power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' α is quantified by A and B as the functions αA : � R×Mn1(A) → Mn1(A) (t, ⃗xAt) → αA(t, ⃗xAt) := [α(t, pt)] ⃗A � and αB : � R×Mn1(B) → Mn1(B) (t, ⃗xBt) → αB(t, ⃗xBt) := [α(t, pt)] ⃗B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So αA(t, ⃗xAt) and αB(t, ⃗xBt) are the row matrices of the components of α(t, pt) in RA and RB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 α is objective covariant iff, for all referentials RA and RB and for all t, αAt(⃗xAt) = αBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xBt)−1 when ⃗xAt = Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' αAt = αBt∗ is the push-forward of αBt by Θt cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) are compatible: If ⃗w is an objective vector field and if α is an objective differential form, then the scalar function α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w is objective: αAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wAt(⃗xAt) = αBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt(⃗xBt) (= (α(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, pt)), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) since αAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wAt(⃗xAt) = (αBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xBt)−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt(⃗xBt)) = αBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Covariant objectivity of tensors A tensor acts on both vector fields and differential forms, and its objectivity is deduced from the previous §.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, let T be a (Eulerian) tensor corresponding to a “physical quantity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The observers A and B describe T as being the functions TA and TB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 T is objective covariant iff, for all referentials RA and RB and for all t, TAt(⃗xAt) = TBt∗(⃗xAt) (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' TAt is the push-forward of TBt by Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Recall: TBt∗(⃗xAt)(α1(⃗xAt), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗w1(⃗xAt)) := TBt(⃗xBt)(α1∗(⃗xBt), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗w1∗(⃗xBt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 72 73 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Non objectivity of the velocities Example 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 (Non covariant objectivity of a differential d⃗w) Let ⃗w be an objective vector field, seen as ⃗wA by A and ⃗wB by B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So ⃗wAt(⃗xAt) =(11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt(⃗xBt) when ⃗xAt = Θt(⃗xBt), thus d⃗wAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xBt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗wBt(⃗xBt) + (d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt(⃗xBt)), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) hence d⃗wAt(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗wBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xBt)−1 + (d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt(⃗xBt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xBt)−1 ̸= dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗wBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xBt)−1 when d2Θt ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) Thus d⃗w is not covariant objective in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' However in classical mechanics for “change of Cartesian referentials” Θt is affine, so d2Θt = 0, and in particular d⃗w is objective when ⃗w is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (d2 ⃗wAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xBt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xBt) + d⃗wAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d2Θt(⃗xBt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d2 ⃗wBt(⃗xBt) + 2 d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗wBt(⃗xBt) + d3Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) Thus d2 ⃗w is not covariant objective in general (but if Θt is affine then d2 ⃗w is objective if ⃗w is).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Non objectivity of the velocities 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Eulerian velocity ⃗v : not covariant (and not isometric) objective Velocity addition formala: With ⃗vBt∗(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(⃗xBt) when ⃗xAt = Θt(⃗xBt), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39), ⃗vAt(⃗xAt) = ⃗vBt∗(⃗xAt) + ⃗vDt(⃗xAt) ̸= ⃗vBt∗(⃗xAt) when ⃗vDt(⃗xAt) ̸= ⃗0, (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) thus a Eulerian velocity field is not covariant objective (and not isometric objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 d⃗v is not objective The velocity addition formula, (⃗vAt − ⃗vDt)(⃗xAt) = ⃗vBt∗(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt) when ⃗xAt = Θt(⃗xBt), gives d(⃗vAt − ⃗vDt)(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xBt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗vBt(⃗xBt) + d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) thus d⃗v is neither covariant objective nor isometric objective because of d⃗vD: d⃗vAt(⃗xAt) = d⃗vBt∗(⃗xAt) + d⃗vDt(⃗xAt) + d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xBt)−1 ̸= d⃗vBt∗(⃗xAt) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 Recall: “Isometric objective” implies The use of the same Euclidean metric in RB and RA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (·, ·)A = (·, ·)B, �ΦRB (motion of RB) is a solid body motion, and Θt is affine (so d2Θt = 0 for all t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 Prove, with Qt the (orthonormal) transition matrix from ( ⃗Ai) to ( ⃗Bi): [d⃗vt]| ⃗B = Qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗vt]| ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Q−1 t + Q′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Q−1 t , written [L]| ⃗B = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]| ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='QT + Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='QT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) (Used in classical mechanics courses, to prove that d⃗v isn’t “isometric objective” because of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='QT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' t0, t ∈ R, pt0 = �Φ(t0, PObj), pt = �Φ(t, PObj) = Φt0 t (pt0), ⃗v(t, pt) = ∂ �Φ ∂t (t, PObj), and F t0 t (pt0) = dΦt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So ⃗v(t, Φt0 t (pt0)) = ∂Φt0 pt0 ∂t (t, pt0), thus d⃗v(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 pt0 (t) = ∂F t0 pt0 ∂t (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30), with F t0 pt0 =noted F, gives [F(t)]|⃗at0 , ⃗ B = Q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F(t)]|⃗at0 , ⃗ A, thus [F ′(t)]|⃗at0 , ⃗ B = Q′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F(t)]|⃗at0 , ⃗ A + Q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F ′(t)]|⃗at0 , ⃗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus [d⃗v(t, pt)]| ⃗ B = [F t0 pt0 ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 pt0 (t)]| ⃗ B = [F t0 pt0 ′(t)]| ⃗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F t0 pt0 (t)]| ⃗ B = (Q′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F(t)]|⃗at0 , ⃗ A + Q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F ′(t)]|⃗at0 , ⃗ A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F(t)]−1 |⃗at0 , ⃗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Q(t)−1 = Q′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Q(t)−1 + Q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F ′(t)]|⃗at0 , ⃗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F(t)]−1 |⃗at0 , ⃗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Q(t)−1 = Q′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Q(t)−1 + Q(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗v(t, pt)]| ⃗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Q(t)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34)- (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 Prove that d2⃗v is “isometric objective” when �ΦRB is a rigid body motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) with ⃗vA − ⃗vD instead of ⃗wA, and ⃗vB instead of ⃗wB give, in an “isometric objective” framework, d2(⃗vAt − ⃗vDt)(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗uBt∗, ⃗wBt∗) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d2⃗vBt(⃗xBt)(⃗uB, ⃗wB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) Here d2⃗vDt = 0 (rigid body motion), thus d2⃗v is “isometric objective”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 73 74 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Lie derivatives are covariant objective 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 d⃗v + d⃗vT is “isometric objective” Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 If �ΦRB is a rigid body motion then d⃗vt + d⃗vT t is “isometric objective” d⃗vAt + d⃗vT At = (d⃗vBt + d⃗vT Bt)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) (Isometric framework: The rate of deformation tensor is independent of an added added rigid motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='QT = I gives Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='QT + ( Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='QT )T = 0, then apply (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 Prove that Ω = d⃗v−d⃗vT 2 is not isometric objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) gives d⃗vT At = d⃗vT Bt∗ + d⃗vT Dt, thus d⃗vAt−d⃗vT At 2 = d⃗vBt∗−d⃗vT Bt∗ 2 + d⃗vDt−d⃗vT Dt 2 ̸= d⃗vBt∗−d⃗vT Bt∗ 2 , even if �ΦRB is a solid body motion (then d⃗vDt−d⃗vT Dt 2 = ⃗ω∧ is a rotation time a dilation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Lagrangian velocities The Lagrangian velocities do not define a vector field, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus asking about the objectivity of Lagrangian velocities is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 The Lie derivatives are covariant objective Framework of § 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular we have the velocity-addition formula ⃗vAt = ⃗vBt∗ + ⃗vDt in RA where ⃗vBt∗(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt) and ⃗xBt = Θt(⃗xAt), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The objectivity under concern is the covariant objectivity (no inner dot product or basis required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Lie derivatives are also called “objective rates” because they are covariant objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Easy proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Scalar functions Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 If f be a covariant objective function, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1), then its Lie derivative L⃗vf is covariant objective: L⃗vAfA = Θ∗(L⃗vBfB), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L⃗vAfA(t, ⃗xAt) = L⃗vBfB(t, ⃗xBt) when ⃗xAt = Θt(⃗xBt), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', DfA Dt (t, ⃗xAt) = DfB Dt (t, ⃗xBt), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ( ∂fA ∂t + dfA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vA)(t, ⃗xAt) = ( ∂fB ∂t + dfB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vB)(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider the motion t → p(t) = �Φ(tPObj) of a particle PObj, and ⃗xA(t) = [−−−−→ OAp(t)]| ⃗A and ⃗xB(t) = [−−−−→ OBp(t)]| ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With f objective, (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) gives fB(t, ⃗xB(t)) = fA(t, Θ(t, ⃗xB(t))) (= fA(t, ⃗xA(t))), thus DfB Dt (t, ⃗xB(t)) = ∂fA ∂t (t, ⃗xA(t)) + dfAt(⃗xA(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (∂Θ ∂t (t, ⃗xB(t)) � �� � ⃗vDt(⃗xAt) + dΘt(⃗xB(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xB(t))) � �� � ⃗vBt∗(⃗xAt) = ∂fA ∂t (t, ⃗xAt) + dfAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vAt(⃗xAt) = DfA Dt (t, ⃗xAt), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) thanks to velocity addiction formula ⃗vAt = ⃗vBt∗ + ⃗vDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Vector fields Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15 Let ⃗w be a covariant objective vector field, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then its Lie derivative L⃗v ⃗w is covariant objective: L⃗vA ⃗wA = Θ∗(L⃗vB ⃗wB), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', when ⃗xAt = Θt(⃗xBt), L⃗vA ⃗wA(t, ⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L⃗vB ⃗wB(t, ⃗xBt), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', (D ⃗wA Dt − d⃗vA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wA)(t, ⃗xAt) = dΘ(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (D ⃗wB Dt − d⃗vB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wB)(t, ⃗xBt), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', (∂ ⃗wA ∂t + d⃗wA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vA − d⃗vA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wA)(t, ⃗xAt) = dΘ(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (∂ ⃗wB ∂t + d⃗wB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vB − d⃗vB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wB)(t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) But the partial, convected, material, and Lie autonomous derivatives are not covariant objective (not 74 75 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Lie derivatives are covariant objective even isometric objective because of the drive velocity ⃗vD): We have (d⃗wAt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗vAt−⃗vDt))(⃗xAt) = (dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗wBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt) + (d2Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt)(⃗xBt), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) (d(⃗vAt−⃗vDt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wAt)(⃗xAt) = (dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗vBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt) + (d2Θt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt)(⃗xBt), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) (d(⃗vAt−⃗vDt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗vAt−⃗vDt))(⃗xAt) = (dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗vBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt) + d2Θt(⃗vBt,⃗vBt))(⃗xBt), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) L0 (⃗vAt−⃗vDt) ⃗wAt(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L0 ⃗vBt ⃗wBt(⃗xBt), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) ∂ ⃗wA ∂t (t, ⃗xAt) + L0 ⃗vD ⃗wAt(⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂ ⃗wB ∂t (t, ⃗xBt), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) D ⃗wA Dt (t, ⃗xAt) − d⃗vDt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wAt(⃗xAt) = dΘt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D ⃗wB Dt (t, ⃗xBt) + d2Θt(⃗vBt, ⃗wBt)(⃗xBt), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) ∂(⃗vA−⃗vD) ∂t (t, ⃗xAt) + L0 ⃗vD(⃗vA−⃗vD)(t, ⃗xAt) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗vB ∂t (t, ⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' • ⃗wAt(Θt(⃗xBt)) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt(⃗xBt) gives d⃗wAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xBt) = d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt(⃗xBt) + dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗wB(⃗xBt), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) thus, with dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt) = (⃗vAt−⃗vDt)(⃗xAt) = ⃗vBt∗(⃗xAt) (velocity-addition formula), d⃗wAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗vAt−⃗vDt)(⃗xAt) = (d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt(⃗xBt) + dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗wBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt), hence (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular d⃗wAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vAt(⃗xAt) ̸= dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗wBt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt)) (the vector field d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v is not objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗vAt−⃗vDt)(Θt(⃗xBt)) = dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt) gives d(⃗vAt−⃗vDt)(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xBt) = d2Θt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xBt) + dΘt(⃗xBt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗vBt(⃗xBt), so, applied to ⃗wBt (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗vBt), we get (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If ⃗xAt = Θt(⃗xB), then ⃗wA(t, Θ(t, ⃗xB)) = dΘ(t, ⃗xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wB(t, ⃗xB), so, with ∂Θ ∂t (t, ⃗xB) = ⃗vΘt(⃗xAt), we get ∂ ⃗wA ∂t (t, ⃗xAt) + d⃗wAt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vΘt(⃗xAt) = d∂Θ ∂t (t, ⃗xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt(⃗xB) + dΘt(⃗xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂ ⃗wB ∂t (t, ⃗xB) = (d⃗vΘt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xB)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wBt(⃗xB) + dΘt(⃗xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂ ⃗wB ∂t (t, ⃗xB), Thus (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) since ⃗vΘ = ⃗vD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) gives (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗vB∗(t, Θ(t, ⃗xB)) = dΘ(t, ⃗xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vB(t, ⃗xB) gives ∂⃗vB∗ ∂t (t, ⃗xAt) + d⃗vB∗(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vΘ(t, ⃗xAt) = ∂dΘ ∂t (t, ⃗xB) � �� � d⃗vΘt(⃗xAt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΘt(⃗xB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vBt(⃗xB) + dΘ(t, ⃗xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂⃗vB ∂t (t, ⃗xB, ) since ∂dΘ ∂t (t, ⃗xB) = d( ∂Θ ∂t )(t, ⃗xB) and ∂Θ ∂t (t, ⃗xB) = ⃗vΘ(t, ⃗xAt) = ⃗vΘt(Θt(⃗xB));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' hence (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Tensors Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16 It T is a covariant objective tensor, then its Lie derivatives are covariant objectives: L⃗vATA = Θ∗(L⃗vBTB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Corollary of (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) to get L⃗v(α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) = (L⃗vα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L⃗v ⃗w);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then use L⃗v(t1 ⊗ t2) = (L⃗vt1) ⊗ t2 + t1 ⊗ (L⃗vt2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 75 76 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Taylor expansions and ubiquity gift 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Taylor expansions and ubiquity gift 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 In Rn with ubiquity Generic formula: f(t) = f(t0) + (t−t0) f ′(t0) + (t−t0)2 2 f ′(t0)2 + o((t−t0)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) In particular f(t) = ⃗w(t, p(t)) gives ⃗w(t, p(t)) = ⃗w(t0, p(t0)) + (t−t0) D ⃗w Dt (t0, p(t0)) + (t−t0)2 2 D ⃗w Dt (t0, p(t0))2 + o((t−t0)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) Problem : ⃗w(t, p(t)) is a vecteur at t at p(t) while ⃗w(t0, p(t0)) is a vecteur at t0 at p(t0), so (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) cannot be written ⃗w(t, p(t)) − � ⃗w(t0, p(t0)) + (t−t0) D ⃗w Dt (t0, p(t0)) + (t−t0)2 2 D ⃗w Dt (t0, p(t0))2� = o((t−t0)2), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) since the left-hand side supposes the ubiquity gift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' in a non-planar manifold (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' a surface in R3 considered on its own), ⃗w(t, pt) ∈ Tpt(Ωt) = the linear tangent space at p(t) = pt, whereas ⃗w(t0, pt0) ∈ Tpt0(Ωt0) = the linear tangent space at p(t0) = pt0, and the tangent spaces Tpt(Ωt) and Tpt0(Ωt0) are distinct at two distinct points in general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the left-hand side of (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In R3 our affine space (our Universe), Tpt(Ωt) and Tpt0(Ωt0) are identified with ⃗R3, and (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) is well defined, and very useful!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 General case By definition, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10), with p(t) = Φt0(t, pt0) = Φt0 t (pt0), L⃗v ⃗w(t0, pt0) = dΦt0 t (pt0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, p(t)) − ⃗w(t0, pt0) t − t0 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) Thus, dΦt0 t (pt0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, p(t)) = ⃗w(t0, pt0) + (t−t0) L⃗v ⃗w(t0, pt0) + o(t−t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) Hence we get the first order Taylor expansion without ubiquity gift: ⃗w(t, p(t)) = dΦt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � ⃗w + (t−t0) L⃗v ⃗w � (t0, pt0) + o(t−t0), (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) both side of the equality being in Tpt(Ωt) (meaningful in any manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17 In Rn, with the gift of ubiquity, (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) gives (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dΦt0(t0+h, pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t0, pt0) =(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) ⃗w(t0, pt0) + h d⃗v(t0, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t0, pt0) + o(h), thus ⃗w(t, pt) (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) = (I + h d⃗v(t0, pt0) + o(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � (⃗w + h D ⃗w Dt − h d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w)(t0, pt0) + o(h) � = (⃗w + h d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + h D ⃗w Dt − h d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w)(t0, pt0) + o(h) = (⃗w + h D ⃗w Dt )(t0, pt0) + o(h), which is (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18 In Rn, at second order, ⃗w(t, p(t)) = dΦt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � (⃗w + hL⃗v ⃗w + h2 2 L⃗v(L⃗v ⃗w))(t0, pt0) + o(h2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) So if the values ⃗w(t0, pt0), L⃗v ⃗w(t0, pt0) and L⃗v(L⃗v ⃗w)(t0, pt0) are known, then ⃗w(t, p(t)) is estimated at second order thanks to the push-forward of (⃗w + hL⃗v ⃗w + h2 2 L⃗v(L⃗v ⃗w))(t0, pt0) by Φt0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 76 77 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Taylor expansions and ubiquity gift Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let dΦt0(t, pt0) = F t0 pt0 (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗g(t) = dΦt0(t, pt0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, p(t)) when p(t) = Φt0(t, pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So L⃗v ⃗w(t0, pt0) = ⃗g ′(t0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And D ⃗w Du (u, p(u)) = d⃗v(u, p(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(u, p(u)) + F t u(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗g ′(u), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus D2 ⃗w Du2 (u, p(u)) = D(d⃗v) Du (u, p(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(u, p(u)) + d⃗v(u, p(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' D ⃗w Du (u, p(u)) + d⃗v(u, p(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t u(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗g ′(u) + F t u(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗g ′′(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus D2 ⃗w Dt2 (t, p(t)) = ( D(d⃗v) Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' D ⃗w Dt + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L⃗v ⃗w)(t, p(t)) + ⃗g ′′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39) we get ⃗g ′′(t) = L⃗v(L⃗v ⃗w)(t, p(t)), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Alternate proof (calculation): (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) gives F t0 pt0 (t) = It0 + h d⃗v(t0, pt0) + h2 2 d⃗γ(t0, pt0) + o(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, omitting the reference to (t0, pt0) to lighten the writing, dΦt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗w + hL⃗v ⃗w + h2 2 L⃗vL⃗v ⃗w + o(h2)) = � I + h d⃗v + h2 2 d(D⃗v Dt ) + o(h2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � ⃗w + hL⃗v ⃗w + h2 2 L⃗vL⃗v ⃗w + o(h2) � (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) The h0 term is I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The h term is L⃗v ⃗w + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = D ⃗w Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The h2 term is the sum of 1 2L⃗vL⃗v ⃗w = 1 2(D2 ⃗w Dt2 − 2 d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D ⃗w Dt − D(d⃗v) Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39), d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L⃗v ⃗w = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D ⃗w Dt − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = 1 2(2d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D ⃗w Dt − 2d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w), 1 2d(D⃗v Dt ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = 1 2(D(d⃗v) Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the sum gives D2 ⃗w Dt2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 77 78 Part V Appendix In this appendix, we tried to give standard results useful in mechanics, results that are scattered in the existing literature, and sometimes difficult to find except in math books (differential geometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The definitions, notations and results are detailed, so that no ambiguity is possible (some notations can be nightmarish when not understood, or misused, or come like a bull in a china-shop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' All the results presented apply to solids, fluids, thermodynamics, general relativity, electromagnetism, quantum mechanics, chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (the same math applies to all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' even applies to mechanical engineers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A Classical and duality notations A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Contravariant vector and basis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Contravariant vector Let (E, +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') =noted E be a real vector space (= a linear space on the field R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 An element ⃗x ∈ E is called a vector, or a “contravariant vector”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A vector is a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So why this name contravariant?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Historical answer: Because of the change of basis formula [⃗x]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|old, see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28), which uses P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, what is a covariant vector?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: From the vector space E, you can build the vector space (an overlay) L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) =noted E∗ = the space of linear forms on E (a linear form is a measuring instrument that gives values to vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then an element ℓ ∈ E∗ will be called a covariant vector, because of the change of basis formula [ℓ]|new = [ℓ]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Basis Definitions: • n vectors ⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en ∈ E are linearly independent iff, for all λ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', λn ∈ R, the equality �n i=1λi⃗ei = ⃗0 implies λi = 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' n vectors ⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en ∈ E span E iff, for all ⃗x ∈ E, ∃λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', λn ∈ R such that ⃗x = �n i=1λi⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A basis in E is a set {⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en} ⊂ E made of n linearly independent vectors which span E, in which case the dimension of E is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Canonical basis Consider the field R of reals and the Cartesian product ⃗Rn = R × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' × R, n times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The canonical basis is ⃗e1 = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', 0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗en = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', 0, 1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) with 0 = the addition identity element used n−1 times, and 1 = the multiplication identity element used once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The 3-D geometric space we live in has no canonical basis: What would the identity element 1 mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1 metre?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1 foot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And there is no “intrinsic” preferred direction to define ⃗e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So the Cartesian product ⃗Rn = R × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' × R and its canonical basis form an abstract mathematical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Cartesian basis (René Descartes 1596-1650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Let n = 1, 2, 3, let Rn be the usual affine space (space of points), and let ⃗Rn = (⃗Rn, +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') be the usual real vector space of bipoint vectors with its usual algebraic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let p ∈ Rn, and let (⃗ei(p)) be a basis at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A Cartesian basis in ⃗Rn is a basis independent of p (the same at all p), and then (⃗ei(p)) =noted (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example of a non Cartesian basis: The polar basis, see example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 (polar coordinate system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And a Euclidean basis is a particular Cartesian basis described in § B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' More generally, a Cartesian basis refers to En = E ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='×E (n-times) where E is a dimension 1 vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 78 79 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Representation of a vector relative to a basis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Representation of a vector relative to a basis We give: the classical notation (non ambiguous), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' used by Arnold [3] and Germain [8], and the duality notation (can be ambiguous because of misuses), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' used by Marsden and Hughes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Both classical and duality notation are equally good, but if you have any doubt, use the classical notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Let ⃗x ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗ei) be a basis in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The components of ⃗x relative to the basis (⃗ei) are the n real numbers x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', xn (classical notation) also named x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', xn (duality notation) such that ⃗x = x1⃗e1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' + xn⃗en � �� � clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = x1⃗e1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' + xn⃗en � �� � dual , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗e = � � x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' xn � � � �� � clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = � � x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' xn � � � �� � dual , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) [⃗x]|⃗e being the column matrix representing ⃗x relative to the basis (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Of course xi = xi for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') And the column matrix [⃗x]|⃗e is simply named [⃗x] if one chosen basis is imposed to all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With the sum sign: ⃗x = n � i=1 xi⃗ei � �� � clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = n � i=1 xi⃗ei � �� � dual (= n � J=1 xJ⃗eJ = n � α=1 xα⃗eα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) (The index in a summation is a dummy index, even if you do not write the sum sign � as can be done with Enstein’s convention: ⃗x = �n j=1xj⃗ej =noted xj⃗ej = xi⃗ei = xJ⃗eJ = xα⃗eα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 In ⃗R2 with ⃗x = 3⃗e1 + 4⃗e2 = �2 i=1 xi⃗ei = �2 i=1 xi⃗ei: We have x1=x1=3 and x2=x2=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And [⃗x]|⃗e = 3[⃗e1]|⃗e +4[⃗e2]|⃗e = �2 i=1 xi[⃗ei]|⃗e = �2 i=1 xi[⃗ei]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular, with δi j = δij := � = 1 if i=j = 0 if i̸=j � the Kronecker symbols, ⃗ej = n � i=1 δij⃗ei � �� � clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = n � i=1 δi j⃗ei � �� � dual , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗e1]|⃗e = � � � � 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 0 � � � � , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', [⃗en]|⃗e = � � � � 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 0 1 � � � � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) that is, the components of ⃗ej are δij with classical notations, and δi j with duality notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the matrices [⃗ej]|⃗e mimic the use of theoretical Cartesian space ⃗Rn = R × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' × R and its canonical basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 The column matrix [⃗x]|⃗e is also called a “column vector”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: A “column vector” is not a vector, but just a matrix (a collection of real numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See the change of basis formula (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) where the same vector is represented by two “column vectors” (two column matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Dual basis Recall: Let E and F be vector spaces and (F(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F), +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') =noted F(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) be the usual real vector space of functions with the internal addition (f, g) → f +g defined by (f +g)(x) := f(x)+g(x) and the external multiplication (λ, f) → λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='f defined by (λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='f)(x) := λ(f(x)), for all f, g ∈ F(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F), x ∈ E, λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='f =noted λf for all f ∈ F(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) and λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Linear forms = “Covariant vectors” Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 The set E∗ := L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) of linear scalar valued functions is called the dual of E: E∗ := L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) = the dual of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) And a linear scalar valued function ℓ ∈ E∗ is called a linear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' More precisely, E∗ as defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) is the algebraic dual of E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' To define the topological dual usually needed with L2 functions in mechanics, E needs to be a Banach space (a vector space equipped with a norm with which E is complete), and E∗ is then the set of continuous linear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (If E is finite dimensional then any linear form is continuous relative to any norm since all norms are equivalent in finite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 79 80 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Dual basis E∗ is a vector space: sub-space of (F(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') (trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Interpretation: It answers the question: What does a function E → R do?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: Like any function, it gives values to vectors: ℓ(⃗u) = the value of ⃗u through ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' That is, a ℓ ∈ E∗ is a measuring tool for vectors: If ⃗u ∈ E then ℓ(⃗u) = real value given by ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Notation: If ℓ ∈ E∗ then ∀⃗u ∈ E, ℓ(⃗u) noted = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) also written ⟨ℓ, ⃗u⟩E∗,E where ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⟩E∗,E is the duality bracket: The dot in ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u is “the distributivity dot” since linearity ℓ(⃗u + λ⃗v) = ℓ(⃗u) + λℓ(⃗v) = distributivity ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗u + λ⃗v) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u + λℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: The dot in ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u is not an inner dot product (since ℓ /∈ E while ⃗u ∈ E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 A linear form ℓ in E∗ is also called a “covariant vector”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Co-variant refers to: 1- The action of a function on a vector, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) (co-variant calculation), and 2- The change of coordinate formula [ℓ]new = [ℓ]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P, see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) (covariant formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: E∗ being a vector space, an element ℓ ∈ E∗ is indeed a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But E∗ has no existence if E has not been specified first since E∗ := L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ℓ ∈ E∗ can’t be confused with a vector ⃗u ∈ E since there is no natural canonical isomorphism between E and E∗ (no “intrinsic representation”), see § T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Misner–Thorne–Wheeler [14], box 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1, insist: “Without it [the distinction between covari- ance and contravariance], one cannot know whether a vector is meant or the very different object that is a linear form.” A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Covariant dual basis (= the functions that give the components of a vector) Notation: If ⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗uk are vectors in E, then Vect{⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗uk} := the vector space spanned by ⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let E be a finite dimensional vector space, and let (⃗ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n be a basis in E Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 Let i ∈ [1, n]N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The scalar projection on Vect{⃗ei} parallel to Vect{⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗ei−1,⃗ei+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en} is the linear form named πei ∈ E∗ with the classical notation, named ei ∈ E∗ with the duality notation, defined by, for all i, j, � clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' : πei(⃗ej) = δij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' πei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = δij, dual not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' : ei(⃗ej) = δi j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = δi j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) Thus, πei = ei being linear, if ⃗x =clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' �n i=1xi⃗ei =dual �n i=1xi⃗ei (classical or duality notations), then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) gives πei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = xi = xi dual = ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' πei = ei gives the i-th component of a vector ⃗x relative to the basis (⃗ei), see figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1: Parallel projections: πe1(⃗x) = x1 and πe2(⃗x) = x2 (dual not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' : e1(⃗x) = x1 and e2(⃗x) = x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: The dual basis (πei) is intrinsic to (⃗ei);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And there can’t be any notion of orthogonality in E here since we can’t use a inner dot product: The functions πei = ei and vectors ⃗x do not belong to a same vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 and definition of the dual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (πei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n = (ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n is a basis in E∗, called the (covariant) dual basis of the basis (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 80 2 f 1 1 2 1 er81 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Dual basis Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If �n i=1λiπei = 0, then 0 = (�n i=1λiπei)(⃗ej) = �n i=1λiπei(⃗ej) = �n i=1λiδij = λj for all j, thus (πei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n is a family of n independent vectors in E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then let ℓ ∈ E∗ and m = � i(ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei)πei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus m ∈ E∗ (since E∗ is a vector space), and m(⃗ej) = � i(ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei)(πei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej) = � i(ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei)δij = (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej), thus m = ℓ, thus ℓ = � i(ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei)πei, thus Vect{(πei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n} span E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (πei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n is a basis in E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus dim E∗ = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Use duality notations if you prefer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 Following example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The size of a child is represented on a wall by a bipoint vector ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And English observer chooses the foot as unit of length, represented by a vertical bipoint vector which he names ⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And then defines the linear form πe : ⃗R → R by πe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus πe is a measuring instrument, which gives s = πe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = the size of the child in foot, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗u = s⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 Let (⃗ai) and (⃗bi) be bases and let (πai) and (πbi) be the dual bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let λ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove: If ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, ⃗bi = λ⃗ai, then ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, πbi = 1 λ πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) (With duality notations, bi = 1 λ ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' πbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj = δij = πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗bj λ = 1 λ πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj for all j (since πai is linear), thus πbi = 1 λ πai, true for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Example: aeronautical units Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 International aeronautical units: Horizontal length = nautical mile (NM), altitude = English foot (ft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Application: An air traffic controller chooses the point O = the position of its control tower, and a plane p is located thanks to the bipoint vector ⃗x = −→ Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the traffic controller chooses ⃗e1 = the vector of length 1 NM oriented South (first runway), ⃗e2 = the vector of length 1 NM oriented Southwest (second runway), ⃗e3 = the vertical vector of length 1 ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus his referential is R = (O, (⃗e1,⃗e2,⃗e3)), and his dual basis (πe1, πe2, πe3) is defined by πei(⃗ej) = δij for all i, j, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' He writes ⃗x = �n i=1xi⃗ei ∈ ⃗Rn, so that x1 = πe1(⃗x) = the distance to the south in NM, x2 = πe2(⃗x) = the distance to the southwest in NM, x3 = πe3(⃗x) = the altitude in ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Here the basis (⃗ei) is not a Euclidean basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This non Euclidean basis (⃗ei) is however vital if you take a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A Euclidean basis is not essential to life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See next remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 The metre is the international unit for NASA that launched the Mars Climate Orbiter probe, and the foot is the international vertical unit for aviation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And for the Mars Climate Orbiter landing procedure, NASA (uses the metre) asked Lockheed Martin (uses the foot) to do the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Result?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Mars Climate Orbiter space probe burned in the Martian atmosphere because of λ ∼ 3 times too high a speed during the landing procedure: One metre is λ ∼ 3 times one foot, and someone forgot it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Although NASA and Lockheed Martin used a Euclidean dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But not the same (one based on a metre, and one based on the foot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Objectivity and covariance can be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Matrix representation of a linear form Let ℓ ∈ E∗, let (⃗ei) be a basis: The components of ℓ are the n reals ℓi := ℓ(⃗ei) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei, and [ℓ]|⃗e = ( ℓ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ℓn ) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) is the row matrix of ℓ, called the matrix of ℓ relative to (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, if ⃗x ∈ E and ⃗x =clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' �n i=1xi⃗ei =dual �n i=1xi⃗ei, then ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = n � i=1 ℓixi dual = n � i=1 ℓixi = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗e (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) with usual matrix computation rules (a 1 ∗ n matrix times a n ∗ 1 matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular for the dual basis (πei) = (ei) (classical and duality notations), [πej]|⃗e = [ej]|⃗e = (0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 0 1 ���� jth position 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 0) (= row matrix = [⃗ej]T |⃗e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) Thus we have, with classical and duality notations, ℓ clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = n � i=1 ℓi πei dual = n � i=1 ℓi ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) 81 82 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Einstein convention Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15 Relative to a basis, a vector is represented by a column matrix, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2), and a linear form by a row matrix, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This enables: The use of matrix calculation to compute ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗e, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11), not to be confused with an inner dot product calculation ⃗x • ⃗y relative to an inner dot product in E for ⃗x, ⃗y ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Not to confuse the “nature of objects”: Relative to a basis, a (contravariant) vector is a mathematical object represented by a column matrix, while a linear form (covariant vector) is a mathematical object represented by a row matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Example: Thermodynamic Consider the Cartesian space ⃗R2 = {(T, P) ∈ R×R} = {(temperature,pressure)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' There is no meaningful inner dot product in this ⃗R2: What would √ T 2+P 2 mean (Pythagoras: Can you add Kelvin degrees and kg/(m·s2)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, in thermodynamics, the (covariant) dual bases are the main ingredient for calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', in the Cartesian product ⃗R2 = R × R consider the basis ( ⃗E1=(1, 0), ⃗E2=(0, 1)) (after a choice of temperature and pressure units);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗X ∈ ⃗R2, ⃗X = T ⃗E1 + P ⃗E2 =noted (T, P), and let (πE1, πE2) = (E1, E2) =noted (dT, dP) be the (covariant) dual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The first principle of thermodynamics tells that the density α of internal energy is an exact differential form: ∃U ∈ C1( ⃗R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' α = dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, at any ⃗X0 = (T0, P0), α( ⃗X0) = dU( ⃗X0) = ∂U ∂T ( ⃗X0) dT + ∂U ∂P ( ⃗X0) dP and [dU( ⃗X0)]| ⃗E = � ∂U ∂T ( ⃗X0) ∂U ∂P ( ⃗X0) � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) (row matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And we have the first order Taylor expansion in the vicinity of ⃗X0, U( ⃗X0 + δ ⃗X) = U( ⃗X0) + dU( ⃗X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='δ ⃗X + o(δ ⃗X) = U(T0, P0) + δT ∂U ∂T (T0, P0) + δP ∂U ∂T (T0, P0) + o((δT, δP)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) Matrix computation: Column matrices for vectors, row matrices for linear forms: [ ⃗E1]| ⃗E = � 1 0 � , [ ⃗E2]| ⃗E = � 0 1 � , [ ⃗X0]| ⃗E = � T0 P0 � , [δ ⃗X]| ⃗E = � δT δP � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) [E1]| ⃗E = [dT]| ⃗E = ( 1 0 ) , [E2]| ⃗E = [dP]| ⃗E = ( 0 1 ) , [dU]| ⃗E = ( ∂U ∂T ∂U ∂P ) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) give dU( ⃗X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='δ ⃗X = � ∂U ∂T ( ⃗X0) ∂U ∂P ( ⃗X0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � δT δP � = ∂U ∂T ( ⃗X0)δT + ∂U ∂P ( ⃗X0)δP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) This is a “covariant calculation” (in particular no inner dot product has been used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Einstein convention A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition When you work with components (after a choice of a basis), the goal is to visually differentiate a lin- ear form from a vector (to visually differentiate covariance from contravariance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Framework: a finite dimension vector space E, dim E = n, and duality notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Einstein Convention: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A basis in E (contravariant) is written with bottom indices: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', (⃗ei) is a basis in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A vector ⃗x ∈ E (contravariant) has its components relative to (⃗ei) (quantification) written with top indices: ⃗x = �n i=1xi⃗ei, and is represented by the column matrix [⃗x]|⃗e = � � x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' xn � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Classical notations: ⃗x = �n i=1xi⃗ei, and column matrix of xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A basis in E∗ = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) (covariant) is written with top indices: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', (ei) ∈ E∗n is the dual basis of the basis (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Classical notations: (πei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A linear form ℓ ∈ E∗ (covariant) has its components relative to (ei) (quantification) written with bottom indices: ℓ = �n i=1ℓiei, and its matrix representation is the row matrix [ℓ]|⃗e = ( ℓ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ℓn ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 82 83 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Change of basis formulas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' You can also omit the sum sign � when there are repeated indices at a different position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' �n i=1xi⃗ei =noted xi⃗ei, and �n i=1Lij⃗ei =noted Li j⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In fact, before computers and word processors, to print �n i=1 was not easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But with LATEX this is no more a problem, so in this manuscript the sum sign � is not omitted (and some confusions are avoided).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16 Einstein’s convention is not mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Arnold doesn’t use it when he doesn’t need it, or when it makes reading difficult, or when it induces misunderstandings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In classical mechanics, Einstein’s convention may induce more confusion than understandings, and may be misused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' so it is better not to use it: Golden rule: Use classical notations when in doubt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Do not mistake yourself 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Einstein’s convention is just meant not to confuse a linear function with a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It only deals with quantification relative to a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Classical notations are as good as duality notations, even you are told that classical notations cannot detect obvious errors in component manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But duality notations can be misused in classical mechanics (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the paradigmatic example of the vectorial dual basis, correctly treated at § F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And thus add confusion to the confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The convention does not admit shortcuts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' with a metric: g(⃗u,⃗v) = �n i,j=1gijuivj shows the observer dependence on a choice of a basis thanks to the gij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And even if gij = δij you cannot write g(⃗u,⃗v) = �n i,j=1uivj: You have to write g(⃗u,⃗v) = �n i,j=1gijuivj: Unmissable in physics because you need to see the metric and bases in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Golden rule: Return to classical notations if in doubt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Einstein’s convention can add confusions, un- truths, misinterpretations, absurdities, misuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Change of basis formulas E being a finite dimension vector space, dim E = n, let (⃗eold,i) and (⃗enew,i) be two bases in E, and let (πold,i) and (πnew,i) be the dual bases in E∗, written (ei old) and (ei new) with duality notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Change of basis endomorphism and transition matrix Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17 The change of basis endomorphism P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) from (⃗eold,i) to (⃗enew,i) is the endo- morphism (= the linear map E → E) defined by, for all j ∈ [1, n]N, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗eold,j = ⃗enew,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18 The transition matrix from (⃗eold,i) to (⃗enew,i) is the matrix P := [P]|⃗eold = [Pij] of the endomorphism P relative to the basis (⃗eold,i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' defined by, for all j, ⃗enew,j = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗eold,j = n � i=1 Pij ⃗eold,i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗enew,j]|⃗eold = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[⃗eold,j]|⃗eold = � � � P1j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Pnj � � � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', [⃗enew,j]|⃗eold is the j-th column of P = [P]|⃗eold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notations: ⃗enew,j = �n i=1P ij ⃗eold,i and P := [P]|⃗eold = [P ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Apart from the classical and notations, you may find other “component type” notations: ⃗enew,j = n � i=1 Pij ⃗eold,i = n � i=1 (Pj)i ⃗eold,i = n � i=1 P i j ⃗eold,i = n � i=1 (Pj)i ⃗eold,i, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Pij = (Pj)i = P ij = (Pj)i are four notations for the i-th component of ⃗ej, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗enew,j]|⃗eold = � � � P1j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Pnj � � � = � � � (Pj)1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Pj)n � � � = � � � P 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' P n j � � � = � � � (Pj)1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Pj)n � � � (= the j-th column of P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) 83 84 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Change of basis formulas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Inverse of the transition matrix The inverse endomorphism Q := P−1 ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19), is given by, for all j ∈ [1, n]N, ⃗eold,j = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗enew,j (= P−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗enew,j), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Q is change of basis endomorphism from (⃗enew,i) to (⃗eold,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And Q := [Q]|⃗enew = [Qij] is the transition matrix from (⃗enew,i) to (⃗eold,i): ⃗eold,j = n � i=1 Qij⃗enew,i, [⃗eold,j]|⃗enew = � � � Q1j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Qnj � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Q is change of basis endomorphism from (⃗enew,i) to (⃗eold,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Use other notation if you prefer: Qij = (Qj)i = Qij = (Qj)i Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19 Q = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗enew,j = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗eold,j = �n i=1Pij⃗eold,i = �n i=1Pij(�n k=1Qki⃗enew,k) = �n k=1(�n i=1QkiPij)⃗enew,k = �n k=1(Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P)kj⃗enew,k for all j, thus (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P)kj = δkj for all j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P = I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20 Prove � [P]|⃗eold = [P]|⃗enew = P, [Q]|⃗enew = [Q]|⃗eold = Q, � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗enew,j = �n i,j=1Pij⃗enew,i (= �n i,j=1P ij⃗enew,i = �n i,j=1(Pj)i⃗enew,i), Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗eold,j = �n i,j=1Qij⃗eold,i (= �n i,j=1Qij⃗eold,i = �n i,j=1(Qj)i⃗eold,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Z = [Zij] = [P]|⃗enew means P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗enew,j = � i Zij⃗enew,i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗enew,j = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (�n i=1Zij⃗enew,i) = �n i=1ZijQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗enew,i = �n i=1Zij(�n k=1Qki⃗enew,k) = �n k=1(�n i=1QkiZij)⃗enew,k = �n k=1(Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Z)kj⃗enew,k for all j, thus (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Z)kj = δkj for all j, k, thus Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Z = I, thus Z = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Idem for Q, thus (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21 P T ̸= P −1 in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', (⃗eold,i) = (⃗ai) is a foot-built Euclidean basis, and (⃗enew,i) = (⃗bi) is a metre-built Euclidean basis, and ⃗bi = λ⃗ai for all i (the basis are “aligned”), so P = λI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus P T = λI and P −1 = 1 λI ̸= P T , since λ = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3048 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus it is essential not to confuse P T and P −1 (not to confuse covariance with contravariance), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the Mars Climate Orbiter crash (remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Change of dual basis Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22 (πnew,i) and (πold,i) being the dual bases of (⃗enew,i) and (⃗eold,i), for all i ∈ [1, n]N, πnew,i = n � j=1 Qijπold,i, and [πnew,i]|⃗eold = ( Qi1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Qin ) (i-th row of Q), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) to compare with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) (matrices of linear forms are row matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notations: ei new = �n j=1Qijej old and [ei new]|⃗eold = ( Qi1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Qin ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' πnew,i(⃗eold,k) =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) πnew,i(� j Qjk⃗enew,j) = � j Qjk πnew,i(⃗enew,j) = � j Qjk δij = Qik, and � j Qijπold,j(⃗eold,k) = � j Qijδjk = Qik, true for all i, k, thus πnew,i = � j Qij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Change of coordinate system for vectors and linear forms Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23 Let ⃗x ∈ E and ℓ ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then [⃗x]|⃗enew = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗eold (contravariance formula for vectors: between column matrices), [ℓ]|⃗enew = [ℓ]|⃗eold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P (covariance formula for linear forms: between row matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) And the scalar value ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x is computed indifferently with one or the other basis (objective result): ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = [ℓ]|⃗eold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗eold = [ℓ]|⃗enew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗enew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) 84 85 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Bidual basis (and contravariance) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗x = � j xj⃗eold,j = � i yi⃗enew,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have ⃗x = � j xj⃗eold,j = � j xj(�n i=1Qij⃗enew,i) = � ij Qijxj⃗enew,i, thus yi = � j Qijxj for all i, thus (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ℓ = � j mjπnew,j = � i ℓiπold,i =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) � ij ℓiPijπnew,j gives mj = � i ℓiPij for all j, thus (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Use duality notations if you prefer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Thus [ℓ]|⃗enew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗enew = ([ℓ]|⃗eold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗eold) = [ℓ]|⃗eold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗eold, hence (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Notation: (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) and ⃗x = � j xj⃗eold,j = � i yi⃗enew,i give yi = �n j=1Qijxj, which means: yi is the function defined by yi(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', xn) = �n j=1Qijxj, thus Qij = ∂yi ∂xj (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', xn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Similarly with Pij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Which is written Qij = ∂yi ∂xj , and Pij = ∂xi ∂yj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) (Use duality notations if you prefer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Qij = ∂yi ∂xj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Bidual basis (and contravariance) Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24 The dual of E∗ is E∗∗ := (E∗)∗ = L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) and is named the bidual of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗∗ is also called the space of contravariant vectors = the space of directional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25 Let (⃗ei) be a basis in E, let (πei) be its dual basis (basis in E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The dual basis (∂i) of (πei) is called the bidual basis of (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Duality notations: (πei) = (ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') (The notation ∂i refers to the derivation in the direction ⃗ei: ∂i(df(⃗x)) = df(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei = ∂f ∂xi (⃗x), see § S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Thus, the linear form ∂i ∈ E∗∗ = L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) are characterized by, for all j, ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='πej = δij = πej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei, so ℓ = n � i=1 ℓiπei iff ℓi = ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ (= ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) Indeed, ∂i(ℓ) = ∂i(�n j=1ℓjπej) = �n j=1ℓj∂i(πej) = �n j=1ℓjδij = ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Duality notation: ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ej = δj i = ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei and ℓ = �n i=1ℓiei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Remark: With the natural canonical isomorphism J : � E → E∗∗ = L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) ⃗u → J (⃗u), where J (⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u, ∀ℓ ∈ E∗ � see (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9), we can identify ⃗u and J (⃗u) (observer independent identification), thus ∂i = J (⃗ei) =noted ⃗ei, and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) reads (usual notation in differential geometry) ⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='πej = δij and ℓi = ⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Bilinear forms A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition Let E and F be vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26 • A bilinear form is a 2-multilinear form β(·, ·) : � E × F → R (⃗u, ⃗w) → β(⃗u, ⃗w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, β(⃗u1 +λ⃗u2, ⃗w) = β(⃗u1, ⃗w)+λβ(⃗u2, ⃗w) and β(⃗u, ⃗w1 +λ⃗w2) = β(⃗u, ⃗w1)+λβ(⃗u, ⃗w2) for all ⃗u, ⃗u1, ⃗u2 ∈ E, ⃗w, ⃗w1, ⃗w2 ∈ F, λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is the set of bilinear forms E × F → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If (ℓ, m) ∈ E∗ × F ∗, then the bilinear form ℓ ⊗ m ∈ L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is defined by (ℓ ⊗ m)(⃗u, ⃗w) = ℓ(⃗u)m(⃗w) (= (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u)(m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w)) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) for all (⃗u, ⃗w) ∈ E × F, and is called an elementary bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The transposed of a bilinear form (Warning: Not to be confused with the subjective definition of a transposed of a linear map which requires inner dot products to be defined, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27 If β ∈ L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) then its transposed is the bilinear form βT ∈ L(F, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) defined by, for all (⃗w, ⃗u) ∈ F × E, βT (⃗w, ⃗u) = β(⃗u, ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) (This definition is observer independent: no basis or inner dot product is required in this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 85 86 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Bilinear forms A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Symmetric and definite positive bilinear forms Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28 Here F = E (no choice), and β ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' β is semi-positive, iff for all ⃗u ∈ E, β(⃗u, ⃗u) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) β is definite positive, iff for all ⃗u ̸= ⃗0, β(⃗u, ⃗u) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) β is symmetric iff βT = β, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all ⃗u,⃗v ∈ E, β(⃗u,⃗v) = β(⃗v, ⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Inner dot product, and metric Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29 • An “inner dot product” (or “scalar inner dot product”, or “inner scalar product”, or “inner product”) in a vector space E is a bilinear form β =noted g =noted g(·, ·) ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) which is symmetric and definite positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And then (for inner dot products) g(·, ·) noted = (·, ·)g noted = g ·, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' g(⃗u, ⃗w) = (⃗u, ⃗w)g noted = ⃗u •g ⃗w, ∀⃗u, ⃗w ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) Then two vectors ⃗u, ⃗w ∈ E are (·, ·)g-orthogonal iff (⃗u, ⃗w)g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the associated norm with (·, ·)g is the function ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||g : E → R+ given by, for all ⃗u ∈ E, ||⃗u||g = � (⃗u, ⃗u)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38) (To prove that it is a norm, use the Cauchy–Schwarz inequality (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') An “semi-inner dot product” (·, ·)g (or “semi-scalar inner dot product”) in a vector space E is a bilinear form β =noted g(·, ·) ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) which is symmetric and semi-positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the associated semi-norm is given by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30 (Cauchy–Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') (·, ·)g being an inner dot product in E, ∀⃗u, ⃗w ∈ E, |(⃗u, ⃗w)g| ≤ ||⃗u||g||⃗w||g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39) And |(⃗u, ⃗w)g| = ||⃗u||g||⃗w||g iff ⃗u and ⃗w are parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let p(λ) = ||⃗u+λ⃗w||2 g = (⃗u+λ⃗w, ⃗u+λ⃗w)g, so p(λ) = aλ2 + bλ + c where a = ||⃗w||2 g, b = 2(⃗u, ⃗w)g and c = ||⃗u||2 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With p(λ) ≥ 0 (since(·, ·)g is positive), we get b2 − 4ac ≥ 0, thus (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39), and p(λ) = 0 iff ⃗u+λ⃗w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31 (Metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') With Rn our usual affine geometric space, n = 1, 2 or 3, and ⃗Rn = the usual associated vector space made of bipoint vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Ω ⊂ Rn be open in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A metric in Ω is a C∞ function g : � Ω → L(⃗Rn, ⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) p → g(p) noted = gp � such that gp is an inner dot product in ⃗Rn at each p ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Particular Case: When the gp is independent of p (general case in continuum mechanics), a metric is simply called a inner dot product (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' a Euclidean metric is called a Euclidean dot product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (In a differentiable manifold Ω, a metric is a C∞ �0 2 � tensor g s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' g(p) is an inner dot product at each p ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A Riemannian metric is a metric s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' g(p) is a Euclidean dot product at each p ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Quantification: Matrice [βij] and tensorial representation dim E = n, dim F = m, β ∈ L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), (⃗ai) is a basis in E which dual basis is (πai), (⃗bi) is a basis in F which dual basis is (πbi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (With duality notations, (πai) = (ai) and (πbi) = (bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32 The components of β ∈ L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) relative to the bases (⃗ai) and (⃗bi) are the nm reals βij := β(⃗ai,⃗bj), and [β]|⃗a,⃗b = [βij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) is the matrix of β relative to the bases (⃗ai) and (⃗bi), simply written [βij] if the bases are implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And if F = E and (⃗bi) = (⃗ai) then [β]|⃗a,⃗b =noted [β]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 86 87 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Linear maps Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33 A bilinear form β ∈ L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is known as soon as the nm scalars βij = β(⃗ai,⃗bj) are known, and, for all (⃗u, ⃗w) ∈ E × F, β(⃗u, ⃗w) = [⃗u]|⃗a T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [β]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗b, written β(⃗u, ⃗w) = [⃗u]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w] , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) so β = n � i=1 m � j=1 βijπai ⊗ πbj, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='42) and a basis in L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is made of the nm functions πai ⊗ πbj, and dim L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) = nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Duality notations: β = �n i=1 �m j=1βijai ⊗ bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' β being bilinear, ⃗u = �n i=1ui⃗ai and ⃗w = �n j=1wj⃗bj give β(⃗u, ⃗w) = �n i,j=1uiwjβ(⃗ai,⃗bj) = �n i,j=1uiβijwj = ([⃗u]|⃗a)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [β]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗b, thus (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular, if the βij are known, then b is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (πai ⊗ πbj)(⃗ak,⃗bℓ) =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) (πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ak)(πbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bℓ) = δikδjℓ (all the elements of the matrix [πai ⊗ πbj]|⃗a,⃗b are zero except the element at the intersection of row i and column j which is equal to 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (πai ⊗ πbj)(⃗u, ⃗w) =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) (πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u)(πbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) = uiwj, thus β(⃗u, ⃗w) = �n i,j=1βijuiwj = �n i,j=1βij(πai ⊗ πbj)(⃗u, ⃗w), thus β := �n i,j=1βij(πai ⊗ πbj), thus the πai ⊗ πbj span L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And � ij λij(πai ⊗ πbj) = 0 implies 0 = (� ij λij(πai ⊗ πbj))(⃗ak,⃗bℓ) = � ij λij(πai ⊗ πbj)(⃗ak,⃗bℓ) = λkℓ = 0 for all k, ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the πai ⊗ πbj are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (πai ⊗ πbj) is a basis in L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) and dim(L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R)) = nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Duality notations: β(⃗u, ⃗w) = �n i,j=1βijuiwj and β := �n i,j=1βij ai ⊗ bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34 dim E = dim F = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [β]|⃗a,⃗b = � 1 2 0 3 � means β(⃗a1,⃗b1) = β11 = 1, β(⃗a1,⃗b2) = β12 = 2, β(⃗a2,⃗b1) = β21 = 0, β(⃗a2,⃗b2) = β22 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And β12 = [⃗a1]T |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[β]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗b2]|⃗b = ( 1 0 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � 1 2 0 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � 0 1 � = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35 Let β ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), let (⃗ai) and (⃗bi) be two bases in A, and let λ ∈ R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove: if, ∀i ∈ [1, n]N, ⃗bi = λ⃗ai, then [β]|⃗b = λ2[β]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43) (A change of unit, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' from foot to metre, has a “big” influence on the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗bi = λ⃗ai give β(⃗bi,⃗bj) = β(λ⃗ai, λ⃗aj) = λ2β(⃗ai,⃗aj) (bilinearity), thus [β]|⃗b = λ2[β]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36 Prove [βT ]⃗b,⃗a = ([β]⃗a,⃗b)T , written [βT ] = [β]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='44) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let[β]⃗a,⃗b = [βij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m and [βT ]⃗b,⃗a = [γij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have γij = βT (⃗bi,⃗aj) = β(⃗aj,⃗bi) = βji, qed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Linear maps A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition Let E and F be vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37 • A function L : E → F is linear iff L(⃗u1 + λ⃗u2) = L(⃗u1) + λL(⃗u2) for all ⃗u1, ⃗u2 ∈ E and all λ ∈ R (distributivity type relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (distributivity notation): L(⃗u) noted = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u, so L(⃗u1 + λ⃗u2) = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(⃗u1 + λ⃗u2) = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u1 + λL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45) NB: This dot notation L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u is a linearity notation (distributivity type notation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It is an “outer” dot product between a (linear) function and a vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It is not an “inner” dot product since L and ⃗u don’t belong to a same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It is not a matrix product since no basis has been introduced yet (no quantification has been done yet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) is the set of linear maps E → F (vector space, subspace of (F(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F), +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If F = E then a linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) is called an endomorphism in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (If F = R then a linear map E → R is called a linear form, and E∗ := L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is the dual of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 87 88 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Linear maps Vocabulary: Let Li(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) be the space of linear invertible linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If E is a finite dimension vector space, dim E = n, then, in algebra, the set (Li(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E), ◦) of linear maps equipped with the composition rule is named GLn(E) = “the linear group” (it is indeed a group, easy check).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the “linear group” of n ∗ n invertible matrices is GLn(Mn) := (Li(Mn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Mn), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Li(Mn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Mn) with the matrix product rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38 (Math exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Let E = (E, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||E) and F = (F, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||F ) be Banach spaces, and let Lic(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) be the space of invertible linear continuous maps E → F, with its usual norm ||L|| = sup||⃗x||E=1 ||L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x||F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Z : � Lic(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) → Lic(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) L → L−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove dZ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M = −L−1 ◦ M ◦ L−1, for all M ∈ Lic(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Recall: In finite dimension, a linear map is always continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider limh→0 Z(L+hM)−Z(L) h = limh→0 (L+hM)−1−L−1 h ( =noted dZ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M if the limit exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With N = L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M we have L + hM = L(I + hN), and (I + hN) is invertible as soon as ||hN|| < 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' h < 1 ||N|| = 1 ||L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M||, its inverse being I − hN + h2N − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Neumann serie);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus I + hN = I − hN + o(h), and (L + hM)−1 = (I + hN)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L−1 = (I − hN + o(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L−1 = L−1 − hN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L−1 + o(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (L+hM)−1−L−1 h = L−1−hN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L−1+o(h)−L−1 h = −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L−1 + o(1) −→h→0 −N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Quantification: Matrices [Lij] = [Lij] dim E = n, dim F = m, L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F), (⃗ai) is a basis in E which dual basis is (πai), (⃗bi) is a basis in F which dual basis is (πbi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (With duality notations, (πai) = (ai) and (πbi) = (bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39 The components of a linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) relative to the bases (⃗ai) and (⃗bi) are the nm reals named Lij (classical notation) = Lij (duality notation), which are the components of the vectors L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj relative to the basis (⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' That is: � � � � � � � � � � � clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' : L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = m � i=1 Lij⃗bi, dual not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = m � i=1 Li j⃗bi, � � � � � � � � � � � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj]|⃗b clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = � � � L1j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Lmj � � � dual = � � � L1j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Lmj � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='46) And [L]|⃗a,⃗b clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = [Lij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n dual = [Li j] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='47) is the matrix of L relative to the bases (⃗ai) and (⃗bi) (so [L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj]|⃗b is the j-th column of [L]|⃗a,⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Particular case: If E = F (so L is an endomorphism) and if (⃗bi) = (⃗ai) then [L]|⃗a,⃗a =noted [L]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40 n = m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|⃗a,⃗b = � 1 2 0 3 � means L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1 = ⃗b1 and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2 = 2⃗b1 + 3⃗b2 (column reading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Here L11=1, L12=2, L21=0, L22=3 (duality notations: L11=1, L12=2, L21=0, L22=3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And L being linear, for all ⃗u ∈ E, ⃗u = �n j=1uj⃗aj = �n j=1uj⃗aj, we get, thanks to linearity, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = m � i=1 n � j=1 Lijuj⃗bi dual = m � i=1 n � j=1 Li juj⃗bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u]|⃗b = [L]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u]|⃗a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='48) Shortened notation: [L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u] = [L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u] when the bases are implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41 A linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) is known as soon as the n vectors L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj are known, j ∈ [1, n]N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the linear maps Lij ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) defined by Lij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aℓ = δjℓ⃗bi (all the elements of the matrix [Lij]|⃗a,⃗b vanish except the element at the intersection of row i and column j which is equal to 1), for i, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', m, constitute a basis ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Duality notations: Lij =noted Lij, and Lij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aℓ = δj ℓ⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') So, dim(L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F)) = nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗u ∈ E and ⃗u = � k uj⃗aj give L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = � j ujL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj, since L is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus L is known iff the n vectors L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj are known for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ak = � i Lik⃗bi together with � ij LijLij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ak = � ij Lijδjk⃗bi = � i Lik⃗bi, for all k, thus L = � ij LijLij, thus the Lij span L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And �m i=1 �n j=1λijLij = 0 implies, for all ℓ, ⃗0 = �m i=1 �n j=1λijLij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aℓ = �m i=1 �n j=1λijδjℓ⃗bi = �m i=1λiℓ⃗bi, thus λiℓ = 0, for all i and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the Lij are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (Lij) i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m is a basis in L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 88 89 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Transposed matrix Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='42 If L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) (endomorphism), if (⃗ai) is a basis in E, prove: if λ ∈ R∗ and ⃗bi = λ⃗ai ∀i ∈ [1, n]N, then [L]|⃗b = [L]|⃗a, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='49) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', a change of unit has not influence on the matrix of an endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Check with the change of basis formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: To compare with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43): Covariance and contravariance should not be confused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = �n i=1Laij⃗ai and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj = �n i=1Lbij⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then �n i=1Lbij⃗bi = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(λ⃗aj) = λL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = λ�n i=1Laij⃗ai = λ�n i=1Laij ⃗bi λ = �n i=1Laij⃗bi, thus Lbij = Laij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Change of basis formula: [L]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P with P = λI here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Trace of an endomorphism Let E be a vector space, dim E = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗u ∈ E and ℓ ∈ E∗ and call L ⃗w,ℓ ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) the endomorphism, called an elementary endomorphism, defined by L ⃗w,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u := ⃗w(ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u) = (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u)⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='50) Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43 The trace of the endomorphism L ⃗w,ℓ is the real Tr(L ⃗w,ℓ) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='51) And the trace operator is the linear map Tr : � L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) → R L → Tr(L) � defined on elementary endomor- phisms ℓ ⊗ ⃗w by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='44 Let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The real Tr(L) is objective (is intrinsic to L), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' is independent of any basis in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (quantification) if (⃗ei) is a basis and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1Lij⃗ei for all j, then Tr(L) = n � i=1 Lii (∈ R), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='52) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', Tr(L) is the trace of the matrix [L]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Duality notations L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1Lij⃗ei and Tr(L) = �n i=1Lii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Tr(L ⃗w,ℓ) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w is a real that can be considered by any observer, and which value is the same for all observers, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29): It is objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗ai) be a basis and (πai) be its (covariant) dual basis, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = � i Lij⃗ai, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L][⃗a = [Lij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And we have (� ik LikL⃗ai,πak).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = � ik LikL⃗ai,πak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='50) � ik Lik⃗ai(πak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj) = � ik Lik⃗aiδkj = � i Lij⃗ai, thus L = � ij LijL⃗ai,πaj (sum of elementary endomorphisms), thus, Tr being linear Tr(L) = � ij LijTrL⃗ai,πaj = � ij Lijδji = � i Lii, thus (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45 Check with the change of basis formula that Tr(L) is an invariant (the same value for all observers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗ai) and (⃗bi) be two bases, P = [Pij] be the transition matrix from (⃗ai) to (⃗bi), Q = P −1, [L][⃗a = [(La)ij], [L][⃗b = [(Lb)ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have [L][⃗b =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='104) P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L][⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Lb)ij = � kℓ Qik(La)kℓPℓj, thus � i(Lb)ii = � ikℓ Qik(La)kℓPℓi = � kℓ(P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Q)ℓk(La)kℓ = � kℓ δℓk(La)kℓ = � k(La)kk, qed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Alternative definition with one-one tensors: see § Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 Transposed matrix The definition can be found in any elementary books, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', Strang [18]: If M = [Mij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n is an m ∗ n matrix then its transposed is the n ∗ m matrix M T = [(M T )ij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m defined by (M T )ij := Mji (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='53) (exchange rows and columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', M = � 1 2 3 4 � gives M T = � 1 3 2 4 � , and (M T )12=M21=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And M is symmetric iff M T = M (this requires m=n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='N = [� k MikNkj] i j gives (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='N)T = [� k MjkNki] i j = [� k(N T )ik(M T )kj] i j = N T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='46 Prove: If M is an n ∗ n invertible matrix then M T is invertible and (M T )−1 = (M −1)T ( =noted M −T );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And if moreover M is symmetric, then M −1 is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M −1 = I gives (M −1)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M T = IT = I, thus M T is invertible with (M T )−1 = (M −1)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Moreover if M = M T then M −1 = (M −1)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 89 90 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A transposed endomorphism: depends on a chosen inner dot product A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 A transposed endomorphism: depends on a chosen inner dot product Not to be confused with the transposed of a matrix, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And not to be confused with the transposed of a bilinear form (observer independent), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular, a transposed of a linear map depends on the observer who use it (depends on the choice of an inner dot product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition (requires an inner dot product: Not objective) Let E be a finite dimensional vector space equipped with an inner dot product g(·, ·) = (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='47 The transpose of an endomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) relative to (·, ·)g is the endomorphism LT g ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) defined by ∀⃗x, ⃗y ∈ E, (LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗x)g = (⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x)g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y) •g ⃗x = ⃗y •g (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54) (It depends on (·, ·)g, see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') If (·, ·)g is an imposed Euclidean dot product (isometric framework) then LT g =noted LT , thus (LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗x)g = (⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x)g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y) • ⃗x = ⃗y • (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='48 (Math exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') The existence and uniqueness of LT g is e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' proved with a basis in E when E is finite dimensional, see next § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' More general proof: Prove: If (E, (·, ·)g) is an infinite dimensional Hilbert space and if L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) is continuous, then LT g exists, is unique, and is continuous (apply the Riesz representation theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗y ∈ E, then let ℓ⃗yg : ⃗x ∈ E → ℓ⃗yg(⃗x) := (⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x)g ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ℓ⃗yg is linear (trivial since L is linear and (·, ·)g is bilinear) and continuous: |ℓ⃗yg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x| ≤ ||⃗y||g||L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x||g ≤ ||⃗y||g||L|| ||⃗x||g gives ||ℓ⃗yg||E∗ ≤ ||L|| ||⃗y||g < ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗ℓ⃗yg ∈ E be the (·, ·)g-Riesz representation of ℓ⃗yg ∈ E∗: ℓ⃗yg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = (⃗ℓ⃗yg, ⃗x)g for all ⃗x, with ||⃗ℓ⃗yg||g = ||ℓ⃗yg||E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have thus defined LT g : ⃗y ∈ E → LT g (⃗y) := ⃗ℓ⃗yg ∈ E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' with (LT g (⃗y), ⃗x)g = (⃗ℓ⃗yg, ⃗x)g = ℓ⃗yg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = (⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x)g, thus LT g is linear (since (·, ·)g is bilinear) and continuous: ||LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y||g = ||⃗ℓ⃗yg||g = ||ℓ⃗yg||E∗ ≤ ||L|| ||⃗y||g gives ||LT g || ≤ ||L||L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='E) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='49 Recall: The transposed βT of a bilinear form β is objective, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33): We don’t need any tool like an inner dot product to define βT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Not to be confused with: The transposed LT g =noted LT of a linear map L is subjective: It depends on a choice of an inner dot products (·, ·)g by an observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular it is dangerous to represent a linear map in a basis with its “bilinear tensorial representation” when dealing with the transposed: L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) is naturally canonically represented by the bilinear form βL ∈ L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), and thus (βL)T ∈ L(E, F ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = n � i=1 Li j⃗bi gives βL = n � i,j=1 Li j⃗bi ⊗ aj, thus (βL)T (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) = n � i,j=1 Lj iai ⊗⃗bj, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='55) while LT ∈ L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) is naturally canonically represented by the bilinear form β(LT ) ∈ L(E∗, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), and LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj = n � i=1 (LT )i j⃗ai gives b(LT ) = n � i,j=1 (LT )i j⃗ai ⊗ bj, thus β(LT ) ̸= (βL)T (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='56) for two reasons: 1- ⃗ai ⊗ bj ̸= ai ⊗ ⃗bj, and 2- LT := LT gh depends on chosen inner dot products (·, ·)g and (·, ·)h by observers in E and F, see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='73): (LT )ij =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='73) �n k,ℓ=1([g]−1)ikLℓk hℓj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' While (βL)T is independent of any inner dot products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (In fact (βL)T ∈ L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is the tensorial representation of the adjoint L∗ ∈ L(F ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗) of L: With L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='bj = �n i=1(L∗)ijai we get � (L∗) = �n i,j=1(L∗)ijai ⊗⃗bj = (βL)T , see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='83).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') So in continuum mechanics it is strongly advised not to use the tensorial notation for linear maps when dealing with transposed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' when using F T the transposed of the deformation gradient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Quantification with bases Let (⃗ei) be a basis in E, let gij := g(⃗ei,⃗ej), so [g]|⃗e := [gij] =noted [g], and let (classical notation) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = n � i=1 Lij⃗ei, LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = n � i=1 (LT g )ij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|⃗e = [Lij] noted = [L], [LT g ]|⃗e = [(LT g )ij] noted = [LT g ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='57) 90 91 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A transposed endomorphism: depends on a chosen inner dot product (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54) gives [⃗x]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y] = [L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗y] for all ⃗x, ⃗y, thus [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [LT g ] = [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' n � k=1 gik(LT g )kj = n � k=1 Lki gkj (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='58) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', [LT g ] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g] , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (LT g )ij = n � k,ℓ=1 ([g]−1)ikLℓkgℓj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='59) To compare with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If and only if (⃗ei) is (·, ·)g-orthonormal then [g] = [δij] and (LT g )ij = Lji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With duality notations, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1Lij⃗ei, LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1(LT g )ij, [L]|⃗e = [Lij], [LT g ]|⃗e = [(LT g )ij], and n � k=1 gik(LT g )k j = n � k=1 Lk i gkj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (LT g )i j = n � k,ℓ=1 ([g]−1)ikLℓ k gℓj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='60) Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='50 The last equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='60)2 is also written (LT g )i j = n � k,ℓ=1 gikLℓ k gℓj when ([g]⃗e)−1 = [gij]−1 noted = [gij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='61) Don’t be fooled by the notation gij, defined by [gij] := [gij]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (It is also the short notation for (g♯)ij, see (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Use classical notations to avoid misuses and misinterpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='51 A bilinear form β ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) satisfies [βT ] = [β]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A linear endomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) satisfies [LT g ] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g] ̸= [L]T in general (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' take [L] = � 0 1 1 0 � and [g] = � 1 0 0 2 � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So do not confuse a bilinear on E (objective) with a linear endomorphism on E (subjective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='52 In ⃗R2, let (⃗e1,⃗e2) be a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let L ∈ L( ⃗R2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗R2) be defined by [L]|⃗e = � 0 1 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Find two inner dot products (·, ·)g and (·, ·)h in ⃗R2 such that LT g ̸= LT h (a transposed endomorphism is not unique, is not intrinsic to L, since it depends on a choice of an inner dot product by an observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Calculations with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='58): Choose (·, ·)g given by [g]|⃗e = � 1 0 0 1 � = [I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus [LT g ]|⃗e = [I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[L]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [I] = � 0 1 1 0 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So LT g = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Choose (·, ·)h given by [h]|⃗e = � 1 0 0 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus [LT h ]|⃗e = [h]−1 |⃗e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [h]|⃗e = � 0 2 1 2 0 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So LT h ̸= L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus LT h ̸= LT g , e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗e2 = LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e1 ̸= LT h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e1 = 1 2⃗e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='53 Prove: If L is invertible then LT g is invertible, and (LT g )−1 = (L−1)T g (written L−T g ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Suppose: ∃⃗y ∈ E, ⃗y ̸= ⃗0, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L being invertible, ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x ∈ E s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = ⃗y, with ⃗x ̸= ⃗0 since ⃗y ̸= ⃗0 and L is linear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y = 0 gives LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = 0, thus (LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, ⃗x)g = 0, thus ||L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x||2 g = 0, thus L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = 0, thus ⃗x = 0 since L is linear bijective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus Ker(LT g ) = {⃗0}, thus LT g is invertible since it is an endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L−1)T g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, ⃗y)g (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54) = ((L−1)T g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y)g (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54) = (⃗x, (L−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y)g = (⃗x, ⃗y)g = (LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (LT g )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, ⃗y)g, true ∀⃗x, ⃗y, thus LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L−1)T g = LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (LT g )−1, thus (L−1)T g = (LT g )−1 since LT g is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54 Special case of proportional inner dot products (·, ·)a and (·, ·)b: ∃λ > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (·, ·)a = λ2(·, ·)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove: LT a = LT b : Two proportional inner dot products give the same transposed endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (LT b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗x)b = (⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x)b = λ2(⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x)a = λ2(LT a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗x)a = (LT a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗x)b, for all ⃗x, ⃗y, so LT b = LT a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Symmetric endomorphism Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='55 An endomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) is (·, ·)g-symmetric iff LT g = L: L (·, ·)g-symmetric ⇐⇒ LT g = L ⇐⇒ (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, ⃗y)g = (⃗x, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y)g, ∀⃗x, ⃗y ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='62) Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='56 The symmetric character of an endomorphism L is not intrinsic to the endomorphism: It depends on (·, ·)g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See exercise A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='52 where L is (·, ·)g-symmetric while it is not (·, ·)h-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 91 92 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A transposed of a linear map: depends on chosen inner dot products A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 The general flat ♭ notation for an endomorphism: Relative to a (·, ·)g Let (·, ·)g be an inner dot product in a vector space E, and let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) (a C0 endomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='57 The bilinear form L♭ g ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) which is (·, ·)g-associated to the endomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) is defined by, for all ⃗u, ⃗w ∈ E, L♭ g(⃗u, ⃗w) := (⃗u, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='63) (L♭ g depends on a choice of a (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') We have thus defined the (·, ·)g-dependent operator: (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' )♭ g = Jg(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') : � L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) → L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) L → Jg(L) := L♭ g, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='64) If (·, ·)g is imposed, then L♭ g =noted L♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (The bilinearity of L♭ g is trivial since L is linear and (·, ·)g is bilinear, and the bilinear form L♭ g continuous as soon as L and (·, ·)g are since |L♭ g(⃗u,⃗v)| ≤ ||g|| ||L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u|| ||⃗v|| ≤ (||g|| ||L||) ||⃗u|| ||⃗v||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='58 With the natural canonical isomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) ≃ TL ∈ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) given by TL(ℓ, ⃗w) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w, and with TL =noted L, the function (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' )♭ g is the change of contravariance to covariance mapping given by L♭ g = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='65) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Recall: The contraction of an elementary �0 2 � tensor ℓ1 ⊗ ℓ2 with an elementary �1 1 � tensor ⃗v ⊗ ℓ3 is the �0 2 � tensor (ℓ1 ⊗ℓ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗v ⊗ℓ3) := (ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v)ℓ1 ⊗ℓ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the contraction on any tensors is the bilinear map defined on elementaty tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, with a basis (⃗ei) in E and its dual basis (ei) in E∗, if g = � ij gijei⊗ej and L = � ij Lij⃗ei ⊗ ej then g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L = � ijk gikLkj⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L)(⃗u, ⃗w) = � ij uiwj(g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L)(⃗ei,⃗ej) = � ijk uiwjgikLkj = � ik uigik(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w)k = � ik uigik(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w)k = g(⃗u, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) = L♭ g(⃗u, ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification: Let (⃗ei) be a basis in E, and, with duality notations motivated by the flat notation “i top changed into i bottom” in the components Lij of L, let gij := g(⃗ei,⃗ej), L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1Lij⃗ei and L♭ g,ij = L♭ g(⃗ei,⃗ej), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' with tensorial notations for calculations g = � ij gijei ⊗ ej, L = � ij Li j⃗ei ⊗ ej, L♭ g = � ij L♭ g,ijei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='66) So [g]|⃗e = [gij], [L]|⃗e = [Lij] and [L♭ g]|⃗e = [L♭ g,ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='65) gives (or see next exercise) [L♭ g] = [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='67) Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='59 Prove (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='67) with components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='63) we get L♭ g,ij = L♭ g(⃗ei,⃗ej) = (⃗ei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej)g = (⃗ei, � k Lk j⃗ek)g = � k Lk jgik = ([g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L])ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='60 A change of variance, here from the �1 1 � type tensor L to the �0 2 � tensor L♭ g, is necessarily observer dependent: There is no natural canonical isomorphism between a vector space E and its dual E∗, see § T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Details: Here fix ⃗w and write ℓg,⃗w(⃗u) = (⃗u, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w)g (= L♭ g(⃗u, ⃗w));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ℓg,⃗w ∈ E∗ is the (·, ·)g- representation function (linear form) of the vector L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ℓg,⃗w = ⃗Rg(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) where ⃗Rg is the (·, ·)g-Riesz- representation operator (the change of variance operator, see (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 A transposed of a linear map: depends on chosen inner dot products This paragraph is needed to define the transposed of the deformation gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Not to be confused with the transposed of a matrix, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And not to be confused with the objective transposed of a bilinear form, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', a transposed of a linear map is not objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 92 93 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A transposed of a linear map: depends on chosen inner dot products A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition (subjective) (E, (·, ·)g) and (F, (·, ·)h) are Hilbert spaces, and L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) (which is supposed to be continuous if E and F are infinite dimensional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', E = ⃗Rn t0, F = ⃗Rn t , L = dΦt0 t (P) ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) = the deformation gradient, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1), (·, ·)g is the foot built Euclidean dot product chosen by the observer who made the measurements at t0, (·, ·)h is the metre built Euclidean dot product chosen by the observer who makes the measurements at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='61 The transposed of L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) relative to (·, ·)g and (·, ·)h is the linear map LT gh ∈ L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) defined by, for all (⃗x, ⃗y) ∈ E × F, (LT gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗x)g = (⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x)h, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='68) where we used the dot notation LT gh(⃗y) =noted LT gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y since LT gh is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This defines the map (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' )T gh : � L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) → L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) L → (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' )T gh(L) := LT gh (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='69) NB: So a linear map has an infinite number of transposed (it depends on inner dot products).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Notation: If (·, ·)g and (·, ·)h are imposed then LT gh =noted LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And if F = E and (·, ·)h = (·, ·)g then LT gh = LT g , see § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Quantification with bases Let (⃗ai) and (⃗bi) be bases in E and F, let gij := g(⃗ai,⃗aj), hij := h(⃗bi,⃗bj), [g]|⃗a = [gij], [h]|⃗b = [hij], and let (classical notation) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = m � i=1 Lij⃗bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|⃗a,⃗b = [Lij] noted = [L], LT gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj = n � i=1 (LT gh)ij⃗ai, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [LT gh]|⃗b,⃗a = [(LT gh)ij] noted = [LT gh].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='70) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='68) gives [⃗x]T |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [LT gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y]|⃗y = ([L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x]|⃗b)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [h]|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗y]|⃗b for all ⃗x, ⃗y, thus, [g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [LT gh]|⃗b,⃗a = ([L]|⃗a,⃗b)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [h]|⃗b and [LT gh]|⃗b,⃗a = [g]−1 |⃗a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ([L]|⃗a,⃗b)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [h]|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Shortened notation: [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [LT ] = [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [h], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' n � k=1 gik(LT gh)kj = m � k=1 Lki hkj, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='71) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [LT ] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [h] , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (LT gh)ij = n � k=1 m � ℓ=1 ([g]−1)ikLℓkhℓj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='72) With duality notations, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1Lij⃗ei, [L]|⃗e = [Lij], LT gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1(LT gh)ij, [LT gh]|⃗e = [(LT gh)ij], and n � k=1 gik(LT gh)k j = n � k=1 Lk i hkj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (LT gh)i j = n � k,ℓ=1 ([g]−1)ikLℓ k hℓj ( noted = n � k,ℓ=1 (gikLℓ k hℓj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='73) (Be careful with the notation ([g]−1)ik =noted gij, see remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='62 Prove: If L is invertible then (LT gh)−1 = (L−1)T hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (LT gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L−1)T hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, ⃗y)g = ((L−1)T hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y)h = (⃗x, L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y)g = (⃗x, ⃗y)g = (LT gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (LT gh)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, ⃗y)g, true ∀⃗x, ⃗y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Deformation gradient symmetric: Absurd The symmetry of a linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) is a nonsense if E ̸= F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' : The gradient of deformation F t0 t (pt0) = dΦt0 t (pt0) =noted F ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) cannot be symmetric since F T ∈ L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Idem for the first Piola–Kirchhoff tensor PKt0 t , which motivates the introduction of the symmetric second Piola–Kirchhoff tensor SKt0 t , see Marsden–Hughes [12] or § M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 93 94 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The adjoint of a linear map (objective) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Isometry Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='63 A linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) is an isometry relative to (·, ·)g and (·, ·)h iff ∀⃗x, ⃗y ∈ E, (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y)h = (⃗x, ⃗y)g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' LT gh ◦ L = IE (identity in E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='74) In particular, an endomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) is a (·, ·)g-isometry iff ∀⃗x, ⃗y ∈ E, (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y)g = (⃗x, ⃗y)g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' LT g ◦ L = IE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='75) Thus, if L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) is an isometry and (⃗ei) is a (·, ·)g-orthonormal basis, then (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei) is a (·, ·)h- orthonormal basis, since (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej)h = (⃗ei,⃗ej)g = δij for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='64 Let ⃗f : E → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove: if ⃗f is an isometry then ⃗f is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='76) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗ei) be a (·, ·)g-orthonormal basis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (⃗f(⃗ei)) is a (·, ·)h-orthonormal basis (since ⃗f is an isometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, if ⃗x = �n i=1xi⃗ei then ⃗f(⃗x) b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = n � i=1 (⃗f(⃗x), ⃗f(⃗ei))h ⃗f(⃗ei) hyp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = n � i=1 (⃗x,⃗ei)g ⃗f(⃗ei) b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = n � i=1 xi ⃗f(⃗ei), thus ⃗f(⃗x+λ⃗y) = n � i=1 (xi + λyi)⃗f(⃗ei) = n � i=1 xi ⃗f(⃗ei) + λ n � i=1 yi ⃗f(⃗ei) = ⃗f(⃗x) + λ⃗f(⃗y), thus ⃗f is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='65 Rn is an affine space, ⃗Rn is the usual associated vector space, and (·, ·)g is an inner dot product in ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition: A distance-preserving function f : p ∈ Rn → f(p) ∈ Rn is a function s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ||−−−−−→ f(p)f(q)||g = ||−→ pq||g, ∀p, q ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='77) Prove: If f is a distance-preserving function, then f is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let O ∈ Rn (an origin) and ⃗f : ⃗x = −→ Op ∈ ⃗Rn → ⃗f(⃗x) := −−−−−−→ f(O)f(p) (vectorial associated function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗x = −→ Op and ⃗y = −→ Oq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then the remarkable identity 2(⃗f(⃗x), ⃗f(⃗y))g = ||⃗f(⃗x)||2 g + ||⃗f(⃗y)||2 g − ||⃗f(⃗x)−⃗f(⃗y)||2 g gives 2(⃗f(⃗x), ⃗f(⃗y))g = ||⃗f(⃗x)||2 g+||⃗f(⃗y)||2 g−||−−−−−→ f(q)f(p)||2 g = ||⃗f(⃗x)||2 g+||⃗f(⃗y)||2 g−||−→ qp||2 g = ||⃗x||2 g+||⃗y||2 g−||⃗x−⃗y||2 g = 2(⃗x, ⃗y)g, thus ⃗f is an isometry, thus ⃗f is linear cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='76), thus f is affine since f(p) = f(O) + ⃗f(−→ Op).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 The adjoint of a linear map (objective) (For mathematicians;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' May produce misunderstandings, misuses, problematic mechanical interpretations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') No inner dot product is required here: A linear map L has only one adjoint L∗ (intrinsic to L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' While L has many transposed LT = LT gh which depend on inner dot products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition E and F are vector spaces, and E∗ = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) and F ∗ = L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) are the dual spaces (made of linear continuous forms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (If E and F are finite dimensional, the continuity is always satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='66 Let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) (linear and continuous);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its adjoint is the linear map L∗ ∈ L(F ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗) canonically defined by L∗ : � F ∗ → E∗ m → L∗(m) := m ◦ L, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='78) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all (⃗x, m) ∈ E × F ∗, (L∗(m))(⃗x) := m(L(⃗x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='79) (The adjoint L∗ cannot be confused with a transposed LT which requires inner dot products, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') The linearity of L∗ is trivial, thus, together with the linearity of m and L, we can use the dot notation: L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m := m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L, and (L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x := m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='80) And ||L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m||E∗ = ||m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L||E∗ ≤ ||m||F ∗||L||L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F ) gives ||L∗||L(F ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='E∗) ≤ ||L||L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F ) < ∞, thus L∗ is continuous (when L is).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 94 95 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Tensorial representation of a linear map A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Quantification E and F are finite dimensional, dim E = n, dim F = m, and (⃗ai) and (⃗bi) are bases in E and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let [L]|⃗a,⃗b =noted [L], [L∗]|b,a =noted [L∗], [m]|b =noted [m] and [⃗x]|⃗a =noted [⃗x] be the matrices relative to the chosen bases: (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='80) gives ([L∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x] = [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x] for all ⃗x ∈ E and m ∈ F ∗, thus, for all m ∈ F ∗ (recall that [m] is a line matrix), thus [L∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [m]T = ([L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [m]T , thus [L∗] = [L]T (transposed matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='81) (Full notation: [L∗]|b,a = ([L]|⃗a,⃗b)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Details: With the dual bases (πai) and (πbi), with L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = �m i=1Lij⃗bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|⃗a,⃗b = [Lij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n , and with L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='πbj = �n i=1(L∗)ijπai, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L∗]|b,a = [(L∗)ij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='80) gives, for all (i, j) ∈ [1, n]N × [1, m]N, (L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='πbj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai = πbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai), thus (L∗)ij = Lji and [L∗] = [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='82) Duality notations (warning: can be misused): L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = �m i=1Lij⃗bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|⃗a,⃗b = [Lij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n , and L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='bj = �n i=1(L∗)i jai, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L∗]|b,a = [((L∗)i j] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m , thus, for all (i, j) ∈ [1, n]N × [1, m]N, (L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai = bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai), thus (L∗)i j = Lj i and [L∗] = [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='83) (Recall: Use classical notations if in doubt, or, preferably, don’t use duality notations here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='67 Reminder: The transposed bT of a bilinear b form is intrinsic to b, and the adjoint L∗ of a linear map L is intrinsic to L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But a transposed LT of a linear form L is not intrinsic to the linear form (it depends on chosen inner dot products): Watch out for the (unfortunate) vocabulary “transpose”!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Relation with the transposed when inner dot products are introduced let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We need inner dot products (·, ·)g and (·, ·)h in E and F to define LT = LT gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' To have a functional relation between L∗ and LT gh, we use the (·, ·)g-Riesz representation mapping ⃗Rg : � E∗ → E ℓ → ⃗Rg(ℓ) = ⃗ℓg � , where ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = (⃗ℓg, ⃗x)g for all ⃗x ∈ E, see (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' idem with F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) (continuous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' For all ⃗x ∈ E and all m ∈ F ∗ we have (L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='79) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x), thus (⃗Rg(L∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m), ⃗x)g = (⃗Rh(m), L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x)h, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='84) thus ((⃗Rg ◦ L∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m), ⃗x)g = ((LT gh ◦ ⃗Rh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ⃗Rg ◦ L∗ = LT gh ◦ ⃗Rh, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' LT gh = ⃗Rg ◦ L∗ ◦ (⃗Rh)−1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E LT gh ←− F ⃗Rg ↑ ↑ ⃗Rh E∗ ←− L∗ F ∗ is a commutative diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='85) Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='68 From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='85), recover (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='71), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [LT gh] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [LT gh] =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='85) [⃗Rg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[L∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗Rh]−1 =(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 Tensorial representation of a linear map A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 A tensorial representation Consider the natural canonical isomorphism (between linear maps E → F and bilinear forms F ∗×E → R) � J : � L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) → L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) L → βL = � J (L) � where βL(m, ⃗u) := m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u), ∀(m, ⃗u) ∈ F ∗ × E, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='86) see § T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And βL is also named L for calculations purposes, see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (NB: It can be dangerous to substitute L with βL, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 95 96 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Change of basis formulas for bilinear forms and linear maps Quantification: Let (⃗ai)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n be a basis in E, (⃗bi)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m be a basis in F which dual basis is (πbi), L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then βL(πbi,⃗ai) = πbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='87) Thus, if L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = m � i=1 Lij⃗bi then βL = m � i=1 n � j=1 Lij⃗bi ⊗ πaj (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='88) Indeed, (� ij Lij⃗bi ⊗ πaj)(πbk,⃗aℓ) = � ij Lij(⃗bi ⊗ πaj)(πbk,⃗aℓ) = � ij Lij(⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='πbk)(πaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aℓ) = � ij Lij(⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='πbk)(πaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aℓ) = � ij Lijδkiδjℓ = Lkℓ = πbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aℓ, so (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='87) gives (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='88).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notations: L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = �m i=1Lij⃗bi and βL = �m i=1 �n j=1Lij⃗bi ⊗ aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Contraction rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If you write L = �m i=1 �n j=1Lij⃗bi ⊗πaj (≃ βL), then the vector L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u ∈ F is computed thanks to the “contraction rule”: L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = ( m � i=1 n � j=1 Lij⃗bi ⊗ πaj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u � �� � contraction := m � i=1 n � j=1 Lij⃗bi(πaj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u) = m � i=1 n � j=1 Lijuj⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='89) (With duality notations: L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = ( m � i=1 n � j=1 Li j⃗bi ⊗ aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u � �� � contraction = m � i=1 n � j=1 Li j⃗bi(aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u) = m � i=1 n � j=1 Li juj⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='69 Warning: The bilinear form βL should not be confused with the linear map L: The domain of definition of βL is F ∗ × E, and βL acts on the two objects ℓ (linear form) and ⃗u (vector) to get a scalar result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' While the domain of definition of L is E, and L acts one object ⃗u to get a vector result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' However, you can use the tensorial notation for L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' only to calculate L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='89).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Warning: Confusion between transposed and adjoint The transposed LT ∈ L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) of a linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) needs inner dot products to be defined, cf (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='68): It is not intrinsic to L, not objective ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' While the transposed bT ∈ L(B, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) of a bilinear form b ∈ L(A, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is intrinsic to L (it does not need inner dot products to be defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So if you represent a linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) by its tensorial representation βL ∈ L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='88), then 1- you know the transposed (βL)T (given by (βL)T (⃗w, ⃗u) = βL(⃗u, ⃗w)), 2- but you cannot deduce the transposed LT ∈ L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) from (βL)T (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', to start with (βL)T is misleading): You need to choose inner dot products, and then use the formula (LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗x)g = (⃗y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x)h where LT := LT gh to get [LT ] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [h] (and [LT ] ̸= [L]T in general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3- In particular: If L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) is symmetric (relative to the chosen inner dot products), then βL ∈ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is never symmetric because E∗ ̸= E !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Recall: there is no natural canonical isomorphism between E and E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 Change of basis formulas for bilinear forms and linear maps A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Notations for transitions matrices for bilinear forms and linear maps Let A and B be finite dimension vector spaces, dim A = n, dim B = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' application to the change of basis formula for the deformation gradient A=⃗Rn t0 → B=⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Let (⃗aold,i) and (⃗anew,i) be two bases in A, and (⃗bold,i) and (⃗bnew,i) be two bases in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let PA and PB be the change of basis endomorphisms from old to new bases, and PA := [PA]|⃗aold = [PAij] and PB := [PB]|⃗bold = [PBij] be the associated transition matrices, and QA = PA −1 and QB = PB −1: ⃗anew,j = PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aold,i = n � i,j=1 PAij⃗aold,i, πanew,j = n � i=1 QAijπaold,i, ⃗bnew,j = PB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bold,i = m � i,j=1 PBij⃗bold,i, πbnew,j = n � i,j=1 QBijπbold,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='90) Duality notations: ⃗anew,j = �n i=1PA i j⃗aold,i and ai new = �n j=1QA i jaj old and ⃗bnew,j = �n i=1PB i j⃗bold,i and bi new = �n j=1QB i jbj old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 96 97 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Change of basis formulas for bilinear forms and linear maps A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Change of coordinate system for bilinear forms ∈ L(A, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) Let g ∈ L(A, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), and, for all (i, j) ∈ [1, n]N × [1, m]N, g(⃗aold,i,⃗bold,j) = Mij, g(⃗anew,i,⃗bnew,j) = Nij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � � � [g]|olds = M = [Mij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m , [g]|news = N = [Nij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='91) Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='70 Change of basis formula: [g]|news = PA T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PB, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' N = PA T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='92) In particular, if A = B and (⃗aold,i) = (⃗bold,i) and (⃗anew,i) = (⃗bnew,i), then PA = PB =noted P, and [g]|new = P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' N = P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='93) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Nij = g(⃗anew,i,⃗bnew,j) = � kℓ PA k iPB ℓ jg(⃗aold,k,⃗bold,ℓ) = � kℓ PA k iMkℓPB ℓ j = � kℓ(PA T )ikMkℓPB ℓ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='71 Prove (objective result): g(⃗u, ⃗w) = [⃗u]T |⃗anew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗bnew = [⃗u]T |⃗aold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='94) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u]T |⃗anew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗bnew = (PA −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u]|⃗aold)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (PA T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (PB −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗bold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Change of coordinate system for bilinear forms ∈ L(A∗, B∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) Let z ∈ L(A∗, B∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), and, for all (i, j) ∈ [1, n]N × [1, m]N, z(ai old, bj old) = M ij, z(ai new, bj new) = N ij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � � � [z]|olds = M = [M ij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m , [z]|news = N = [N ij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='95) Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='72 Change of basis formula: [z]|news = PA −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [z]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PB −1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' N = PA −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PB −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='96) In particular, if A = B and (⃗aold,i) = (⃗bold,i) and (⃗anew,i) = (⃗bnew,i), then PA = PB =noted P, and [z]|new = P −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [z]old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P −1 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' N = P −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='97) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Nij = z(ai new, bj new) = � kℓ QA k iQB ℓ jz(ak old, bℓ old) = � kℓ QA k iM kℓQB ℓ j = � kℓ(QA T )ikM kℓQB ℓ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Change of coordinate system for bilinear forms ∈ L(B∗, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) (Toward linear maps L ∈ L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B) ≃ L(B∗, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) thanks to the natural canonical isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Let T ∈ L(B∗, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), and, for all (i, j) ∈ [1, n]N × [1, m]N, T(bi old,⃗aold,j) = M i j, T(bi new,⃗anew,j) = N i j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � � � [T]|olds = M = [M i j] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m , [T]|news = N = [N i j] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='98) Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='73 Change of basis formula: [T]|news = PB −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [T]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' N = QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='99) In particular, if A = B and (⃗aold,i) = (⃗bold,i) and (⃗anew,i) = (⃗bnew,i), then PA = PB =noted P, and [T]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [T]old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' N = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' N i j = n � k,ℓ=1 Qi kM k ℓP ℓ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='100) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' N ij = T(bi new,⃗anew,j) = � kℓ QB i kPA ℓ jT(bi old,⃗aold,j) = � kℓ QB i kM ijPA ℓ j 97 98 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Change of basis formulas for bilinear forms and linear maps A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Change of coordinate system for tri-linear forms ∈ L(A∗, A, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) (Toward d2⃗u: For a vector field ⃗u ∈ Γ(U) ≃ T 1 0 (U), ⃗u(p) ∈ ⃗Rn, its differential satisfies d⃗u(p) ∈ L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn) ≃ L(Rn∗, ⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), and d2⃗u(p) ∈ L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn)) ≃ L(Rn∗, ⃗Rn, ⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), see § S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Consider a tri-linear form T ∈ L(A∗, A, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), and M i jk = T(ai old,⃗aold,j,⃗aold,k), N i jk = T(ai new,⃗anew,j,⃗anew,k), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [T]|⃗aold = [M i jk], [T]|⃗anew = [N i jk].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='101) Then N i jk = n � λ,µ,ν=1 Qi λP µ j P ν k M λ µν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='102) Indeed � λµν M λ µν⃗aold,λ ⊗ aµ old ⊗ aν old = � λµνijk M λ µνQi λP µ j P ν k⃗anew,i ⊗ aj new ⊗ ak new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Change of coordinate system for linear maps ∈ L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B) Notation of § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let L ∈ L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B) be a linear map, and let, for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, � � � � � � � � � � � L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aold,j = m � i=1 Mij⃗bold,i = m � i=1 M i j⃗bold,i i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|olds = M = [Mij] = [M i j] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗anew,j = m � i=1 Nij⃗bnew,i = m � i=1 N i j⃗bnew,i i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|news = N = [Nij] = [N i j] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='103) with classical and duality notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='74 Change of bases formula: [L]|news = PB −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' N = PB −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='104) In particular, if A = B, if (⃗aold,i) = (⃗bold,i), (⃗anew,i) = (⃗bnew,i), then PA = PB =noted P and [L]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' N = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Nij = n � k,ℓ=1 QikMkℓPℓj, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='105) with Q = P −1, and with duality notations N ij = � kℓ QikM kℓP ℓj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗anew,j = � i N ij⃗bnew,i = � ik N ijPB k i⃗bold,k = � k(PB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='N)kj⃗bold,k and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗anew,j = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(� i PA i j⃗aold,i) = � i PA i j � k M ki⃗bold,k = � k(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PA)kj⃗bold,k, for all j, thus PB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='N = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='75 Prove: ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = [ℓ]|⃗bnew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[L]|news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u]|⃗anew = [ℓ]|⃗bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[L]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u]|⃗aold (objective result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='106) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ℓ]|⃗bnew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[L]|news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u]|⃗anew = ([ℓ]|⃗bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (PB −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[L]|olds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (PA −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u]|⃗aold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='76 Bilinear forms in L(A, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) and endomorphisms in L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A) behave differently: The formulas (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='93) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='105) should not be confused since P −1 ̸= P T in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', if an English observer uses a Euclidean (old) basis (⃗ai) = (⃗aold,i) in foot, if a French observer uses a Euclidean (new) basis (⃗bi) = (⃗anew,i) in metre, and if (simple case) ⃗bi = λ⃗ai for all i (change of unit), then [L]|new = [L]|old, while [g]|new = λ2 ���� >10 [g]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='107) Quite different results!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P ̸= P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P for a general change of basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See the Mars Climate Orbiter crash, remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14, where someone forgot that 1 foot ̸= 1 metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B Euclidean Frameworks Time and space are decoupled (classical mechanics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Rn is the geometric affine space, n = 1, 2, 3, and ⃗Rn is the associated vector space made of “bi-point vectors”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 98 99 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Euclidean basis B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Euclidean basis Manufacturing of a Euclidean basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' An observer chooses a unit of measure (foot, metre, a unit of length used by Euclid, the diameter a of pipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') and makes a “unit rod” of length 1 in this unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Postulate: The length of the rod does not depend on its direction in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Space dimension n = 1: This rod models a vector ⃗e1 which makes a basis (⃗e1) called the Euclidean basis relative to the chosen unit of measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Space dimension n ≥ 2: The observers makes three rods of length 3, 4 and 5, and makes a triangle (A, B, C) with A, B and C are the vertices and A not on the side on length 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Pythagoras: 32 + 42 = 52 gives: The triangle (A, B, C) is said to have a right angle at A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Two vectors ⃗u and ⃗w in ⃗Rn are orthogonal iff the triangle (A, B, C) can be positioned such that ⃗ AB and ⃗ AC are parallel to ⃗u and ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A basis (⃗ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n is Euclidean relative to the chosen unit of measurement iff the ⃗ei are two to two orthogonal and their length is 1 (relative to the chosen unit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 An English observer defines a Euclidean basis (⃗ai) using the foot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A French observer defines a Euclidean basis (⃗bi) using the metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have 1 foot = µ metre, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3048, and 1 metre = λ foot, λ = 1 µ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) (µ = 0, 3048 is the official length in metre for the English foot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', the bases are “aligned” iff, for all i, ⃗bi = λ⃗ai (change of measurement unit), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) thus the transition matrix from (⃗ai) to (⃗bi) is P = λI, thus P T = P, P −1 = 1 λI and P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P = λ2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The bases used in practice are not all Euclidean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13, especially if you fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Euclidean dot product Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 An observer who has built) his Euclidean basis (⃗ei), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' § B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The associated Euclidean dot product is the bilinear form g(·, ·) = (·, ·)g ∈ L(⃗Rn, ⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) defined by (gij =) g(⃗ei,⃗ej) = δij, ∀i, j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|⃗e = [δij] = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) In other words, (·, ·)g := n � i=1 πei ⊗ πei = n � i=1 ei ⊗ ei, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) with classical and duality notations, (πei) = (ei) being the dual basis of (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And if you want to use the Einstein convention you have to write (·, ·)g := �n i,j=1δijei ⊗ ej: You cannot avoid writing δij = gij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, for all ⃗x, ⃗y ∈ ⃗Rn, with ⃗x = �n i=1xi⃗ei and ⃗y = �n i=1yi⃗ei (classical notations), (⃗x, ⃗y)g = n � i=1 xiyi = [⃗x]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗y]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) With duality notations, ⃗x = �n i=1xi⃗ei, ⃗y = �n i=1yi⃗ei and (⃗x, ⃗y)g = �n i=1xiyi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And if you want to use the Einstein convention then write (⃗x, ⃗y)g := �n i,j=1δijxiyj: You cannot avoid writing δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 The associated norm is ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||g := � (·, ·)g, and the length of a vector ⃗x relative to the chosen Euclidean unit of measurement is ||⃗x||g := � (⃗x, ⃗x)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus with the Euclidean basis (⃗ei) (used to build (·, ·)g), if ⃗x = �n i=1xi⃗ei, then ||⃗x||g = ��n i=1x2 i is the length of ⃗x relative to the chosen Euclidean unit of measure (Pythagoras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (With duality notations ||⃗x||g = ��n i=1(xi)2, and if you want to use the Einstein convention: ||⃗x||g = ��n i,j=1δijxixj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Definition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 The angle θ(⃗x, ⃗y) between two vectors ⃗x, ⃗y ∈ ⃗Rn − {⃗0} is defined by cos(θ(⃗x, ⃗y)) = ( ⃗x ||⃗x||g , ⃗y ||⃗y||g )g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) (With a calculator, this formula gives θ(⃗x, ⃗y) = arccos(( ⃗x ||⃗x||g , ⃗y ||⃗y||g )g) a value in [0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 99 100 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Change of Euclidean basis B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Change of Euclidean basis Let (⃗ai) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' English observer basis built with the foot) and (⃗bi) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' French observer basis built with the metre) be Euclidean bases in ⃗Rn, and let (·, ·)g and (·, ·)h be the associated Euclidean dot products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Two Euclidean dot products are proportional Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 If λ = ||⃗b1||g, then ||⃗bi||g = λ for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n (change of unit) and (·, ·)g = λ2(·, ·)h, and ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||g = λ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' By definition of a Euclidean basis, the length of the rod that enabled to define (⃗bi) is independent of i, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' § B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1, thus ||⃗bi||g = ||⃗b1||g for all i, and here ||⃗bi||g =noted λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ||⃗bi||2 g = λ2 = λ2||⃗bi||2 h for all i, since ||⃗bi||2 h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And if i ̸= j then (⃗bi,⃗bj)g = 0 = (⃗bi,⃗bj)h since ⃗bi and ⃗bj form a right angle (Pythagoras), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence (⃗bi,⃗bj)g = λ2(⃗bi,⃗bj)h for all i, j, thus (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Continuation of example B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1: (·, ·)a = �n i=1ai ⊗ ai is the English Euclidean dot product (foot), and (·, ·)b = �n i=1bi ⊗ bi is the French Euclidean dot product (metre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) give: (·, ·)a = λ2(·, ·)b and ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||a = λ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||b, with λ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28 and λ2 ≃ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) In particular, if ⃗w is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ||⃗w||b = 1 (its length is 1 metre), then ||⃗w||a = λ (its length is λ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28 foot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Counterexample : non existence of a Euclidean dot product 1- Thermodynamic: Let T be the temperature and P the pressure, and consider the Cartesian vector space {(T, P)} = {(temperature,pressure)} = R × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' There is no associated Euclidean dot product: An associated norm would give ||(T, P)|| = √ T 2 + P 2 ∈ R which is meaningless (incompatible dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- Polar coordinate system ⃗q = (r, θ) ∈ R × R: There is no Euclidean norm √ r2 + θ2 for ⃗q that is physically meaningful (incompatible dimensions), see example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Euclidean transposed of the deformation gradient Let n ∈ {1, 2, 3} and consider a linear map L ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L = F t0 t (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (·, ·)G be a Euclidean dot product in ⃗Rn t0 (used in the past by someone), and let (·, ·)g and (·, ·)h be Euclidean dot products in ⃗Rn t (the actual space where the results are obtained by two observers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', (·, ·)g built with a foot and (·, ·)h built with a metre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let LT Gg and LT Gh be the transposed of L relative to the dot products, that is, LT Gg and LT Gh in L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) are characterized by, for all ( ⃗X, ⃗y) ∈ ⃗Rn t0 × ⃗Rn t , cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='68), (LT Gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗X)G = (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗X, ⃗y)g and (LT Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗X)G = (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗X, ⃗y)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 if (·, ·)g = λ2(·, ·)h then LT Gg = λ2LT Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) NB: Do not forget λ2, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 (Mars Climate Orbiter crash).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (LT Gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗X)G (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) = (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗X, ⃗y)g (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10)1 = λ2(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗X, ⃗y)h (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) = λ2(LT Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗X)G for all ⃗X ∈ ⃗Rn t0 and all ⃗y ∈ ⃗Rn t , thus LT Gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y = λ2LT Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y for all ⃗y ∈ ⃗Rn t , thus (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 The Euclidean transposed for endomorphisms Let n ∈ {1, 2, 3} and consider an endomorphism L ∈ L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L = d⃗vt(p) ∈ L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) the differential of the Eulerian velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (·, ·)g and (·, ·)h be dot products in ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let LT g and LT h be the transposed of L relative to (·, ·)g and (·, ·)h, that is, LT g and LT h in L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) are the endomorphisms defined by, for all ⃗x, ⃗y ∈ ⃗Rn t , cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54), (LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗x)g = (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, ⃗y)g, and (LT h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗x)h = (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, ⃗y)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) 100 101 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Unit normal vector, unit normal form Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 if (·, ·)g = λ2(·, ·)h then LT g = LT h noted = LT ∈ L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) (an endomorphism type relation): Thus we can speak of “the Euclidean transposed of an endomorphism”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗x)g (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) = (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, ⃗y)g hyp = λ2(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x, ⃗y)h (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) = λ2(LT h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗x)h hyp = (LT h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗x)g for all ⃗x, ⃗y ∈ ⃗Rn, thus LT g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y = LT h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y for all ⃗y ∈ ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Unit normal vector, unit normal form The results in this § are not objective: We need a Euclidean dot product (need a unit of length: Foot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Meter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') to get a unit (Euclidean) normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Framework (·, ·)g is a Euclidean dot product (needed to define Euclidean orthonormality) and, for all ⃗u, ⃗w ∈ ⃗Rn, (⃗u, ⃗w)g noted = ⃗u •g ⃗w (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) (or =noted ⃗u • ⃗w when one chosen Euclidean dot product is imposed to all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ω is a regular open bounded set in Rn, n = 2 or 3, and Γ := ∂Ω is its regular surface (dimension n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If p ∈ Γ then TpΓ is the tangent plane at p to Γ, and a basis (⃗β1(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗βn−1(p)) in TpΓ is known (usually obtained thanks to a coordinate system describing Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And, to lighten the writings, (⃗β1(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗βn−1(p)) is written (⃗β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗βn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Unit normal vector Call ⃗ng(p) the unit outward normal vector at p ∈ Γ at TpΓ relative to (·, ·)g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So ⃗ng(p) •g ⃗βi(p) = 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n−1, and ||⃗ng(p)||g = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ng is defined on Γ by (up to its sign) ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n−1, ⃗βi •g ⃗ng = 0, and ⃗ng •g ⃗ng = 1 (= ||⃗ng||2 g), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', at any p ∈ Γ, ⃗ng(p) is orthogonal to the hyperplane Vect{⃗β1(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗βn−1(p)} and ⃗ng(p) is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So (⃗β1(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗βn−1(p),⃗ng(p)) is a basis at p in ⃗Rn, written in short (⃗β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗βn−1,⃗ng).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Drawing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, for all ⃗w ∈ ⃗Rn, if ⃗w = �n−1 i=1 wi⃗βi + wn⃗ng (classical notations) then wn = ⃗w •g ⃗ng = the normal component of ⃗w at p at Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) (wn depends on (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') (Duality notations: ⃗w = �n−1 i=1 wi⃗βi + wn⃗ng and wn = ⃗w •g ⃗ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Exercice B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 Let (⃗ai) be a basis in ⃗Rn, ⃗βj = �n i=1Bij⃗ai for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n−1, and ⃗ng = �n i=1ni⃗ai, and gij = g(⃗ai,⃗aj) for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' What equations satisfy the nj?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And particular case (⃗ai) is (·, ·)g-orthonormal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) gives [⃗βi]T |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ng]|⃗a = 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n−1 (so n−1 equations), with [⃗ng]T |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ng]|⃗a = 1 (so 1 equation), and ⃗ng is obtained up to its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If (⃗ai) is (·, ·)g-orthonormal, then �n j=1Bijnj = 0 for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n−1, with �n i=1n2 i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 Let (⃗ai) be a Euclidean basis in foot, (⃗bi) a Euclidean basis in metre, (·, ·)a and (·, ·)b the associated Euclidean dot products, so (·, ·)a = λ2(·, ·)b with λ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗na(p) and ⃗nb(p) be the corresponding unit outward normal vectors, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- Prove (up to the sign): ⃗nb = λ⃗na, and (⃗w,⃗na)a = λ(⃗w,⃗nb)b ∀⃗w ∈ ⃗Rn (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) 2- Then let ⃗na = �m i=1nai⃗ai and ⃗nb = �m i=1nbi⃗bi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove: If, ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, ⃗bi = λ⃗ai then ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, nai = nbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) So the vectors ⃗na and ⃗nb are different (λ > 1), and their respective components are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' relative to different bases!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And of course 1 = ||⃗na||2 a = �n i=1(nai)2 = �n i=1(nbi)2 = ||⃗nb||2 b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗na(p) ∥ ⃗nb(p), since the vectors are Euclidean and orthogonal to TpΓ cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||a = λ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||b cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8), thus ||⃗nb||b = 1 = ||⃗na||a = λ||⃗na||b = ||λ⃗na||b, so ⃗nb = ±λ⃗na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And they both are outward vectors, so ⃗nb = +λ⃗na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (⃗w,⃗na)a = λ2(⃗w,⃗na)b = λ2(⃗w, ⃗nb λ )b = λ(⃗w,⃗nb)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And if ⃗bi = λ⃗ai (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) gives �n i=1ni b⃗bi = λ�n i=1ni a⃗ai = �n i=1ni a(λ⃗ai) = �n i=1ni a⃗bi, then ni a = ni b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 101 102 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Integration by parts (Green–Gauss–Ostrogradsky) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Unit normal form n♭ associated to ⃗n (For mathematicians;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' May produce misunderstandings and lack of mechanical interpretations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Don’t forget: n♭ is obtained after ⃗n has been defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') At p ∈ Γ, once you have computed ⃗ng(p), you can define the associated unit normal form n♭ g(p) ∈ Rn∗: It is the linear form defined by n♭ g(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w := ⃗ng(p) •g ⃗w for all ⃗w ∈ ⃗Rn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' on Γ, for all ⃗w ∈ ⃗Rn, n♭ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w := ⃗ng •g ⃗w (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) ( =noted ⃗n • ⃗w if one chosen Euclidean dot product is imposed to all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus [n♭ g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w] = [⃗ng]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification: Let (⃗ei) be a basis in ⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) gives [n♭ g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗e = [⃗ng]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗e simply written [n♭ g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w] = [⃗ng]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w] if the basis (⃗ei) is imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, with duality notations to justify the ♭ notation, with (ei) the dual basis of (⃗ei), let ⃗ng = n � i=1 ni g⃗ei and n♭ g = n � i=1 ngiei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ni g and ngi are the components of ⃗ng and n♭ g relative to the basis (⃗ei) and (ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Since (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) gives n♭ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei := ⃗ng •g ⃗ei for all i, we get, for all i, nig = n � j=1 gijnj g (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) Particular case (⃗ei) is a (·, ·)g-Euclidean basis, then nig = ni g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Classical notations: ⃗ng = �n i=1(⃗ng)i⃗ei, dual basis (πei), n♭ g = �n i=1(n♭ g)iπei, (n♭ g)i = �n j=1gij(⃗ng)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: In physics don’t forget to write the gij in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) even if gij = δij, since you need to see the chosen metric and basis (and verify the Einstein convention), although (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) is simply written ni = �n j=1gijnj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Integration by parts (Green–Gauss–Ostrogradsky) Let Ω be a regular bounded open set in Rn and Γ = ∂Ω its frontier, let ϕ ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), let (⃗ei) be a Euclidean basis and (·, ·)g ites associated Euclidean dot product, let ∂ϕ ∂xi (p) := dϕ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei (usual notation), let ⃗ng(p) = ⃗n(p) = �n i=1ni(p)⃗ei (classical notations) be the unit outward normal at p ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, � p∈Ω ∂ϕ ∂xi (p) dΩ = � p∈Γ ϕ(p)ni(p) dΓ, in short � Ω ∂ϕ ∂xi dΩ = � Γ ϕni dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) Thus, for any v ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), with ϕv instead of ϕ in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21), we get the integration by parts formula (Green formula): � Ω ∂ϕ ∂xi v dΩ = − � Ω ϕ ∂v ∂xi dΩ + � Γ ϕvni dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) Thus, for any ⃗v ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn) (vector field), with ⃗v(p) = �n i=1vi(p)⃗ei) we get � Ω ∂ϕ ∂xi vi dΩ = − � Ω ϕ ∂vi ∂xi dΩ + � Γ ϕvini dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) Thus, with the gradient vector ⃗ gradϕ(p) = �n i=1 ∂ϕ ∂xi⃗ei and with div⃗v = �n i=1 ∂vi ∂xi , we get the Gauss– Ostrogradsky formula: � Ω ⃗ gradϕ • ⃗v dΩ = − � Ω ϕ div⃗v dΩ + � Γ ϕ⃗v • ⃗n dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) (And � Γ ϕ⃗v • ⃗n dΓ gives the flux through Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Exercice B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 Use the differential dϕ instead of the gradient ⃗ gradϕ (which is the (·, ·)g-Riesz represen- tation vector of dϕ) to express (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Is the use of n♭ useful in that case?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � Ω dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v dΩ = − � Ω ϕ div⃗v dΩ + � Γ ϕ⃗v • ⃗n dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Since n♭ depends on ⃗n (definition), there is no reason that justifies the use of n♭ (unless you want to introduce useless notations here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 102 103 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The symmetric and antisymmetric parts of d⃗v C Rate of deformation tensor and spin tensor Let �Φ : [t1, t2] × Obj → Rn be a regular motion, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5), and let ⃗v : C → ⃗Rn be the Eulerian velocity field, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4), that is, ⃗v(t, p) = ∂Φ ∂t (t, PObj) when p = �Φ(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its differential d⃗v is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' At t, an observer chooses a unit of measurement (foot, metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') and builds the associated Euclidean dot product (·, ·)g in ⃗Rn t , cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' § B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (We loose the objective point of view here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the same (·, ·)g is used at all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The symmetric and antisymmetric parts of d⃗v With the imposed chosen Euclidean dot product (·, ·)g in ⃗Rn t , we can consider the transposed endomor- phism d⃗vt(p)T g =noted d⃗vt(p)T ∈ L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ), which is defined by, for all ⃗w1, ⃗w2 ∈ ⃗Rn t vectors at p, (d⃗vt(p)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, ⃗w2)g = (⃗w1, d⃗vt(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2)g (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have thus defined d⃗vT t : � Ωt → L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) p → d⃗vT t (p) := d⃗vt(p)T (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) Other usual notations (definitions): d⃗vt(p)T =noted d⃗v(t, p)T =noted d⃗vT (t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The (Eulerian) rate of deformation tensor, or stretching tensor, is the (·, ·)g-symmetric part of d⃗v: D = d⃗v + d⃗vT 2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ∀(t, p) ∈ � t∈R ({t} × Ωt), D(t, p) = d⃗v(t, p) + d⃗v(t, p)T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) The (Eulerian) spin tensor is the (·, ·)g-antisymmetric part of d⃗v: Ω = d⃗v − d⃗vT 2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ∀(t, p) ∈ � t∈R ({t} × Ωt), Ω(t, p) = d⃗v(t, p) − d⃗v(t, p)T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) (So d⃗v = D + Ω with D the rate of deformation tensor and Ω = ⃗ω∧ a rotation times a dilation, see the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') NB: The same notation is used for the set of points Ωt = Φt0 t (Ωt0) ⊂ Rn and for the spin tensor Ωt = d⃗vt−d⃗vT t 2 : The context removes ambiguities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Quantification with a basis With a basis (⃗ei) in ⃗Rn t , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) gives [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗vT ]|⃗e = [d⃗v]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|⃗e, and [d⃗vT ]|⃗e = [g]−1 |⃗e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗v]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) In particular, if (⃗ei) is a (·, ·)g-orthonormal basis, then [d⃗vT ]|⃗e = [d⃗v]T |⃗e (orthonormal basis case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus for the endomorphisms D and Ω, and with the above Euclidean framework and its Euclidean orthonormal basis, we have D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1Dij⃗ei and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1Ωij⃗ei with Dij = 1 2( ∂vi ∂xj + ∂vj ∂xi ) and Ωij = 1 2( ∂vi ∂xj − ∂vj ∂xi ), that is, [D]|⃗e = [d⃗v]|⃗e + [d⃗v]T |⃗e 2 and [Ω]|⃗e = [d⃗v]|⃗e − [d⃗v]T |⃗e 2 (Euclidean framework).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) Duality notations: D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1Di j⃗ei, Di j = 1 2( ∂vi ∂xj + ∂vj ∂xi ) and Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1Ωij⃗ei, Ωij = 1 2( ∂vi ∂xj − ∂vj ∂xi ), so with Di j = Dj i and Ωij = −Ωji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 103 104 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Affine motions and rigid body motions D Interpretation of the rate of deformation tensor We are interested in the evolution of the deformation gradient F(t) := F t0 pt0 (t) along the trajectory of a particle PObj which was at pt0 at t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So: Let ⃗A = ⃗a(t0, pt0) and ⃗B = ⃗b(t0, pt0) be vectors at t0 at pt0 in Ωt0, and consider their push-forwards by the flow Φt0 t (the transported vectors), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the vectors at t at p(t) = Φt0 pt0 (t) given by ⃗a(t, p(t)) := F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗A and ⃗b(t, p(t)) := F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) and figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' They define the function (⃗a,⃗b)g : � C → R (t, pt) → (⃗a,⃗b)g(t, pt) := (⃗a(t, pt),⃗b(t, pt))g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) Proposition D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The rate of deformation tensor D = d⃗v+d⃗vT 2 gives (half) the evolution rate between two vectors deformed by the flow, that is, along trajectories, D(⃗a,⃗b)g Dt = 2(D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a,⃗b)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' f(t) := (⃗a(t, p(t)),⃗b(t, p(t)))g = (F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗A, F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗B)g gives f ′(t) = (F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗A, F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗B)g + (F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗A, F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗B)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) And F ′(t) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, with ⃗a(t, p(t)) = F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗A and ⃗b(t, p(t)) = F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗B, f ′(t) = (d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗A, F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗B)g + (F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗A, d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗B)g = (d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a(t, p(t)),⃗b(t, p(t)))g + (⃗a(t, p(t)), d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗b(t, p(t)))g = ((d⃗v(t, p(t)) + d⃗v(t, p(t))T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a(t, p(t)),⃗b(t, p(t)))g, (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3), since f(t) = (⃗a,⃗b)g(t, p(t)) gives f ′(t) = D(⃗a,⃗b)g Dt (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E Rigid body motions and the spin tensor Choose a Euclidean dot product (·, ·)g (required to characterize a rigid body motion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Result: A rigid body motion is a motion whose Eulerian velocity satisfies d⃗v + d⃗vT = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', D = 0 (Eulerian approach independent of any initial time t0 chosen by some observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But the usual classical introduction to rigid body motion relies on some initial time t0 (Lagrangian approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, to begin with, let us do it with the Lagrangian approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Recall: T the first order Taylor expansion of Φt0 t in the vicinity of a pt0 ∈ Ωt0 is Φt0 t (qt0) = Φt0 t (pt0) + F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−−−→ pt0qt0 + o(−−−→ pt0qt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Affine motions and rigid body motions E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Affine motions Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Φt0 is an affine motion (understood “affine motion in space”) iff Φt0 t is an “affine motion”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' iff Φt0 t is a C1 diffeomorphism (in space), and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) reads, for all pt0, qt0 ∈ Ωt0 and all t ∈ [t1, t2], Φt0 t (qt0) = Φt0 t (pt0) + F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−−−→ pt0qt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) Marsden–Hughes notations: Φ(Q) = Φ(P) + F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−−→ PQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 and definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If Φt0 is an affine motion, then F t0 t (pt0) is independent of pt0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all t ∈]t1, t2[ and all pt0 ∈ Ωt0 and all qt0 ∈ Ωt0, F t0 t (pt0) = F t0 t (qt0) noted = F t0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) And then dF t0 t (pt0) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' d2Φt0 t (pt0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And for all t ∈]t1, t2[, Φt is an affine motion: For all τ ∈]t1, t2[ and all pt, qt ∈ Ωt, Φt τ(qt) = Φt τ(pt) + F t τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−−→ ptqt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) And �Φ is said to be an affine motion (understood “affine motion in space”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 104 105 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Affine motions and rigid body motions Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' qt0 = pt0 + −−−→ pt0qt0 gives Φt0 t (qt0) = Φt0 t (pt0 + −−−→ pt0qt0) = Φt0 t (pt0) + dΦt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−−−→ pt0qt0, and, similarly, Φt0 t (pt0) = Φt0 t (qt0 +−−−→ qt0pt0) = Φt0 t (qt0)+dΦt0 t (qt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−−−→ qt0pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (addition) Φt0 t (qt0)+Φt0 t (pt0) = Φt0 t (pt0)+ Φt0 t (qt0) + (dΦt0 t (pt0) − dΦt0 t (qt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−−−→ pt0qt0, thus (dΦt0 t (pt0) − dΦt0 t (qt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−−−→ pt0qt0 = 0, true for all pt0, qt0, thus dΦt0 t (pt0) − dΦt0 t (qt0) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus d2Φt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ut0 = limh→0 dΦt0 t (pt0+h⃗ut0)−dΦt0 t (pt0) h = limh→0 dΦt0 t −dΦt0 t h = 0 for all pt0 and all ⃗ut0, thus d2Φt0 t (pt0) = 0 for all pt0, thus d2Φt0 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) gives (Φt τ ◦ Φt0 t )(pt0) = Φt0 τ (pt0), thus, with pt = Φt0 t (pt0), we get dΦt τ(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΦt0 t (pt0) = dΦt0 τ (pt0), thus dΦt τ(pt) = dΦt0 τ (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΦt0 t (pt0)−1, and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) gives dΦt τ(pt) = dΦt0 τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΦt0 t −1 noted = dΦt τ (independent of pt), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) thus (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Corollary E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 If �Φ is affine then, ⃗vt is affine for all t, and ⃗V t0 t is affine for all t0, t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all pt ∈ Ωt we have d⃗vt(pt) = d⃗vt (independent of pt), and for all pt0 ∈ Ωt0 we have d⃗V t0 t (pt0) =noted d⃗V t0 t (independent of pt0): For all qt ∈ Ωt and all qt0 ∈ Ωt0, � ⃗vt(qt) = ⃗vt(pt) + d⃗vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−−→ ptqt, ⃗V t0 t (qt0) = ⃗V t0 t (pt0) + d⃗V t0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−−−→ pt0qt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) gives Φt0(t, qt0) = Φt0(t, pt0) + F t0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−−−→ pt0qt0, and the derivation in time gives (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6)2, then (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6)1 thanks to pt = Φt0 t (pt0), qt = Φt0 t (qt0) and −−−→ pt0qt0 = (F t0 t )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−−→ ptqt, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 In R2, with a basis ( ⃗E1, ⃗E2) in ⃗Rn t0 and a basis (⃗e1,⃗e2) ∈ ⃗Rn t , then F t0 t given by [F t0 t ]| ⃗E,⃗e = � 1 + t 2t2 3t3 et � derives from the affine motion [−−−−−−−−−−−→ Φt0 t (pt0)Φt0 t (qt0)]|⃗e = � 1 + t 2t2 3t3 et � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [−−−→ pt0qt0]| ⃗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Rigid body motion A Euclidean dot product (·, ·)g in ⃗Rn t is chosen, the same at all time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Φ := Φt0 t and F := F t0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Recall: If P ∈ Ωt0 and p = Φ(P) (∈ Ωt) then the transposed of the linear map F(P) ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) relative to (·, ·)g is the linear map F T (p) := F(P)T ∈ L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) defined by F T (p) := F(P)T : � ⃗Rn t → ⃗Rn t0 ⃗wp → F T (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wp s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F T (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wp, ⃗UP )g = (⃗wp, F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗UP )g, ∀⃗UP ∈ ⃗Rn t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) We have thus defined the function F T : Ωt → L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Particular case: For an affine motion, since F is independent of P, we get F T is independent of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 A rigid body motion is an affine motion �Φ such that, for all t0, t ∈ R, P ∈ Ωt0, ⃗UP , ⃗WP ∈ ⃗Rn t0, and with p = Φt0 t (P), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗UP , F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗WP )g = (⃗UP , ⃗WP )g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗UP , ⃗WP )g = (⃗UP , ⃗WP )g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F = I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) (Angles and lengths are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') In other words, with the Cauchy strain tensor C ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) defined by C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F, the motion is rigid iff it is affine and C = I , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F −1 = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 If Φt0 is a rigid body motion, if ( ⃗Ai) is a (·, ·)g-Euclidean basis in ⃗Rn t0, if P ∈ Ωt0, if t ∈ [t0, T] and p = Φt0 t (P), and if ⃗ai(t, p) = F t0(t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ai for all i, then ⃗ai(t, p) =noted ⃗ai,t is independent of p, and (⃗ai,t) is a (·, ·)g-Euclidean basis with the same orientation than ( ⃗Ai) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 105 106 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Representation of the spin tensor Ω: vectors, and pseudo-vectors Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Φt0 t is affine, thus, for all t, P, F t0 t (P) = F t0 t (independent of P), thus ⃗ai,t(p) = F t0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ai ∈ ⃗Rn t is independent of p, this at all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let t be fixed and ⃗ai,t =noted ⃗ai (= F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We get (⃗ai,⃗aj)g = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ai, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Aj)g = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ai, ⃗Aj)g = (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ai, ⃗Aj)g = ( ⃗Ai, ⃗Aj)g = δij for all i, j, thus (⃗ai) is (·, ·)g-orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And det(⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗an) = det(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗An) = det(F) det( ⃗A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗An) = det(F) since ( ⃗Ai) is a (·, ·)g- orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And, Φt0 t being a diffeomorphism, t → det(F t0 t ) is continuous, does not vanish, moreover with det(F t0 t0 ) = det(I) = 1 > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus det(F t0 t ) > 0 for all t, hence det(⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗an) > 0: The bases have the same orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 In R2, a rigid body motion is given by F t0 t = � cos(θ(t)) − sin(θ(t)) sin(θ(t)) cos(θ(t)) � with θ a regular function s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' θ(t0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Let �Φ be a rigid body motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove (F T )′(t) = (F ′(t))T , and F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F ′ is antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let t ∈ R, p(t) = Φt0 t (P), ⃗U, ⃗W ∈ ⃗Rn t0 and ⃗w(t, p(t)) = F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And recall that the function F T : t → F T (t) is defined (as usual) by F T (t) := (F(t))T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have (F(t)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, p(t)), ⃗U)g = (⃗w(t, p(t)), F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ((F T )′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, p(t))+F T (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' D ⃗w Dt (t, p(t)), ⃗U)g = ( D ⃗w Dt (t, p(t)), F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U)g+(⃗w(t, p(t)), F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U)g, which simplifies into ((F T )′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, p(t)), ⃗U)g = (⃗w(t, p(t)), F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U)g = ((F ′(t))T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(t, p(t)), ⃗U)g, thus (F T )′(t) = (F ′(t))T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) reads F T (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t) = It0, thus (F T )′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t)+F T (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F ′(t) = 0, thus (F ′)T (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t)+F T (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F ′(t) = 0, thus F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F ′ is antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Alternative definition of a rigid body motion: d⃗v + d⃗vT = 0 The stretching tensor Dt = d⃗vt+d⃗vT t 2 and the spin tensor Ωt = d⃗vt−d⃗vT t 2 have been defined in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3)-(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 If �Φ is a rigid body motion, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8), then the endomorphism d⃗vt ∈ L( ⃗ Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ Rn t ) is antisymmetric at all t: d⃗vt = Ωt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) Conversely, if d⃗vt + d⃗vT t = 0 at all t, then �Φ is a rigid body motion (here no initial time is required).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So the relation « d⃗vt + d⃗vT t = 0 for all t » gives an equivalent definition to the definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let F(t) := F t0 pt0 (t) and F T (t) := F(t)T and V (t) := ⃗V t0 pt0 (t) = (Φt0 pt0 )′(t) = ⃗v(t, pt) (the Lagrangian and Eulerian velocities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) gives (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F T )′(t) = 0 = F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t)T + F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F T )′(t) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) = F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t)T + (F ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t)T )T = dV (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t)−1 + (dV (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t)−1)T (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) = d⃗v(t, pt) + d⃗v(t, pt)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Conversely, suppose d⃗v + d⃗vT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) gives D(⃗a,⃗b)g Dt = 0, thus (⃗a,⃗b)g(t, pt) = (⃗a,⃗b)g(t0, pt0) for all t, t0 and all pt0 = Φt0 t (pt), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗A, F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗B)g = ( ⃗A, ⃗B)g for all t, t0, all pt0 and all ⃗A, ⃗B ∈ ⃗Rn t0: Thus �Φ is a rigid body motion, cf (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Representation of the spin tensor Ω: vectors, and pseudo-vectors We are dealing here with concepts that are sometimes misunderstood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Framework: Rn = R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Reminder The determinant det|⃗e associated with a basis (⃗ei) in R3 is the alternating multilinear form defined by det|⃗e(⃗e1,⃗e2,⃗e3) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The algebraic volume (or signed volume) limited by three vectors ⃗u1, ⃗u2, ⃗u3 is det|⃗e(⃗u1, ⃗u2, ⃗u3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the (positive) volume is | det|⃗e(⃗u1, ⃗u2, ⃗u3)|, see § K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let A and B be two observers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A=English and B=French), let (⃗ai) be a Euclidean basis chosen by A (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' based on the foot), let (⃗bi) be a Euclidean basis chosen by B (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' based on the metre), see § B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 106 107 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Representation of the spin tensor Ω: vectors, and pseudo-vectors Let λ = ||⃗b1||a > 0 (change of unit of length coefficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The relation between the determinants is: det |⃗a = ±λ3 det |⃗b with � � � � � + if det |⃗a (⃗b1,⃗b2,⃗b3) > 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' if the bases have the same orientation), − if det |⃗a (⃗b1,⃗b2,⃗b3) < 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' if the bases have opposite orientation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) In particular, if A and B use the same unit of length (or if A uses two (·, ·)g-Euclidean basis (⃗ai) and (⃗bi)), then λ = 1 and det|⃗a = ± det|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With an imposed Euclidean dot product (·, ·)g: An endomorphism L is (·, ·)g-antisymmetric iff ∀⃗u,⃗v, (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u,⃗v)g + (⃗u, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v)g = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' LT = −L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Definition of the vector product (cross product) Let (⃗ei) be a (·, ·)g-orthonormal basis, let ⃗u,⃗v ∈ ⃗R3, and let ℓ⃗e,⃗u,⃗v ∈ L( ⃗R3, R) be the linear form defined by ℓ⃗e,⃗u,⃗v : � � � ⃗R3 → R ⃗z → ℓ⃗e,⃗u,⃗v(⃗z) := det |⃗e (⃗u,⃗v, ⃗z) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) (the algebraic volume of the parallelepiped limited by ⃗u,⃗v, ⃗z in the Euclidean chosen unit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 The vector product, or cross product, ⃗u ∧e ⃗v of two vectors ⃗u and ⃗v is the (·, ·)g-Riesz representation vector of ℓ⃗e,⃗u,⃗v, that is, ⃗u ∧e ⃗v ∈ ⃗R3 is characterized by ℓ⃗e,⃗u,⃗v(⃗z) = (⃗u ∧e ⃗v, ⃗z)g for all ⃗z ∈ ⃗R3, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∀⃗z ∈ ⃗R3, (⃗u ∧e ⃗v, ⃗z)g = det |⃗e (⃗u,⃗v, ⃗z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) NB: ⃗u ∧e ⃗v depends on (·, ·)g since we need a (·, ·)g-Euclidean basis (⃗ei) (and depends on the orientation of (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have thus defined the bilinear cross product operator ∧e : � ⃗R3 × ⃗R3 → ⃗R3 (⃗u,⃗v) → ∧e(⃗u,⃗v) := ⃗u ∧e ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) (The bilinearity is trivial thanks to the multilinearity of the determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') If one Euclidean basis is imposed by one observer to all the other observers, then ⃗u ∧e ⃗v is written ⃗u ∧ ⃗v (non objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Calculation of the vector product ⃗u = �3 i=1 ui⃗ei, ⃗v = �3 i=1 vi⃗ei and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) give (⃗u ∧e ⃗v,⃗e1)g = det |⃗e (⃗u,⃗v,⃗e1) = det � � u1 v1 1 u2 v2 0 u3 v3 0 � � = det � u2 v2 u3 v3 � = u2v3 − u3v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) Similar calculation for (⃗u ∧e ⃗v,⃗e2)e and (⃗u ∧e ⃗v,⃗e3)e, thus ⃗u ∧e ⃗v = 3 � i=1 (ui+1vi+2 − ui+2vi+1)⃗ei, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u ∧e ⃗v]|⃗e = � � u2v3 − u3v2 u3v1 − u1v3 u1v2 − u2v1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) with the generic notation w4 := w1 and w5 = w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (In particular ⃗ei ∧e ⃗ei+1 = ⃗ei+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 1- ⃗u ∧e ⃗v = −⃗v ∧e ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- ⃗u ∥ ⃗v iff ⃗u ∧e ⃗v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3- If ⃗u and ⃗v are independent then ⃗u ∧e ⃗v is orthogonal to the linear space Vect{⃗u,⃗v} generated by ⃗u and ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4- ⃗u ∧e ⃗v depends on the unit of measurement and on the orientation of (⃗ei): If (·, ·)a and (·, ·)b are two Euclidean dot products, let λ > 0 such that (·, ·)a = λ2(·, ·)b, and then ⃗u ∧a ⃗v = ±λ⃗u ∧b ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) 107 108 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Representation of the spin tensor Ω: vectors, and pseudo-vectors Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- det|⃗e(⃗u,⃗v, ⃗z) = − det|⃗e(⃗v, ⃗u, ⃗z) (since det|⃗e is alternated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- If ⃗u ∥ ⃗v then det|⃗e(⃗u,⃗v, ⃗z) = 0 = (⃗u ∧e ⃗v, ⃗z)e, so ⃗u ∧e ⃗v ⊥g ⃗z, for all ⃗z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And if ⃗u ∧e ⃗v = 0 then (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) gives ⃗u ∥ ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3- If ⃗z ∈ Vect{⃗u,⃗v} then det|⃗e(⃗u,⃗v, ⃗z) = 0 = (⃗u ∧e ⃗v,⃗z)g thus ⃗u ∧e ⃗v ⊥g ⃗z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4- (⃗u ∧a ⃗v, ⃗z)a (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) = det |⃗a (⃗u,⃗v, ⃗z) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) = ±λ3 det |⃗b (⃗u,⃗v, ⃗z) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) = ±λ3(⃗u ∧b ⃗v, ⃗z)b = ±λ3 1 λ2 (⃗u ∧b ⃗v, ⃗z)a, true for all ⃗z, thus (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 Prove that ⃗u ∧e ⃗v is a contravariant vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It is a vector (Riesz representation vector) in ⃗R3, so it is contravariant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Or calculation: It satisfies the contravariance change of basis formula, see (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Antisymmetric endomorphism represented by a vector Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 Let (⃗ei) be a chosen (·, ·)g-Euclidean basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If an endomorphism Ω ∈ L( ⃗R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗R3) is (·, ·)g-antisymmetric then there exists a unique vector ⃗ωe ∈ ⃗R3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all ⃗y, ⃗z ∈ ⃗R3, (Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y, ⃗z)g = det |⃗e (⃗ωe, ⃗y,⃗z), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', there exists a unique vector ⃗ωe ∈ ⃗R3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all ⃗y, ⃗z ∈ ⃗R3, Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y = ⃗ωe ∧e ⃗y , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) And [Ω]|⃗e = � � 0 −c b c 0 −a −b a 0 � � iff [⃗ωe]|⃗e = � � a b c � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) In particular Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ωe = ⃗0 (= ⃗ωe ∧e ⃗ωe), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ωe is an eigenvector associated with the eigenvalue 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ω is antisymmetric, thus [Ω]|⃗e is given as in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular [Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e1]|⃗e = [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗e1]|⃗e = � � 0 c −b � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Calculation of the components of ⃗ωe if it exists: Let ⃗ω = ω1⃗e1 + ω2⃗e2 + ω3⃗e3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' thus [⃗ω ∧ ⃗e1]|⃗e = � � 0 ω3 −ω2 � �, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18), thus ω3 = c and ω2 = b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Idem with ⃗e2 so that ω1 = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus if it exists ⃗ω is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ⃗ωe given in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) satisfies (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21): It exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 Let (·, ·)a and (·, ·)b be two Euclidean dot products (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' in foot and metre), let (⃗ai) and (⃗bi) be Euclidean associated bases, let ||⃗b1||a = λ (change of unit coefficient), so (·, ·)a = λ2(·, ·)b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Suppose [Ω]|⃗a = � � 0 −c b c 0 −a −b a 0 � �, thus [⃗ωa]|⃗a = � � a b c � �, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (change of representation vector for Ω): If (⃗bi) and (⃗ai) have the same orientation, then ⃗ωb = λ⃗ωa, If (⃗bi) and (⃗ai) have opposite orientation, then ⃗ωb = −λ⃗ωa, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', if ⃗bi = λ⃗ai for all i (change of unit, same orientation) then ⃗ωb = λ⃗ωa, and if ⃗b1 = −λ⃗a1, ⃗b2 = λ⃗a2, ⃗b3 = λ⃗a3 (change of unit, opposite orientation) then ⃗ωb = −λ⃗ωa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: The formula ⃗ωb = ±λ⃗ωa is a change of vector formula, not a change of basis formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Apply (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Interpretation of ⃗ωe: Suppose [Ω]|⃗e = α � � 0 −1 0 1 0 0 0 0 0 � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So Ω is the rotation with angle π 2 in the horizontal plane composed with the dilation with ratio α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And [⃗ωe]|⃗e = α � � 0 0 1 � � = α⃗e3 is orthogonal to the horizontal plane and gives the rotation axis and the dilation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 108 109 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Pseudo-cross product, and pseudo-vector Exercice E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15 Let Ω s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [Ω]|⃗e = � � 0 −c b c 0 −a −b a 0 � � (see (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Find a direct orthonormal basis (⃗bi) (relative to (⃗ei)) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [Ω]|⃗b = √ a2+b2+c2 � � 0 −1 0 1 0 0 0 0 0 � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗b3 = ⃗ωe ||⃗ωe||e , that is, [⃗b3]|⃗e = 1 √ a2+b2+c2 � � a b c � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then choose ⃗b1 ⊥ ⃗b3, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗b1]|⃗e = 1 √ a2+b2 � � −b a 0 � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then choose ⃗b2 = ⃗b3 ∧e ⃗b1, that is, [⃗b2]|⃗e = 1 √ a2+b2 1 √ a2+b2+c2 � � −ac −bc a2 + b2 � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (⃗bi) is a direct orthonormal basis, and the transition matrix from (⃗ei) to (⃗bi) is P = � [⃗b1]|⃗e [⃗b2]|⃗e [⃗b3]|⃗e � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And [Ω]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P (change of basis formula), with P −1 = P T (change of orthonormal basis), thus [Ω]|⃗b = P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P With [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗b1]|⃗e = 1 √ b2+c2 � � 0 −c b c 0 −a −b a 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � � −b a 0 � � = 1 √ b2+c2 � � −ac −bc a2 + b2 � � = √ a2+b2+c2[⃗b2]|⃗e (expected), [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗b2]|⃗e = 1 √ b2+c2 1 √ a2+b2+c2 � � 0 −c b c 0 −a −b a 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � � −ac −bc a2 + b2 � � = 1 √ b2+c2 1 √ a2+b2+c2 � � bc2 + b(a2 + b2) −ac2 − a(a2 + b2) abc − abc � � = − √ a2+b2+c2[⃗b1]|⃗e (expected), and [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗b3]|⃗e = [⃗0] (expected since ⃗b3 ∥ ⃗ωe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P = √ a2+b2+c2 � [⃗b2]|⃗e −[⃗b1]|⃗e [⃗0]|⃗e � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P)ij = [⃗bi]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗bj]|⃗e gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Curl Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16 If ⃗v is a C1 vector field, if (⃗ei) is a Euclidean basis in ⃗R3, and if ⃗v = �3 i=1 vi⃗ei, then the curl (or rotational) of ⃗v relative to (⃗ei) is the vector field ⃗ curle⃗v = ⃗ rote⃗v given by ⃗ curle⃗v = 3 � i=1 ( ∂vi+2 ∂xi+1 − ∂vi+1 ∂xi+2 )⃗ei, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗ curle⃗v]|⃗e = � � ∂v3 ∂x2 − ∂v2 ∂x3 ∂v1 ∂x3 − ∂v3 ∂x1 ∂v2 ∂x1 − ∂v1 ∂x2 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17 Let Ω(t, pt) = d⃗v(t,pt)−d⃗v(t,pt)T 2 , and let ⃗ωe(t, pt) be the associated vector relative to the Euclidean basis (⃗ei), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then ⃗ωe = 1 2 ⃗ curle⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) gives [Ω]|⃗e = 1 2 � � 0 ∂v1 ∂x2 − ∂v2 ∂x1 ∂v1 ∂x3 − ∂v3 ∂x1 0 ∂v2 ∂x3 − ∂v3 ∂x2 0 � �, with [Ω]|⃗e antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) gives (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Pseudo-cross product, and pseudo-vector Framework: M31 the space of 3∗1 matrices, so we leave the vector framework to enter the matrix world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18 A column matrice is also called a pseudo-vector, or a column vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19 � � x1 x2 x3 � � noted = [⃗x] and � � y1 y2 y3 � � noted = [⃗y] being two matrices in M31, their pseudo-cross product is � � x1 x2 x3 � � ⟲∧ � � y1 y2 y3 � � := � � x2y3 − x3y2 x3y1 − x1y3 x1y2 − x2y1 � � noted = [⃗x] ⟲∧[⃗y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) Thus the pseudo-cross product of two pseudo-vectors is a pseudo-vector (is a matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 109 110 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Examples E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Antisymmetric matrix represented by a pseudo-vector Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20 Let A = [Aij] = � � 0 −c b c 0 −a −b a 0 � � be an antisymmetric matrix (Aji = −Aij for all i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The pseudo-vecteur ⟲ω associated to A is the column matrix ⟲ω := � � a b c � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗y] = ⟲ω ⟲∧[⃗y] , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � � y1 y2 y3 � � = ⟲ω ⟲∧ � � y1 y2 y3 � � , for all matrix [⃗y] = � � y1 y2 y3 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Antisymmetric endomorphism and its pseudo-vectors representations Let R3 be our usual affine space, (·, ·)g be a Euclidean dot product, and (⃗ei) be a (·, ·)g-Euclidean associated basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Ω be an antisymmetric endomorphism relative to (·, ·)g, so ΩT = −Ω, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus [Ω]|⃗e is an antisymmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Call ⟲ω the associated pseudo-vector, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27), for all ⃗y ∈ ⃗R3, [Ω]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗y]|⃗e = ⟲ω ⟲∧[⃗y]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) This formula is widely used in mechanics, and unfortunately sometimes noted Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗y = ⃗ω ∧ ⃗y (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ): Be careful: (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) is not a vectorial formula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This is just a formula for matrix calculations which gives false result if a change of basis is considered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', with (⃗a1,⃗a2,⃗a3) be a (·, ·)g-Euclidean basis, and (⃗b1,⃗b2,⃗b3) = (−⃗a1,⃗a2,⃗a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So (⃗bi) is also a (·, ·)g-Euclidean basis, but with a different orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- Vector approach: Let P be the transition matrix from (⃗ai) to (⃗bi), so P = � � −1 0 0 0 1 0 0 0 1 � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let [Ω]|⃗a = � � 0 −c b c 0 −a −b a 0 � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, Ω being an endomorphism, the change of basis formula gives [Ω]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [Ω]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P = � � −1 0 0 0 1 0 0 0 1 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � � 0 −c b c 0 −a −b a 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � � −1 0 0 0 1 0 0 0 1 � � = � � 0 c −b −c 0 −a b a 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) Thus the vectors ⃗ωa and ⃗ωb are given by (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22): [⃗ωa]|⃗a = � � a b c � � , [⃗ωb]|⃗b = � � a −b −c � � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � ⃗ωa = a⃗a1 + b⃗a2 + c⃗a3, ⃗ωb = a⃗b1 − b⃗b2 − c⃗b3, � thus ⃗ωb = −⃗ωa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) 2- Matrix approach (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) gives [Ω]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗y] = ⟲ωa ⟲∧[⃗y] and [Ω]|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗y] = ⟲ωb ⟲∧[⃗y], with ⟲ωa = � � a b c � � and ⟲ωb = � � a −b −c � � , so ⟲ωa ̸= − ⟲ωb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) And ⟲ω does not represent a single vector either, since it does not satisfy the vector change of basis formula ⟲ωb ̸= P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⟲ωa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ⟲ω is not a vector (is not tensorial): It is just a matrix (called a “pseudo-vector”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Examples E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Rectilinear motion Let �Φ : [t1, t2] × Obj → Rn be a C1 motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let t0 ∈]t1, t2[ and PObj ∈ Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 110 111 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Examples Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21 The motion of PObj is rectilinear iff, for all t0, t ∈ [t1, t2], �ΦPObj (t) − �ΦPObj (t0) t−t0 ∥ �ΦPObj ′(t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) And the motion is rectilinear uniform iff, for all t0, t ∈ [t1, t2], �ΦPObj (t) = �ΦPObj (t0) + (t−t0) �ΦPObj ′(t0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' p(t) = p(t0) + (t−t0) ⃗V t0(t0, p(t0)) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) when p(t) = �Φ(t, PObj), that is, the trajectory is traveled at constant velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Circular motion Let ( ⃗E1, ⃗E2) be a Euclidean basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let t0 ∈ [t1, t2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A motion Φt0 is a circular motion iff −−−−−→ OΦt0 P (t) = x(t) ⃗E1 + y(t) ⃗E2, [−−−−−→ OΦt0 P (t)]| ⃗E = � x(t) = a + R cos(θ(t)) y(t) = b + R sin(θ(t)) � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) for some R > 0 (called the radius), some a, b ∈ R, and some function θ : R → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And � a b � = OC ∈ R2 is the center of the circle and θ(t) is the angle at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the particle PObj (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' �Φ(t0, PObj) = P) stays on the circle with center OC and radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The circular motion is uniforme iff, for all t, θ′′(t) = 0, that is, ∃ω0 ∈ R, ∀t ∈ [t1, t2], θ(t) = ω0t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Notation: ⃗ϕt0 P (t) = R cos(θ(t) ⃗E1 + R sin(θ(t)) ⃗E2, so [⃗ϕt0 P (t)]| ⃗E = � R cos(θ(t)) R sin(θ(t)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) Thus the Lagrangian velocity of a circular motion is ⃗V t0 P (t) = (Φt0 t )′(t) = (⃗ϕt0 P )′(t), so [⃗V t0 P (t)]| ⃗E = Rθ′(t) � − sin(θ(t)) cos(θ(t)) � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) and ⃗V t0 P (t) is orthogonal to ⃗ϕt0 P (t) (the radius vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the Lagrangian acceleration is ⃗Γ t0 P (t) = Rθ′′(t) � − sin(θ(t)) cos(θ(t)) � + R(θ′(t))2 � − cos(θ(t)) − sin(θ(t)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) Consider ⃗er(t) = ⃗ϕt0 P (t) ||⃗ϕt0 P (t)|| = � cos(θ(t)) sin(θ(t)) � , and ⃗eθ(t) = � − sin(θ(t)) cos(θ(t)) � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38) thus (⃗er(t),⃗eθ(t)) is an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then : ⃗V t0 P (t) = Rθ′(t)⃗eθ(t), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39) and : ⃗Γ t0 P (t) = −R(θ′(t))2 ⃗er(t) + Rθ′′(t)⃗eθ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', in R3 and a motion in he “horizontal” plane given by (⃗e1,⃗e2), the vertical line being given by ⃗E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Here ⃗V t0 P (t) = ⃗ω(t) ∧ ⃗ϕt0 P (t), where ⃗ω(t) = ω(t)⃗e3 and ω(t) = θ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) And ⃗Γ t0(t) = d⃗ω dt (t) ∧ ⃗ϕt0 P (t) + ⃗ω(t) ∧ ⃗V t0 P (t) (= Rdω dt (t)⃗eθ(t) − ω2(t)R⃗er(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='42) 111 112 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Examples E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Motion of a planet (centripetal acceleration) Illustration: Obj is e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' a planet from the solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗e1,⃗e2,⃗e3) be a Euclidean basis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' fixed relative to stars an (⃗e1,⃗e2) define the ecliptic plane), (·, ·)g be the Euclidean associated dot product, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='|| the Euclidean associated norm, O an origin in R3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the center of the Sun), and R = (O, (⃗ei)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider a motion �Φ of Obj in R, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let t0 ∈ [t1, t1], and consider Φt0 =noted Φ or ⃗ϕ t0 =noted ⃗ϕ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22 The motion of a particle PObj is a centripetal acceleration motion iff the particle is not static and, at all time, its acceleration vector ⃗A(t) points to a fixed point F (focus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We will take the focus F as the origin of the referential, that is, O := F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, for all t ∈ [t1, t2], −−−−−→ OΦP (t) ∥ ⃗AP (t), that is, −−−−−→ OΦP (t) ∧ ⃗AP (t) = ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43) Remark E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23 A rectilinear motion is a centripetal acceleration motion, but such a motion is usually excluded in the definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24 The motion of a planet from the solar system is a centripetal acceleration motion: An elliptical motion of focus the center of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25 The second Newton’s law of motion � ⃗f = m⃗γ (Galilean referential) gives: If � ⃗f is, at all time, directed to a unique point F, then the motion is a centripetal acceleration motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Φ be a centripetal acceleration motion, let O be the focus, and let ⃗ϕP (t) := −−−−−→ OΦP (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So the Lagrangian velocity and acceleration are ⃗VP (t) = dΦP dt (t) = d⃗ϕP dt (t), and ⃗AP (t) = d2ΦP dt2 (t) = d2⃗ϕP dt2 (t), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='44) and ⃗ϕP (t) ∧ ⃗AP (t) = ⃗0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26 The areolar velocity at t is the vector ⃗Z(t) = 1 2 ⃗ϕP (t) ∧ ⃗VP (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45) Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27 If Φ is a centripetal acceleration motion, then the areolar velocity is contant, that is, d⃗Z dt (t) = ⃗0 pour tout t, so ⃗Z(t) = ⃗Z(t0), ∀t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='46) That is, the position vectors sweep equal areas in equal times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ⃗Z(t0) = ⃗0 iff Φ is a rectilinear motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If ⃗Z(t0) ̸= ⃗0 then : ⃗ϕP (t) and ⃗VP (t) are orthogonal to ⃗Z(t0) at all time t, The motion of the particle PObj takes place in the affine plane orthogonal to ⃗Z(t0) passing through O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗VP (t) never vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45) and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43) give 2 d⃗Z dt (t) = d⃗ϕP dt (t)∧⃗VP (t)+⃗ϕ(t)∧ d⃗VP dt (t) = ⃗VP (t)∧⃗VP (t)+⃗ϕ(t)∧ ⃗AP (t) = ⃗0+⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ⃗Z is constant, ⃗Z(t) = ⃗Z(t0) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, if ⃗Z(t0) ̸= ⃗0 then ⃗Z(t) ̸= ⃗0 pour tout t, and ⃗Z(t) = 1 2 ⃗ϕP (t) ∧ ⃗VP (t) gives that ⃗ϕP (t) et ⃗VP (t) are orthogonal to ⃗Z(t0) for all t, thus ⃗AP (t) is orthogonal to ⃗Z(t0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Taylor expansion reads ⃗ϕP (t) = ⃗ϕP (t0) + ⃗VP (t0)(t−t0) + � t τ=t0 ⃗AP (τ)(t−τ)2 dτ, with ⃗VP (t0) and ⃗AP (τ) ⊥ ⃗Z(t0) for all τ, thus ⃗ϕP (t) − ⃗ϕP (t0) ⊥ ⃗Z(t0) for all τ, that is −−−→ Op(t) − −−→ OP = −−−→ Pp(t) ⊥ ⃗Z(t0) for all τ, Thus p(t) belongs to the affine plane containing P orthogonal to ⃗Z(t0), for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And −−→ OP = ⃗ϕP (t0) ⊥ ⃗Z(t0), thus O belong to the same plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Z(t) = ⃗Z(t0) ̸= ⃗0 implies ⃗VP (t) ̸= ⃗0 for all t, and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45) gives: (⃗ϕP (t), ⃗VP (t), ⃗Z(t0)) is a positively- oriented basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Since ⃗ϕP and ⃗V are continuous and do not vanish, since ⃗Z(t0) ̸= ⃗0, we get: PObj “turns around ⃗Z(t0)” and keeps its direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 112 113 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Examples If ⃗Z(t) = ⃗0 then ⃗ϕP (t) ∥ ⃗VP (t) for all t, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45), so ⃗VP (t) = f(t)⃗ϕP (t) where f is some scalar function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ⃗VP (t) = ⃗ϕP ′(t) gives ⃗ϕP ′(t) = f(t)⃗ϕP (t), thus ⃗ϕP (t) = ⃗ϕP (t0)eF (t) where F is a primitive of f s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F(t0) = 0, thus ⃗ϕP (t) ∥ ⃗ϕP (t0), so −−−−−→ OΦP (t) ∥ −−−−−−→ OΦP (t0), for all t: The motion is rectilinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Non rectilinear motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') The area swept by ⃗ϕP (t) is, at first order, the area of the triangle whose sides are ⃗ϕP (t) and ⃗ϕP (t + τ) (“anglular sector”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, with τ close to 0, let ⃗St(τ) = 1 2 ⃗ϕP (t) ∧ ⃗ϕP (t + τ), and St(τ) = ||⃗St(τ)||, (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='47) the vectorial an scalar area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With ⃗ϕP (t+τ) = ⃗ϕP (t) + ⃗VP (t)τ + o(τ) we get ⃗St(τ) = 1 2 ⃗ϕP (t) ∧ (⃗VP (t)τ + o(τ)), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='48) Since ⃗St(0) = 0 we get ⃗St(τ)−⃗S(0) τ = 1 2 ⃗ϕP (t) ∧ ⃗VP (t) + o(1), then d⃗St dτ (0) = 1 2 ⃗ϕP (t) ∧ ⃗VP (t) = ⃗Z(t) = ⃗Z(t0), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='49) thanks to (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='46), thus d⃗St dτ (0) = d⃗St0 dτ (0), ∀t ∈ [t0, T], (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='50) that is, the rate of variation of ⃗St is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And with ||⃗St(∆τ)||2 = (⃗St(∆τ), ⃗St(∆τ)) we get d||⃗St||2 dτ (∆τ) = 2(d⃗St dτ (∆τ), ⃗St(∆τ)), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='51) so, since ⃗St(0) = 0, d||⃗St||2 dτ (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='52) Therefore the function t → ||⃗St(0)||2 = St(0)2 is constant, thus t → St(0) est constant, and dSt dτ (0) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28 Give a parametrization of the swept area, and redo the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let r(t) = ||⃗ϕP (t)||, θ(t) = � p(t)OP (angle), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='53) then ⃗ϕP (t) = � � r(t) cos(θ(t)) r(t) sin(θ(t)) 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54) Thus ⃗VP (t) = � � r′(t) cos(θ(t) − r(t))θ′(t) sin(θ(t)) r′(t) sin(θ(t) + r(t))θ′(t) cos(θ(t)) 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='55) With (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45) we get ⃗Z(t) = 1 2 � � 0 0 r2(t)θ′(t) � � , with r2(t)θ′(t) = r2(t0)θ′(t0) (constant), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='56) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A parametrization of the swept area is then ⃗A : � [0, 1] × [t0, T] → R3 (ρ, t) → ⃗A(ρ, t) � , ⃗A(ρ, t) = � � ρ r(t) cos(θ(t)) ρ r(t) sin(θ(t)) 0 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='57) Therefore, the tangent associated vectors are ∂ ⃗A ∂ρ (ρ, t) = � � r(t) cos(θ(t)) r(t) sin(θ(t)) 0 � � , ∂ ⃗A ∂t (ρ, t) = � � ρr′(t) cos(θ(t) − ρr(t))θ′(t) sin(θ(t)) ρr′(t) sin(θ(t) + ρr(t))θ′(t) cos(θ(t)) 0 � � , (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='58) 113 114 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Examples hence the vectorial and scalare element areas are d⃗σ = (∂ ⃗A ∂ρ ∧ ∂ ⃗A ∂t )dρdt = � � 0 0 ρr2θ′ dρdt � � , dσ = ρr2θ′ dρdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='59) Therefore the area between t0 and t is A(t) = A(t0) + � 1 ρ=0 � t τ=t0 ρr2(τ)θ′(τ) dρdτ = 1 2 � t τ=t0 r(τ)2θ′(τ) dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='60) Hence A′(t) = r(t)2θ′(t) = r(t0)2θ′(t0) (= constant = ||⃗Z(t0)||), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='61) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29 Prove the Binet formulas (non rectilinear central motion): VP (t)2 = Z2 0 � 1 r2 + (d 1 r dθ )2� (t), ⃗ΓP (t) = −Z2 0 r2 �1 r + d2 1 r dθ2 � (t)⃗er(t), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='62) for the energy and the acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27 tells that Φ is a planar motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='53) and ⃗er(t) = � cos(θ(t)) sin(θ(t)) � we have ⃗ϕ(t) = r(t)⃗er(t) (in the plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗eθ(t) = � − sin(θ(t)) cos(θ(t)) � , thus ⃗V (t) = dr dt (t)⃗er(t) + r(t)d⃗er dt (t) = r′(t)⃗er(t) + r(t)θ′(t)⃗eθ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ⃗er(t) ⊥ ⃗eθ(t) gives V 2(t) = (r′(t))2 + (r(t)θ′(t))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Since θ′(t) ̸= 0 for all t (non rectilinear central motion) Let s(θ(t)) = r(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let us suppose that θ is C1, thus θ′ > 0 or θ′ < 0, and θ : t → θ(t) defines a change of variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And r′(t) = s′(θ(t))θ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='61) and θ′(t) = Z0 r2(t) give V 2(t(θ)) = (s′(θ))2 Z2 0 r4(t) + r2(t) Z2 0 r4(t) = Z2 0((s′(θ))2 s4(θ) + 1 s2(θ)) = Z2 0[ �d 1 s dθ (θ) �2 + 1 s2(θ)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus r(t) = s(θ) and dr dθ := ds dθ give the first Binet formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then ⃗Γ(t) = r′′(t)⃗er(t) + r′(t)d⃗er dt (t) + (r′(t)θ′(t) + r(t)θ′′(t))⃗eθ(t) + r(t)θ′(t)d⃗eθ dt (t), with d⃗er dt ∥ ⃗eθ, and d⃗eθ dt (t) = −θ′(t)⃗er(t), and ⃗eθ ⊥ ⃗Γ (central motion), we get ⃗Γ(t) = (r′′(t) − r(t)(θ′(t))2)⃗er(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And r′(t) = s′(θ)θ′(t) = s′(θ) Z0 r2(t) = Z0 s′(θ) s2(θ) = −Z0 d 1 s dθ (θ), thus r′′(t) = −Z0 d2 1 s dθ2 (θ) θ′(t) = − Z2 0 r2(t) d2 1 s dθ2 (θ), which is the second Binet formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 114 115 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Riesz representation theorem F Riesz representation theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The Riesz representation theorem Framework: (E, (·, ·)g) is Hilbert space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E is a vector space equipped with an inner dot product (·, ·)g such that, with the associated norm defined by||⃗v||g := � (⃗v,⃗v)g, (E, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||g) is a complete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And E∗ = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is the space of the linear and continuous forms on E (the space of linear “measuring tools”) equipped with its norm ||ℓ||E∗ := sup ||⃗x||g=1 |ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have the easy statement: ∀⃗v ∈ E (vector), ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='vg ∈ E∗ (linear continuous form) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = (⃗v, ⃗x)g, ∀⃗x ∈ E, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) and ||vg||E∗ = ||⃗v||g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Usual notation in finite dimension: vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = ⃗v •g ⃗x, or simply v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = ⃗v • ⃗x if a chosen (·, ·)g is imposed to all observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Indeed: Define vg : E → R by vg(⃗x) = (⃗v, ⃗x)g for all ⃗x ∈ E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The definition domain of vg is E and vg is trivially linear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the Cauchy–Schwarz inequality gives |vg(⃗x)| = |(⃗v, ⃗x)g| ≤ ||⃗v||g ||⃗x||g for all ⃗x ∈ E, thus ||vg||E∗ ≤ ||⃗v||g < ∞, thus vg is continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And |vg(⃗v)| = |(⃗v,⃗v)g| = ||⃗v||g ||⃗v||g, thus ||vg||E∗ ≥ ||⃗v||g, thus ||vg||E∗ = ||⃗v||g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Riesz representation theorem concerns the converse: If you choose an inner dot product (·, ·)g in E (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' English of French), then you can represent a “measuring instrument” ℓ ∈ E∗ by a vector ⃗ℓg ∈ E: Theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 (Riesz representation theorem, and definition) (E, (·, ·)g) being a Hilbert space, ∀ℓ ∈ E∗ (linear continuous form), ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ℓg ∈ E (vector) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = (⃗ℓg, ⃗x)g, ∀⃗x ∈ E, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) and ||⃗ℓg||g = ||ℓ||E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ⃗ℓg is called the (·, ·)g-Riesz representation vector of ℓ (depends on g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Usual notation in finite dimension: vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = ⃗v •g ⃗x, or simply v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = ⃗v • ⃗x if a chosen (·, ·)g is imposed to all observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Easy in finite dimension: With a basis (⃗ei), if [ℓ]|⃗e = ( ℓ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ℓn ) (row matrix since ℓ is a linear form) then (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) gives [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗e = [⃗ℓg]T ⃗e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗e, thus [⃗ℓg]⃗e = [g]−1 |⃗e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ℓ]T |⃗e (column matrix), thus ⃗ℓg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' General case (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' with E = L2(Ω) and the finite element method): If ℓ = 0 then ⃗ℓg = ⃗0 (trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Suppose ℓ ̸= 0: Thus Kerℓ = ℓ−1({0}) ̸= {⃗0} (the kernel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If dim E = 1, it is trivial (exercise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Suppose dim E ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Since ℓ is continuous, its kernel Kerℓ = ℓ−1({0}) is closed in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, if ⃗x ∈ E, then its (·, ·)g- orthogonal projection ⃗x0 ∈ Kerℓ on Kerℓ exists, is unique, and is given by: ∀⃗y0 ∈ Kerℓ, (⃗x −⃗x0, ⃗y0)g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (So ⃗x − ⃗x0 ⊥g Kerℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Choose a ⃗x /∈ Kerℓ (possible since ℓ ̸= 0), and let ⃗n := ⃗x−⃗x0 ||⃗x−⃗x0||g ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So ⃗n is a (·, ·)g- orthonormal vector to Kerℓ, and (Kerℓ)⊥ = Vect{⃗n} since dim(Kerℓ)⊥ = 1 (in finite dimension cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the Dimension Formula which states that the dimension of the domain of a linear map is the sum of the dimension of its range and the dimension of its kernel, and in infinite dimension see next exercise F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And E = Kerℓ ⊕ (Kerℓ)⊥ since both vector spaces are closed (an orthogonal is always closed in a Hilbert space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus if ⃗x ∈ E then ⃗x = ⃗x0 + λ⃗n ∈ Kerℓ ⊕ (Kerℓ)⊥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (⃗x,⃗n)g = λ and ℓ(⃗x) = 0 + λℓ(⃗n) = (⃗x,⃗n)gℓ(⃗n) = (⃗x, ℓ(⃗n)⃗n)g (bilinearity of (·, ·)g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ⃗ℓg := ℓ(⃗n)⃗n satisfies (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And if ⃗ℓg1 and ⃗ℓg2 satisfy (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) then (⃗ℓg1 − ⃗ℓg2, ⃗x)g = 0 for all ⃗x ∈ E, thus ⃗ℓg1 − ⃗ℓg2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ⃗ℓg is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the Cauchy–Schwarz theorem give ||ℓ||E∗ := sup||⃗x||g=1 |ℓ(⃗x)| = sup||⃗x||g=1 |(⃗ℓg, ⃗x)g| = ||⃗ℓg||g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rg is an isomorphism between Banach spaces: linearity since (⃗Rg(ℓ + λm), ⃗x)g = (ℓ + λm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x + λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = (⃗Rg(ℓ), ⃗x)g +λ(⃗Rg(m), ⃗x)g = (⃗Rg(ℓ)+λ⃗Rg(m), ⃗x)g for all ⃗x gives ⃗Rg(ℓ+λm) = ⃗Rg(ℓ)+λ⃗Rg(m), bijectivity thanks to (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2), and the norm is kept since ||⃗ℓg||g = ||ℓ||E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Prove: If ℓ ∈ E∗−{0} then dim(Kerℓ)⊥ = 1 (= dim(Im(ℓ)) = dim R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider the restriction ℓ|Kerℓ⊥ : � (Kerℓ)⊥ → R ⃗x → ℓ|Kerℓ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It is linear (since ℓ is), and thus one to one: Indeed it is onto since ℓ ̸= 0, and it is one to one since if ℓ|Kerℓ⊥(⃗x) = 0 = ℓ(⃗x) then ⃗x ∈ (Kerℓ)⊥ � Kerℓ = {⃗0}, thus ⃗x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus dim(Kerℓ)⊥ ≤ dim(Im(ℓ)) = 1: Indeed, if ⃗z1, ⃗z2 ∈ (Kerℓ)⊥−{⃗0} then ℓ|Kerℓ⊥(⃗z1) ∈ R and ℓ|Kerℓ⊥(⃗z2) ∈ R, thus ∃λ ∈ R s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ℓ|Kerℓ⊥(⃗z2) = λℓ|Kerℓ⊥(⃗z1), thus ℓ|Kerℓ⊥(⃗z2 − λ⃗z1) = 0, thus ⃗z2 − λ⃗z1 = ⃗0 since ℓ|Kerℓ⊥ is one to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ⃗n ∈ (Kerℓ)⊥ gives dim(Kerℓ)⊥ ≥ 1 (above proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus dim(Kerℓ)⊥ = 1 = Vect{⃗n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 115 116 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Riesz representation operator F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The Riesz representation operator The Riesz representation theorem F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 gives the (·, ·)g-Riesz representation operator ⃗Rg : � E∗ → E ℓ → ⃗Rg(ℓ) := ⃗ℓg, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗Rg(ℓ),⃗v)g = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v, ∀⃗v ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) So ⃗Rg transforms a « covariant ℓ » into a « contravariant ⃗ℓg » thanks to the tool (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB (fundamental): ⃗Rg is a isomorphism between the Banach spaces (E, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||g) and (E∗, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||E∗), but ⃗Rg is not canonical since it requires a man made tool (an inner dot product chosen by some observer) to be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (An isomorphism E ↔ E∗ can never be canonical, see § T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') And with G the set of inner dot products in E, we have thus defined the Riesz representation mapping ⃗R : � G × E∗ → E (g, ℓ) → ⃗R(g, ℓ) := ⃗ℓg = ⃗Rg(ℓ) = ⃗ℓ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) So ⃗R has two inputs: A choice (·, ·)g by an observer for the first slot, a linear form for the second slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Quantification with a basis Here E is finite dimensional, dim E = n, ℓ ∈ E∗ (a linear form), (·, ·)g is an inner dot product, (⃗ei) is a basis, (πei) is the dual basis (classical notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let gij = g(⃗ei,⃗ej), ℓ = n � i=1 ℓiπei, ⃗ℓg = n � i=1 (⃗ℓg)i⃗ei, ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='πej = n � i=1 Rij⃗ei, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|⃗e = [gij], [ℓ]|πe = ( ℓ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ℓn ) (row matrix), [⃗ℓg]|⃗e = � � � (⃗ℓg)1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗ℓg)n � � � (column matrix), [⃗Rg]πe,⃗e = [Rij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Duality notations: ℓ = �n i=1ℓiei, ⃗ℓg = �n i=1ℓi g⃗ei, ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ej = �n i=1Rij⃗ei, [⃗Rg]e,⃗e = [Rij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 [⃗ℓg] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ℓ]T and [⃗Rg] = [g]−1 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗ℓg)i = n � j=1 ([g]−1)ij(ℓ)j = n � j=1 (⃗Rg)ij(ℓ)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) Full matrix notation: [⃗ℓg]|⃗e = ([g]|⃗e)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ([ℓ]|πe)T , and [⃗Rg]|πe,⃗e = ([g]|⃗e)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notation to see the change of variance induced by (·, ·)g (bottom index for ℓ, top index for ⃗ℓg): ℓi g = n � j=1 Rijℓj, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ℓi g = �n j=1gijℓj when ([g]−1)ij =noted [gij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (In particular, if (⃗ei) is a (·, ·)g-orthonormal basis, then [⃗Rg] = [g]−1 = I and ℓi g = ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) gives [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗e = [⃗ℓg]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗e for all ⃗x, thus [ℓ]|⃗e = [⃗ℓg]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|⃗e, thus [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ℓg]|⃗e = [ℓ]T |⃗e (since [g]|⃗e = [g]T |⃗e), thus [⃗ℓg] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ℓ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ =(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) ⃗ℓg gives �n j=1(ℓ)j ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='πj = �n i=1(⃗ℓg)i⃗ei, thus �n i,j=1(ℓ)jRij⃗ei = �n i=1(⃗ℓg)i⃗ei, thus �n j=1Rij(ℓ)j = (⃗ℓg)i for all i, thus [⃗Rg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ℓ]T = [⃗ℓg].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus [⃗Rg] = [g]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 If a chosen inner dot product (·, ·)g is imposed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Euclidean foot based) and if duality notations are used, then a usual notation for ⃗ℓg is ℓ♯, since ⃗ℓg = ⃗Rg(ℓ) = �n i=1ℓi⃗ei with a top index for ℓi: the index i has been raised through ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) read (isometric framework) ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = ℓ♯ • ⃗x and [ℓ♯]|⃗e = [g]−1 |⃗e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ℓ]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) We won’t use this notation (we deal with objectivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 116 117 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Change of Riesz representation vector, and Euclidean case F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Change of Riesz representation vector, and Euclidean case For one linear form ℓ ∈ E∗, two observers with their inner dot products (·, ·)g and (·, ·)h get two Riesz representation vectors ⃗ℓg = ⃗Rg(ℓ) and ⃗ℓh = ⃗Rh(ℓ) given by, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2): ∀⃗x ∈ E, (⃗ℓg, ⃗x)g = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = (⃗ℓh, ⃗x)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 For any basis (⃗ei) in E, we have the change of representation vector formula: [h]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ℓh]|⃗e = [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ℓg]|⃗e, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ℓh]|⃗e = [h]−1 |⃗e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ℓg]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) In particular (for the Euclidean case), with λ > 0: If (·, ·)g = λ2(·, ·)h then ⃗ℓh = λ2⃗ℓg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) Conversely, if ⃗ℓh = λ2⃗ℓg for all linear forms ℓ ∈ E∗, then (·, ·)g = λ2(·, ·)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, a linear form ℓ cannot be identified with a Riesz representation vector (which one: ⃗ℓg?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ℓh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In other words, a Riesz representation vector ⃗Rg(ℓ) is not objective, is not intrinsic to a linear form ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10)-(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) is a “change of vector formula” (one linear form gives two vectors relative to two inner dot products);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It is not a “change of basis formula” (for one vector and its two sets of components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) gives [⃗x]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ℓg]|⃗e = [⃗x]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[h]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ℓh]|⃗e for all ⃗x, hence [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ℓg]|⃗e = [h]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ℓh]|⃗e, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular λ2(·, ·)h = (·, ·)g give λ2(⃗ℓg, ⃗x)h = (⃗ℓg, ⃗x)g =(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9)(⃗ℓh, ⃗x)h for all ⃗x, hence λ2⃗ℓg = ⃗ℓh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Converse: λ2⃗ℓg = ⃗ℓh for all ℓ gives λ2(⃗ℓg, ⃗x)h = (⃗ℓh, ⃗x)h (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) = (⃗ℓg, ⃗x)g, for all ⃗x and for all ℓ, thus for all ⃗ℓg thanks to the isomorphism ⃗Rg : E∗ → E, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3), thus λ2(·, ·)h = (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 If (·, ·)g and (·, ·)h are the Euclidean dot products made with the foot and the metre then, with (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9), (·, ·)g = λ2(·, ·)h =⇒ ⃗ℓh = λ2⃗ℓg, with λ2 > 10 : (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) ⃗ℓg (English) and ⃗ℓh (French) are quite different!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A Riesz representation vector is subjective, and certainly not “canonical” (a word that you may find in books where.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' nothing is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', aviation: If you do want to use a Riesz representation vector to represent a ℓ ∈ Rn∗, it is vital to know which Euclidean dot product is in use, see also remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 (Mars Climate Orbiter Crash).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Recall: The foot is the international unit of altitude for aviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 If f ∈ C1(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) and p ∈ Rn, the differential of f at p is the linear form df(p) ∈ Rn∗ defined by, for all ⃗w ∈ ⃗Rn, df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w := lim h→0 f(p + h⃗w) − f(p) h (definition independent of any inner dot product), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) see (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If you can choose an inner dot product (·, ·)g then the gradient ⃗ gradgf(p) is the (·, ·)g-Riesz representation vector of df(p): ⃗ gradgf(p) := ⃗Rg(df(p)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = ⃗ gradgf(p) •g ⃗w, ∀⃗w ∈ ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) And (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) gives ⃗ gradhf(p) = λ2 ⃗ gradgf(p) with λ2 > 10 (English vs French) : (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) The gradient is very dependent on the observer (the gradient is subjective, the differential is objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 The “gradient” is observer dependent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We already had this observer dependence for the usual derivative in the 1-D case f : x ∈ R → f(x) ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Question: What does f ′(x) = 3 mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 11- For one observer, it means f ′(x) = limh→0 f(x+h)−f(x) h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' but.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' where in the departure space this observer has chosen a basis vector ⃗a of length 1 for him (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' length 1 foot) which he calls 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, with no abusive notations, his derivative f ′(x) is in fact f ′ a(x) = limh→0 f(x+h⃗a)−f(x) h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 117 118 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A Riesz representation vector is contravariant 12- For some other observer, it means f ′(x) = limh→0 f(x+h)−f(x) h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' but.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' where in the departure space this observer has chosen a basis vector ⃗b of length 1 for him (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' length 1 metre) which he calls 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, with no abusive notations, his derivative f ′(x) is in fact f ′ b(x) = limh→0 f(x+h⃗b)−f(x) h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 13- Both observer use the same formula f ′(x) = limh→0 f(x+h)−f(x) h but get different results!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In- deed, if ⃗b = λ⃗a, then = lim h→0 f(x + h⃗b) − f(x) h = lim h→0 f(x + hλ⃗a) − f(x) h = λ lim h→0 f(x + (hλ)⃗a) − f(x) (hλ) = λ lim k→0 f(x + k⃗a) − f(x) k thus f ′ b(x) = λf ′ a(x), with λ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) with foot and metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quite different results!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (In fact f ′(x) = opposite side adjacent side depends on the length of the adjacent side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Remark F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 We insist on the subjectivity of the gradient: 20- The differential of f at a point x along a vector ⃗w ∈ ⃗R is df(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = limh→0 f(x+h⃗w)−f(x) h and is objective: The observers all use this same formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 21- An observer chooses a Euclidean dot product (·, ·)g (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' based on the foot), then represent df(x) by its (·, ·)g-Riesz representation vector ⃗Rg(df(x)) =noted ⃗ gradgf(x) called the gradient of f at x relative to (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 22- Another observer chooses a Euclidean dot product (·, ·)h (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' based on the metre), then represent df(x) by its (·, ·)h-Riesz representation vector ⃗Rh(df(x)) =noted ⃗ gradhf(x) called the gradient of f at x relative to (·, ·)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 23- Both observer use the same formula df(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = limh→0 f(x+h⃗w)−f(x) h to get a different result: ⃗ gradhf = λ2 ⃗ gradgf, because they use different measuring tools (one based on the foot, the other on the metre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 24- Recall: The gradient depends on a choice of a Euclidean unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 In (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) we have f ′ b(x) = λf ′ a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the 1-D gradient gives gradbf(x) = λ2gradaf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' To define a gradient gradaf we need a Euclidean dots products (·, ·)a built from a basis (⃗a) in ⃗R, while to define f ′ a we need a unit of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Details: (⃗a) and (⃗b) are two bases in ⃗R with ⃗b = λ⃗a, thus (·, ·)a = λ2(·, ·)b (since 1 = (⃗a,⃗a)a = (⃗b,⃗b)b = (λ⃗a, λ⃗a)b = λ2(⃗a,⃗a)b gives (⃗a,⃗a)a = λ2(⃗a,⃗a)b and (⃗a) is a basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And we have f ′ b(x) =(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) λf ′ a(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' df(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗b = λdf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a, thus (gradfb(x),⃗b)b = λ(gradfa(x),⃗a)a, thus (gradfb(x),⃗b)b = λλ2(gradfa(x), ⃗b λ)b = (λ2gradfa(x),⃗b)b, thus gradfb(x) = λ2gradfa(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 1- Prove that (·, ·)g = λ2(·, ·)h gives ||⃗ℓh||g = λ||⃗ℓh||h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- Does it contradict the Riesz representation theorem which gives ||ℓ||Rn∗ = ||⃗ℓg||Rn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- ⃗ℓh =(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) λ2⃗ℓg gives ||⃗ℓh||h = λ2||⃗ℓg||h = λ||⃗ℓg||g since ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||h = λ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- No, since ||ℓ||Rn∗ := sup||⃗x||Rn =1 |ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x| depends on the norm ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||Rn chosen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Here ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||Rn is either ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||g or ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus if you write ||ℓ||Rn∗ =noted ||ℓ||g∗ if you use the norme ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||g, then ||ℓ||h∗ = sup⃗v∈⃗Rn |ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v| ||⃗v||h = sup⃗v∈⃗Rn |ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v| 1 λ ||⃗v||g = λ sup⃗v∈⃗Rn |ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v| ||⃗v||g = λ||ℓ||g∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 A Riesz representation vector is contravariant ⃗ℓg is a vector in E, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2), so it is contravariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' To be convinced: Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 Check: [⃗ℓg]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ℓg]|old (contravariance formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider two bases (⃗eold,i) and (⃗enew,i) in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With the change of basis formulas [⃗x]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|old and [g]|new = P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) gives (with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='94)), for all ⃗x, [⃗x]T |old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ℓg]|old = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x = [⃗x]T |new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ℓg]|new = ([⃗x]T |old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P −T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ℓg]|new = [⃗x]T |old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ℓg]|new), (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) thus [⃗ℓg]|old = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[⃗ℓg]|new since [g] is invertible (an inner dot product is positive definite), thus (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 118 119 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' What is a vector versus a (·, ·)g-vector?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 • Dont forget: A representation vector ⃗ℓg is not intrinsic to the linear form ℓ because it depends on a (·, ·)g (depends on a observer: foot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' metre?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=')).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Reminder: there is no natural canonical isomorphism between E and E∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' it is impossible to identify a linear form with a vector, see § T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ℓg is incompatible with the use of push-forwards, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' § 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ℓg is incompatible with the use of Lie derivatives, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 What is a vector versus a (·, ·)g-vector?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- Originally, a vector was a bipoint vector ⃗v = −−→ AB in ⃗R3 used to represent of a “material object”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the height of a child is represented on a wall by a vertical bipoint vector ⃗x starting from the ground up to a pencil line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The vector ⃗x is objective: Any observer uses this same vector to get the height of the child.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' and then use “their subjective unit” (foot, metre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') to give a value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- Then (mid 19th century), the concept of vector space was introduced: It is a quadruplet (E, +, K, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') where + is an inner law, (E, +) is a group, K is a field, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' is a external law on E (called a scalar multiplication) compatible with + (see any math book).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And then the concept of scalar inner dot product (in a vector space) was introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3- We can then get non “material” vectors (“subjectively built vectors”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' : start with our usual vector space ⃗Rn of bi-point vectors, then consider its dual (Rn∗, +, R, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') =noted Rn∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, for a given ℓ ∈ Rn∗ (a given measuring device), consider two observers: An English observer with his foot built Euclidean dot product (·, ·)g, and a French observer with with his metre built Euclidean dot product (·, ·)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' These observers build their own artificial Riesz representation vectors ⃗ℓg = ⃗Rg(ℓ) ∈ ⃗Rn and ⃗ℓh = ⃗Rh(ℓ), cf (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' They remark that ⃗ℓg ̸= ⃗ℓh: These constructions are very subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4- Then, with differential geometry, a vector ⃗v has been redefined: It is a “tangent vector”, which means that there exists a C1 curve c : s ∈ [a, b] → c(s) ∈ E such that ⃗v is defined at a p = c(s) ∈ Im(c) by ⃗v(p) := ⃗c ′(s) (so a vector is part of a vector field, here defined along the range of c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (This definition of a tangent vector is applicable to “tangent vectors to a surface” i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' tangent vectors to a manifold, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' § 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1,2-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Then it is shown that ⃗v is equivalent to ∂ ∂⃗v = the directional derivative in the direction ⃗v (natural canonical isomorphism between E and E∗∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' For other equivalent definitions of vectors, see Abraham–Marsden [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 The “(·, ·)g-dual vectorial bases” of one basis (and warnings) Framework: E is a finite dimensional vector space, dim E = n (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E = ⃗R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' An observer chooses an inner dot product (·, ·)g (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', in ⃗R3, a foot-built Euclidean dot product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence the results will be subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (⃗ei) is some basis in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 A basis and its many associated “dual vectorial basis” Definition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 The (·, ·)g-dual vectorial basis (or (·, ·)g-vectorial dual basis, or (·, ·)g-dual basis) of the basis (⃗ei) is the basis (⃗eig) in E defined by ∀j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, (⃗eig,⃗ej)g = δij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗eig •g ⃗ej = δij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) NB: A vectorial dual basis is not unique: It depends on the chosen inner dot product, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: Pay attention to the notations: ⃗eig is a contravariant vector (⃗eig ∈ E), so, even if you use the Einstein convention, the index i in ⃗eig must be a bottom index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (πei) be the (covariant) dual basis of the basis (⃗ei), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the πei ∈ E∗ are the objective (the same for all observers) linear forms defined by πei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = δij for all j, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15 (Equivalent definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') The (·, ·)g-dual vectorial basis of the basis (⃗ei) is the basis (⃗eig) in E made of the (·, ·)g-Riesz representative vectors of the πei, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗eig := ⃗Rg(πei) , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗eig •g ⃗v = πei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v, ∀⃗v ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) where ⃗Rg is the (·, ·)g-Riesz operator, see (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With duality notations, (ei) is the dual basis and ⃗eig := ⃗Rg(ei), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗eig,⃗v)g = ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v for all ⃗v ∈ E where here the position of the index i is bottom on the left and up on the right, since ⃗Rg changes a covariant vector (a linear form) into a contravariant vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 119 120 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The “(·, ·)g-dual vectorial bases” of one basis (and warnings) Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16 Prove that the vectors ⃗eig satisfy the contravariant change of basis formula [⃗eig]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗eig]|old (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) (the ⃗ejg are indeed “contravariant vectors”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' • First answer: ⃗eig is a vector in E, thus it is contravariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Second answer: Apply (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) since ⃗eig is a Riesz-representation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Third answer = direct computation: Consider two bases (⃗ai) and (⃗bi), and the transition matrix P from (⃗ai) to (⃗bi), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗bj = �n i=1Pij⃗ai for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) and the change of basis formulas for the vectors ⃗ei and the bilinear form (·, ·)g give [⃗ej]T |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗eig]|⃗a = (⃗eig,⃗ej)g = [⃗ej]T |⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗eig]|⃗b = (P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ej]|⃗a)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗eig]|⃗a = [⃗ej]T |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗eig]|⃗a for all i, j, thus [⃗eig]|⃗a = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[⃗eig]|⃗b, thus (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17 Consider two inner dot products (·, ·)a and (·, ·)b (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', a foot-built and a metre-built Euclidean dot product), and a basis (⃗ei) in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Call (⃗eia) and (⃗eib) the (·, ·)a and (·, ·)b-dual vectorial bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove: (·, ·)a = λ2(·, ·)b =⇒ ⃗eib = λ2⃗eia, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', λ2 > 10 with foot and metre built Euclidean bases: ⃗eib is very different from ⃗eia !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A dual vectorial basis highly depends on an observer: A vectorial dual basis is not intrinsic to (⃗ei) (not objective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) gives (⃗eib,⃗ej)b = δij = (⃗eia,⃗ej)a = λ2(⃗eia,⃗ej)b, thus (⃗eib − λ2⃗eia,⃗ej)b = δij, for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18 If (⃗ei) is a (·, ·)g-orthonormal basis we trivially get ⃗eig = ⃗ei for all i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', (⃗eig) = (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='This particular case is not compatible with joint work by an English (foot) and French (metre) observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Components of ⃗ejg in the basis (⃗ei) Proposition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19 The components of ⃗ejg in the basis (⃗ei) are given by, for any j ∈ [1, n]N, [⃗ejg]|⃗e = [⃗Rg]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ej]|⃗e = ([g]|⃗e)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ej]|⃗e = the j-th column of ([g]|⃗e)−1, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the i-th component of ⃗ejg is ([g]−1 |⃗e )ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the matrix of g(·, ·) in the basis (⃗eig) is the inverse of the matrix of g(·, ·) in the basis (⃗ei): ([g(⃗eig,⃗ejg)] =) [g]|(⃗eig) = [g]|(⃗ei) −1 (= ([g(⃗ei,⃗ej)])−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' First proof of (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) (straight forward calculation): (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) gives, for all i, j, [⃗ej]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗eig]|⃗e = δij = [⃗ej]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ei]|⃗e, thus [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗eig]|⃗e = [⃗ei]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) Second proof of (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23): Apply (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) (generic Riesz representation result) to get (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, [g]|⃗e being symmetric we have [g]|⃗e−1 symmetric, and g(⃗eig,⃗ejg) = [⃗eig]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ejg]|⃗e = [⃗ei]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ej]|⃗e = [⃗ei]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ej]|⃗e = ([g]|⃗e−1)ij, thus (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20 ⃗R2, [g]|⃗e = � 1 0 0 2 � , thus [g]−1 |⃗e = � 1 0 0 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ⃗e1g = ⃗e1, ⃗e2g = 1 2⃗e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21 Warning: When ([g]−1 |⃗e )ij =noted gij then (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) reads ⃗ejg = n � i=1 gij⃗ei, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) where the Einstein convention is not satisfied: The Einstein convention is satisfied with ⃗ejg = n � i=1 (⃗ejg)i⃗ei noted = n � i=1 (Pj)i⃗ei (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) (the components of vectors have up indices), and this can be verified with (⃗eig,⃗ej)g =(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) δij which gives �n k,ℓ=1(Pi)kgkj = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And in (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) the scalars gij is just another name for (Pj)i, nothing more (nothing to do with the Einstein convention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 120 121 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The “(·, ·)g-dual vectorial bases” of one basis (and warnings) We insist:In other words: M = [g]|⃗e = [Mij] is a matrix, and its inverse is the matrix M −1 = [Mij]−1: A matrix is just a collection of scalars (has nothing to do with the Einstein convention), and its inverse is also a collection of scalars, and you do not change this fact by calling M −1 =noted [M ij] (the use of up indices is irrelevant for matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And because (Pj)i equals ([g]−1 |⃗e )ij =noted gij, some people rename ⃗ejg as ⃗e j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' to get ⃗e j = �n i=1gij⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But doing so they despise Einstein’s convention, despite eventual claims: They confuse covariance and contravariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' and add confusion to the confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: Recall: If in trouble with a notation which comes as a surprise (the notation gij here), use classical notations: Then no misuse of Einstein’s convention and no possible misinterpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular here ⃗ejg is a (contravariant) vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Multiple admissible notations for the components of ⃗ejg Let P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) be the change of basis endomorphism from (⃗ei) to (⃗eig): defined by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = ⃗ejg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And let P = [P]|⃗e (the associated transition matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It gives multiple admissible (non confusing) notations for the components of ⃗ejg relative to the basis (⃗ei): ⃗ejg = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = n � j=1 Pij⃗ei = n � j=1 (Pj)i⃗ei � �� � clas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = n � j=1 (Pj)i⃗ei = n � j=1 P i j⃗ei � �� � dual , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the i-th component of the vector ⃗ejg has the names Pij = (Pj)i = (Pj)i = P ij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' P = [P]|⃗e = [Pij] = [(Pj)i] = [(Pj)i] = [P ij] (four different notations for the same matrix), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∀j, [⃗ejg]|⃗e = [P]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗ej]|⃗e = � � � P1j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Pnj � � � = � � � (Pj)1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Pj)n � � � = � � � P 1j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' P nj � � � = � � � (Pj)1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Pj)n � � � (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) = the j-th column of [P]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' You can choose any notation, depending on your current need or mood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 (Huge) differences between “the (covariant) dual basis” and “a dual vectorial basis” 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A basis (⃗ei) has an infinite number of vectorial dual bases (⃗eig), as many as the number of inner dot products (·, ·)g (as many as observers), see (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' While a basis (⃗ei) has a unique intrinsic (covariant) dual basis (πei) noted = (ei), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7): Two observers who consider the same basis (⃗ei) have the same (covariant) dual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' πei = ei is covariant, while ⃗ei and ⃗eig are contravariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And there is no transition matrix between (⃗ei) and (πei) = (ei), since ⃗ei ∈ E and πei = ei ∈ E∗ don’t live in the same vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If you fly, it is vital to use the dual basis (πei) = (ei): It is possibly fatal if you confuse foot and metre at takeoff and at landing (if you survived takeoff) because of the choice of different Euclidean dot product (·, ·)g or (·, ·)h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the Mars Climate Orbiter crash, remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Einstein’s convention can help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' only if it is really followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 About the notation gij = shorthand notation for (g♯)ij Definition F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22 The Riesz associated inner dot product g♯ ∈ L(E∗, E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is the bilinear form defined by, for all ℓ, m ∈ E∗, g♯(ℓ, m) := g(⃗ℓg, ⃗mg), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (ℓ, m)g♯ := (⃗ℓg, ⃗mg)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) where ⃗ℓg = ⃗Rg(ℓ) and ⃗mg = ⃗Rg(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus g♯(·, ·) =noted (·, ·)g♯ is indeed an inner dot product in E∗: trivial check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification: (⃗ei) is a basis in E and (ei) is its dual basis (duality notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) gives: (g♯)ij := g♯(ei, ej) = g(⃗eig,⃗ejg), thus [g♯]|e = [(g♯)ij] = [gij]−1 = [g]−1 |⃗e , (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And [(g♯)ij] shorthand = notation [gij] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) Classical notations: [g♯]|e = [(g♯)ij] = [g♯(πei, πej)] = [g(⃗eig,⃗ejg)] = [gij]−1 = ([g]|⃗e)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 121 122 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Goal Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23 How do we compute g♯(ℓ, m) with matrix computations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ℓ = �n i=1ℓiei and m = �n j=1mjej give g♯(ℓ, m) = �n i,j=1ℓimjg♯(ei, ej) = �n i,j=1ℓi(g♯)ijmj = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g♯]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [m]T |⃗e = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]−1 |⃗e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [m]T |⃗e (a linear form is represented by a row matrix,).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) tells that the �2 0 � tensor g♯ ∈ L(E∗, E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) was created from the �0 2 � tensor g = (·, ·)g ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) using twice the (·, ·)g-Riesz representation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- Show that if you use the (·, ·)g-Riesz representation theorem just once you get the �1 1 � tensor g♮ ∈ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) ≃ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) given by g♮ = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) 2- Reciprocal: What is the �0 2 � tensor g♭ ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) that you create from the identity I ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) when using the (·, ·)g-Riesz representation theorem once?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3- Summary: �I = g♮ gives (�I)♭ = g♭ = g and (�I)♯ = g♯ Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- g♮ ∈ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is defined by g♮(ℓ, ⃗w) = (⃗ℓg, ⃗w)g for all (ℓ, ⃗w) ∈ E∗ × E, where ⃗ℓg is the (·, ·)g-Riesz representation vector of ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus g♮(ℓ, ⃗w) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w, for all (ℓ, ⃗w) ∈ E∗×E, hence g♮ ∈ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is naturally canonically associated with the identity I ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- The identity operator I ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) (observer independent) is naturally canonically associated with the �1 1 � tensor �I ∈ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) defined by �I(ℓ, ⃗w) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w for all (ℓ, ⃗w) ∈ E∗ × E, thus �I = g♮.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G Cauchy–Green deformation tensor C = F T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F Framework: �Φ : � R × Obj → Rn (t, PObj) → �Φ(t, PObj) � is a motion of Obj, Ωτ = �Φ(τ, PObj) is the configuration of Obj at any τ, t0 and t are fixed, Φ := Φt0 t : � Ωt0 → Ωt P → p = Φ(P) � is the associated motion between t0 and t, and F(P) := dΦ(P) : � � � � � ⃗ Rn t0 → ⃗Rn t ⃗W → ⃗w = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W := lim h→0 Φ(P+h ⃗W) − Φ(P) h � � � � � is the deformation gradient at P between t0 and t, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='0 Goal Construction of C (summary of Cauchy’s approach): 1- At t0, consider two vectors ⃗W1 and ⃗W2, 2- at t, they are distorted by the motion and become the vectors F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1 and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3- Then choose a Euclidean dot product (·, ·)g =noted · ·, the same at all t (to simplify);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4- Then, by definition of the transposed, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1) • (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2) = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1) • ⃗W2: You have got the Cauchy strain tensor C := F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 5- Then (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1) • (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2) − ⃗W1 • ⃗W2 = ((C−I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1) • ⃗W2 gives a measure of the deformation with ⃗W2 as a reference, measure that is used to build first order constitutive laws for the stress (Cauchy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Transposed F T: Inner dot products required We first give the functional definition of F T (qualitative);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then we get the usual matrix representation of F T relative to observers (quantification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition of the function F T At t0, a past observer chose an inner dot product (·, ·)G in ⃗Rn t0, and at t the present observer chooses an inner dot product (·, ·)g in ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' By definition, the transposed of the linear map F(P) ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) relative to (·, ·)G and (·, ·)g is the linear map F(P)T Gg ∈ L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) defined by, for all ⃗UP ∈ ⃗Rn t0 (vector at P) and ⃗wp ∈ ⃗Rn t (vector at p), (F(P)T Gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wp, ⃗UP )G = (F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗UP , ⃗wp)g, in short (F T Gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w, ⃗U)G = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U, ⃗w)g , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) 122 123 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Transposed F T : Inner dot products required see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This defines F T Gg(p) := F(P)T Gg when p = Φ(P): F T Gg : � � � Ωt → L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) p → F T Gg(p) := F(P)T Gg � � � , so in short (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) •G ⃗U = ⃗w •g (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U) , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) without forgetting that F T := F T Gg depends on (·, ·)G and (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = ⃗z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗z = ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗z, which dots are inner dot products?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' What does F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2 = ⃗W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2 mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' No choice: ( ⃗W, ⃗z) ∈ ⃗Rn t0 × ⃗Rn t , so (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗z) •G ⃗W = ⃗z •g (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W) = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W) •g ⃗z = ⃗W •G (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' No choice: ⃗W1, ⃗W2 ∈ ⃗Rn t0, so (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1) •g (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2) = ⃗W1 •G (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 More generally, on a surface Ω (a manifold), (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) is defined for all (⃗UP , ⃗wp) ∈ TP Ωt0 × TpΩt, where Tpτ Ωτ is the tangent space at Ωτ at pτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Quantification with bases (matrix representation) Classical notations: (⃗ai) is a basis in ⃗Rn t0, and (⃗bi) is a basis in ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Marsden–Hughes duality notations: ( ⃗EI) is a basis in ⃗Rn t0 and (⃗ei) is a basis in ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the reference to the points P and p is omitted to lighten the writings (use the full notation of § G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 if in doubt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let [G] := [(⃗ai,⃗aj)G], [g] := [(⃗bi,⃗bj)g], [F]|⃗a,⃗b = [Fij] =noted [F], [F T ]|⃗b,⃗a = [(F T )ij] =noted [F T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) gives [⃗U]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w] = [F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w], thus [⃗U]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w] = [⃗U]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w], for all ⃗U, ⃗w, thus [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F T ] = [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F T ] = [G]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) Remark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 If (⃗ai) and (⃗bi) are (·, ·)G and (·, ·)g-orthonormal bases, then [C] = [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But recall: If you need to work with a coordinate system, then the bases in use are the coordinate system bases which are not orthonormal in general, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [G]−1 ̸= I and [g]−1 ̸= I in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Use classical notation, then Marsden duality notations, to express (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) with components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Classical notations: Gij = G(⃗ai,⃗aj), gij = g(⃗bi,⃗bj), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [G]⃗a = [Gij], [g]|⃗b = [gij], and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = n � i=1 Fij⃗bi, F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj = n � i=1 (F T )ij⃗ai, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F]|⃗a,⃗b = [Fij], [F T ]|⃗b,⃗a = [(F T )ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) Then (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj,⃗ai)G =(G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1)(⃗bj, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai)g gives (�n k=1(F T )kj⃗ak,⃗ai)G = (⃗bj, �n k=1Fki⃗bk)g, thus �n k=1(F T )kj(⃗ak,⃗ai)G = �n k=1Fki(⃗bj,⃗bk)g with Fki = ([F]T )ik, thus n � k=1 Gik(F T )kj = n � k=1 ([F]T )ikgkj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F T )ij = n � k,ℓ=1 ([G]−1)ikFℓkgℓj, (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) for all i, j, thus (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Marsden notations: GIJ = G( ⃗EI, ⃗Ej), gij = g(⃗ei,⃗ej), F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗EJ = �n i=1F i J⃗ei, F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n I=1(F T )I j ⃗EI, thus n � K=1 GIK(F T )K j = n � k=1 F k Igkj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F T )I j = n � K,k=1 GIKF k Kgkj where [GIJ] := [GIJ]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Remark: F ∗ (For mathematicians: F ∗ doesn’t seem to be very useful in mechanics, apart from making simple things difficult, and playing games with components and duality notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 The adjoint of the linear map F ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) (acting on vectors) is the linear map F ∗ ∈ L(⃗Rn∗ t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn∗ t0 ) (acting on functions) canonically defined by, for all m ∈ ⃗Rn∗ t , F ∗(m) := m ◦ F, written F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F (∈ ⃗Rn∗ t0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) 123 124 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Cauchy–Green deformation tensor C So, for all (m, ⃗W) ∈ ⃗Rn∗ t × ⃗Rn t0, (F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W (∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) Quantification (matrix representation): (πai) and (πbi) are the covariant dual bases of (⃗ai) and (⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (F ∗)ij be the components of F ∗ relative to these dual bases: F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='πbj = n � I=1 (F ∗)ijπai, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F ∗]|πb,πa = [(F ∗)ij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) gives (F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='πbj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai = πbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai, thus ∀i, j, (F ∗)ij = Fji , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F ∗]|πb,πa = ([F]|⃗a,⃗b)T , in short [F ∗] = [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) Marsden duality notations: F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ej = �n I=1(F ∗)IjEI gives (F ∗)Ij = F jI for all I, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Interpretation of F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' As usual in classical mechanics, we use Euclidean dot products, here (·, ·)G in ⃗Rn t0 and (·, ·)g in ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then we use the (·, ·)G-Riesz representation vector ⃗RG(F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m) ∈ ⃗Rn t0 of F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m ∈ ⃗Rn∗ t0 , and the (·, ·)g-Riesz representation vector ⃗Rg(m) ∈ ⃗Rn t of m ∈ ⃗Rn∗ t , so, for all m ∈ ⃗Rn∗ t and ⃗W ∈ ⃗Rn t0, (F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = ⃗RG(F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m) •G ⃗W, and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W) = ⃗Rg(m) •g F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Rg(m)) •G ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) Thus (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) gives ⃗RG(F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='m) = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Rg(m), thus ⃗RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F ∗ = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Rg, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F ∗ = ⃗RG −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) Remark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 The definition of F ∗ is intrinsic to F (objective), while the definition of F T is not intrinsic to F (not objective) since it needs inner dot products (observer choices) to be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Cauchy–Green deformation tensor C G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition of C Consider vectors ⃗Wi ∈ ⃗Rn t0 at P, i = 1, 2, and their push forwards ⃗wi toward p = Φ(P), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗wi = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Wi, (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) short notation for ⃗wi(p) = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Wi(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With the chosen inner dot products (·, ·)G in ⃗Rn t0 and (·, ·)g in ⃗Rn t , we get (⃗w1(p), ⃗w2(p))g = (F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1(P), F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2(P))g=(G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2)(F T Gg(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1(P), ⃗W2(P))G when p = Φ(P), written in short: (⃗w1, ⃗w2)g = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2)g = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F � �� � C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G, (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 The (right) Cauchy–Green deformation tensor at P ∈ Ωt0 relative to (·, ·)G and (·, ·)g, is the endomorphism CGg(P) ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) defined by CGg(P) := F T Gg(p) ◦ F(P), in short C := F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) So C = F T ◦ F : ⃗W F −→ F( ⃗W) F T −→ F T (F( ⃗W)) = C( ⃗W), (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) with F and F T linear, thus C is linear and C( ⃗W) is written C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) tells that C is characterized by, for all ⃗W1, ⃗W2 ∈ ⃗Rn t0, ⃗w1 •g ⃗w2 = (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1) •G ⃗W2 = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1) •g (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) Moreover, (·, ·)g being symmetric (inner dot product), C is a (·, ·)G-symmetric endomorphism in ⃗Rn t0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all ⃗W1, ⃗W2 ∈ ⃗Rn t0, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G = ( ⃗W1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2)G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1) •G ⃗W2 = ⃗W1 •G (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2), (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) since (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2)g = ( ⃗W1, F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 124 125 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Time Taylor expansion of C G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Quantification (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) gives [C] = [F T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F], with [F T ] =(G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3)[G]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g], thus [C] = [G]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F] , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) short notation for [CGg]|⃗a = [G]−1 |⃗a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ([F]|⃗a,⃗b)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F]|⃗a,⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Use classical notation, then duality notations, to express (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) with components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Classical notations: F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = n � i=1 Fij⃗bi and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = n � i=1 Cij⃗ai, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F]|⃗a,⃗b = [Fij] and [C]|⃗a = [Cij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16)-(G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) give (⃗ai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj)G = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj)g, so (⃗ai, � k Ckj⃗ak)G = (� k Fki⃗bk, � ℓ Fℓj⃗bℓ)g, thus � k Ckj(⃗ai,⃗ak)G = � kℓ Fki(⃗bk,⃗bℓ)gFℓj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' n � k=1 GikCkj = n � k,ℓ=1 Fki gkℓFℓj = n � k,ℓ=1 ([F]T )ik gkℓFℓj, so [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [C] = [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F] , (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) so Cij = �n k,ℓ,m=1([G]−1)imFkm gkℓFℓj = �n k,ℓ,m=1([G]−1)im([F]T )mk gkℓFℓj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notations: F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗EJ = n � i=1 F i J⃗ei and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗EJ = n � I=1 CI J ⃗EI, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F]| ⃗ E,⃗e = [F i J] and [C]| ⃗ E = [CI J], and n � K=1 GIKCK J = n � k,ℓ=1 F k I gkℓF ℓ J, and CI J = n � k,ℓ,M=1 GIMF k M gkℓF ℓ J when [GIJ] := [GIJ]−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) Exercice G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 (·, ·)G is a Euclidean dot product in foot, (·, ·)g is a Euclidean dot product in metre, so (·, ·)g = µ2(·, ·)G with µ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3048;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (⃗ai) = (⃗bi) is a (·, ·)G-orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove [C] = µ2[F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [C]|⃗a =(G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) [G]−1 |⃗a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F]T |⃗a,⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F]|⃗a,⃗a gives [C]|⃗a = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F]T |⃗a,⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='µ2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[F]|⃗a,⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Shorten notation = (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Time Taylor expansion of C Here we use a unique inner dot product (·, ·)G = (·, ·)g at all time (to compare results in the vicinity of t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Moreover we use an orthonormal basis (to lighten the notations), thus, in short, [C] = [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' P is fixed, Ct0 t (P) =noted C(t), and [C(t)] = [F(t)]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F(t)] (since [G] = [g] = I here), and ⃗V t0 t (P) =noted ⃗V (t) and ⃗At0 t (P) =noted ⃗A(t) are the Lagrangian velocities and accelerations, and ⃗v(t, p) and ⃗γ(t, p) are the Eulerian velocities and accelerations at t at p = Φt0 t (t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With Lagrangian variables (used to define C): F(t+h) = F(t) + h d⃗V (t) + h2 2 d ⃗A(t) + o(h2) gives [C(t+h)] = [F(t+h)]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F(t+h)] = [F T + h d⃗V T + h2 2 d ⃗AT + o(h2)](t)[F + h d⃗V + h2 2 d ⃗A + o(h2)](t) = [C(t) + h ([F T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗V ] + [d⃗V ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F])(t) � �� � =[(Ct0 P )′(t)] =noted [C′(t)] +h2 2 ([F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d ⃗A] + 2[d⃗V ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗V ] + [d ⃗A]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F])(t) � �� � =[(Ct0 P )′′(t)] =noted [C′′(t)] )(t) + o(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) (As usual with Lagrangian variables, we have three times involved: t0, t and t+h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') In particular [Ct0 P (t0+h)] = I + ([d⃗V ] + [d⃗V ]T )(t0) + h2 2 ([d ⃗A] + 2[d⃗V ]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗V ] + [d ⃗A]T )(t0) + o(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) Abusively written Ct0 P (t0+h) = I + (d⃗V + d⃗V T )(t0) + h2 2 (d ⃗A + 2d⃗V T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗V + d ⃗AT )(t0) + o(h2), but don’t forget it is a matrix meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 125 126 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark: C♭ With Eulerian variables: With p(t) = Φt0(t, P), we have d⃗V t0(t, P) = d⃗v(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t) and d ⃗At0(t, P) = d⃗γ(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t), thus writing d⃗v := d⃗v(t, p(t)) and d⃗γ := d⃗γ(t, p(t)) (for short), Ct0 P (t+h) = Ct0 P (t) + h (F T (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗v + d⃗vT )(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t)) + h2 2 (F T (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗γ + 2d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + d⃗γT )(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t)) + o(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) abusive notation of [Ct0 P (t+h)] = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (matrices relative to a basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 F ′′ = d ⃗A is easy to interpret, but C′′ = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗A + 2d⃗V T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗V + d ⃗AT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗A + d⃗V T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗V ) + (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗A + d⃗V T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗V )T is not that easy to interpret (and in not linear in ⃗V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We already had a problem with the composition of flows: The formula F t0 t2 = F t1 t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t1 is simple (determinism), but the formula Ct0 t2 = (F t0 t2 )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t2 = (F t0 t1 )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F t1 t2 )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t1 t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t1 = (F t0 t1 )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Ct1 t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t1 is “not that simple” (̸= Ct1 t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Ct0 t1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Indeed, to consider C instead of F amounts to consider the “motion squared”, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, ⃗W)g = ||F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W||2 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Since C′(t0) = d⃗V (t0) + d⃗V (t0)T this may have little consequences for linear approximation near t0, but ultimately not small consequences for second-order approximations (and large deformations) if C′′ is used to make constitutive laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The consideration of Lie derivatives may be an interesting alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Remark: C♭ For mathematicians: May produce errors, misuses, covariance-contravariance confusion, see next § G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' For the general ♭ notation see § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition of C♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 At P ∈ Ωt0, the bilinear form C♭ Gg(P) =noted C♭ ∈ L(⃗Rn t0, ⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) associated with the linear map CGg(P) =noted C ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) is defined by, for all ⃗W1, ⃗W2 ∈ ⃗Rn t0 vectors at P, C♭( ⃗W1, ⃗W2) := ( ⃗W1, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2)G (= (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2)g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) Then C♭ is a bilinear symmetric form (trivial) and is a metric in ⃗Rn t0 when F t0 t =noted F is a diffeo- morphism (usual hypothesis), but not a Euclidean one (it is iff C = I i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' for rigid body motions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification: (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) gives [ ⃗W2]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [C♭].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗W1] = [ ⃗W2]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗W1] for all ⃗W1, ⃗W2 since C♭ and (·, ·)G are symmetric, thus [C♭] = [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [C] (= [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) Exercice G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 Use duality notations to express (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) with components, and explain the flat ♭ notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) gives C♭( ⃗EJ, ⃗EI) := (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗EJ, ⃗EI)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus with C♭( ⃗EI, ⃗EJ) = CIJ and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗EJ = � I CI J ⃗EI we get CJI = � K CK J( ⃗EK, ⃗EI)G = � K CK JGKI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And C♭ and (·, ·)G are symmetric, thus CIJ = � K GIKCK J, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [C♭]| ⃗ E = [G]| ⃗ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[C]| ⃗ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) The flat notation C♭ is due to: The top index I in CI J has been transformed into a bottom index in CIJ in C♭, which characterizes a change of variance because of the use of an inner dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) also gives CIJ = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗EI, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗EJ)g = � kℓ F k IF ℓ J(⃗bk,⃗bℓ)g, thus CIJ = � kℓ F k IgkℓF ℓ J = � kℓ (F T )I kgkℓF ℓ J, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [C♭]| ⃗ E = ([F]| ⃗ E,⃗e)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F]| ⃗ E,⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' and remarks about C♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' and Jaumann C♭ can also be defined only with (·, ·)g by, for all ⃗W1, ⃗W2 ∈ ⃗Rn t0, C♭ g( ⃗W1, ⃗W2) := (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2)g, (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', C♭ g := g∗ =noted C♭.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So we can also say that C♭ g is the pull-back of the metric (·, ·)g by Φ, see (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 126 127 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Stretch ratio and deformed angle However C♭ = C♭ g is useless in itself: C♭ is not a Euclidean dot product (it is a metric defined at each P by C♭ g(P)( ⃗W1, ⃗W2) := (F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2)g for all ⃗W1, ⃗W2 ∈ ⃗Rn t0 vectors at P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In fact, C♭ is only useful to characterize a deformation if the value C♭( ⃗W1, ⃗W2) can be compared with the initial value ( ⃗W1, ⃗W2)G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' if a Euclidean dot product (·, ·)G was introduced in ⃗Rn t0: This is why C♭ is classically defined from C, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' You may want to use the infinitesimal strain tensor ε = F +F T 2 − I, or the Green–Lagrange defor- mation tensor E = 1 2(C − I), obtained from F T := F T Gg (essential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' There is no objective “trace” for a �0 2 � tensor like C♭, while Tr(C) is objective since C is an endo- morphism (≃ a �1 1 � tensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Lie derivatives of a second order tensor depends on the type of the tensor, and the Lie derivative of the �1 1 � tensor like C gives the Jaumann derivative, which is usually preferred to the Lie derivative of the �0 2 � tensor like C♭ which is the lower convected Lie derivative, see remark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So the introduction and use of C♭ in mechanics mostly complicate things unnecessarily, and interferes with basic understandings like the distinction between covariance and contravariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 Interpretation issue (with Jaumann).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2D = d⃗v + d⃗vT gives 2 DD Dt = D(d⃗v) Dt + D(d⃗v)T Dt = d⃗γ + d⃗γT − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v − d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗vT , thus, with (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) and keeping in mind the matrix meaning, C′′(t) = F(t)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (2DD Dt + d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗vT + 2d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v)(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t) = 2F(t)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (DD Dt + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D)(t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) The DD Dt + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D term looks like a lower-convected Lie derivative, but with d⃗vT instead of d⃗v∗, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='58);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So you may find (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) written as C′′ = 2F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L⃗vD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But you get disappointing results when using the the lower convected Lie derivative (Jaumann is usually preferred).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In fact, it is L⃗vD♭ (lower convected Lie derivative) that should be used, where D♭ g := d⃗v♭ g+(d⃗v♭ g)T 2 , to get (C♭)′′ = 2F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L⃗vD♭ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Stretch ratio and deformed angle Here (·, ·)g = (·, ·)G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' at t0 and t we use the same Euclidean dot product, to be able to compare the lengths relative to the same unit of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (If (·, ·)g ̸= (·, ·)G then use (·, ·)g = µ2(·, ·)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Stretch ratio The stretch ratio at P ∈ ⃗Rn t0 between t0 and t for a ⃗WP ∈ ⃗Rn t0 is defined by λ( ⃗WP ) := ||⃗wp||G || ⃗WP ||G = ||FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗WP ||G || ⃗WP ||G (= ||FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ( ⃗WP || ⃗WP ||G )||G) (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) where ⃗wp = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗WP is the deformed vector by the motion at p = Φ(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', in short ∀ ⃗W ∈ ⃗Rn t0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' || ⃗W|| = 1, λ( ⃗W) := ||F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) (You may find: λ(d ⃗X) = ||F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d ⃗X|| with d ⃗X a unit vector(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This notation should be avoided, see § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Deformed angle Recall: The angle θt0 = � ( ⃗W1, ⃗W2) formed by two vectors ⃗W1 and ⃗W2 in ⃗ Rn t0−{⃗0} at P ∈ Ωt0 is given by cos(θt0) = ⃗ W1 || ⃗ W1||G ⃗ W2 || ⃗ W2||G (= ( ⃗ W1 || ⃗ W1||G , ⃗ W2 || ⃗ W2||G )G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With the deformed vectors ⃗wi = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Wi at p = Φt0 t (P), the deformed angle is θt defined by cos(θt) := � (⃗w1, ⃗w2) = ⃗w1 ||⃗w1|| ⃗w2 ||⃗w2|| = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1 ||F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1|| F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2 ||F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2|| (= (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1) • ⃗W2 ||⃗w1|| ||⃗w2|| ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) 127 128 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Decompositions of C G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Decompositions of C G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Spherical and deviatoric tensors Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 The deformation spheric tensor is Csph = 1 nTr(C) I, (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) with Tr(C) = the trace of the endomorphism C (there is no “trace” for the �0 2 � tensor C♭).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15 The deviatoric tensor is Cdev = C − Csph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) (So Tr(Cdev) = 0 , and C = Csph + Cdev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Rigid motion The deformation is rigid iff, for all t0, t, (F t0 t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t = I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ct0 t = I, written C = I = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) Thus, after a rigid body motion, lengths and angles are left unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Diagonalization of C Proposition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16 C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F being symmetric positive, C is diagonalizable, its eigenvalues are positive, and ⃗ Rn t0 has an orthonormal basis made of eigenvectors of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (C(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G = (F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2)g = ( ⃗W1, C(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2)G, thus C is (·, ·)G-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W1)G = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1)g = ||F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1||2 g > 0 when ⃗W1 ̸= ⃗0, since F invertible (Φt0 t is supposed to be a diffeomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus C est (·, ·)G-symmetric definite positive real endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17 Let λi be the eigenvalues of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then the √λi are called the principal stretches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the associated eigenvectors give the principal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Mohr circle This § deals with general properties of 3 ∗ 3 symmetric positive endomorphism, like Ct0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider ⃗R3 with a Euclidean dot product (·, ·)R3 and a (·, ·)R3-orthonormal basis (⃗ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let M : ⃗R3 → ⃗R3 be a symmetric positive endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus M is diagonalizable in a (·, ·)R3- orthonormal basis (⃗e1,⃗e2,⃗e3), that is, ∃λ1, λ2, λ3 ∈ R, ∃⃗e1,⃗e2,⃗e3 ∈ ⃗R3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei = λi⃗ei and (⃗ei,⃗ej)R3 = δij, so [M]|⃗e = diag(λ1, λ2, λ3) = � � λ1 0 0 0 λ2 0 0 0 λ3 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38) And the orthonormal basis (⃗e1,⃗e2,⃗e3) is ordered s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' λ1 ≥ λ2 ≥ λ3 (> 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let S be the unit sphere in R3, that is the set {(x, y, z) : x2 + y2 + z2 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its image M(S) by M is the ellipsoid {(x, y, z) : x2 λ2 1 + y2 λ2 2 + z2 λ2 3 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then consider ⃗n = � i ni⃗ei s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ||⃗n||R3 = 1: [⃗n]|⃗e = � � n1 n2 n3 � � with n2 1 + n2 2 + n2 3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39) Thus its image ⃗A = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n ∈ M(S) satisfies ⃗A = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n, [ ⃗A]|⃗e = � � λ1n1 λ2n2 λ3n3 � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) Then define An = ( ⃗A,⃗n)R3, ⃗A⊥ = ⃗A − An⃗n, A⊥ := || ⃗A⊥||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) So ⃗A = An⃗n + ⃗A⊥ ∈ Vect{⃗n} ⊗ Vect{⃗n}⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Remark: ⃗A⊥ is not orthonormal to the ellipsoid M(S), but is orthonormal to the initial sphere S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 128 129 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Green–Lagrange deformation tensor E Mohr Circle purpose: To find a relation: A⊥ = f(An), (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='42) relation between “the normal force An” (to the initial sphere) and the “tangent forceA⊥” (to the initial sphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39), (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) and An = (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n,⃗n)R3 give � � � � � n2 1 + n2 2 + n2 3 = 1, λ1n2 1 + λ2n2 2 + λ3n2 3 = An λ2 1n2 1 + λ2 2n2 2 + λ2 3n2 3 = || ⃗A||2 = A2 n + A2 ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43) This is linear system with the unknowns n2 1, n2 2, n2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The solution is � � � � � � � � � � � � � � � � � n2 1 = A2 ⊥ + (An − λ2)(An − λ3) (λ1 − λ2)(λ1 − λ3) , n2 2 = A2 ⊥ + (An − λ3)(An − λ1) (λ2 − λ3)(λ2 − λ1) , n2 3 = A2 ⊥ + (An − λ1)(An − λ2) (λ3 − λ1)(λ3 − λ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='44) The n2 i being non negative, and with λ1 > λ2 > λ3 ≥ 0, we get � � � � � A2 ⊥ + (An − λ2)(An − λ3) ≥ 0, A2 ⊥ + (An − λ3)(An − λ1) ≤ 0, A2 ⊥ + (An − λ1)(An − λ2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45) Then let x = An and y = A⊥, and consider, for some a, b ∈ R, the equation y2 + (x − a)(x − b) = 0, so (x − a+b 2 )2 + y2 = (a−b)2 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This is the equation of a circle centered at ( a+b 2 , 0) with radius |a−b| 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45)2 tells that An and A⊥ are inside the circle centered at ( λ1+λ3 2 , 0) with radius λ1−λ3 2 , and (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45)1,3 tell that An and A⊥ are outside the other circles (adjacent and included in the first, drawing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18 What happens if λ1 = λ2 = λ3 > 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then � � � � � � � � � � � � � n2 1 + n2 2 + n2 3 = 1, n2 1 + n2 2 + n2 3 = An λ1 , n2 1 + n2 2 + n2 3 = A2 n + A2 ⊥ λ2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � � � � � � � � � � � � � Thus An = λ1 and A2 n + A2 ⊥ = λ2 1, thus A⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Here C = λ1I, and we deal with a dilation: A⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19 What happens if λ1 = λ2 > λ3 > 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then � � � � � � � n2 1 + n2 2 + n2 3 = 1, λ1(1 − n2 3) + λ3n2 3 = An, λ2 1(1 − n2 3) + λ2 3n2 3 = A2 n + A2 ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � � � � � � � Thus An = λ1 − (λ1 − λ3)n2 3 ∈ [λ3, λ1], and A⊥ = ±(λ2 1 − (λ2 1 − λ2 3)n2 3 − A2 n) 1 2 , with A2 n + A2 ⊥ a point on the circle with radius λ2 1(1 − n2 3) + λ2 3n2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Green–Lagrange deformation tensor E (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) gives (⃗w1, ⃗w2)g = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2)g = (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, ⃗W)G at p = Φ(P), thus (⃗w1, ⃗w2)g − ( ⃗W1, ⃗W2)G = ((C − I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='46) 129 130 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Small deformations (linearization): The infinitesimal strain tensor ε Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20 The Green–Lagrange tensor (or Green–Saint Venant tensor) at P relative to t0 and t is the endomorphism Et0 t (P) ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) defined by Et0 t (P) := Ct0 t (P) − It0 2 , in short E = C − I 2 (= F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F − I 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='47) (In particular E = 0 for rigid body motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') And Et0 t : Ωt0 → L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) is the Green–Lagrange tensor relative to t0 and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The 1 2 because (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') corresponds to the “motion squared”, see the following linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And we get the time Taylor expansion of Et0 P (t) = 1 2(Ct0 P (t) − It0) with p(t) = Φt0 P (t) and (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25): Et0 P (t+h) = F t0 P (t)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � h d⃗v + d⃗vT 2 + h2 2 (d⃗γ + d⃗γT 2 + d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v) � (t, p(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 P (t) + o(h2) = F t0 P (t)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � h D + h2 ((DD Dt + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D)(t, p(t))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 P (t) + o(h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='48) G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Small deformations (linearization): The infinitesimal strain tensor ε G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Landau notations big-O and little-o Reminder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let f, g : R → R and x0 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' f = O(g) near x0 ⇐⇒ ∃C > 0, ∃η > 0, ∀x s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' |x − x0| < η, |f(x)| < C|g(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='49) and f is said to be “comparable with g” near x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If |g| > 0 then the conclusion reads |f(x)| |g(x)| < C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And f = O(xn) near x=0 iff |f(x)| |xn| < C near x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And f = o(g) near x0 ⇐⇒ ∀ε > 0, ∃η > 0, ∀x s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' |x − x0| < η, |f(x)| < ε|g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='50) and f is said to be “negligible compared with g near x0”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If |g| > 0 then the conclusion reads |f(x)| |g(x)| −→x→x0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And f = o(xn) near x=0 iff |f(x)| |xn| −→x→0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Definition of the infinitesimal strain tensor ε The motion is supposed to be C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Along a trajectory, with F t0 P (t0) = I we have, near t0, F t0 P (t0+h) = I + O(h), (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='51) thus F t0 P (t0+h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = ⃗W + O(h) for all ⃗W ∈ ⃗Rn t0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', near t0, ||⃗w − ⃗W|| = O(h) when ⃗w = F t0 P (t0+h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='52) This supposes the use of a unique inner dot product (·, ·)G = (·, ·)g at all time, and (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='52) means ||⃗w − ⃗W||g = O(h) near t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21 If (·, ·)g is an inner dot product, the same at all time, and if (⃗ei) is a (·, ·)g-orthonormal basis, the same at all time, then the infinitesimal strain tensor at P is the matrix defined by [ε(P)]|⃗e = [F(P)]|⃗e + [F(P)]T |⃗e 2 − [I], (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='53) abusively written in short, ε := F + F T 2 − I (matrix meaning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54) (And more precisely, at P ∈ Ωt0 and between t0 and t, [εt0 t (P)]|⃗e = [F t0 t (P )]|⃗e+[F t0 t (P )]T |⃗e 2 − [I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') So ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ W +F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ W 2 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W means [ε]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗W]|⃗e = [F ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗ W ]|⃗e+[F ]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗ W ]|⃗e 2 − [ ⃗W]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 130 131 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition Remark G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22 ε in (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54) cannot be a tensor (cannot be a function) since F t0 t (P) : ⃗ Rn t0 → ⃗ Rn t and F t0 t (P)T : ⃗ Rn t → ⃗ Rn t0 and It0 : ⃗ Rn t0 → ⃗ Rn t0 don’t have the same definition domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So ε is not a function, is not a tensor: It is a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But is called “the infinitesimal strain tensor”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23 The Green–Lagrange tensor E = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −I 2 ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) satisfies near t0: E = ε + o(t−t0) (= F + F T 2 − I + o(t−t0)) (matrix meaning), (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='55) which means [E] = [ε] + o(t−t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, “for small deformations” we write E ≃ ε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E ≃ F +F T 2 − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Interpretation: (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='55) is a linearization of E, since we keep the linear part of the “quadratic” E = 1 2(F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F − I) given by (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, ⃗U)g = 1 2 � (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U)g − ( ⃗W, ⃗U)g � for all ⃗U, ⃗W ∈ ⃗Rn t0 (“motion squared” cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the (F·, F·)g term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A (·, ·)g-orthonormal basis being chosen, [F T ] =(G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3)[F]T , thus [C] = [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F], thus 2[E] = [C] − [I] = [F]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F] − [I] = ([F]T − [I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ([F] − [I) + [F]T + [F] − 2[I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='56) Then, near t0 and with h = t−t0, (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='51) gives ([F]T − [I]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ([F] − I]) = O(h)O(h) = O(h2), thus 2[E] = [F]T + [F] − 2[I] + O(h), thus (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' H Finger tensor F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F T (left Cauchy–Green tensor) Finger’s approach is consistent with the foundations of relativity (Galileo classical relativity or Einstein general relativity): We can only do measurements at the current time t, and we can refer to the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' There is a lot of misunderstandings, as was the case for the Cauchy–Green deformation tensor C, due to the lack of precise definitions: Definition domain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Value domain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Points at stake (p or P)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Euclidean dot product (English?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' French?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=')?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Covariance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Contravariance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition Let �Φ be motion, t0 ∈ R, Φt0 the associated motion, P ∈ Ωt0, t ∈ R, and F t0 t (P) := dΦt0 t (P) ∈ L( ⃗ Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ Rn t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And let (·, ·)G and (·, ·)g be Euclidean dot products in ⃗ Rn t0 and ⃗ Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The Finger tensor bt0 t (pt), or left Cauchy–Green deformation tensor, at t at pt relative to t0 is the endomorphism ∈ L( ⃗ Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ Rn t ) defined by, with P = Φt0 t −1(pt), bt0 t (pt) := F t0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F t0 t )T Gg(pt) written in short b = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F T , (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' is defined by (bt0 t (pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, ⃗w2)g = (F t0 t (P)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, F t0 t (P)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2)G = ((F t0 t )T (pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, (F t0 t )T (pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2)G, for all ⃗w1, ⃗w2 vectors at pt ∈ Ωt, written in short (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, ⃗w2)g = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) (To compare with C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') And the Finger tensor relative to t0 is bt0 : � � � � � C = � t ({t} × Ωt) → L( ⃗ Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ Rn t ) (t, pt) → bt0(t, pt) := bt0 t (pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) NB: bt0 looks like a Eulerian function, but isn’t, since it depends on a t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Other definition found: Bt0 t := bt0 t ◦ (Φt0 t )−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Bt0 t (P) := bt0 t (pt) = F t0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (P)T , written B = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) Pay attention: Bt0 t (P) ∈ L( ⃗ Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ Rn t ) is an endomorphism at t at pt, not at t0 at P: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', Bt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt(pt) = bt0 t (pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt(pt) is meaningful, while Bt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Wt0(P) is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The push-forward by Φ := Φt0 t of the Cauchy–Green deformation tensor C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F is Φ∗(C) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −1 = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F T = b, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15): It is the Finger tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So the endomorphism C in ⃗Rn t0 is the pull-back of the endomorphism b in ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (However a push-forward and a pull-back don’t depend on any inner dot product while the transposed F T does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 131 132 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' b−1 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 b−1 With pull-backs (towards the virtual power principle at t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With pt = Φt0 t (P) and ⃗Wi(P) = (F t0 t (P))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wi(pt): ( ⃗W1, ⃗W2)G = (F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2)G = (F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, ⃗w2)g = (b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, ⃗w2)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) So b−1 := (bt0 t )−1 is useful: (bt0 t )−1 : � Ωt → L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) pt → (bt0 t )−1(pt) = F t0 t (P)−T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (P)−1 = Ht0 t (pt)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Ht0 t (pt) (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) with pt = Φt0 t (P) and Ht0 t (pt) = (F t0 t (P))−1 cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus we can define (bt0)−1 : � � � � � � t ({t} × Ωt) → L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) (t, pt) → (bt0)−1(t, pt) := (bt0 t )−1(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) Remark: (bt0)−1 looks like a Eulerian function, but isn’t, since it depends on t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In short: b−1 = HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='H, to compare with C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F, (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) and with ⃗w = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, to compare with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w, (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) and with ⃗Wi = F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗wi = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Wi, (b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, ⃗w2)g = ( ⃗W1, ⃗W2)G, to compare with (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G = (⃗w1, ⃗w2)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) Remark H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 pt = Φt0 t (P) and b(pt) = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(P)T and C(P) = F(P)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(P) give b(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(P) = F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='C(P), (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) written b = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus b−1 = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='C−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −1, so Φt0∗ t b−1 = F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F = F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −T = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F)−1 = C−1, (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the pull-back of b−1 is C−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' b−1 is the push-forward of C−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Time derivatives of b−1 With (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) let (bt0)−1 =noted b−1 = HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, along a trajectory, and with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45), we get Db−1 Dt = DHT Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='H + HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='DH Dt = −d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='H − HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v = − b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v − d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) Exercice H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Prove (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) with (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) gives D Dt(b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, ⃗w2)g = 0 = ( Db−1 Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, ⃗w2)g + (b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' D ⃗w1 Dt , ⃗w2)g + (b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, D ⃗w2 Dt )g, and ⃗wi(t, p(t)) = F t0(t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Wt0(P) gives D ⃗wi Dt = d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wi, thus ( Db−1 Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, ⃗w2)g +(b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, ⃗w2)g +(b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w2)g = 0, thus (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Prove (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) with F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F = It0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' b−1 = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F T )−1 = F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −1 gives F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F = It0, thus (F T )′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F + F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Db−1 Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F + F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F ′ = 0, thus F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F + F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Db−1 Dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F + F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F = 0, thus (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 132 133 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Euler–Almansi tensor a H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Euler–Almansi tensor a Euler–Almansi approach is consistent with the foundations of relativity (Galileo relativity or Einstein general relativity): We can only do measurements at the current time t, and we can refer to the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' At t in Ωt, consider the Finger tensor b = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F T and its inverse b−1 = F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F T = HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='H cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Euler–Almansi tenor at pt ∈ Ωt is the endomorphism at0 t (pt) ∈ L( ⃗ Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ Rn t ) defined by at0 t (pt) = 1 2(It − bt0 t (pt)−1) = 1 2(It − H(pt)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='H(pt)), (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) written a = 1 2(I − b−1) = 1 2(I − HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='H), (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) to compare with the Green–Lagrange tensor E = 1 2(C − I) = 1 2(F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F − I) ∈ L( ⃗ Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ Rn t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark: at0 looks like a Eulerian function, but isn’t, since it depends on t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) gives (⃗wi = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Wi) (⃗w1, ⃗w2)g − ( ⃗W1, ⃗W2)G = 2(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, ⃗w2)g, (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) to compare with (⃗w1, ⃗w2)g − ( ⃗W1, ⃗W2)G = 2(E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (This also gives (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w1, ⃗w2)g = (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') And (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) gives F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F = E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' a = F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −1, (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) standing for F t0 t (P)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='at0 t (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (P) = Et0 t (P) when p = Φt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 at0 t is not the push-forward of Et0 t by Φt0 t (the push-forward is F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Time Taylor expansion for a (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) gives Da Dt = b−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v + d⃗vT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='b−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Almansi modified Infinitesimal strain tensor �ε We are at t (present time) and remember the past: We prefer a definition of a infinitesimal strain tensor �ε from the Euler–Almansi tensor a, instead of ε from Euler–Lagrange tensor E, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' § G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Same Euclidean framework as in § G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2, and matrix meaning again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have I − b−1 = I − HT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='H = −(I − HT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I − H) + 2I − HT − H where H stands for Ht0 t (pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, for small displacement we get I − b−1 = 2I − HT − H + O(h), so a(t, p(t)) = �ε(t, p(t)) + O(h) where �ε := I − H + HT 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) And, with t = t0 + h we have F t0(t, P) = I + (t−t0) d⃗v(t, P) + o(t−t0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35), thus we have Ht0(t, p(t)) = F t0(t, P)−1 = I − (t−t0) d⃗v(t, P) + o(t−t0) when p(t) = Φt0(t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus F t0(t, P) − I = I − Ht0(t, p(t)) + O(t−t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) Therefore, for small displacements (|t − t0| << 1): a(t, p(t)) ≃ �ε(t, p(t)) ≃ εt0(t, P) (matrix meaning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) I Polar decomposition, elasticity and objectivity I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Polar decompositions of F (“isometric objectivity”) The motion is supposed regular, t0, t ∈ R, pt0 ∈ Ωt0, F := F t0 t (pt0) (= dΦt0 t (pt0)), (·, ·)G and (·, ·)g are Euclidean dot products in ⃗Rn t0 and ⃗Rn t , and C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Here the covariant objectivity is abandoned due to the need for inner dot products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 133 134 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Polar decompositions of F (“isometric objectivity”) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 F = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U (right polar decomposition) The endomorphism C being (·, ·)G-symmetric definite positive (the motion is supposed to be regular), ∃α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', αn ∈ R∗ + (the eigenvalues), ∃⃗c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗cn ∈ ⃗Rn t0 (associated eigenvectors), such that, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ci = αi⃗ci and (⃗ci) is a (·, ·)G-orthonormal basis in ⃗Rn t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) So, if (⃗ai) is a (·, ·)G-Euclidean basis then D = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [C]⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P, where D = diag(α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', αn) = [C]⃗c and P −1 = P T , P = [Pij] being the transition matrix from (⃗ai) to (⃗ci), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' defined by ⃗cj = � i Pij⃗ai for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, define the endomorphism U ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0), called the right stretch tensor, by, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ci = √αi ⃗ci, and U noted = √ C, (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) the √αi being called the principal stretches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, define the linear map R ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ), called the rotation map, by R := F ◦ U −1 noted = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U −1, (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) so that F = R ◦ U noted = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U, called the right polar decomposition of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) Proposition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 We have: C = U ◦ U noted = U 2, U is symmetric definite positive, R−1 = RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) And the right polar decomposition F = R ◦ U is unique : (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) If F = �R◦ �U where �U ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) is symmetric definite positive and �R ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) satisfies �R−1 = �RT , then �U = U and �R = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) yields (U ◦ U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗cj = λ⃗cj = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗cj for all j, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1), thus U ◦ U = C ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (U T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ci,⃗cj)G = (⃗ci, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗cj)G = (⃗ci, √αj⃗cj)G = √αj(⃗ci,⃗cj)G = √αjδij = √αiδij = √αi(⃗ci,⃗cj)G = (√αi⃗ci,⃗cj)G = (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ci,⃗cj)G for all i, j, thus U T = U (symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then RT ◦ R = U −T ◦ F T ◦ F ◦ U −1 = U −T ◦ C ◦ U −1 = U −1 ◦ (U ◦ U) ◦ U −1 = It0 identity in ⃗Rn t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Details: (RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, ⃗Z)G = (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Z)g = (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Z)g = (F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Z)G = (U 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Z)G = (U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, ⃗Z)G = ( ⃗W, ⃗Z)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Thus R−1 = RT ∈ L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0), thus R ◦ RT = R ◦ R−1 = It identity in ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And F = �R ◦ �U = R ◦ U gives F T ◦ F = �U T ◦ �RT ◦ �R ◦ �U = U T ◦ �RT ◦ �R ◦ U, thus F T ◦ F = �U T ◦ �U, with F T ◦ F = U T ◦ U, thus �U ◦ �U = U ◦ U = √ C, thus �U = U (uniqueness of the positive square root eigenvalues).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence �R = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 F = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U (shifted right polar decomposition for covariant objectivity) In fact we need to be more specific if the gift of ubiquity is prohibited: Since we work with the affine space Rn, consider the Marsden’s shifter S := St0 t (pt0) : � Tpt0(Ωt0) noted = ⃗Rn t0 → Tpt(Ωt) noted = ⃗Rn t ⃗wt0,pt0 → (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0,pt0 )(t, pt) = ⃗wt0,pt0 where pt = Φt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) NB: 1- S looks like the algebraic identity if you have time and space ubiquity gift (otherwise it is not), 2- S is not a topological identity since it changes the norms in the general case: You consider ||⃗wt0,pt0 ||G at t0 and ||S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wt0,pt0 ||g = ||⃗wt0,pt0 ||g at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, let R0 ∈ L(Tpt0(Ωt0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Tpt0(Ωt0)) =noted L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) be the endomorphism defined by, in short, R0 := S−1 ◦ R noted = S−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R, so R = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R0 (= S ◦ R0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) Full notations: (R0)t0 t,Gg(pt0) := (St0 t )−1(Rt0 t,Gg(pt0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 134 135 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Polar decompositions of F (“isometric objectivity”) Proposition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The endomorphism R0 = S−1 ◦ R is a rotation operator in (⃗Rn t0, (·, ·)G): R−1 0 = RT 0 in (⃗Rn t0, (·, ·)G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) And F = S ◦ R0 ◦ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) Interpretation: F is composed of: The pure deformation U (endomorphism in ⃗Rn t0), the rotation R0 (endomorphism in ⃗Rn t0), and the shift operator S : ⃗Rn t0 → ⃗Rn t (from past to present time and position).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (RT 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2, ⃗W1)G = (R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G (definition of the transposed) = (S−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G (definition of R0) = (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G (S is the algebraic identity) = (RT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2, ⃗W1)g (definition of RT ) = (R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2, ⃗W1)g (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) ) = (R−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W2, ⃗W1)G (S is the algebraic identity), (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) true for all ⃗W1, ⃗W2 ∈ ⃗Rn t0, thus RT 0 = R−1 0 in (⃗Rn t0, (·, ·)G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) and (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) give (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Let D = diag(αi), let (⃗ai) be a Euclidean basis in ⃗Rn t0, let P be the transition matrix from (⃗ai) to (⃗ci), so [C]|⃗a = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove [U]|⃗a = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' √ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Case (⃗ai) = ( ⃗Ei) is a (·, ·)g-orthonormal basis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The n equations (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) (for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n), read as the matrix equation [C]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D since [⃗cj]⃗a is the j-th column of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And he n equations (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) (for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n), read as the matrix equation [U]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' √ D since [⃗cj]⃗a is the j-th column of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Instead of R0 ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8), you may prefer to consider �R0 ∈ L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) defined by R = �R0 ◦ S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', �R0 = R ◦ S−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 F = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R (left polar decomposition) Same steps than for the right polar decomposition, but with pull-backs (with F −1 instead of F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let pt = Φt0 t (pt0) ∈ Ωt, let bt0 t (pt) := F t0 t (pt0) ◦ (F t0 t )T (pt) ∈ L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ), written b = F ◦ F T (the left Cauchy–Green deformation tensor also called the Finger tensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The endomorphism b being symmetric definite positive: ∃β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', βn ∈ R∗ + (the eigenvalues), ∃⃗d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗dn ∈ ⃗Rn t (associated eigenvectors), such that, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗di = βi ⃗di, and (⃗di) is a (·, ·)g-orthonormal basis in ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) Then, define the unique endomorphism V ∈ L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ), called the left stretch tensor, by, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗di = � βi ⃗di, and V noted = � b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) (Full notation: V t0 t,Gg(pt) = � bt0 t (pt)Gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Then define the linear map Rℓ ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) by Rℓ := V −1 ◦ F noted = V −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F, (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) so that F = V ◦ Rℓ noted = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Rℓ, called the left polar decomposition of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) Proposition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 We have: 1- b = V ◦ V noted = V 2, V is symmetric definite positive, R−1 ℓ = RT ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) And the left polar decomposition F = V ◦ R is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- Rℓ = R and V = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1 (so U and V are similar), thus U and V have the same eigenvalues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', αi = βi for all i, and ⃗di = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ci for all i gives a relation between eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 135 136 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Linear elasticity: A Classical “tensorial” approach Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- Use F −1 and b−1 = (F −1)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F −1), instead of F and C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F, to get F −1 = R−1 ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U −1 ℓ , cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus F = Uℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Rℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then name Uℓ = V to get (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) and (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Rℓ = F = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U = (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R, thus, by uniqueness of the right polar decomposition, V = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1 (so U and V are similar) and Rℓ = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, with (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12), βi ⃗di = V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗di = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗di), thus with ⃗ci = R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗di, then (⃗ci) is an orthonormal basis in ⃗Rn t0 and βiR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ci = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ci = αiR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ci gives βi = αi and the ⃗ci are eigenvectors of U, for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Linear elasticity: A Classical “tensorial” approach I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Classical approach (“isometric objectivity”), and an issue With the infinitesimal strain “tensor” (which is not a tensor but a matrix) ε = F + F T 2 − I, (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) the homogeneous isotropic elasticity constitutive law reads (matrix equation for the stress) (σ(Φ) =) σ = λTr(ε)I + 2µε, (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) where λ, µ are the Lamé coefficients and σ is the Cauchy stress “tensor”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Issue: Recall: Adding F and F T to make ε is functionally a mathematical nonsense since F : ⃗Rn t0 → ⃗Rn t and F T : ⃗Rn t → ⃗Rn t0 and I is some identity operator: σ is not a tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular the meaning of Tr(ε) is questionable (since ε is not an endomorphism and Tr(ε) means Tr([ε]) = Tr([F ])+Tr([F T ]) 2 − n), as well as the meaning of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n = 1 2(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n + F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n), or the meaning of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n = λTr(ε)⃗n + 2µε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) since ⃗n has to be defined at (t0, pt0) for F and at (t, pt) for F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Cauchy’s approach: ⃗n is defined at (t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') So, despite the eventual claims, neither ε nor σ are tensors (they don’t have a functional meaning): They only have a questionable matrix meaning (observer dependent) [ε] := [F ]+[F ]T 2 − [I] and [σ] = λTr([ε])[I] + 2µ[ε], and [σ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗n] = λTr([ε])[⃗n] + 2µ[ε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) Remark I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 To justify the name “tensor” applied to ε, you may read: “For small displacements the Eulerian variable pt and the Lagrangian variable pt0 can be confused”: pt ≃ pt0 (so Ωt0 and Ωt are “almost equal”, so F(pt0) + F T (pt) can be considered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Which means that you use the zero-th order Taylor expansions pt = Φt0 pt0 (t) = pt0 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But then, you cannot also use the first (or higher) order Taylor expansion in following calculations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' you cannot use velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 A functional (tensorial) formulation (“isometric objectivity”) Question: Can the constitutive law (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) be modified into a tensorial expression (a functional expression)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposal for a yes: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider the “right polar decomposition” F = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U where U ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Green Lagrange tensor E = C−I 2 (endomorphism in ⃗Rn t0) then reads, with (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5), E = U 2−It0 2 = (U−It0)2 + 2(U − It0) 2 (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) (the Green–Lagrange tensor is independent of the rotation R), thus, with U − It0 = O(h) (small defor- mation approximation), we get the modified infinitesimal strain tensor at pt0 ∈ Ωt0 �ε = U−It0 ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0), (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) endomorphism in ⃗Rn t0 (to compare with ε which is not a function, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the previous §).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Full notation �εt0 t,Gg(pt0) = U t0 t,Gg(pt0)−It0(pt0) in L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') And, for all ⃗W ∈ ⃗Rn t0 we get �ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W − ⃗W = R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w − ⃗W ∈ ⃗Rn t0, when ⃗w = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W (push-forward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) Interpretation: From ⃗w = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W ∈ ⃗Rn t (the deformed by the motion), remove the “rigid body rotation” to get R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W ∈ ⃗Rn t0, to which you remove the initial ⃗W to obtain �ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W ∈ ⃗Rn t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 136 137 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Linear elasticity: A Classical “tensorial” approach In particular ||�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W||G = ||(U−It0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W||G measures the relative elongation undergone by ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And you can then apply R to get back into ⃗Rn t at pt: R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W − R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = ⃗w − R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W ∈ ⃗Rn t , when ⃗w = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = (push-forward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, at pt0 ∈ Ωt0, consider the stress tensor �Σ(Φ) =noted �Σ ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) (functionally well) defined by �Σ = λTr(�ε)It0 + 2µ�ε = λTr(U−It0)It0 + 2µ(U−It0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) (The trace Tr(�ε) is well defined since �ε is an endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Then for any ⃗W ∈ ⃗Rn t0 you get in ⃗Rn t0, at pt0 ∈ Ωt0, �Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = λTr(�ε) ⃗W + 2µ�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = λTr(U−It0) ⃗W + 2µ(U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W− ⃗W) (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) (functionally well defined in ⃗Rn t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then rotate and shift with R to get into ⃗Rn t at pt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = λTr(�ε)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W + 2µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = λTr(U−It0)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W + 2µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(U−It0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = λTr(U−It0)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W + 2µ(F − R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, = λTr(U−It0)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W + 2µ(⃗w − R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W), where ⃗w = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) You have defined the two point tensor (functionally well defined) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�Σ = λTr(�ε)R + 2µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�ε ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then you can propose the constitutive law with the stress tensor (the symmetric endomorphism) in ⃗Rn t given by (�σ(Φ) =) �σ = R ◦ �Σ ◦ R−1 noted = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1 ∈ L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) (Functionally well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') And, for all vector fields ⃗w defined in Ωt, you get the (functionally well defined) vector field �σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w ∈ ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) Interpretation of (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29)-(I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30): Shift and rigid rotate backward by applying R−1, apply the elastic stress law with Σ which corresponds to a rotation free motion (Noll’s frame indifference principle), then shift and rigid rotate forward by applying R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Detailed expression for (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29)-(I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30): With Tr(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1) = Tr(�ε) (see exercise I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8), we get, at (t, pt), �σ = λTr(�ε) It + 2µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1 = λTr(U−It0) It + 2µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U−It0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1 = λTr(U−It0) It + 2µ(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1−It).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) And for any ⃗w ∈ ⃗Rn t , and with ⃗w = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W, you get �σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = λTr(�ε) ⃗w + 2µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = λTr(U−It0) ⃗w + 2µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(U−It0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = λTr(U−It0) ⃗w + 2µ(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w−⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) To compare with the classical functionally meaningless (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Doing so, you avoid the use of the Piola–Kirchhoff tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Prove: Tr(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1) = Tr(�ε) = � i(αi−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (NB: �ε is an endomorphism in ⃗Rn t0 while R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1 is an endomorphism in ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' det|⃗e(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1 − λIt) = det|⃗e(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(�ε−λIt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1) = det| ⃗ E,⃗e(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' det| ⃗ E(�ε−λI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' det|⃗e, ⃗ E(R−1) = det| ⃗ E(�ε−λI) for all Euclidean bases ( ⃗Ei) and (⃗ei) in ⃗Rn t0 and ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (With L = �ε and components, Tr(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1) = � i(R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R−1)i i = � ijk Ri jLj k(R−1)k i = � jk(R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='R)k j Lj k = � jk δk j Lj k = � j Lj j = Tr(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 137 138 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Elasticity with a covariant objective approach?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 Elongation in R2 along the first axis : origin O, same Euclidean basis ( ⃗E1, ⃗E2) and Eu- clidean dot product at all time, ξ > 0, t ≥ t0, L, H > 0, P ∈ [0, L] × [0, H], [−−→ OP]| ⃗E = � X0 Y0 � , and [−−−−−−→ OΦt0 t (P)]| ⃗E = � X0 + ξ(t−t0)X0 Y0 � = � X0(κ+1) Y0 � = � x y � = [−→ Op]| ⃗E, where κ = ξ(t−t0) > 0 for t > t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- Give F, C, U = √ C and R = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Relation with the classical expression ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- Spring −−→ OP = −−→ Oct0(s) = X0 ⃗E1+Y0 ⃗E2+s ⃗W, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [−−→ OP]| ⃗E = [−−→ Oct0]| ⃗E = � X0+sW1 Y0+sW2 � | ⃗E with s ∈ [0, L] and ⃗W = W1 ⃗E1 + W2 ⃗E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Give the deformed spring, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' give p = ct(s) = Φt0 t (ct0(s)), and ⃗ct′, and the stretch ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- [F] = [dΦ] = � κ+1 0 0 1 � , same Euclidean dot product and basis at all time, thus [F T ] = [F]T = [F], then [C] = [F T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [F] = [F]2 = � (κ+1)2 0 0 1 � , thus [U] = [F] = � κ+1 0 0 1 � , thus [R] = [I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' All the matrices are given relative to the basis ( ⃗Ei), thus F, C, U, R (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E1 = (κ+1)2 ⃗E1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E2 = ⃗E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Since R = I and [ε] = [�ε], (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) gives the usual result [σ] = λTr([ε])I + 2µ[ε], cf (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) (matrix meaning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- −−−−→ Oct(s) = −−−−−−−−−→ OΦt0 t (ct0(s)) = � (X0+sW1)(κ+1) Y0+sW2 � | ⃗ E , thus ⃗ct ′(s) = � W1(κ+1) W2 � | ⃗ E , stretch ration W 2 1 (κ+1)2+W 2 2 W 2 1 +W 2 2 at (t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 Simple shear in R2 : [−−−−−−→ OΦt0 t (P)]| ⃗E = � X + ξ(t−t0)Y Y � =noted � X + κY Y � = � x y � = [−→ Op]| ⃗E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Same questions, and moreover give the eigenvalues of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- [F] = � 1 κ 0 1 � , [C] = � 1 0 κ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � 1 κ 0 1 � = � 1 κ κ κ2+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Eigenvalues: det(C − λI) = λ2 − (2+κ2)λ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Discriminant ∆ = (2+κ2)2 − 4 = κ2(κ2+4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Eigenvalues α± = 1 2(2+κ2 ± κ √ κ2+4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (We check that α± > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Eigenvectors ⃗v±(main directions of deformations) given by (1−α±)x+κy = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', y = 1 2(κ± √ κ2+4)x, thus, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗v± = � 2 κ ± √ κ2+4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (We check that ⃗v+ ⊥ ⃗v−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') With P the transition matrix from ( ⃗E1, ⃗E2) to ( ⃗v+ ||⃗v+||, ⃗v− ||⃗v−||) and D = diag(α+, α−) we get C = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P −1 (with P −1 = P T since here ( ⃗v+ ||⃗v+||, ⃗v− ||⃗v−||) is an orthonormal basis), thus U = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' √ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P −1 (we check that U T = U and U 2 = C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And R = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- −−−−→ Oct(s) = −−−−−−−−−→ OΦt0 t (ct0(s)) = � (X0+sW1) + κ(Y0+sW2) Y0+sW2 � , thus [⃗ct ′(s)] = � W1 + κW2 W2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Stretch ratio (W1+κW2)2+W 2 2 W 2 1 +W 2 2 at (t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Second functional formulation: With the Finger tensor The above approach uses the push-forward: It uses F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' you arrive with your memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' You may prefer to use the pull-back, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' use F −1 (you remember the past which is Cauchy’s point of view): Then you use F −1 = R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='V −1 the right polar decomposition of F −1, and you consider the tensor ��εt = V −1−It ∈ L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ), (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) and (σt(Φ) =) σt = λTr(��εt)It + 2µ��εt, and σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗nt = λTr(��εt)⃗nt + 2µ��εt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) (Quantities functionally well defined: Give a tensorial approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Elasticity with a covariant objective approach?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In § I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 you need to start with Euclidean dot products, so from the start the result can’t be covariant objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Can you start without Euclidean dot products to set up general laws?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposal: Hypothesis: The Cauchy stress ⃗w is a Eulerian vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then we could use the (covariant objective) Lie derivative which characterizes the rate of stress, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' § 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5: With a particle PObj ∈ Obj, with ⃗v(τ, pτ) = ∂�Φ ∂τ (τ, PObj) its Eulerian velocity at τ at 138 139 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The displacement vector ⃗U pτ = �Φ(τ, PObj), the Lie derivative of a Eulerian vector field ⃗w along ⃗v is, at (t, pt), L⃗v ⃗w(t, pt) = lim τ→t ⃗w(τ, pτ) − ⃗wt∗(τ, pτ) τ − t = (∂ ⃗w ∂t + d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w)(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) Hence the proposal, with the virtual power principle to measure the rate of stress (see https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10780v1 for a full description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- Hypotheses: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1- Suppose that n Eulerian vector fields ⃗wj (“force fields”), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, enable to characterize a material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (In fact, for elasticity problems it could be better to replace vector fields ⃗wj with 1-forms αi to characterize the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2- With a basis (⃗ei) chosen in ⃗Rn t , with (ei) its (covariant) dual basis in ⃗Rn∗ t , assume that the internal power density at (t, pt) is given by (at first order): pint(⃗v) = n � j=1 ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L⃗v ⃗wj = n � j=1 ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (∂ ⃗wj ∂t + d⃗wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v − d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) (At second order you can add second order Lie derivatives as L⃗v(L⃗v ⃗wj), similarly for higher orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 2- Then, so that this pint satisfies the frame invariance hypothesis, choose a Euclidean dot prod- uct (·, ·)g in ⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, first, the internal power has to vanish if d⃗v = 0, thus we are left with pint(⃗v) = − n � j=1 ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wj = −τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗v, where τ = n � j=1 ⃗wj ⊗ ej, (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) defined at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (The j-th column of [τ]|⃗e is [⃗wj]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') And, second, the internal power vanishes if d⃗v +d⃗vT = 0 (rotation), thus we are left with pint(⃗v) = −τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗v + d⃗vT 2 = −σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗v where σ = τ + τ T 2 , (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38) this pint(⃗v) = −σ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗v being the usual expression of the internal power at first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) may be applied to orthotropic elasticity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' for a material which fibers at some time t0 are along ⃗e1, in a 2-D case for simplicity: 1- With an elongation type motion (Φe) given by [(Fe)(pt0)] = [d(Φe)(pt0)] = � 1+α11(pt0) 0 0 1−α22(pt0) � you measure the Young moduli in the direc- tions ⃗e1 and ⃗e2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- And with a shear type motion given by [(Fs)(pt0)] = [d(Φs)(pt0)] = � 1 γ12 0 1 � you measure the shear modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' For more complex material, you may need more vectors ⃗wj to describe the constitutive law, that is, (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) may be considered with �m i=1ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L⃗v ⃗wj with m > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 The Lie approach is different from the usual classic approach: 1- The classic approach looks for an order two stress tensor [σ] as a function of the deformation gradient [F], cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- The Lie approach begins with the internal power (which measures forces), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36), which then enable to build τ and the σ (the stress tensor), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37)-(I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', application to visco-elasticity: With the Lie approach, you automatically use Lie derivative of vector fields (and/or of differential forms), instead of Lie derivative of order 2 tensor fields (which does not seem to give good result, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the Maxwell visco-elastic type laws, as well as footnote1 page 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' J Displacement J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The displacement vector ⃗U In Rn, let pt = Φt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then the bi-point vector ⃗Ut0 t (pt0) = Φt0 t (pt0) − It0(pt0) = pt − pt0 = −−→ pt0pt (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) is called the displacement vector at pt0 relative to t0 and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This defines the map ⃗Ut0 t : � Ωt0 → ⃗Rn pt0 → ⃗Ut0 t (pt0) := pt − pt0 = −−→ pt0pt when pt = Φt0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) 139 140 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The differential of the displacement vector Remark J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 ⃗Ut0 t (pt0) doesn’t define a vector field (it is not tensorial), because ⃗Ut0 t (pt0) = pt−pt0 = −−→ pt0pt is a bi-point vector which is neither in ⃗Rn t0 or in ⃗Rn t since pt0 ∈ Ωt0 and pt ∈ Ωt (it requires time and space ubiquity gift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular, it makes no sense on a non-plane surface (manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' More at § J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 For elastic solids in Rn, the function ⃗Ut0 is often considered to be the unknown (to be computed);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But the “real” unknown is the motion Φt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And it is sometimes confused with the extension of a spring 1-D case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But see figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 where ||⃗wt0(pt0)|| represents the initial length and ||⃗wt0∗(t, pt)|| represents the current length of the spring, while the length of the displacement vector ⃗Ut0 t = pt − pt0 can be very long for a very small elongation ||⃗wt0∗(t, pt)|| − ||⃗wt0(pt0)|| of the spring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The differential of the displacement vector The differential of ⃗Ut0 t at pt0 is d⃗Ut0 t (pt0) = dΦt0 t (pt0) − It0 = F t0 t (pt0) − It0, written d⃗U = F − I, (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) thus isn’t defined as a function, because F t0 t (pt0) : ⃗Rn t0 → R while It0 : ⃗Rn t0 → ⃗Rn t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So d⃗Ut0 t (pt0) as to be understood as a matrix: With [⃗Ut0 t (pt0)] = [−−→ pt0pt] = [−−−−−−→ OΦt0 t (pt0)] − [−−→ Opt0], (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) relative to an origin O and a unique basis at all time, compute [d⃗Ut0 t (pt0)] = [dΦt0 t (pt0)] − I, abusively written d⃗U = dΦ − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, with ⃗W ∈ ⃗Rn t0 , d⃗U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W − ⃗W, which means = [F t0 t (pt0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗W] − [ ⃗W].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) Thus we have defined (matrix meaning) ⃗Ut0 : � [t0, T] × Ωt0 → ⃗Rn (t, pt0) → ⃗Ut0(t, pt0) := ⃗Ut0 t (pt0), and ⃗Ut0 pt0 : � [t0, T] → ⃗Rn t → ⃗Ut0 pt0 (t) := ⃗Ut0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Deformation “tensor” ε (matrix), bis (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) gives (matrix meaning) F t0 t (pt0) = It0 + d⃗Ut0 t (pt0), written F = I + d⃗U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) Therefore, Cauchy–Green deformation tensor C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F reads, in short, (matrix meaning) C = I + d⃗U + d⃗UT + d⃗UT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗U (matrix meaning), (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [Ct0 t (pt0)] = [It0] + [d⃗Ut0 t (pt0)] + [d⃗Ut0 t (pt0)]T + [d⃗Ut0 t (pt0)]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗Ut0 t (pt0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the Green–Lagrange deformation tensor E = C−I 2 , cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='47), reads, in short, (matrix meaning) E = d⃗U + d⃗UT 2 + 1 2d⃗UT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗U (matrix meaning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) Thus the deformation tensor ε, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54), reads (matrix meaning) ε = E − 1 2(d⃗U)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗U, (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) with ε the “linear part” of E (small displacements: we only used the first order derivative dΦt0 t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 140 141 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Small displacement hypothesis, bis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Small displacement hypothesis, bis (Usual introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Let pt = Φt0 t (pt0), ⃗Wi ∈ ⃗ Rn t0, ⃗wi(pt) = F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Wi(pt0) ∈ ⃗Rn t (the push-forwards), written ⃗wi = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then define (matrix meaning) ⃗∆i := ⃗wi − ⃗Wi = dU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Wi, and ||⃗∆||∞ = max(||⃗∆1||Rn, ||⃗∆2||Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) Then the small displacement hypothesis reads (matrix meaning): ||⃗∆||∞ = o(|| ⃗W||∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) Thus ⃗wi = ⃗Wi + ⃗∆i (with ⃗∆i “small”) and the hypothesis (·, ·)g = (·, ·)G (same inner dot product at t0 and t) give (⃗w1, ⃗w2)G − ( ⃗W1, ⃗W2)G = (⃗∆1, ⃗W2)G + (⃗∆2, ⃗W1)G + (⃗∆1, ⃗∆2)G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) gives 2(E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G = 2(ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G + (d⃗UT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G, And (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) gives (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G = (ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗W1, ⃗W2)G + O(||⃗∆||2 ∞), (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) so Et0 t is approximated by εt0 t , that is, Et0 t ≃ εt0 t (matrix meaning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Displacement vector with differential geometry J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The shifter We give the steps, see Marsden–Hughes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The complexity introduced is due to the small displacement hypothesis applied to the Green–Lagrange tensor E = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −I 2 which linearization gives ε = F +F T 2 − I (the classical approach “squares the motion” to get E, then “linearizes” E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' to get back to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' with a spurious F T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let P ∈ Ωt0, ⃗WP ∈ ⃗Rn t0, pt = Φt0 t (P) ∈ Ωt, and ⃗wpt = F t0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗WP ∈ ⃗Rn t (push-forward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Affine case Rn (continuum mechanics): With pt = Φt0 t (P), the shifter is: � St0 t : � Ωt0 × ⃗Rn t0 → Ωt × ⃗Rn t (P, ⃗ZP ) → � St0 t (P, ⃗ZP ) = (pt, St0 t (⃗ZP )) with St0 t (⃗ZP ) = ⃗ZP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) (The vector is unchanged but the time and the application point have changed: A real observer has no ubiquity gift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So: St0 t ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) and [St0 t ]|⃗e = I identity matrix, (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) the matrix equality being possible after the choice of a unique basis at t0 and at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (simplified notation) � St0 t (P, ⃗ZP ) =noted St0 t (⃗ZP ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then the deformation tensor ε at P can be defined by εt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Z(P) = (St0 t )−1(F t0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Z(P)) + F t0 t (P)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (St0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Z(P)) 2 − ⃗Z(P), (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) in short: ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Z = (St0 t )−1(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Z)+F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (St0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Z) 2 − ⃗Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In a manifold: Ω is a manifold (like a surface in R3 from which we cannot take off).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let TP Ωt0 be the tangent space à P (the fiber at P), and TptΩt be the tangent space à pt (the fiber at pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In general TP Ωt0 ̸= TptΩt (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' on a sphere “the Earth”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The bundle (the union of fibers) at t0 is TΩt0 = � P ∈Ωt0 ({P} × TP Ωt0), and the bundle at t is TΩt = � pt∈Ωt({pt} × TptΩt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then the shifter � St0 t : � TΩt0 → TΩt (P, ⃗ZP ) → � St0 t (P, ⃗ZP ) = (pt, St0 t (⃗ZP )), (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) is defined such that ⃗ZP ∈ TP Ωt0 “as little distorted as possible” along a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', on a sphere, if the path is a geodesic, if θt0 is the angle between ⃗ZP and the tangent vector to the geodesic at P, then θt0 is also the angle between St0 t (⃗ZP ) and the tangent vector to the geodesic at pt, and St0 t (⃗ZP ) has the same length than ⃗ZP (at constant speed in a car you think the geodesic is a straight line, although St0 t (⃗ZP ) ̸= ⃗ZP : the Earth is not flat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 141 142 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Alternating multilinear form J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The displacement vector (Affine space framework, Ωt0 open set in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Let P ∈ Ωt0, ⃗WP ∈ ⃗Rn t0, pt = Φt0 t (P) ∈ Ωt, and dΦt0 t = F t0 t ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Define δ� ⃗Ut0 t : � � � Ωt0 × ⃗Rn t0 → Ωt × L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) (P, ⃗ZP ) → δ� ⃗Ut0 t (P, ⃗ZP ) = (pt, δ ⃗Ut0 t (⃗ZP )) with δ ⃗Ut0 t (⃗ZP ) = (F t0 t − St0 t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ZP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) Then δ� ⃗Ut0 t = F t0 t − St0 t is a two-point tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And Ct0 t = (F t0 t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t = (δUt0 t + St0 t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (δUt0 t + St0 t ) = I + (St0 t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='δUt0 t + (δUt0 t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='St0 t + (δUt0 t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='δUt0 t , (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) since (St0 t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='St0 t = I identity in TΩt0: Indeed, ((St0 t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='St0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗A, ⃗B)Rn = (St0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗A, St0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗B)Rn = ( ⃗A, ⃗B)Rn, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14), for all ⃗A, ⃗B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then the Green–Lagrange tensor is defined on Ωt0 by Et0 t = 1 2(Ct0 t − It0) = (St0 t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='δUt0 t + St0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (δUt0 t )T 2 + 1 2(δUt0 t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='δUt0 t , (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) to compare with (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' K Determinants K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Alternating multilinear form Let E be a vector space, and let L(E, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) =noted L(En;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) be the set of multilinear forms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' m ∈ L(En;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) iff m(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗x + λ⃗y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') = m(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') + λm(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) for all ⃗x, ⃗y ∈ E and all λ ∈ R and for all “slot”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular, m(λ1⃗x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', λn⃗xn) = (� i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n λi) m(⃗x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗xn), for all λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', λn ∈ R and all ⃗x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗xn ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 If n = 1 then a 1-alternating multilinear function is a linear form, also called a 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If n ≥ 2 then Aℓ : � En → R (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) → Aℓ(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) � ∈ L(En;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is a n-alternating multilinear form iff, for all ⃗u,⃗v ∈ E, Aℓ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗u, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗v, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') = −Aℓ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗v, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗u, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='), (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) the other elements being unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If n = 1, the set of 1-forms is Ω1(E) = E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If n ≥ 2, the set of n-alternating multilinear forms is Ωn(E) = {m ∈ L(En;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) : m = Aℓ is alternating}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) If Aℓ, Bℓ ∈ Ωn(E) and λ ∈ R then Aℓ + λBℓ ∈ Ωn(E) thanks to the linearity for each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus Ωn(E) is a vector space, sub-space in (F(En;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Leibniz formula Particular case dim E=n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Aℓ ∈ Ωn(E) (a n-alternating multilinear form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Recall (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Cartan [5]): 1- A permutation σ : [1, n]N → [1, n]N is a bijective map (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' one-to-one and onto);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Sn be the set of permutations of [1, n]N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- A transposition τ : [1, n]N → [1, n]N is a permutation that exchanges two elements, that is, ∃i, j s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' τ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') = (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='), the other elements being unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3- A permutation is a composition of transpositions (theorem left as an exercise, of see Cartan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And a permutation is even iff the number of transpositions is even, and a permutation is odd iff the number of transpositions is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Based on: The parity (even or odd) of a permutation is an invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4- The signature ε(σ) = ±1 of a permutation σ is +1 if σ is even, and is −1 if σ is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 142 143 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Determinant of vectors Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 (Leibniz formula) Let Aℓ ∈ Ωn(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n =noted (⃗ei) be a basis in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' For all vectors ⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn ∈ E, with ⃗vj = �n i=1vi j⃗ei for all j, Aℓ(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) = c � σ∈Sn ε(σ) n � i=1 vσ(i) i = c � τ∈Sn ε(τ) n � i=1 vi τ(i) (with c := Aℓ(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) Thus if c = Aℓ(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en) is known, then Aℓ is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus dim(Ωn(E)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Classic not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' : ⃗vj = �n i=1vij⃗ei, Aℓ(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) = c � σ∈Sn ε(σ) �n i=1 vσ(i),i = c � τ∈Sn ε(τ) �n i=1 vi,τ(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let F := F([1, n]N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [1, n]N) =noted [1, n][1,n]N N be the set of functions i : � [1, n]N → [1, n]N k → ik = i(k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Aℓ being multilinear, Aℓ(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) = �n j1=1 vj1 1 Aℓ(⃗ej1,⃗v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) (“the first column” development).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' By recurrence we get Aℓ(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) = �n j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',jn=1 vj1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='vjn n Aℓ(⃗ej1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗ejn) = � j∈F �n k=1 vj(k) k Aℓ(⃗ej(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗ej(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And Aℓ(⃗ei1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗ein) ̸= 0 iff i : k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n} → i(k) = ik ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n} is one-to-one (thus bijective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus Aℓ(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) = � σ∈Sn �n i=1 vσ(i) i Aℓ(⃗eσ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗eσ(n)) = � σ∈Sn ε(σ) �n i=1 vσ(i) i Aℓ(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en), which is the first equality in (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then � σ∈Sn ε(σ) �n i=1 vσ(i) i = � σ∈Sn ε(σ) �n i=1 vσ(σ−1(i)) σ−1(i) since σ is bijectif, thus � σ∈Sn ε(σ) �n i=1 vσ(i) i = � τ∈Sn ε(τ −1) �n i=1 vi τ(i), thus the second equality in (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) since ε(τ)−1 = ε(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (See Cartan [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Determinant of vectors Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 (⃗ei)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n being a basis in E, the alternating multilinear form det|⃗e ∈ Ωn(E) defined by det |⃗e (⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en) = 1 (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) is called the determinant relative to (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And, with prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 (here c = 1), det |⃗e (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) = � σ∈Sn ε(σ) n � i=1 vσ(i) i = � τ∈Sn ε(τ) n � i=1 vi τ(i) (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) is called the determinant of the vectors ⃗vi relative to (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And we write Ωn(E) = Vect{det |⃗e } (the 1-D vector space spanned by det|⃗e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) Thus, if Aℓ ∈ Ωn(E) then Aℓ = Aℓ(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en) det |⃗e , (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) thus if (⃗bi) is another basis then ∃c ∈ R, det |⃗b = c det |⃗e , with c = det |⃗b (⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Change of measuring unit: If (⃗ai) is a basis and ⃗bj = λ⃗aj for all j, prove ∀j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, ⃗bj = λ⃗aj =⇒ det |⃗a = λn det |⃗b (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) (relation between volumes relative to a change of measuring unit in the Euclidean case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' det |⃗a (⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗bn) = det |⃗a (λ⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', λ⃗an) multi = linear λn det |⃗a (⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗an) (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) = λn (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) = λn det |⃗b (⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 det|⃗e(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) ̸= 0 iff (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) is a basis, or equivalently, det|⃗e(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) = 0 iff ⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn are linearly dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If one of the ⃗vi is = ⃗0 then det|⃗e(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) = 0 (multilinearity), and if �n i=1ci⃗vi = 0 and one of the ci ̸= 0 and then a ⃗vi is a linear combination of the others thus det|⃗e(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) = 0 (since det|⃗e is alternate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus det|⃗e(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) ̸= 0 ⇒ the ⃗vi are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And if the ⃗vi are independent then (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) is a basis, thus det|⃗v(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) = 1 ̸= 0, with det|⃗v = c det|⃗e, thus det|⃗e(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 143 144 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Determinant of a matrix Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 In R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗v1 = �2 i=1 vi 1⃗ei and ⃗v2 = �2 j=1 vj 2⃗ej (duality notations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove: det |⃗e (⃗v1,⃗v2) = v1 1v2 2 − v2 1v1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Development relative to the first column (linearity used for the first vector ⃗v1 = v1 1⃗e1 + v2 1⃗e2): det|⃗e(⃗v1,⃗v2) = det|⃗e(v1 1⃗e1 + v2 1⃗e2,⃗v2) = v1 1 det|⃗e(⃗e1,⃗v2) + v2 1 det|⃗e(⃗e2,⃗v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (linearity used for the second vector ⃗v2 = v1 2⃗e1 + v2 2⃗e2): det|⃗e(⃗v1,⃗v2) = 0 + v1 1v2 2 det(⃗e1,⃗e2) + v2 1v1 2 det(⃗e2,⃗e1) + 0 = v1 1v2 2 − v2 1v1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 In R3, with ⃗vj = �3 i=1 vi j⃗ei, prove: det(⃗v1,⃗v2,⃗v3) = 3 � i,j,k=1 εijkvi 1vj 2vk 3, (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) where εijk = 1 2(j−i)(k−j)(k−i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' εijk = 1 if (i, j, k) = (1, 2, 3), (3, 1, 2) or (2, 3, 1) (even signature), εijk = −1 if (i, j, k) = (3, 2, 1), (1, 3, 2) and (2, 1, 3) (odd signature), and εijk = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Development relative to the first column (as in exercise K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Result = v1 1v2 2v3 3 + v1 2v2 3v3 1 + v1 3v2 1v3 2 − v3 1v2 2v1 3 − v3 2v2 3v1 1 − v3 3v2 1v1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Determinant of a matrix Let M = [Mij] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n be a n2 real matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗Rn = R × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' × R (Cartesian product n-times) with its canonical basis ( ⃗Ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗vj ∈ ⃗Rn, ⃗vj = �n i=1Mij ⃗Ei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So M = � [⃗v1]| ⃗E, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', [⃗vn]| ⃗E � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 The determinant of the matrix M = � [⃗v1]| ⃗E, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', [⃗vn]| ⃗E � is det(M) := det | ⃗E (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 Let M T be the transposed matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', (M T )ij = Mji for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then det(M T ) = det(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' det[Mij] = det ⃗E (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) = � σ∈Sn ε(σ) n � i=1 vσ(i) i = � τ∈Sn ε(τ) n � i=1 vi τ(i) = det[Mji].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Volume Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 Let (⃗ei) be a Euclidean basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider a parallelepiped in Rn which sides are vectors ⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its algebraic volume relative to (⃗ei) is algebraic volume = det |⃗e (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) And its volume relative to (⃗ei) is (non negative) volume = ��det |⃗e (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', if n = 1 and ⃗v = v1⃗e1, then det|⃗e(⃗v) = v1 is the algebraic length of ⃗v (relative to the unit of measurement given by ⃗e1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And | det|⃗e(⃗v)| = |v1| is the length of ⃗v (the norm of ⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (The volume function (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) → ��det|⃗e(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) �� is not a multilinear form, because the absolute value function is not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', if n = 2 or 3, see exercises K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗ei) be a Cartesian basis and (ei) = (dxi) be the dual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Cartan [6], det |⃗e noted = e1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∧ en = dx1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∧ dxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) And, for integration, the volume element (non negative) uses a Euclidean basis (⃗ei) and is dΩ(⃗x) = | det |⃗e | = |dx1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∧ dxn| noted = dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) Thus the volume of a parallelepiped at ⃗x which sides are given by δx1⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', δxn⃗un is dΩ(⃗x)(δx1⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', δxn⃗un) = |δx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='δxn| | det|⃗e(⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗un)|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the volume of a polygonal domain Ω = 144 145 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Determinant of an endomorphism �N i=1 Pi where Pi is a parallelepiped which sides are given by δxi,1⃗ui,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', δxi,n⃗ui,n is |Ω| = N � i=1 | det |⃗e (⃗ui,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗ui,n)|δxi,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='δxi,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) And thus (Riemann approach), the volume of a regular domain Ω is written |Ω| = � Ω dΩ = � ⃗x∈Ω | det |⃗e (⃗ui,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗ui,n)| dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) In particular, since any regular volume Ω can be approximated with cubes as small as wished, |Ω| = �N i=1 |δxi,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='δxi,n det|⃗e(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en)| = �N i=1 |δxi,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='δxi,n| gives |Ω| = � Ω dΩ = � ⃗x∈Ω dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 Let Ψ : ⃗q = (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', qn) ∈ [a1, b1] × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' × [an, bn] → ⃗x = (x1 = Ψ1(⃗q), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', xn = Ψn(⃗q)) ∈ Ω be a parametric description of a domain Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove dΩ(⃗x) = |JΨ(⃗q)| dq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dqn (= | det |⃗e (⃗p1(⃗x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗pn(⃗x))| dq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dqn), (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) where (⃗pi(x)) = ( ∂Ψ ∂qi (⃗q)) is the parametric basis at ⃗x = Ψ(⃗q) and JΨ(⃗q) = det|⃗e[dΨ(⃗q)]|⃗e is the Jacobian matrix of Ψ at ⃗q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And thus |Ω| = � ⃗q |JΨ(⃗q)| dq1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Polar coordinates for illustration purpose (immediate generalization): Consider the disk Ω parametrized with the polar coordinate system Ψ : ⃗q = (ρ, θ) ∈ [0, R] × [0, 2π] → ⃗x = (x = ρ cos θ, y = ρ sin θ) ∈ R2 where a Euclidean basis (⃗e1,⃗e2) has been used in R2 (so ⃗x = ρ cos θ⃗e1 + ρ sin θ⃗e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The associated polar ba- sis at ⃗x = Ψ(⃗q) is (⃗p1(⃗x) = ∂Ψ ∂ρ (ρ, θ), ⃗p2(⃗x) = ∂Ψ ∂θ (ρ, θ)), so [⃗p1(⃗x)]|⃗e = � cos θ sin θ � and [⃗p2(⃗x)]|⃗e = � −ρ sin θ ρ cos θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus det|⃗e(⃗p1(⃗x), ⃗p2(⃗x)) = ρ (> 0 here), thus dΩ = |ρ| dρdθ = ρ dρdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the volume is |Ω| = � ⃗x∈Ω dΩ = � R ρ=0 � 2π θ=0 ρ dρdθ (= πR2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 What is the “volume element” on a regular surface Σ in R3, called the “surface element”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗e1,⃗e2,⃗e3) be a Euclidean basis in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We need a regular parametric description Ψ : (u, v) ∈ [a1, b2]× [a2, b2] → ⃗x = Ψ(u, v) = x1(u, v)⃗e1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' + x3(u, v)⃗e3 of the geometric surface Σ = Im(Ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ⃗t1(⃗x) = ∂Ψ ∂u (u, v) and ⃗t2(⃗x) = ∂Ψ ∂v (u, v) are tangent vectors at Σ at ⃗x = Ψ(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hence a normal unit vector is ⃗n(⃗x) = ⃗t1(⃗x)∧⃗t2(⃗x) ||⃗t1(⃗x)∧⃗t2(⃗x)||, and thus det|⃗e(⃗t1,⃗t2,⃗n) = ||⃗t1(⃗x) ∧ ⃗t2(⃗x)|| is the area of the parallelogram which sides are given by ⃗t1 and ⃗t2 (volume with height 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the surface element at ⃗x = Ψ(u, v) is dΣ(⃗x) = || ∂Ψ ∂u (u, v) ∧ ∂Ψ ∂v (u, v)|| dudv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus |Σ| = � ⃗x∈Σ dΣ(⃗x) = � b1 u=a1 � b2 v=a2 || ∂Ψ ∂u (u, v) ∧ ∂Ψ ∂v (u, v)|| dudv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Determinant of an endomorphism K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition and basic properties Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 The determinant of an endomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) relative to a basis (⃗ei) is � det |⃗e (L) := det |⃗e (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗en).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) This define � det|⃗e : L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (If the context is not ambiguous, then � det|⃗e =noted det|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 Let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- If L = I the identity, then � det|⃗e(I) = 1 for all basis (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- For all ⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn ∈ E, det |⃗e (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vn) = � det |⃗e (L) det |⃗e (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) 3- If L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1Lij⃗ei, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|⃗e = [Lij], then � det |⃗e (L) = det([L]|⃗e) = det([Lij]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) 4- For all M ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E), and with M ◦ L =noted M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L (thanks to linearity), � det |⃗e (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L) = � det |⃗e (M) � det |⃗e (L) = � det |⃗e (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) 5- L is invertible iff � det|⃗e(L) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 145 146 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Determinant of an endomorphism 6- If L is invertible then � det |⃗e (L−1) = 1 � det|⃗e(L) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) 7- If (·, ·)g is an inner dot product in E and LT g is the (·, ·)g transposed of L (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', (LT g ⃗w, ⃗u)g = (⃗w, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u)g for all ⃗u, ⃗w ∈ E) then � det |⃗e (LT g ) = � det |⃗e (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) 8- If (⃗ei) and (⃗bi) are two (·, ·)g-orthonormal bases in ⃗Rn t (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' two Euclidean basis for the same measuring unit), then det|⃗b = ± det|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- � det|⃗e(I) =(K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) det|⃗e(I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗en) =(K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) det|⃗e(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en) = 1, true for all basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- Let m : (⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) → m(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) := det|⃗e(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vn): It is a multilinear alternated form, since L is linear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus m =(K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) m(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en) det|⃗e;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With m(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en) =(K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) � det|⃗e(L), thus (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3- Apply (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) with M = [L]|⃗e to get (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4- det|⃗e((M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗en) = det|⃗e(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗en)) =(K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) � det|⃗e(M) det|⃗e(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗en).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 5- If L is invertible, then 1 = � det|⃗e(I) = � det|⃗e(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L−1) = � det|⃗e(L) � det|⃗e(L−1), thus � det|⃗e(L) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If � det|⃗e(L) ̸= 0 then det|⃗e(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗en) ̸= 0, thus (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗en) is a basis, thus L is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 6- (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) gives 1 = � det|⃗e(I) = � det|⃗e(L−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L) = � det|⃗e(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � det|⃗e(L−1), thus (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 7- (LT g ⃗w, ⃗u)g = (⃗w, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u)g gives [g]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [LT g ]|⃗e = ([L]|⃗e)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [g]|⃗e, thus det([g]|⃗e) det([LT g ]|⃗e) = det(([L]|⃗e)T ) det([g]|⃗e), and det([g]|⃗e) ̸= 0 (exercise), thus (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 8- Let P be the change of basis endomorphism from (⃗ei) to (⃗bi), and P be the transition matrix from (⃗ei) to (⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Both basis being (·, ·)g-orthonormal, P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P = I, thus det(P) = ±1 = � det|⃗e(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And det|⃗e(⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗bn) = det|⃗e(P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗en) = � det|⃗e(P) det|⃗e(⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗en) = � det|⃗e(P) det|⃗b(⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗bn), thus det|⃗e = � det|⃗e(P) det|⃗b = ± det|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15 Two (·, ·)g-orthonormal bases (⃗ei) and (⃗bi) have the same orientation iff det|⃗b = + det|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16 Prove � det|⃗e(λL) = λn � det|⃗e(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � det |⃗e (λL) = det |⃗e (λL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', λL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗en) = λn det |⃗e (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗en) = λn � det |⃗e (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The determinant of an endomorphism is objective Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17 Let (⃗ai) and (⃗bi) be bases in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The determinant of an endomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) is objective (observer independent, here basis independent): (det([L]|⃗a) =) � det |⃗a (L) = � det |⃗b (L) (= det([L]|⃗b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) NB: But the determinant of n vectors is not objective, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) (compare the change of basis formula for vectors [⃗w]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗a with the change of basis formula for endomorphisms [L]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗ai) and (⃗bi) be bases in E, and P be the transition matrix from (⃗ai) to (⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The change of basis formula [L]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P and (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) give det([L]|⃗b) = det(P −1) det([L]|⃗a) det(P) = det([L]|⃗a), thus (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) gives (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18 Let (⃗ai) and (⃗bi) be bases in E, and P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) be the change of basis endomorphism from (⃗ai) to (⃗bi) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = ⃗bj for all j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove det |⃗a (⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗bn) = � det |⃗a (P), thus det |⃗a = � det |⃗a (P) det |⃗b , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' det |⃗b = det|⃗a � det|⃗a(P) , (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' det |⃗a (⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗bn) = det |⃗a (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗an) (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) = � det |⃗a (P) det |⃗a (⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗an) = � det |⃗a (P) 1 = � det |⃗a (P) det |⃗b (⃗b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗bn), thus (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) and det|⃗a = � det|⃗a(P) det|⃗b and det|⃗a(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn) = � det|⃗a(P) det|⃗b(⃗v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 146 147 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Determinant of a linear map K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Determinant of a linear map (Needed for the deformation gradient F t0 t (P) = dΦt0 t (P) : ⃗Rn t0 → ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Let A and B be vector spaces, dim A = dim B = n, and (⃗ai) and (⃗bi) be bases in A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition and first properties Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19 The determinant of a linear map L ∈ L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B) relative to the bases (⃗ai) and (⃗bi) is � det |⃗a,⃗b (L) := det |⃗b (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) (And � det|⃗a,⃗b(L) =noted det(L) if the bases are implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Thus, (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) gives, with L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = �n i=1Lij⃗bi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [L]|⃗a,⃗b = [Lij]: � det |⃗a,⃗b (L) = det([L]|⃗a,⃗b) = det([Lij]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20 Let ⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗un ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then det |⃗b (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗un) = � det |⃗a,⃗b (L) det |⃗a (⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗un).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' m : (⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗un) ∈ An → m(⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗un) := det|⃗b(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗un) ∈ R is a multilinear alternated form, since L is linear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And m(⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗an) = det|⃗b(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗an) =(K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) � det|⃗a,⃗b(L) = � det|⃗a,⃗b(L) det|⃗a(⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',⃗an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus m = � det|⃗a,⃗b(L) det|⃗a, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9), thus (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Corollary K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21 Let A, B, C be vector spaces such that dim A = dim B = dim C = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗ai), (⃗bi), (⃗ci) be bases in A, B, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let L : A → B and M : B → C be linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, with M ◦ L =noted M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L (thanks to linearity), � det |⃗a,⃗c(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L) = � det |⃗a,⃗b (L) � det |⃗b,⃗c (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � det |⃗a,⃗c(M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L) = det |⃗c (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗an)) = � det |⃗b,⃗c (M) det |⃗b (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗an) = � det |⃗b,⃗c (M) � det |⃗a,⃗b (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Jacobian of a motion, and dilatation Let �Φ be a motion, let t0, t ∈ R, let Φt0 t be the associated motion, let F t0 t (pt0) := dΦt0 t (pt0) : ⃗Rn t0 → ⃗Rn t the deformation gradient at pt0 ∈ Ωt0 relative to t0 and t, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ( ⃗Ei) be a Euclidean basis in ⃗ Rn t0 and (⃗ei) be a Euclidean basis in ⃗ Rn t for all t ≥ t0, and [F t0 t (pt0)]| ⃗E,⃗e = [Fij(pt0)], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ej = �n i,j=1Fij(pt0)⃗ei for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22 The “volume dilatation” at pt0, relative to the Euclidean bases ( ⃗Ei) in ⃗ Rn t0 and (⃗ei) in ⃗ Rn t , is J| ⃗E,⃗e(Φt0 t )(pt0) := � det | ⃗E,⃗e (F t0 t (pt0)) (= det |⃗e (F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗En) = det([Fij(pt0)])), (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) usually written J| ⃗E,⃗e := det([F]| ⃗E,⃗e) (or simply J = det(F) when everything is implicit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, at t0 at pt0, (pt0, ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗En) is a unit parallelepiped which volume is 1 relative to the unit of mea- surement chosen in ⃗Rn t0, and, at t at pt = Φt0 t (pt0), J| ⃗E,⃗e(Φt0 t )(pt0) = det|⃗e(F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗En) is the volume of the parallelepiped (pt, F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', F t0 t (pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗En) relative to the unit of measurement chosen in ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Interpretation: With t2 > t1 ≥ t0, and [⃗ei) is the basis at t1 and t2: Dilatation if J| ⃗E,⃗e(Φt0 t2)(pt0) > J| ⃗E,⃗e(Φt0 t1)(pt0) (volume increase), 147 148 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Dilatation rate contraction if J| ⃗E,⃗e(Φt0 t2)(pt0) < J| ⃗E,⃗e(Φt0 t1)(pt0) (volume decrease), and incompressibility if J| ⃗E,⃗e(Φt0 t2)(pt0) = J| ⃗E,⃗e(Φt0 t1)(pt0) for all t (volume conservation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular, if (⃗ei) = ( ⃗Ei) then J|⃗e,⃗e(Φt0 t0)(pt0) = 1, and if t > t0, then Dilatation if J|⃗e,⃗e(Φt0 t )(pt0) > 1 (volume increase), contraction if J|⃗e,⃗e(Φt0 t )(pt0) < 1 (volume decrease), and incompressibility if J|⃗e,⃗e(Φt0 t )(pt0) = 1 for all t (volume conservation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23 Let ( ⃗Ei) be a Euclidean basis in ⃗Rn t0, and let (⃗ai) and (⃗bi) be two Euclidean bases in ⃗Rn t for the same Euclidean dot product (·, ·)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove: J| ⃗E,⃗a(Φt0 t (P)) = ±J| ⃗E,⃗b(Φt0 t (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' P being the transition matrix from (⃗ai) to (⃗bi), det(P) = ±1 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) gives [F]| ⃗ E,⃗a = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[F]| ⃗ E,⃗b, thus det([F]| ⃗ E,⃗a) = ± det([F]| ⃗ E,⃗b), thus det|⃗a(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗En) = ± det|⃗b(F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Determinant of the transposed Let (A, (·, ·)g) and (B, (·, ·)h) be finite dimensional Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let L ∈ L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B) (a linear map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Recall: The transposed LT gh ∈ L(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A) is defined by, for all ⃗u ∈ A and all ⃗w ∈ B, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='68) (LT gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w, ⃗u)g := (⃗w, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) Let (⃗ai) be a basis in A and (⃗bi) be a basis in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then � det([LT gh]|⃗b,⃗a) = det([L]|⃗a,⃗b)det([(·, ·)g]|⃗a) det([(·, ·)h]|⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38) Indeed, (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) gives [(·, ·)g]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [LT gh]|⃗b,⃗a = ([L]|⃗a,⃗b)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [(·, ·)h]|⃗b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Dilatation rate A unique Euclidean basis (⃗ei) at all time is chosen, and (·, ·)g is the associated inner dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 ∂Jt0 ∂t (t, pt0) = Jt0(t, pt0) div⃗v(t, pt) A regular motion �Φ is considered, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5), and the Eulerian velocity is ⃗v(t, pt) = ∂�Φ ∂t (t, PObj) at pt = �Φ(t, PObj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let t0 be given;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The associated motion Φt0 is given by Φt0(t, pt0) = �Φ(t, PObj) =noted pt when pt0 = �Φ(t0, PObj), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1), and is supposed to be at least C2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Lagrangian velocity is ⃗V (t, pt0) = ∂Φt0 ∂t (t, pt0), and the Eulerian velocity satisfies ⃗v(t, pt) = ∂Φt0 ∂t (t, pt0) when pt = Φt0(t, pt0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let F t0(t, pt0) = dΦt0(t, pt0) = F t0 t (pt0) = dΦt0 t (pt0), and consider the Jacobian Jt0 t (pt0) = det |⃗e (F t0 t (pt0)) = Jt0(t, pt0), (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39) Lemma K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24 ∂Jt0 ∂t (t, pt0) satisfies, with pt = Φt0 t (pt0), ∂Jt0 ∂t (t, pt0) = Jt0(t, pt0) div⃗v(t, pt) (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) (value to be considered at t at pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular, �Φ is incompressible iff div⃗v(t, pt) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let O be a origin in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let −−−→ OΦt0 = �n i=1Φi⃗ei, ⃗V t0 = �n i=1V i⃗ei, ⃗v = �n i=1vi⃗ei, F t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ej = dΦt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ej = �n i=1 ∂Φi ∂Xj ⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let [F t0]| ⃗E,⃗e =noted F, Jt0 =noted J and [dΦi]| ⃗E = � ∂Φi ∂X1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂Φi ∂Xn � =noted dΦi 148 149 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Dilatation rate (row matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus J = det F = det � � dΦ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dΦn � �, thus (a determinant is multilinear) ∂J ∂t = det � � � � � � ∂(dΦ1) ∂t dΦ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dΦn � � � � � � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' + det � � � � � dΦ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dΦn−1 ∂(dΦn) ∂t ) � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With Φt0 C2, thus ∂(dΦi) ∂t (t, pt0) Swhartz = d(∂Φi ∂t )(t, pt0) = dV i(t, pt0) = dvi(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t, pt0), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus det � � � � � � ∂(dΦ1) ∂t dΦ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dΦn � � � � � � = det � � � � � � � n � i=1 ∂v1 ∂xi dΦi dΦ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dΦn � � � � � � � det is = alternating det � � � � � � ∂v1 ∂x1 dΦ1 dΦ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dΦn � � � � � � = ∂v1 ∂x1 det � � � � dΦ1 dΦ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dΦn � � � � = ∂v1 ∂x1 J Idem for the other terms, thus ∂J ∂t (t, pt0) = ∂v1 ∂x1 (t, pt) J(t, pt0) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' + ∂vn ∂xn (t, pt) J(t, pt0) = div⃗v(t, pt) J(t, pt0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25 div⃗v(t, pt) is the dilatation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Leibniz formula Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26 (Leibniz formula) Under regularity assumptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' hypotheses of the Lebesgue theorem to be able to derive under � ) we have d dt �� pt∈Ωt f(t, pt) dΩt � = � pt∈Ωt �Df Dt + f div⃗v � (t, pt) dΩt = � pt∈Ωt �∂f ∂t + df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + f div(⃗v) � (t, pt) dΩt = � pt∈Ωt �∂f ∂t + div(f⃗v) � (t, pt) dΩt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Z(t) := � p∈Ωt f(t, p) dΩt = � P ∈Ωt0 f(t, Φt0(t, P)) Jt0(t, P) dΩt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (The Jacobian is positive for a regular motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Then (derivation under � ) Z′(t) = � P ∈Ωt0 Df Dt (t, pt) Jt0(t, P) + f(t, pt)∂Jt0 ∂t (t, P) dΩt0 = � P ∈Ωt0 (Df Dt (t, pt) + f(t, pt) div⃗v(t, pt))Jt0(t, P) dΩt0, thanks to (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And div(f⃗v) = df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v + f div⃗v gives (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Corollary K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27 With (⃗u, ⃗w)g =noted ⃗u • ⃗w (in the given Euclidean framework), d dt � Ωt f(t, pt) dΩt = � Ωt ∂f ∂t (t, pt) dΩt + � ∂Ωt (f⃗v • ⃗n)(t, pt) dΓt, (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='42) sum of the temporal variation within Ωt and the flux through the surface ∂Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Apply (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 149 150 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂J/∂F = J F −T K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 ∂J/∂F = J F −T K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Meaning of ∂ det ∂Mij ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Mnn = {M = [Mij] ∈ Rn2} be the set of n ∗ n matrices, and consider the function Z := det : � Mnn → R M = [Mij] → Z(M) := det(M) = det([Mij]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43) Question: What does ∂Z ∂Mij (M) mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: It is the “standard meaning” of a directional derivative ∂f ∂xi (⃗x) = df(⃗x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' where here f = Z, thus ⃗x =noted M is a matrix (a vector in Mnn), and (⃗ei) is the canonical basis (mij) in Mnn (all the elements of the matrix mij vanish but the element at intersection of line i and column j which equals 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So: ∂Z ∂Mij (M) := dZ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='mij = lim h→0 Z(M + hmij) − Z(M) h (∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='44) K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Calculation of ∂ det ∂Mij Proposition K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28 ∀i, j, ∂Z ∂Mij (M) = Z(M) (M −T )ij, written ∂Z ∂M = Z M −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂Z ∂Mij (M) := limh→0 det(M+hmij)−det(M) h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The development of the determinant det(M + hmij) relative to the column j gives det(M + h[mij]) = det(M) + h cij (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='46) where cij is the (i, j)-th cofactor of M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ∂Z ∂Mij (M) = limh→0 Z(M+hmij)−Z(M) h = cij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And since M −1 = 1 det(M)[cij]T , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [cij] = det(M)M −T , we get ∂Z ∂Mij (M) = det(M)(M −T )ij, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 ∂J/∂F = J F −T usually written [ ∂J ∂Fij ] = J F −T Setting of § K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8: With F := dΦ(pt0) we have F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ej = �n i=1Fij⃗ei where Fij = ∂Φi ∂Xj (pt0), and JΦ,pt0, ⃗E,⃗e noted = J : � � � � � L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) → R F → J(F) := det([Fij]) (= det([ ∂Φi ∂Xj (pt0)]), (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='47) so, J(F) is the Jacobian � det| ⃗E,⃗e(dΦ(pt0)) of Φ at pt0 relative to ( ⃗Ei) and (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45) gives: Corollary K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29 ∀i, j, ∂J ∂Fij (F) = J(F) ([F]−T )ij, written ∂J ∂F = J F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='48) K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Interpretation of ∂J ∂Fij ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The first derivations into play are along the directions ⃗Ej at t0: The Fij = ∂Φi ∂Xj (pt0) := dΦi(pt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Question: ∂J ∂Fij is the usual notation for a directional derivative, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' § K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So ∂J ∂Fij is the derivative in which direction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: 1- “Identify” F ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) with the tensor �F ∈ L(⃗Rn∗ t , ⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) given by �F(ℓ, ⃗U) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, if F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ej = �n i=1Fij⃗ei then �F = �n i,j=1Fij⃗ei ⊗ πEj, relative to a basis ( ⃗Ei) and its covariant dual basis (πEi) in ⃗Rn t0 and a basis (⃗ei) and ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- Define the function � det ⃗E,⃗e = �J : � � � L(⃗Rn∗ t , ⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) → R �F → �J( �F) := J(F) = det ⃗E,⃗e (F) = det([Fij]) � � �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 150 151 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Transformed parallelepiped 3- Then it is meaningful to differentiate �J along the direction ⃗ei ⊗ πEj ∈ L(⃗Rn∗ t , ⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) to get ∂ �J ∂Fij ( �F) := lim h→0 �J|⃗e, ⃗E( �F + h⃗ei ⊗ πEj) − �J|⃗e, ⃗E( �F) h ( noted = ∂J ∂Fij (F));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='49) This is a derivation in both directions πEj in ⃗Rn t0 (past at pt0) and ⃗ei in ⃗Rn t (present at pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' What does this derivative mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (The answer is unknown to the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') L Transport of volumes and areas Here Rn = R3 the usual affine space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let t0, t ∈ R, and Φt0 t : R × Ωt0 → Ωt, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let FP = dΦt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (·, ·)g be a Euclidean dot product in ⃗Rn (English, French.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='), with ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||g the associated norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Transformed parallelepiped The Jacobian of Φt0 t at P relative to a (·, ·)g-Euclidean bases is defined in (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35): With FP = F t0 t (P), JP = J(P) := det |⃗e (F t0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', F t0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗En)), and JP > 0 (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) the motion being supposed regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, if (⃗U1P , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗UnP ) is a parallelepiped at P at t0, if ⃗uip = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗UiP , then (⃗u1p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗unp) is a parallelepiped at p at t which volume is det |⃗e (⃗u1p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗unp) = JP det |⃗e (⃗U1P , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗UnP ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Transformed volumes Riemann integrals and (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) give the change of variable formula: For any regular function f : Ωt → R, � pt∈Ωt f(pt) dΩt = � P ∈Ωt0 f(Φt0 t (P)) |J(P)| dΩt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) (See (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18): dΩt is a positive measure: It is not a multilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') In particular, |Ωt| = � pt∈Ωt dΩt(pt) = � P ∈Ωt0 |J(P)| dΩt0(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) (With J(P) > 0 for regular motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Transformed parallelogram Consider two independent vectors ⃗U1P , ⃗U2P ∈ ⃗Rn t0 at t0 at P, and the vectors ⃗u1p = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U1P and ⃗u2p = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U2P at t at p = Φt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Since Φt0 t is a diffeomorphism, ⃗u1p and ⃗u2p are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then choose a Euclidean dot product (·, ·)g (English, French.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') to be able to use the vectorial product, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15), the same at all time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then the areas of the parallelograms are ||⃗U1P ∧ ⃗U2P ||g and ||⃗u1p ∧ ⃗u2p)||g, (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) and unit normal vectors to the quadrilaterals are ⃗NP = ⃗U1P ∧ ⃗U2P ||⃗U1P ∧ ⃗U2P ||g ∈ ⃗Rn t0, and ⃗np = ⃗u1p ∧ ⃗u2p ||⃗u1p ∧ ⃗u2p||g ∈ ⃗Rn t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) Proposition L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 If ⃗u1p = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U1P and ⃗u2p = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U2P , then ⃗u1p ∧ ⃗u2p = JP F −T P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' �⃗U1P ∧ ⃗U2P � , and ||⃗u1p ∧ ⃗u2p||g = JP ||F −T P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗U1P ∧ ⃗U2P )||g, (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) since JP > 0 (for regular motions), and ⃗np = F −T P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗NP ||F −T P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗NP ||g (̸= FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗NP in general).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) 151 152 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Transformed surface Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗WP ∈ ⃗ Rn t0 and ⃗wp = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗WP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then the volume of the parallelepiped (⃗u1p, ⃗u2p, ⃗wp) is (⃗u1p ∧ ⃗u2p, ⃗wp)g = det(⃗u1p, ⃗u2p, ⃗wp) = det(FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U1P , FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗U2P , FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗WP ) = det(FP ) det(⃗U1P , ⃗U2P , ⃗WP ) = JP (⃗U1P ∧ ⃗U2P , ⃗WP )g = JP (⃗U1P ∧ ⃗U2P , F −1 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wp)g = JP (F −T P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗U1P ∧ ⃗U2P ), ⃗wp)g, (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) for all ⃗wp, thus (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7), thus ⃗u1p∧⃗u2p ||⃗u1p∧⃗u2p||g = JP F −T P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗U1P ∧⃗U2P ) JP ||F −T P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗U1P ∧⃗U2P )||g , thus (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Transformed surface L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Deformation of a surface A parametrized surface Ψt0 in Ωt0 and the associated geometric surface St0 are defined by Ψt0 : � [a, b] × [c, d] → Ωt0 (u, v) → P = Ψt0(u, v) � and St0 = Im(Ψt0) ⊂ Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) (It is also represented after a choice of an origin O by the vector valued parametrized surface ⃗rt0 = −−−→ OΨt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') The transformed parametric surface is Ψt := Φt0 t ◦ Ψt0 and the associated geometric surface is St: Ψt := Φt0 t ◦ Ψt0 : � [a, b] × [c, d] → Ωt0 (u, v) → p = Ψt(u, v) = Φt0 t (Ψt0(u, v)) = Φt0 t (P) � and St = Φt0 t (St0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) (After a choice of an origin O, the associated vector valued parametrized surface is ⃗rt = −−→ OΨt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Let ( ⃗E1, ⃗E2) be the canonical basis in the space R × R ⊃ [a, b] × [c, d] = {(u, v)} of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The surface Ψt0 is supposed to be regular, that is, Ψt0 is C1 and, for all P = Ψt0(u, v) ∈ St0, the tangents vectors ⃗T1P and ⃗T2P at P are independent, that is, ⃗T1P := dΨt0(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E1 noted = ∂Ψt0 ∂u (u, v), ⃗T2P := dΨt0(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E2 noted = ∂Ψt0 ∂v (u, v), � � � � � and ⃗T1P ∧ ⃗T2P ̸= ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) And the tangent vectors at St at p = Φt0 t (P) at t are � � � � � ⃗t1p := dΨt(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E1 = ∂Ψt ∂u (u, v), so ⃗t1p = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗T1P (= dΦt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂Ψt0 ∂u (u, v)), ⃗t2p := dΨt(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗E2 = ∂Ψt ∂v (u, v), so ⃗t2p = FP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗T2P (= dΦt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂Ψt0 ∂v (u, v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) These vectors are independent since Φt0 t is a diffeomorphism and Ψt0 is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In facts, we used tangent vectors to curves and their push-forwards, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 and § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Euclidean dot product and unit normal vectors Then choose a Euclidean dot product (·, ·)g (English, French.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='), to be able to use the vectorial product, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15), the same at all time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then the scalar area elements dΣP at P at St0 relative to Ψt0, and dσp at p at St relative to Ψt, are � � � � � dΣP := ||∂Ψt0 ∂u (u, v) ∧ ∂Ψt0 ∂v (u, v)||g du dv (= ||⃗T1P ∧ ⃗T2P ||g du dv), dσp := ||∂Ψt ∂u (u, v) ∧ ∂Ψt ∂v (u, v)||g du dv (= ||⃗t1p ∧ ⃗t2p||g du dv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) And the areas of St0 and St are � � � � � � � � � |St0| = � P ∈St0 dΣP := � b u=a � d v=c ||∂Ψt0 ∂u (u, v) ∧ ∂Ψt0 ∂v (u, v)||g du dv, |St| = � p∈St dσp := � b u=a � d v=c ||∂Ψt ∂u (u, v) ∧ ∂Ψt ∂v (u, v)||g du dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) (See (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18): dΣP and dσp are positive measures: They are not multilinear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 152 153 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Piola identity And the unit normal vectors ⃗NP at St0 at P at t0 and ⃗np at St at p at t are � � � � � � � � � � � ⃗NP = ∂Ψt0 ∂u (u, v) ∧ ∂Ψt0 ∂v (u, v) || ∂Ψt0 ∂u (u, v) ∧ ∂Ψt0 ∂v (u, v)||g (= ⃗T1P ∧ ⃗T2P ||⃗T1P ∧ ⃗T2P ||g ) ⃗np = ∂Ψt ∂u (u, v) ∧ ∂Ψt ∂v (u, v) || ∂Ψt ∂u (u, v) ∧ ∂Ψt ∂v (u, v)||g (= ⃗t1p ∧ ⃗t2p ||⃗t1p ∧ ⃗t2p||g = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) Then the vectorial area elements d⃗ΣP at P at St0 = Im(Ψt0) relative to ⃗rt0 and d⃗σp at p at St = Im(Ψt) relative to Ψt are � � � � � d⃗ΣP := ⃗NP dΣP = ∂Ψt0 ∂u (u, v) ∧ ∂Ψt0 ∂v (u, v) du dv (= ⃗T1P ∧ ⃗T2P du dv) d⃗σp := ⃗np dσp = ∂Ψt ∂u (u, v) ∧ ∂Ψt ∂v (u, v) du dv (= ⃗t1p ∧ ⃗t2p du dv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) (Useful to get the flux through a surface: � Γ ⃗f • ⃗n dσ = � Γ ⃗f • d⃗σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') (NB: d⃗ΣP and d⃗σp are not multilinear since dΣP and dσp are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Relations between surfaces ⃗t1p ∧ ⃗t2p = JP F −T P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗T1P ∧ ⃗T2P ), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7), gives ∂Ψt ∂u (u, v) ∧ ∂Ψt ∂v (u, v) = JP F −T P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (∂Ψt0 ∂u (u, v) ∧ ∂Ψt0 ∂v (u, v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) This gives the relation between vectorial and scalar area elements, ⃗n dσp = d⃗σp = JP F −T P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗ΣP = JP F −T P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗NP dΣP , and dσp = JP ||F −T P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗NP ||g dΣP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) (Check with (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Piola identity Reminder: Let M = [M i j] be a 3∗3 matrix function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We use the usual divergence in continuum mechanics (non objective) given by divM := � � � ∂M 1 1 ∂X1 + ∂M 1 2 ∂X2 + ∂M 1 3 ∂X3 ∂M 2 1 ∂X1 + ∂M 2 2 ∂X2 + ∂M 2 3 ∂X3 ∂M 3 1 ∂X1 + ∂M 3 2 ∂X2 + ∂M 3 3 ∂X3 � � � = � � � � �n j=1 ∂M 1 j ∂Xj �n j=1 ∂M 2 j ∂Xj �n j=1 ∂M 3 j ∂Xj � � � �, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And if Cof(M) is the matrix of cofactors (in R3: Cof(M)i j = M i+1 j+1M i+2 j+2 − M i+1 j+2M i+2 j+1), then M −1 = 1 det M Cof(M)T , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', (det M)M −1 = Cof(M)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) The framework being Euclidean, we use a Euclidean basis and the associated matrix, and thus (matrix meaning) J(P)F(P)−T = Cof(F(P)) noted = Cof(F)(P), written JF −T = Cof(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) Proposition L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 (Piola identity) In R3, we have div(JF −T )(P) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∀i, n � j=1 ∂Cof(F)i j ∂Xj (P) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) Also written �n j=1 ∂ ∂Xj (J ∂Xi ∂xj ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) is a just a matrix computation since we used the divergence of a matrix (we used components relative to a given basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We are in R3, thus Cof(F)i j = F i+1 j+1F i+2 j+2 − F i+1 j+2F i+2 j+1, and F = [dΦt] = [ ∂ϕi ∂Xj ], that is, F i j = ∂ϕi ∂Xj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ∂Cof(F)i j ∂Xj = ∂2ϕi+1 ∂Xj∂Xj+1 ∂ϕi+2 ∂Xj+2 + ∂ϕi+1 ∂Xj+1 ∂2ϕi+2 ∂Xj∂Xj+2 − ∂2ϕi+1 ∂Xj∂Xj+2 ∂ϕi+2 ∂Xj+1 − ∂ϕi+1 ∂Xj+2 ∂2ϕi+2 ∂Xj∂Xj+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, for all i = 1, 2, 3, we get �n j=1 ∂Cof(F )i j ∂Xj = 0 (the terms cancel each other out two by two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 153 154 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Piola transformation L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Piola transformation Let ⃗u be a vector field in Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The goal is to find a vector field ⃗UPiola in Ωt0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' for all open subset ωt ⊂ Ωt with ωt0 = Φt0 t −1(ωt) ⊂ Ωt0, � ∂ωt0 ⃗UPiola • ⃗N dΣ = � ∂ωt ⃗u • ⃗n dσ, (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) or � ωt0 div(⃗UPiola) dΩt0 = � ωt div(⃗u) dΩt, (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � P ∈ωt0 div(⃗UPiola)(P) dΩt0 = � P ∈ωt0 div(⃗u)(Φt0 t (P)) J(P) dΩt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) (The motion is supposed to be regular, so J(P) > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus we want div⃗UPiola(P) = J(P) div⃗u(p) when p = Φt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) Definition L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 The Piola transform is the map � � � C∞(Ωt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Rn) → C∞(Ωt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Rn) ⃗u → ⃗UPiola, ⃗UPiola(P) := J(P)F(P)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u(p) when p = Φt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) (So ⃗UPiola(P) = J(P)Φ∗(⃗u)(P) where Φ∗(⃗u)(P) = F(P)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u(p) = the pull-back with Φ = Φt0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proposition L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 With p = Φt0 t (P), ⃗UPiola = �n i=1U i Piola⃗ei and ⃗u = �n i=1ui⃗ei we get div⃗UPiola(P) = J(P) div⃗u(p), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' n � i=1 ∂U i Piola ∂Xi (P) = J(P) n � i=1 ∂ui ∂xi (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' div(τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) =(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='61) � div(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗w gives div((JF −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗u ◦ Φt0 t ))(P) = (div(JF −T )(P), ⃗u(p))g + (J(P)F(P)−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. (d⃗u(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(P))=(L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22)0+J(P)(F(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(P)−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗u(p) = J(P) I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗u(p) = J(P)div⃗u(p), which gives (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' M Work and power M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definitions M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Work (Thermodynamic like approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') The elementary work is a differential form α, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' α = dU (internal energy density), α = δW = (elementary work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider a regular curve c : t ∈ [t0, T] → c(t) ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And let ⃗v(t, c(t)) := ⃗c ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The work of α along the curve is � c α := � T t=t0 α(t, c(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗c ′(t) dt noted = � T t=t0 α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗c = � T t=t0 α(t, c(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v(t, c(t)) dt noted = � T t=t0 α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', W t0 T (α, c) = � c δW = work along c of the differential form α = δW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then consider an object Obj and its motion �Φ : (t, PObj) → p(t) = �Φ(t, PObj) = �ΦPObj (t) ∈ Rn, the curves cPObj = �ΦPObj : t ∈ [t0, T] → p(t) = �ΦPObj (t) ∈ Rn, and the Eulerian velocities ⃗v(t, p(t)) = �ΦPObj ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The work for Obj and a Eulerian differential form α along �Φ is the sum of work of α of all particles, formally � PObj ∈Obj( � cPObj αPObj ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So with the associated motion Φt0(t, pt0) = �Φ(t, PObj) = p(t) = Φt0 pt0 (t) when pt0 = �Φ(t0, PObj), and with Ωt = �Φ(t, Obj), W t0 T (�Φ) := � pt0∈Ωt0 � T t=t0 α(t, Φt0 pt0 (t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v(t, Φt0 pt0 (t)) dt dΩt0 = � T t=t0 � pt∈Ωt α(t, pt)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v(t, pt) dΩt dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) (The last equality if Fubini theorem can be applied, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' if α is C0 and Φt0 is C1, Obj being bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 154 155 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Piola–Kirchhoff tensors Exercice M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 If α is a stationary and exact differential form, α = dU, then prove that � c dU = U(c(T)) − U(c(t0)) noted = ∆U (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) only depends on the extremities c(t0) and c(T) of the curve c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � c dU = � T t=t0 dU(c(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗c ′(t) dt = � T t=t0 d(U◦c) dt (t) dt = [U ◦ c]T t0 = U(c(T)) − U(c(t0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark (continuum mechanics): An observer chooses a Euclidean dot product (·, ·)g = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' •g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' • .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (if (·, ·)g is imposed and implicit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And if he chooses to represent a linear form αt(pt) with its (·, ·)g-Riesz representation vector ⃗ft(pt) (observer dependent), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8), then � c α = � T t=t0 α(t, c(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗c ′(t) dt = � T t=t0 ⃗f • d⃗c = � T t=t0 ⃗f • ⃗v dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 And its associated power density Definition: The power density of a differential form α along �Φ is the Eulerian function ψ := α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v : � � � � � C = � t∈[t0,T ] ({t} × Ωt) → R (t, p) → ψ(t, p) = α(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) And the power at t is Pt(�Φ) = P(t, �Φ) := � p∈Ωt ψ(t, p) dΩt = � p∈Ωt αt(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vt(p) dΩt noted = P(t,⃗vt) = Pt(⃗vt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) Remark: With a Euclidean dot product (·, ·)g, then with the (·, ·)g-Riesz representation vector ⃗f of α (observer dependent) we get ψ = ⃗f • ⃗v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ψ(t, p) = ⃗f(t, p) •g ⃗v(t, p) (= (⃗f(t, p),⃗v(t, p))g), (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) which gives P(t, �Φ) := � p∈Ωt ⃗f(t, p) • ⃗v(t, p) dΩt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Piola–Kirchhoff tensors Consider a regular Eulerian velocity field ⃗v, so d⃗v is an endomorphism (identified with a �1 1 � tensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then we need another endomorphism τ (identified with a �1 1 � to get the objective double contraction τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗v := Tr(τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v), (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) which means τ(t, p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗v(t, p) := Tr(τ(t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v(t, p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification: With a basis (⃗ei) at t and τ = � ij τ i j⃗ei ⊗ ej and d⃗v = � jk vj |k⃗ej ⊗ ek (the endomor- phisms have been written like �1 1 � tensors for calculation purpose), τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗v = � ijk τ i jvj |k⃗ei ⊗ ek and τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗v = n � i,j=1 τ i jvj |i (objective value), (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) see (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then choose a Euclidean dot product (·, ·)g, to be able to use the double matrix product τ : d⃗v := n � i,j=1 τ i jvi |j = [τ]|⃗e T : [d⃗v]|⃗e (subjective value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Objective internal power for the stress: function of d⃗v Usual hypothesis for the internal stress in a material: At first order, the power density is of the type ψ = τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ψ(t, p) = τ(t, p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗v(t, p), ∀(t, p) ∈ C, (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) thus the power at t is Pt(⃗vt) = � p∈Ωt ψ(t, p) dΩt = � p∈Ωt τ t(p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗vt(p) dΩt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) 155 156 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Piola–Kirchhoff tensors M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The first Piola–Kirchhoff tensor The Piola–Kirchhoff approach consists in transforming Eulerian quantities into Lagrangian quantities to refer to the initial configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) gives Pt(⃗vt) = � P ∈Ωt0 ψt(Φt0 t (P)) |Jt0 t (P)| dΩt0 = � P ∈Ωt0 τ t(Φt0 t (P)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗vt(Φt0 t (P)) Jt0 t (P) dΩt0 (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) (the Jacobian Jt0 t (P) = det(F t0 t (P)) of Φt0 t at P is positive for a regular motion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Lagrangian velocity ⃗V t0(t, P) = ⃗vt(Φt0 t (P)) satisfies d⃗V t0 t (P) = d⃗vt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (P) where pt = Φt0(t, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus τ t(pt) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗vt(pt) = τ(pt) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. (d⃗V t0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (P)−1) = (F t0 t (P)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='τ(pt)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗V t0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) Quantification: Choose a basis and a Euclidean dot product (·, ·)g, thus Pt(⃗vt) = � P ∈Ωt0 (Jt0 t (P)τ(pt)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (P)−T � �� � PKt0 t (P ) ) : d⃗V t0 t (P) dΩt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) Definition M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The first Piola–Kirchhoff (two point) tensor at P ∈ Ωt0, relative to t0, t and a basis (⃗ei), is the linear map PKt0 t (P) ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) defined by PKt0 t (P) = Jt0 t (P) σt(Φt0 t (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (P)−T , where σ = τ T , (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) abusively written PK = J σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) Hence Pt(⃗vt) = � Ωt0 PKt0 t (P) : d⃗V t0 t (P) dΩt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) Remark M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 The Piola–Kirchhoff tensor is not that easy to master: Everything is quite simple in a Eulerian framework (the configuration at t where the laws are expressed to begin with), but then everything is made more complicated when expressed in an initial configuration (at t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, when the Piola–Kirchhoff tensor is used to introduce the Lie derivatives (Eulerian type), it makes the Lie derivative quite a mysterious mathematical object, see footnote page 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Continuation of the remark: With the pull-backs, (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) reads (Jt0 t (P) being positive) P(t, �Φ) = � pt∈Ωt ψt(pt) dΩt = � P ∈Ωt0 � (Φt0 t )∗ψt � (P) � (Φt0 t )∗dΩt � , (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) since ((Φt0 t )∗dΩt) = Jt0 t (P) dΩt0 and ((Φt0 t )∗ψt)(P) = ψt(pt) (scalar valued functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It gives the Piola–Kirchhoff tensor (pull-back to the initial configuration) since (Φt0 t )∗(αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vt)(pt) = (αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vt)(Φt0 t (P)) = αt(Φt0 t (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗vt(Φt0 t (P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 The second Piola–Kirchhoff tensor The first Piola–Kirchhoff tensor PK may confuse Eulerian and Lagrangian variables, linear maps and endomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And PK(pt0) is not symmetric: It can’t be since PK(pt0) ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) is not an endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' To get a symmetric tensor, the second Piola–Kirchhoff tensor is defined: Definition M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 The second Piola–Kirchhoff tensor is the endomorphism SKt0 t (P) ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) defined by, in short, SK = F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PK = JF −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) Full notation: SKt0 t (P) = (F t0 t (P))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PKt0 t (P) = Jt0 t (P)(F t0 t (P))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='σt(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F t0 t (P))−T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So if σt(p) ∈ L(⃗Rn t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) is symmetric then SKt0 t (P) ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 156 157 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Classical hyper-elasticity: ∂W/∂F Thus, with the pull-back of the endomorphism d⃗vt ∈ L(⃗Rn t , ⃗Rn t ): ((Φt0 t )∗d⃗vt)(P) = F t0 t (P)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗vt(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (P), (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) and with d⃗vt(pt) = d⃗V t0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t (P)−1 and σt(p) symmetric (so SKt0 t is symmetric), Pt(⃗vt) = � Ωt0 PKt0 t : d⃗V t0 t dΩt0 = � Ωt0 (F t0 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='SKt0 t ) : d⃗V t0 t dΩt0 = � Ωt0 ([F t0 t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [SKt0 t ]) : [d⃗V t0 t ]T dΩt0 = � Ωt0 SKt0 t : ((d⃗V t0 t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t ) dΩt0 = � Ωt0 SKt0 t : (F t0 t T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗V t0 t + d(⃗V t0 t )T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F t0 t 2 ) dΩt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) Remark M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 It is a “chosen time derivative” of SK(t) = J(t)F(t)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='σ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F(t)−T that leads to some kind of Lie derivative as explain in books in continuum mechanics, as in footnote page 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Classical hyper-elasticity: ∂W/∂F E and F are finite dimensional spaces, dim E = n, dim F = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition Reminder: Consider a function � W : � L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) → R L → � W(L) (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) Its differential d� W : � L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) → L(L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) L → d� W(L) � is defined at L by, in a direction M, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3), d� W(L)(M) = lim h→0 � W(L + hM) − � W(L) h noted = ∂� W ∂L (L)(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) Also written d� W(L)(M) = d� W(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M since d� W(L) is linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 � W : F ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) → � W(F) ∈ R (real valued function), with F := F t0 t (pt0) the deformation gradient at ∈ Ωt0 at t at pt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus d� W(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M = limh→0 � W (F +hM)−� W (F ) h =noted ∂� W ∂F (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M ∈ R is the derivative of � W at F in a direction M ∈ L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 m = n, endomorphisms L ∈ L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn), and � W(L) := Tr(L) (the trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Here dTr(L)(M) = limh→0 Tr(L+hM)−Tr(L) h = Tr(M) (the trace is linear), thus dTr(L) = Tr for all L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Expression with bases (quantification): The ∂W/∂Lij Let (⃗ai) and (⃗bi) be bases in E and F, with (πai) the (covariant) dual basis of (⃗ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (Lij) i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n = (⃗bi ⊗ πaj) be the associated basis in L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the Lij are defined by Lij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aℓ = δjℓ⃗bi for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', m and j, ℓ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41 (all the elements of the matrix [Lij]|⃗a,⃗b vanish except the element at the intersection of row i and column j which equals 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The derivation of � W at L in a direction Lij is d� W(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Lij = lim h→0 � W(L + hLij) − � W(L) h noted = ∂� W ∂Lij (L) (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) (usual notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Associated matrix relative to the chosen bases: [d� W(L)]|Lij := [ ∂� W ∂Lij ] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',m j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n noted = [d� W(L)]|⃗a,⃗b ( noted = [d� W(L)ij]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) So if M = �m i=1 �n j=1MijLij then d� W(L)(M) = � ij Mij d� W(L)(Lij) since d� W(L) is linear, so d� W(L)(M) = n � i,j=1 ∂� W ∂Lij (L)Mij = [d� W(L)]|⃗a,⃗b : [M]|⃗a,⃗b, (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) double matrix contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notations: d� W(L)(M) = � ij ∂� W ∂Li j (L)M i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 157 158 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Classical hyper-elasticity: ∂W/∂F Remark M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 The notation [M]| ⃗E,⃗e : [d� W(L)]| ⃗E,⃗e is just a matrix product, since M = L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ Rm) and d� W(L) ∈ L(L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ Rm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) are different kinds of mathematical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 Continuing example M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 with (⃗ei) = ( ⃗Ei): Then � W(L) = Tr(L) gives d� W(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M = Tr(M) = � i Mii, thus ∂� W ∂Lij (L) = δij for all i, j, thus [d� W(L)]|⃗e = [I] = [ ∂Tr ∂Lij (L)] (identity matrix), and we recover dTr(L)(M) = [ ∂Tr ∂Lij (L)] : [M] = [I] : [M] = �n i=1Mii = Tr(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 Continuing example M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7: The meaning of the derivation ∂� W ∂Fij is intriguing: It is a derivation in the direction Lij =noted ⃗ei ⊗ πEj, where (⃗ei) is a basis at p = Φt0 t (P) in ⃗Rn t and (πEj) is the dual basis of a basis ( ⃗Ej) at P in ⃗Rn t0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂� W ∂Fij (F) = d� W(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='Lij =noted d� W(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗ei ⊗ πEj) is a derivation “at the same time” in the directions ⃗ei (at (t, p)) and πEj (at (t0, P)), where F stands for F t0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Motions and ω-lemma Generalization of (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) to C1 functions, with UE open subset in a affine space which associated vector space is E, � W : � UE × L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) → R (P, L) → � W(P, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) At P, let � WP (L) := � W(P, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The differential d� WP (L) =noted ∂2� W(P, L) in a direction M ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) is ∂2� W(P, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M := d(� WP )(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M = lim h→0 � W(P, L + hM) − � W(P, L) h noted = ∂� W ∂L (P, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) With a motion Φ := Φt0 t : Ωt0 → Ωt define f : � C1(Ωt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ωt) → C0(Ωt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) Φ → f(Φ) := � W(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', dΦ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' )), (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) a function of Φ which only depends on its first (covariant) gradient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, for all P ∈ Ωt0, f(Φ)(P) = � W(P, dΦ(P)) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) (This kind of relation is generally deduced after application of the frame invariance principle, and the hypothesis of dependence on only the first order derivative dΦ = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Lemma M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 (ω-lemma) For all Φ, Ψ ∈ C1(Ωt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ωt), df(Φ)(Ψ) = ∂2� W(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', dΦ)(dΨ) noted = ∂� W ∂F (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', dΦ)(dΨ) , (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all P ∈ Ωt0, df(Φ)(Ψ)(P) = ∂� W ∂F (P, dΦ(P))(dΨ(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With bases ( ⃗Ei) and (⃗ei) in ⃗Rn t0 and ⃗Rn t and dΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Ej = �n i=1 ∂Ψi ∂Xj ⃗ei, we get df(Φ)(Ψ) = n � i,j=1 ∂� W ∂Fij (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', dΦ) ∂Ψi ∂Xj (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') noted = [ ∂� W ∂Fij ] : [ ∂Ψi ∂Xj ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) Marsden notations: df(Φ)(Ψ) = �n i,J=1 ∂� W ∂F i J ∂Ψi ∂XJ = [ ∂� W ∂F i J ] : [ ∂Ψi ∂XJ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' C1(Ωt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ωt) is a vector space, so df(Φ)(Ψ) = limh→0 f(Φ+hΨ)−f(Φ) h ∈ C0(Ωt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Ωt), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for any P ∈ Ωt0 we have df(Φ)(Ψ)(P) = limh→0 f(Φ+hΨ)(P )−f(Φ)(P ) h = limh→0 � W (P,dΦ(P )+h dΨ(P ))−� W (P,dΦ(P )) h = d ⃗WP (dΨ(P), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) 158 159 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Classical hyper-elasticity: ∂W/∂F M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Application to classical hyper-elasticity: PK = ∂W/∂F Let (·, ·)g be a unique Euclidean dot product in ⃗Rn t at all times t, and let ( ⃗Ei) and (⃗ei) be Euclidean bases at t0 and at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let σt(p) = the Cauchy stress tensor at t at p = Φt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let PK(P) = J(P) σt(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F −T (P) = the first Piola–Kirchhoff (two point) tensor at P, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Since PK depends on Φ, the full notation is PK = PK(Φ) given by PK(Φ)(P) = J(P) σt(Φ(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΦ(P)−T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) Definition M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 If there exists a function � PK such that PK reads PK(Φ)(P) = � PK(P, dΦ(P)) (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) then � PK is called a constitutive function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (First order hypothesis: � PK only depends on dΦ = F the first order derivative of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Definition M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 The material is hyper-elastic iff there exists a function � W : � Ωt0 × L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t ) → R (P, L) → � W(P, L) � such that (PK(Φ) =) � PK(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', dΦ) = ∂� W ∂F (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', dΦ), written � PK = ∂� W ∂F , (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) that is, � PK(P, F(P)) = ∂� W ∂F (P, F(P)) for all P ∈ Ωt0, where F = dΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With bases ( ⃗EI) and (⃗ei) in ⃗Rn t0 and ⃗Rn t , and (EI) the dual basis of ( ⃗EI), and PK = �n i,J=1PKi J⃗ei⊗EJ, [PK(Φ)]| ⃗E,⃗e = [� PK(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', F)]| ⃗E,⃗e = [∂� W ∂F (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', F)]| ⃗E,⃗e, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [PKi J] = [ ∂� W ∂F i J (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', F)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) Thus, for any (virtual) motion Ψ : Ωt0 → Ωt, with (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) and (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27), � PK(dΦ)(dΨ) = ∂� W ∂F (dΦ)(dΨ) = � iJ ∂� W ∂F i J (F) ∂Ψi ∂XJ noted = [� PK] : [dΨ], (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38) that is, � PK(dΦ)(dΨ)(P) = � iJ ∂� W ∂F i J (P, F t0 t (P)) ∂Ψi ∂XJ (P) for all P ∈ Ωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15 With a unique Euclidean dot product (·, ·)g both in ⃗Rn t0 and ⃗Rn t , with Euclidean bases ( ⃗Ei) ∈ ⃗Rn t0 and (⃗ei) ∈ ⃗Rn t , and with (Ei) the dual basis of ( ⃗Ei), with C = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F, prove (derivation in the direction ⃗ei ⊗ EJ): ∂C ∂F i J (F) = � K F i K ⃗EJ ⊗ EK + � K F i K ⃗EK ⊗ EJ (= dC(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗ei ⊗ EJ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39) ∂ √ C ∂F (F) = 1 2 �� C(F) �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂C ∂F (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) ∂ √ C ∂C = 1 2( √ C)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let F = � iJ F i J⃗ei ⊗ EJ, so F T = � Ij(F T )I j ⃗EI ⊗ ej = � Ij F j I ⃗EI ⊗ ej, and C = � IJ CI J ⃗Ei ⊗ EJ = F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F = � IJ � k(F T )I kF k J ⃗EI ⊗ EJ = � IJ � k F k I F k J ⃗EI ⊗ EJ = C(F), so CI J = � k F k I F k J = CI J(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And C(F + h⃗ei ⊗ EJ) = (F + h⃗ei ⊗ EJ)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F + h⃗ei ⊗ EJ) = (F T + h ⃗EJ ⊗ ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F + h⃗ei ⊗ EJ) = C(F) + h ( ⃗EJ ⊗ ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='F + h F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗ei ⊗ EJ) + h2 ⃗EJ ⊗ Ei = C(F) + h ( � K F i K ⃗EJ ⊗ EK + � K (F T )K i ⃗EK ⊗ EJ) + h2 ⃗EJ ⊗ EJ (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='42) Thus (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And C(F + h⃗ei ⊗ EJ) − C(F) = ( √ C(F + h⃗ei ⊗ EJ) + √ C(F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ( √ C(F + h⃗ei ⊗ EJ) − √ C(F)) gives dC(F)(⃗ei ⊗ EJ) = 2 √ C(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d √ C(F)(⃗ei ⊗ EJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ∂ √ C ∂F i J (F) = 1 2( � C(F))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂C ∂F i J (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (C + h⃗ei ⊗ ej) − C = ( √ C + h⃗ei ⊗ ej + √ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ( √ C + h⃗ei ⊗ ej − √ C), divided by h, gives ⃗ei ⊗ ej = 2 √ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' limh→0 √ C+h⃗ei⊗ej− √ C h = 2 √ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d √ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(⃗ei ⊗ ej), thus L = 2 √ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(d √ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L) for all L (linearity of d √ C), thus d √ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L = 1 2( √ C)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 159 160 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Classical hyper-elasticity: ∂W/∂F M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Corollary (hyper-elasticity): SK = ∂W/∂C With the symmetry of the second Piola–Kirchhoff tensor SK = F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='PK, we deduce SKt0 t (Φt0 t )(P) = � SKt0 t (P, F t0 t (P)) (constitutive function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And we deduce the existence of a function � W : � Ωt0 × L(⃗Rn t0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn t0) → R (P, L) → � W(P, L) � such that, � SKt0 t (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', C) = ∂� W ∂C (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43) (See Marsden and Hughes [12] for details and the thermodynamical hypotheses required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') N Conservation of mass Let ρ(t, p) = ρt(p) be the (Eulerian) mass density at t at p ∈ Ωt, supposed to be > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The mass m(ωt) of a subset ωt ⊂ Ωt is m(ωt) = � p∈ωt ρt(p) dωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) Conservation of mass principle (no loss nor production of particles): For all ωt0 ⊂ Ωt0 and all t, m(ωt) = m(ωt0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � p∈ωt ρt(p) dωt = � P ∈ωt0 ρt0(P) dωt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) Proposition N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 If (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) then, with Jt0 t (P) = det(dΦt0 t (P)) (positive Jacobian the motion being sup- posed regular) and p = Φt0 t (P), ρt(p) = ρt0(P) Jt0 t (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The change of variable formula gives � p∈ωt ρt(p) dωt = � P ∈ωt0 ρt(Φt0 t (P)) Jt0 t (P) dωt0, thus (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) gives ρt(Φt0 t (P))Jt0 t (P) = ρt0(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 ⃗v = ⃗v(t, pt) being the Eulerian velocity at (t, pt) ∈ R × Ωt, (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) gives Dρ Dt + ρ div⃗v = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂ρ ∂t + div(ρ⃗v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) Thus, for all ωt ⊂ Ωt, � ωt ∂ρ ∂t dωt = − � ∂ωt ρ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n dσt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) gives d dt( � p(t)∈ωt ρ(t, p(t)) dωt) = 0, and Leibniz formula (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) applied for all ωt gives (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then the Green formula � Ωt div(ρ⃗v) dΩt = � ∂Ωt ρ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n dσt gives (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Use (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) to prove (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' J(t, P)ρ(t, Φ(t, P)) = ρt0(P) give, with pt = Φ(t, P), ∂J ∂t (t, P) ρ(t, pt) + J(t, P) �∂ρ ∂t (t, pt) + dρ(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΦ(t, P) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ∂J ∂t (t, P) = J(t, P) div⃗v(t, p), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40), gives (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 160 161 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Framework O Balance of momentum O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Framework �Φ : [t0, T] × Obj → Rn is a regular motion, Ωt = �Φ(t, Obj), Γt = ∂Ωt (the boundary), ⃗v is the Eulerian velocity field, ωt is a regular sub domain in Ωt and ∂ωt is its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' An observer chooses a Euclidean basis (⃗ei) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' made with the foot or the metre) and call (·, ·)g the associated Euclidean dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And ⃗n(t, p) = ⃗nt(p) is the outer unit normal at t at p ∈ ∂ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' All the functions are assumed to be regular enough to validate the following calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ρ : � � � � � � t∈[t0,T ] ({t} × Ωt) → R (t, pt) → ρ(t, pt) � � � � � (a mass density), let ⃗f : � � � � � � t∈[t0,T ] ({t} × Ωt) → ⃗Rn (t, pt) → ⃗f(t, pt) � � � � � (a body force density), and let ⃗T : � � � � � � t∈[t0,T ] ({t} × ∂ωt × ⃗ Rn t ) → ⃗Rn (t, pt,⃗n(pt)) → ⃗T(t, pt,⃗n(pt)) � � � � � (a surface force density) defined for any regular subset ωt ⊂ Ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Master balance law Definition O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The balance of momentum is satisfied by ρ, ⃗f and ⃗T iff, for all regular open subset ωt in Ωt, d dt( � ωt ρ⃗v dΩt) = � ωt ⃗f dΩt + � ∂ωt ⃗T∂ωt dΓt (master balance law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) (It is in fact a linearity hypothesis, see theorem O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Thus, with (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41), � ωt D(ρ⃗v) Dt + ρ⃗v div⃗v dΩt = � ωt ⃗f dΩt + � ∂ωt ⃗T∂ωt dΓt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) And with the conservation of mass hypothesis, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4), we get � ωt ρD⃗v Dt dΩt = � ωt ⃗f dΩt + � ∂ωt ⃗T∂ωt dΓt, (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) with D⃗v Dt = ⃗γ = the Eulerian acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Cauchy theorem ⃗T = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n (stress tensor σ) Theorem O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 (Cauchy first law: Cauchy stress tensor) If the master balance law (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) is satis- fied, then ⃗T is linear in ⃗n, that is, there exists a Eulerian endomorphism σ, identified to a Eulerian tensor σ ∈ T 1 1 (Ωt), called the Cauchy stress tensor, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' on all ∂ωt, in short ⃗T = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n, (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) where ⃗n is the unit outward normal to ∂ωt (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗T(t, pt) = σ(t, pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n(t, pt) for all t and pt ∈ ∂ωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The proof is based on: Lemma O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Let ϕ : � Ω → R p → ϕ(p) � ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) and ψ : � Ω × ⃗R3 → R (p, ⃗w) → ψ(p, ⃗w) � ∈ C1(Ω, ⃗R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If ∀ω open in Ω, � p∈ω ϕ(p) dΩ = � p∈∂ω ψ(p,⃗n(p)) dΓ (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) (no dependence on the curvature or on higher derivatives since at any p ∈ ∂ω, ψ only depends on ⃗n(p)), then ∃⃗k ∈ C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗R3) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ψ = (⃗k,⃗n)g, and ϕ = div⃗k, (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ψ depends linearly on ⃗n, and ϕ is a divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 161 162 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Cauchy theorem ⃗T = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n (stress tensor σ) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Lemma O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') (This proof is standard: We recall it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Let p ∈ Ω ⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider the tetrahedral defined by its vertices p, p + (h1, 0, 0), p + (0, h2, 0) and p + (0, 0, h3), with hi > 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (On each face of a tetrahedron, the unit normal vector is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Let Σ1 the side which outer unit normal is − ⃗E1: It area is σ1 = 1 2h2h3 (square triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Idem for Σ2 and Σ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Σ be the fourth side: its area is σ = 1 2 � h2 2h2 3 + h2 3h2 1 + h2 1h2 2 and its outer unit normal is ⃗n = 1 2σ(h2h3, h3h1, h1h2) (see exercise O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5), that is ⃗n = (n1, n2, n3) with ni = σi σ pour i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The volume of the tetrahedral is 1 6h1h2h3 =noted ℓ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let M := supp∈Ω |ϕ(p)|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have M < ∞, since ϕ is continuous in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) give Mℓ3 ≥ | � ∂ωt ψ(p,⃗n(p)) dΓ|, so � ∂ωt ψ(p,⃗n(p)) dΓ = O(ℓ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) And ψ being continuous, the mean value theorem applied on Σi gives: There exists pi ∈ Σi s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � Σi ψ(p,⃗n(p)) dΓ = σiψ(pi,⃗ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus � ∂ωt ψ(p,⃗n(p)) dΓ = � σ1ψ(p1, − ⃗E1) + σ2ψ(t, p2, − ⃗E2) + σ3ψ(p3, − ⃗E3) + σψ(p4,⃗n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, Ψ being continous, (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) gives σ1ψ(p1, − ⃗E1) + σ2ψ(p2, − ⃗E2) + σ3ψ(p3, − ⃗E3) + σψ(p4,⃗n) = O(ℓ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) We flatten the tetrahedron on the yz face by taking h2 = h3 =noted h and h1 = h2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus σ1 = 1 2h2, σ2 = o(h2), σ3 = o(h2), σ ∼ σ1, ℓ3 = 1 6h4, with ⃗n ∼ −⃗n1 = ⃗E1 and pi ∼ p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then ψ(p, − ⃗E1) + ψ(p, + ⃗E1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) Idem with xz and xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And for a fixed tetrahedron with h1, h2, h3 given, consider the smaller tetrahedron with εh1, εh2, εh3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then as ε → 0 (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) with (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) give ψ(p,⃗n) = − σi σ ψ(p, − ⃗E1) − σ2 σ ψ(p, − ⃗E2) − σ3 σ ψ(p, − ⃗E3) = 3 � i=1 niψ(p, ⃗Ei), since ni = σi σ pour i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The same steps can be done for any (inclined) tetrahedron (or apply a change of variable to get back to the above tetrahedron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ψp is a linear map in ⃗np, that is, there exists a linear form αp s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ψp(⃗np) = αp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗np for any p ∈ ∂ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the Riesz representation theorem gives: ∃⃗kp s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' αp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗np = (⃗kp,⃗np)g =noted ⃗kp • ⃗np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Apply Lemma O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 component by component with ⃗ϕ = ρ D⃗v Dt − ⃗f = �n i=1ϕi⃗ei, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Corollary O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 With divσ := �n i=1(�n j=1 ∂σij ∂xj )⃗ei (definition of “the matrix divergence” see (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='65)), � � � ⃗f + divσ = ρD⃗v Dt in Ωt, σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n = ⃗T on Γt (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) (matrix meaning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (With duality notations, divσ := �n i=1(�n j=1 ∂σi j ∂xj )⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Apply the divergence Formula to (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Consider a triangle T in R3 which vertices are A = (h1, 0, 0), B = (0, h2, 0), C = (0, 0, h3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove that ⃗n = (h2h3, h3h1, h1h2) is orthogonal to T and that σ = 1 2 � h2 2h2 3 + h2 3h2 1 + h2 1h2 2 is its area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider the parametric surface ⃗r(t, u) = A + t ⃗ AB + u ⃗ AC for t, u ∈ [0, 1] describing the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ⃗n = ∂⃗r ∂t ∧ ∂⃗r ∂u = ⃗ AB ∧ ⃗ AC = � � −h1 h2 0 � � ∧ � � −h1 0 h3 � � = � � h2h3 h3h1 h1h2 � � is orthonormal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And dσ = || ∂⃗r ∂t ∧ ∂⃗r ∂u||dudt = � h2 2h2 3 + h2 3h2 1 + h2 1h2 2dudt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus σ = � 1 t=0 � 1 u=0 dσ = � h2 2h2 3 + h2 3h2 1 + h2 1h2 2 is twice the aera of the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 162 163 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Tensorial product and multilinear forms P Balance of moment of momentum Definition P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The balance of moment of momentum is satisfied by ρ, ⃗f and ⃗T iff for all regular sub-open set ωt ⊂ Ωt d dt � ωt ρ −−→ OM ∧ ⃗v dΩt = � ωt ρ −−→ OM ∧ ⃗f dΩt + � ∂ωt −−→ OM ∧ ⃗T dΓt, (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) equality called the master balance of moment of momentum law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (This excludes e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Cosserat continua materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Theorem P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 (Cauchy second law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') If the master balance law (so ⃗T = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n) and the master balance of moment of momentum law are satisfied then σ is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Standard proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Let ⃗x = −−→ OM = � i xi ⃗Ei, and ⃗T = � i Ti ⃗Ei = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗n = � ij σijnj ⃗Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (first component) (⃗x ∧ ⃗T)1 = x2T3 − x3T2 = x2(σ31n1 + σ32n2 + σ33n3) − x3(σ21n1 + σ22n2 + σ23n3) = (x2σ31 − x3σ21)n1 + (x2σ32 − x3σ22)n2 + (x2σ33 − x3σ23)n3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus � ∂ωt(⃗x ∧ ⃗T)1 dΓt = � ωt ∂(x2σ31−x3σ21) ∂x1 + ∂(x2σ32−x3σ22) ∂x2 + ∂(x2σ33−x3σ23) ∂x3 dΩt = � ωt x2(divσ)3 + x3(divσ)2 + σ32 − σ23 dωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) gives ρ D⃗v Dt − ⃗f = divσ, thus ⃗x ∧ (ρ⃗γ − ⃗f) = ⃗x ∧ divσ, so the first component of ⃗x ∧ (ρ⃗γ − ⃗f) is x2(divσ)3−x3(divσ)2, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) gives � ωt σ32−σ23 dωt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' True for all ωt, thus σ32−σ23 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Idem for the other components: σ is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Q Uniform tensors in Lr s(E) Uniform tensors enable to define without ambiguity the “objective contraction rules”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Uniform tensors are scalar valued multilinear functions acting on both vectors and linear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' NB: In classical mechanics courses, what is called a “tensor” generally not a tensor but a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' you may encounter the expression “Euclidean tensor” which means: The matrix representation of “something” with respect to a Euclidean basis (based on the foot, metre,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') chosen by some observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (An “Euclidean tensor” is a non-sense, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' can you define a “Euclidean vector”?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Tensorial product and multilinear forms Let A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', An be n finite dimension vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And A∗ i = L(Ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) the set of linear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Tensorial product of functions Let f1 : A1 → R, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', fn : An → R be n functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Their tensorial product is the function f1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⊗ fn : A1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' × An → R defined by (separate variable function) (f1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⊗ fn)(⃗x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗xn) = f1(⃗x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='fn(⃗xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n = 2 and A1 = A2 = R and (cos ⊗ sin)(x, y) = cos(x) sin(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Tensorial product of linear forms: multilinear forms Let L(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', An;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) be the set of R-multilinear forms on the Cartesian product A1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' × An, that is, the set of the functions M : A1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' × An → R s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n, all ⃗xi, ⃗yi ∈ Ai and all λ ∈ R, M(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗xi + λ⃗yi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') = M(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗xi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') + λ M(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗yi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='), (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) the other variables being unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition: An elementary tensor is multilinear form M = ℓ1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. ⊗ ℓn, with ℓi ∈ A∗ i for all i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So ∀(⃗xi)i∈N∗ ∈ n � i=1 Ai, (ℓ1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⊗ ℓn)(⃗x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗xn) = (ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='(ℓn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗xn) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) (The dot in ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗xi is not an inner dot product: It is the duality “outer product” ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗xi := ℓi(⃗xi), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 163 164 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Uniform tensors in L0 s(E) Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Uniform tensors in L0 s(E) Let E be a real vector space, with dim(E) = n ∈ N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In this section we consider the first overlay on E made of multilinear forms M on E, called the uniform tensors of type 0 s or of type �0 s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', M ∈ L0 1(E) a linear form, M ∈ L0 2(E) an inner dot product, M ∈ L0 n(E) a determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Notations for quantification purposes: (⃗ei) is a basis in E, (πei) is its (covariant) dual basis (basis in E∗ = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R)), (∂i) is its bidual basis (basis in E∗∗ = L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition of type �0 s � uniform tensors L0 0(E) := R, and if s ∈ N∗ then L0 s(E) := L(E × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' × E � �� � s times ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) is called the set of uniform tensors of type �0 s � on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Example: Type �0 1 � uniform tensor = linear forms A type �0 1 � uniform tensor is an element of L0 1(E) = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) = E∗: It is a linear form ℓ ∈ L0 1(E) = E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification: With ℓi := ℓ(⃗ei) we have, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10), ℓ = n � i=1 ℓiπei, and [ℓ]|πe = ( ℓ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ℓn ) noted = [ℓ]|⃗e (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) (row matrix for a linear form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notations: (ei) is the covariant dual basis and ℓ = �n i=1ℓiei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, if ⃗v ∈ E, ⃗v = �n i=1vi⃗ei, then ⃗v is represented by [⃗v]|⃗e = � � v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' vn � � (column matrix for a vector), and the matrix calculation rules give ℓ(⃗v) = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗v]|⃗e = ( ℓ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ℓn ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � � v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' vn � � = n � i=1 ℓivi noted = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) Duality notations: ⃗v = �n i=1vi⃗ei and ℓ(⃗v) = �n i=1ℓivi, and Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Example: Type �0 2 � uniform tensor A type �0 2 � uniform tensor is an element of L0 2(E) = L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R): It is a bilinear form T ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification: Let Tij := T(⃗ei,⃗ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, with ⃗v = �n i=1vi⃗ei and ⃗w = �n i=1wi⃗ei, T(⃗v, ⃗w) = n � i,j=1 Tijviwj = [⃗v]T |⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[T]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗e, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' T = n � i,j=1 Tijπei ⊗ πej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) Duality notations: T(⃗v, ⃗w) = �n i,j=1Tijviwj, and Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' An elementary uniform tensor in L0 2(E) is a tensor T = ℓ ⊗ m, where ℓ, m ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And so, for all ⃗v, ⃗w ∈ E, (ℓ ⊗ m)(⃗v, ⃗w) = (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v)(m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Example: Determinant The determinant is a alternating �0 n � uniform tensor, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Uniform tensors in Lr s(E) In this section we consider an over-overlay on E: The multilinear forms acting on both vectors (∈ E) and functions ∈ E∗ (linear forms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 164 165 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Uniform tensors in Lr s(E) Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definition of type �r s � uniform tensors Let r, s ∈ N s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' r + s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The set of multilinear forms Lr s(E) := L(E∗ × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' × E∗ � �� � r times , E × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' × E � �� � s times ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) is called the set of uniform tensors of type �r s � on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The case r = 0 has been considered at § Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' When r ≥ 1, a tensor T ∈ Lr s(E) is a functional: Its domain of definition contains a set of functions (the set E∗ = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Example: Type �1 0 � uniform tensor: Identified with a vector A uniform �1 0 � tensor is a element T ∈ L1 0(E) = L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) = L(L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) = E∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With the natural canonical isomorphism J : � E → E∗∗ = L1 0(E) ⃗w → J (⃗w) = w, defined by w(ℓ) := ℓ(⃗w), ∀ℓ ∈ E∗, (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) and prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5, w noted = ⃗w, so w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ noted = ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ (= ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) So a �1 0 � type uniform tensor w is identified (natural canonical) to the vector ⃗w = J −1(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Interpretation: E∗∗ is the set of directional derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Indeed, if E is an affine space, if E is the associated vector space, if p ∈ E, and if f is a differentiable function at p, then w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='df(p) =(Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w is the directional derivative along ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark: In differential geometry, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='df is written ⃗w(f), so ⃗w(f)(p) := df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w, the definition of a vector being a directional derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification: For all i, j, ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='πej = δij = πej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei, thus ∂i = J (⃗ei) noted = ⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) Duality notations: ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ej = δj i = ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', if f is a C1 function then df(p) = �n i=1f|i(p) πei (= �n i=1f|i(p) ei) and ∂i(df(p)) = df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei = f|i(p) noted = ∂i(f)(p) noted = ⃗ei(f)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Example: Type �1 1 � uniform tensor An elementary uniform tensor in L1 1(E) is a tensor T = u ⊗ β, where u ∈ E∗∗ and β ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And, with ⃗u = J−1(u) ∈ E, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10), we also write T = ⃗u ⊗ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, for all ℓ ∈ E∗ and ⃗w ∈ E (u ⊗ β)(ℓ, ⃗w) = u(ℓ)β(⃗w) = ℓ(⃗u)β(⃗w) noted = ⃗u(ℓ)β(⃗w) noted = (⃗u ⊗ β)(ℓ, ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) Quantification: Let T(πei,⃗ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So T = n � i,j=1 Tij ⃗ei ⊗ πej, and [T]|⃗e = [Tij], (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) [T]|⃗e = [Tij] being the matrix of T relative to the basis (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notations: T(ei,⃗ej) = T ij, [T]|⃗e = [T ij], T = �n i,j=1T ij⃗ei ⊗ ej, and Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus with ℓ ∈ E∗, ℓ = �n i=1ℓiei ∈ E∗, and ⃗w ∈ E, ⃗w = �n i=1wi⃗ei ∈ E, (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) gives T(ℓ, ⃗w) = n � i,j=1 Tij⃗ei(ℓ)πej(⃗w) = n � i,j=1 Tijℓiwj = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[T]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗e (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) ([ℓ]|⃗e is a row matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notations: T(ℓ, ⃗w) = �n i,j=1T ijℓiwj and Einstein convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 165 166 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exterior tensorial products Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Example: Type �1 2 � uniform tensor The same steps are applied to any tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', if T ∈ L1 2(E), then with duality notations, T ijk = T(ei,⃗ej,⃗ek) and T = n � i,j,k=1 T i jk⃗ei ⊗ ej ⊗ ek, and T(ℓ, ⃗u, ⃗w) = n � i,j,k=1 T i jkℓiujwk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Exterior tensorial products Let T1 ∈ Lr1 s1(E) and T2 ∈ Lr2 s2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Their tensorial product is the tensor T1 ⊗ T2 ∈ Lr1+r2 s1+s2(E) defined by (T1 ⊗ T2)(ℓ1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ℓ2,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗u1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗u2,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') := T1(ℓ1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗u1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=')T2(ℓ2,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗u2,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) Particular case: with λ ∈ L0 0(E) = R and T ∈ Lr s(E), λ ⊗ T = T ⊗ λ := λT ∈ Lr s(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) Example Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 let T1, T2 ∈ L1 1(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification: Let T1 = �n i,j=1(T1)i j⃗ei ⊗ ej and let T2 = �n k,m=1(T2)k m⃗ek ⊗ em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then T1 ⊗ T2 = �n i,j,k,m=1(T1)i k(T2)j m⃗ei ⊗ ⃗ej ⊗ ek ⊗ em ∈ L2 2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Alternative definition: T1 �⊗T2 := �n i,j,k,m=1(T1)i j(T2)k m⃗ei ⊗ ej ⊗ ⃗ek ⊗ em ∈ L(E∗, E, E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And we get back to the previous definition thanks to the natural canonical isomorphism �J : L(E∗, E, E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) → L(E∗, E∗, E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) = L2 2(E) defined by �J( �T) = T where T(ℓ, m,⃗v, ⃗w) = �T(ℓ,⃗v, m, ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Contractions Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Contraction of a linear form with a vector Let ℓ ∈ L0 1(E) = E∗ and ⃗w ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Their contraction is the value ℓ(⃗w) linearity = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w noted = ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) And with a basis (⃗ei) and its dual basis (πei), ℓ = �n i=1ℓiπei and ⃗w = �n i=1wi⃗ei give ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = n � i=1 ℓiwi = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗e = n � i=1 wiℓi = ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ = Tr(⃗w ⊗ ℓ), (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) where Tr is the objective trace operator Tr : L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) ≃ L1 1(E) → R (defined by Tr(⃗ei ⊗ πej) = δi j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notations: ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = �n i=1ℓiwi, and Einstein convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Use the change of coordinate formulas to prove that the computation ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w in (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) gives a result independent of the basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let P be the change of basis matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So [⃗w]new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]old and [ℓ]new = [ℓ]old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28), thus [ℓ]new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]new = ([ℓ]old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]old) = [ℓ]old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]old = [ℓ]old[⃗w]old (= ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Contraction of a �1 1 � tensor and a vector Let ℓ ∈ E∗ and ⃗u ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The contraction of the elementary tensor ⃗w ⊗ ℓ ∈ L1 1(E) with ⃗u is defined by: (⃗w ⊗ ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u ���� contraction = (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u)⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) Thus, if (⃗ei) is a basis in E and (πei) is the dual basis, and T = �n i,j=1Tij⃗ei ⊗ πej ∈ L1 1(E) and ⃗u = �n j=1uj⃗ej ∈ E, then T = n � i,j=1 Tij⃗ei ⊗ ej =⇒ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = n � i,j=1 Tijuj j⃗ei (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) because πej(⃗u) = uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notations: T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = �n i,j=1T i juj⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 166 167 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Contractions Then, with the natural canonical isomorphism (L1 1(E) =) L(E, E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) ≃ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E), see (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7), any endomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) defined by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1Lij⃗ei can be written, for calculation purpose, �L = n � i,j=1 Lij⃗ei ⊗ πej noted = L, which means L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) = n � i=1 Lijuj⃗ei (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) when ⃗u = � i uj⃗ej, since πej(⃗u) = uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notations: L = �n i,j=1Lij⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Contractions of uniform tensors More generally, the contraction of two tensors, if meaningful, is defined thanks to (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20): Let T1 ∈ Lr1 s1(E), T2 ∈ Lr2 s2(E), ℓ ∈ E∗ and ⃗u ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 The objective contraction of T1 ⊗ ℓ ∈ Lr2 s2+1(E) and ⃗u ⊗ T2 ∈ Lr2+1 s2 (E) is the tensor (T1 ⊗ ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗u ⊗ T2) ∈ Lr1+r2 s1+s2 given by (T1 ⊗ ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗u ���� contraction ⊗T2) := (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u) T1 ⊗ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) In particular (T1 ⊗ ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u) T1 (as in (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22)), and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗u ⊗ T2) = (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u) T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the objective contraction of T1 ⊗ ⃗u ∈ Lr2+1 s2 (E) and ℓ ⊗ T2 ∈ Lr2 s2+1(E) is the tensor (T1 ⊗ ⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (ℓ ⊗ T2) ∈ Lr1+r2 s1+s2 given by (T1 ⊗ ⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (ℓ ⊗ T2) = (⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ) T1 ⊗ T2 (= (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u) T1 ⊗ T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) Quantification with a basis (⃗ei), examples to avoid cumbersome notations: Example Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Let T ∈ L1 1(E) = L1 0+1(E), T = �n i,j=1T i j⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With ⃗w ∈ E ∼ E∗∗ = L1 0(E), ⃗w = �n j=1wj⃗ej, (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) gives T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w ∈ L1 0(E) ∼ E and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = n � i,j=1 T i jwj⃗ei, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w]|⃗e = [T]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗e (column matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) (Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Indeed, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = �n i,j,k=1T i jwk(⃗ei ⊗ ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek = �n i,j,k=1T i jwk⃗ei(ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek) = �n i,j,k=1T i jwk⃗ei(δj k) = �n i,j=1T i jwj⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With ℓ ∈ E∗ = L0 1(E), ℓ = �n i=1ℓiei, (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) gives ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T ∈ L0 1(E) = E∗ and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T = n � i,j=1 ℓiT i jej, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T]|⃗e = [ℓ]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [T]|⃗e (row matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) (Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Indeed ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T = (�n i=1ℓiei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (�n j,k=1T k j ⃗ek⊗ej) = �n i,j,k=1ℓiT k j (ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek)ej = �n i,j=1ℓiT i jej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Let S, T ∈ L1 1(E), S = �n i,k=1Si k⃗ei ⊗ ek and T = �n j,k=1T k j ⃗ek ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T = n � i,j,k=1 Si kT k j ⃗ei ⊗ ej, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T]|⃗e = [S]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [T]|⃗e (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) (Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Indeed S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T = (�n i,k=1Si k⃗ei ⊗ ek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (�n j,m=1 T m j ⃗em ⊗ ej) = �n i,j,k,m=1Si kT m j ⃗ei(ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗em) ⊗ ej = �n i,j,k=1Si kT k j ⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Let T ∈ L1 2(E), T = �n i,j,k=1T i jk⃗ei ⊗ ej ⊗ ek, and ⃗u, ⃗w ∈ E ∼ L1 0(E), ⃗w = �n i=1wi⃗ei and ⃗u = �n i=1ui⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = n � i,j,k=1 T i jkwk⃗ei ⊗ ej ∈ L1 1(E), and (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = n � i,j,k=1 T i jkwkuj⃗ei noted = T(⃗u, ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) (Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') So [T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w]|⃗e = [�n k=1T i jkwk] i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=',n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And with ℓ ∈ E∗, ℓ = �n i=1ℓiei, ((T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ = n � i,j,k=1 T i jkwkujℓi = T(ℓ, ⃗u, ⃗w) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T(⃗u, ⃗w) = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) 167 168 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Contractions Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Objective double contractions of uniform tensors Definition Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Let S, T ∈ L1 1(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And let (⃗ei) be a basis in E, (ei) its dual basis, S = �n i,j=1Si j⃗ei ⊗ ej and T = �n i,j=1T i j⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The double objective contraction S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. T of S and T is defined by S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. T = n � i,j=1 Si jT j i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) (Einstein convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proposition Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. T defined in (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) is an invariant: It is the trace Tr(LS ◦ LT ) of the endo- morphisms LS, LT ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) naturally canonically associated to S and T (given by ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u := S(ℓ, ⃗u) and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u := T(ℓ, ⃗u) for all (⃗u, ℓ) ∈ E × E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So the real value �n i,j=1Si jT j i has the same real value regardless of the chosen basis (⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Which is not the case of the term to term matrix multiplication S : T = �n i,j=1Si jT i j, see next § Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 and example Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗ai) and (⃗bi) be two bases and P = [P i j] be the transition matrix from (⃗ai) to (⃗bi), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ⃗bj = �n i=1P i j⃗ai for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Q = [Qi j] := P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then bi = �n i=1Qi jai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let S = � ij(Sa)i j⃗ai ⊗ aj = � ij(Sb)i j⃗bi ⊗ bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So [(Sb)i j] = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [(Sa)i j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P (change of basis formula for �1 1 � tensors identified with endomorphisms), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Sb)i j = � km Qi k(Sa)k mP m j for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Idem with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus � i,j(Sb)i j(Tb)j i = � i,j,k,m,α,β Qi k(Sa)k mP m j Qj α(Ta)α βP β i = � i,j,k,m,α,β(Sa)k m(Ta)α βP β i Qi kP m j Qj α = � k,m,α,β(Sa)k m(Ta)α βδβ k δm α = � k,m(Sa)k m(Ta)m k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 More generally, the objective double contractions S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. T of uniform tensors, is obtained by applying the objective simple contraction twice consecutively, when applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', T1 ⊗ ℓ1,1 ⊗ ℓ1,2 and ⃗u2,1 ⊗ ⃗u2,2 ⊗ T2 give (T1 ⊗ ℓ1,1 ⊗ ℓ1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗u2,1 � �� � first ⊗⃗u2,2 ⊗ T2) = (ℓ1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2,1)(T1 ⊗ ℓ1,1) ⊗ (⃗u2,2 � �� � second ⊗T2) = (ℓ1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2,1)(ℓ1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2,2) T1 ⊗ T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) Example Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 Let S ∈ L1 2(E), T ∈ L2 1(E), S = �n i,j,k=1Si jk⃗ei ⊗ej ⊗ej, T = �n α,β,γ=1 T αβ γ ⃗eα ⊗⃗eβ ⊗eγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T = n � i,j,k,β,γ=1 Si jkT kβ γ ⃗ei ⊗ ej ⊗ ⃗eβ ⊗ eγ, and S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. T = n � i,j,k,γ=1 Si jkT kj γ ⃗ei ⊗ eγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) (Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Exercice Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 If S ∈ L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), T ∈ L(F, G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) and U ∈ L(G, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) then prove S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U) = (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. U = (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='S) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. T (circular permutation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If S = � Si j⃗ai ⊗ bj, T = � T i j⃗bi ⊗ cj and U = � U i j⃗ci ⊗ aj, then T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U = � T i kU k j ⃗bi ⊗ aj, thus S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='U) = � Si mT m k U k i , and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T = � Si kT k j ⃗ai ⊗ cj, so (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. U = � Si kT k mU m i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the second equality thanks to the symmetry of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. U = U 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T) = (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='S) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. T with the previous calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We define in the same way the triple objective contraction (apply the simple contraction three times consecutively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', with (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) we get S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' T = n � i,j,k=1 Si jkT kj i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) (Einstein’s convention is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 168 169 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Kronecker (contraction) tensor, trace Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Non objective double contraction: Double matrix contraction The engineers often use the double matrix contraction of second order tensors defined by (term to term multiplication): If S = [Sij] = [Si j] and T = [Tij] = [T i j] then S : T := n � i,j=1 SijTij = n � i,j=1 Si jT i j noted = Tr(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) Einstein’s convention is not satisfied, and the result is observer dependent for associated endomorphism: Example Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 Let (⃗ei) be a basis, let S ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) given by [S]⃗e = � 0 4 2 0 � (so S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e1 = 2⃗e2 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e2 = 4⃗e1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then the double matrix contraction (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) gives S : S = [S]⃗e : [S]⃗e = 4 ∗ 4 + 2 ∗ 2 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38) Change of basis: let ⃗b1 = ⃗e1 and ⃗b2 = 2⃗e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The transition matrix from (⃗ei) to (⃗bi) is P = � 1 0 0 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus [S]⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [S]⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P = � 1 0 0 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � 0 8 2 0 � = � 0 8 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus S : S = [S]⃗b : [S]⃗b = 8 ∗ 8 + 1 ∗ 1 = 65 ̸= 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39) To be compared with the double objective contraction: [S]⃗e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. [S]⃗e = 4∗2+2∗4 = 16 = [S]⃗b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. [S]⃗b = S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. S (observer independent result = objective result).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So it is absurd to use S : S (double matrix contraction) if you need objectivity: Recall that the foot is the international vertical unit in aviation, and thus the use of the double objective contraction is vital, while the use of the double matrix contraction can be fatal (really).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Also see the Mars climate orbiter probe crash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 Let S ∈ L0 2(E) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' a metric), let (⃗ai) be a Euclidean basis in foot, and let (⃗bi) = (λ⃗ai) be the related euclidean basis in metre (change of unit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Give [S]|⃗a : [S]|⃗a and [S]|⃗b : [S]|⃗b and compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (The simple and double objective contractions are impossible here since S and T are not compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let S = �n i,j=1Sa,ijai ⊗ aj = �n i,j=1Sb,ijbi ⊗ bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Since (⃗bi) = (λ⃗ai) we have bi = 1 λai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus �n i,j=1Sa,ijai ⊗ aj = �n i,j=1Sa,ijλ2bi ⊗ bj, thus λ2Sa,ij = Sb,ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus [S]|⃗b : [S]|⃗b = n � i,j=1 (Sb,ij)2 = λ4 n � i,j=1 (Sa,ij)2 = λ4[S]|⃗a : [S]|⃗a, (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) with λ4 ≥ 100: Quite a difference isn’t it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Kronecker (contraction) tensor, trace Definition Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15 The Kronecker tensor is the �1 1 � uniform tensor δ ∈ L1 1(E) defined by ∀(ℓ, ⃗u) ∈ E∗ × E, δ(ℓ, ⃗u) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) And the Kronecker symbols relative to a basis (⃗ei) are the reals defined by, calling (πei) the dual basis, δij := δ(πei,⃗ej) = � 1 if i = j, 0 if i ̸= j, � i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' δ := n � i=1 πei ⊗ ei, [δ] = [δj] = [I] (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='42) (identity matrix whatever the basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notations: δi j := δ(ei,⃗ej), δ := �n i=1 ⃗ei ⊗ ei and [δ] = [δi j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16 The trace of a �1 1 � uniform tensor T ∈ L1 1(E) is � Tr(T) = δ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. T (= Tr(LT )) (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43) (with the natural canonical isomorphism T ∈ L1 1(E) ≃ LT ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) given by T(ℓ,⃗v) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus � Tr(T) = �n i=1T ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular � Tr(δ) = n, and � Tr(⃗v ⊗ ℓ) = � i viℓi = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v when ⃗v = � i vi⃗ei and ℓ = � j ℓjej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 169 170 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Introduction, module, derivation R Tensors in T r s (U) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Introduction, module, derivation Let A and B be any sets, and let F(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B) be the set of functions A → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The “plus” inner operation and the “dot” outer operation are defined by, for all f, g ∈ F(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B), all λ ∈ R and all p ∈ A, � (f + g)(p) := f(p) + g(p), and (λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='f)(p) := λ f(p), λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='f noted = λf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) (F(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B), +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', R) is thus a vector space on the field R (see any elementary course) called F(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But the field R is “too small” to define a tensor which can be seen as “a linear tool that satisfies the change of coordinate system rules”: Example R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Fundamental counter-example: Derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let U be an open set in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The derivation d : ⃗w ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn) → d⃗w ∈ C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(⃗Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn)) is R-linear: In particular d(λ⃗w) = λ(d⃗w) for all λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='but d doesn’t satisfy the change of coordinate system rules, see (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So a derivation it not a tensor (it is a “spray”, see Abraham–Marsden [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In fact, one requirement for T to be a tensor is, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' with T = ⃗w a vector field: For all ϕ ∈ C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), and all ⃗w ∈ Γ(U) (C∞-vector field), T(ϕ⃗w) = ϕ T(⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) While d(ϕ⃗w) ̸= ϕ d(⃗w), because d(ϕ⃗w) = ϕ d⃗w + dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) Thus the elementary R-linearity requirement “T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (λ⃗w) = λ(T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) for all λ ∈ R is not sufficient to charac- terize a tensor: The R-linearity has to be replaced by the C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R)-linearity, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus we will have to replace a real vector space (V, +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', R) over the field R with the “module” (V, +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R)) over the ring C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), which mainly amounts to consider (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) for all λ = ϕ ∈ C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark: The use of a module is very similar to the use of a vector space, but for the use of the inverse: all real λ ̸= 0 has a multiplicative inverse in R (namely 1 λ), but a function f ∈ C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' “f ̸= 0 and f vanishes at one point” doesn’t have a multiplicative inverse in C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Field of functions and vector fields Framework of classical mechanics: U is an open set in an affine space E which associated vector is E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the definition of tensors is done at a fixed time t (concerns the space variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' As before, the approach is first qualitative, then quantitative with a basis (⃗ei(p)) and its dual basis (πei(p)) = (ei(p)), at any p ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Field of functions Let f ∈ C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) be a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The associated function field is �f : � U → U × R p → �f(p) := (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' f(p)), (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) and p is called the base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So Im �f = {(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' f(p)) : p ∈ U} is the graph of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition: T 0 0 (U) := { �f : f ∈ C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R)} = {field of functions} = the set of �0 0 � type tensor on U, (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) or the set of tensors of order 0 on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Abusive short notations (to lighten the writings): �f(p) noted = f(p), and T 0 0 (U) noted = C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) but keep the base point in mind (no ubiquity gift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In T 0 0 (U), the internal sum is defined by, for all �f, �g ∈ T 0 0 (U) with �f(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' f(p)) and �g(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' g(p)), ( �f + �g)(p) := (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (f + g)(p)) (= (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' f(p) + g(p))), (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) and the external multiplication on the ring C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is defined by, for all ϕ ∈ C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), (ϕ �f)(p) := (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (ϕf)(p)) (= (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ϕ(p)f(p))) (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) (the base point p remains unchanged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (T 0 0 (U), +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') is a module over the ring C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 170 171 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Differential forms R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Vector fields Let ⃗w ∈ C∞(U, E) be a vector valued function (at least Lipschitzian, to get integral curves, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Cauchy– Lipschitz theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The associated vector field is �⃗w : � U → U × E p → �⃗w(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗w(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) So Im�⃗w = {(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗w(p)) : p ∈ U} is the graph of ⃗w, and the definition of �⃗w tells that the vector ⃗w(p) has to be drawn at p (the base point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Abusive short notation: �⃗w(p) noted = ⃗w(p) instead of �⃗w(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗w(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) It lightens the notations, but keep the base point in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let Γ(U) := the set of vector fields on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) More precisely, we will use the following full definition of vector fields (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Abraham–Marsden [1]): A vector field is built from tangent vectors to curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It makes sense on non planar surfaces, and more generally on differential manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Differential forms The basic concept is that of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A first over-layer is made of differential forms (which “measure vector fields”): Definition R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Let α � U → E∗ p → α(p) � (so α(p) is a linear form at p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The associated differential form (also called a 1-form) is “the field of linear forms” defined by �α : � U → U × E∗ p → �α(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' α(p)) ( = “a pointed linear form at p”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) And p is called the base point, and Im�α = {(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' α(p)) : p ∈ U} is the graph of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, if �α ∈ Ω1(U) (differential form) and �⃗w ∈ Γ(U) (vector field), then �α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�⃗w ∈ T 0 0 (U) (field of scalar valued functions) satisfies �α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�⃗w : � U → U × R p → (�α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='�⃗w)(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w)(p)) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' α(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w(p)) ∈ U × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) Short notation: �α(p) noted = α(p), instead of �α(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' α(p)), (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) but keep the base point in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And Ω1(U) := the set of differential forms U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Tensors A second over-layer is introduced with the tensors with are “functions defined on vector fields and on differential forms” (which “measure vector fields and differential forms”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let r, s ∈ N, r+s ≥ 1, and let T : � U → Lr s(E) p → T(p) � (so T(p) is a uniform �r s � tensor for each p, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And consider the associated function �T : � U → U × Lr s(E) p → �T(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' T(p)) (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) Abusive short notation: �T(p) noted = T(p) instead of �T(p) = (p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' T(p)), (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) but keep the base point in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 171 172 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' First Examples Definition R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 (Abraham–Marsden [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') �T is a tensor of type �r s � iff T is C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R)-multilinear (not only R-multilinear), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all f ∈ C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), all z1, z2 vector field or differentiable form where applicable, and all p ∈ U, � T(p)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', z1(p) + z2(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') = T(p)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', z1(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') + T(p)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', z2(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='), and T(p)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', f(p)z1(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') = f(p) T(p)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', z1(p), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='), (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) written in short � T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', z1 + z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') = T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') + T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='), and T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', fz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') = f T(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) And T r s (U) := the set of �r s � type tensors on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) (Recall: T 0 0 (U) := C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) the set of function fields, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Remark R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Definition in differential geometry lessons: A tensor is a section of a certain bundle over a manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' For classical mechanics, definition R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 gives an equivalent definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 First Examples R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Type �0 1 � tensor = differential forms If T ∈ T 0 1 (U) then T(p) ∈ E∗, so T = α ∈ Ω1(U) is a differential form: T 0 1 (U) ⊂ Ω1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Converse: Does a differential form α ∈ Ω1(U) defines a �0 1 � type tensor on U?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Yes: We have to check (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18), which is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So α ∈ T 0 1 (U), so Ω1(U) ⊂ T 0 1 (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus T 0 1 (U) = Ω1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Type �1 0 � tensor (identified to a vector field) Let T ∈ T 0 1 (U), so T(p) ∈ L1 0(E) = L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) = E∗∗ for all p ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, thanks to the natural canonical isomorphism E∗∗ ≃ E, T(p) can be identified to a vector, thus T 0 1 (U) ⊂ Γ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Converse: Does a vector field ⃗w ∈ Γ(U) defines a �1 0 � type tensor on U?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Yes: We have to check (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18), which is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So Γ(U) ⊂ T 1 0 (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus T 1 0 (U) ≃ Γ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 A metric is a �0 2 � tensor Let T ∈ T 0 2 (U), so T(p) ∈ L0 2(E) for all p ∈ U, and T(⃗u, ⃗w) ∈ T 0 0 (U) for all ⃗u, ⃗w ∈ Γ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 A metric g on U is a �0 2 � type tensor on U such that, for all p ∈ E, g(p) =noted gp is an inner dot product on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 �1 1 � tensor, identification with fields of endomorphisms Let T ∈ T 1 1 (U), so T(p) ∈ L1 1(E) for all p ∈ U, and T(α, ⃗w) ∈ T 0 0 (U) for all α ∈ Ω1(U) and ⃗w ∈ Γ(U) (so T(p)(α(p), ⃗w(p)) ∈ R for all p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The associated field of endomorphisms on U is �LT : � U → U × L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) p → �LT (p) = (p, LT (p)) � where LT (p) is identified with T(p) thanks to the natural canonical isomorphism L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) ≃ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) = L1 1(E) given by ∀ℓ ∈ E∗, ∀⃗w ∈ E, ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (LT (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) = T(p)(ℓ, ⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Unstationary tensor Let t ∈ [t1, t2] ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (Tt)t∈[t1,t2] be a family of �r s � tensors, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then T : t → T(t) := Tt is called an unstationary tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the set of unstationary tensors is also noted T r s (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', a Eulerian velocity field is a �1 0 � unstationary vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 172 173 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Differential S Differential, its eventual gradients, divergences S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Differential The definition of the differential of a function is observer independent: All observers have the same definition (qualitative: no man made tool required, like a basis or an inner dot product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Framework Classical Framework: E are F affine spaces associated with vector spaces E and F, and ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||E and ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||F are norms in E and F such that (E, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||E) and (F, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||F ) are complete (we need “limit that stay in the space as h → 0”, ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' U is an open set in E, and Φ : � U → F p → pF = Φ(p) � is a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If applicable, E and/or F can be replaced by E and/or F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (The definitions can be generalized to manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Reminder: Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Let p ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The function Φ is said to be continuous at p iff Φ(q) −→ q→p Φ(p) relative to the considered norms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', ||Φ(q) − Φ(p)||F −→||q−p||E→0 0, also written (Landau notation): Near p, Φ(q) = Φ(p) + o(1), (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) called “the zero-th order Taylor expansion of Φ near p”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In other words: ∀ε > 0, ∃η > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∀q ∈ E s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ||q − p||E < η we have ||Φ(q) − Φ(p)||F < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) is the set of functions that are continuous at all p ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Directional derivative and differential (observer independent) Let p ∈ U, ⃗u ∈ E, and let f : R → F defined by f(h) := Φ(p + h⃗u) (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 The function Φ is differentiable at p in the direction ⃗u iff f is derivable at 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' iff the limit f ′(0) = limh→0 Φ(p+h⃗u)−Φ(p) h =noted dΦ(p)(⃗u) exists in F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' iff, near p, Φ(p + h⃗u) = Φ(p) + h dΦ(p)(⃗u) + o(h), (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) equation called the first order Taylor expansion of Φ at p in the direction ⃗u (it is the first order Taylor expansion of f near p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then dΦ(p)(⃗u) is called the directional derivative of Φ at p in the direction ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And if, for all ⃗u ∈ E, dΦ(p)(⃗u) exists (in F) then Φ is called Gâteaux differentiable at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Prove: If Φ is Gâteaux differentiable at p then dΦ(p) is homogeneous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', dΦ(p)(λ⃗u) = λ dΦ(p)(⃗u) for all ⃗u ∈ E and all λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' limh→0 Φ(p+h(λ⃗u))−Φ(p) h = λ limh→0 Φ(p+λh⃗u)−Φ(p) λh = λ limk→0 Φ(p+k⃗u)−Φ(p) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 If Φ is Gateaux differentiable and if moreover dΦ(p) is linear and continuous at p, then Φ is said to be differentiable at p (or Fréchet differentiable at p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So Φ(q) = Φ(p) + h dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−→ pq + o(||−→ pq||E), (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) since then dΦ(p)(⃗u) =noted dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u for all ⃗u ∈ E (linearity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the affine function affp : q → affp(q) := Φ(p) + dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='−→ pq is the affine approximation of Φ at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (So, the graph of affp is the tangent plane of Φ at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Φ : U → F is said to be differentiable in U iff Φ is differentiable at all p ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then its differential is the map dΦ : � U → L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) p → dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) And C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) is the set of differentiable functions ψ such that dΦ ∈ C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And C2(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) is the set of differentiable functions ψ such that dΦ ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And Ck(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) is the set of differentiable functions ψ such that dΦ ∈ Ck−1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. 173 174 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A basis and the j-th partial derivative Proposition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 The differentiation (or derivation) operator d : � C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) → C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F)) Φ → dΦ � is R-linear (“a derivation is linear”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' d(Φ + λΨ)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = limh→0 (Φ+λΨ)(p+h⃗u)−(Φ+λΨ)(p) h = limh→0 Φ(p+h⃗u)−Φ(p)+λΨ(p+h⃗u)−λΨ(p) h = limh→0 Φ(p+h⃗u)−Φ(p) h + λ limh→0 Ψ(p+h⃗u)−Ψ(p) h = dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u + λdΨ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = (dΦ(p) + λdΨ(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u for all p and ⃗u, thus d(Φ + λΨ) = dΦ + λdΨ for all λ ∈ R and Φ, Ψ ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Prove: if f ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) (scalar values) and Φ ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) then, for all ⃗u ∈ E, d(fΦ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = (df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u)Φ + f(dΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u) (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) (and we also write d(fΦ) = Φ ⊗ df + f dΦ for a use with contraction rules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' d(fΦ)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = lim h→0 f(p+h⃗u)Φ(p+h⃗u) − f(p)Φ(p) h = lim h→0 f(p+h⃗u)Φ(p+h⃗u) − f(p)Φ(p+h⃗u) h + f(p)Φ(p+h⃗u) − f(p)Φ(p) h = lim h→0 f(p+h⃗u) − f(p) h (Φ(p) + o(1)) + lim h→0 f(p)Φ(p+h⃗u) − Φ(p) h = (df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u)Φ(p) + f(p)(dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) Tensorial writing: d(fΦ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = (Φ ⊗ df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u + (f dΦ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u, thanks to the contraction rule which gives (Φ ⊗ df).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u + (f dΦ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = Φ(df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u) + f(dΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 In differential geometry, the definition of a tangent map is defined by, with definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4: TΦ : � U × E → F × F (p, ⃗u) → TΦ(p, ⃗u) = (Φ(p), dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) The two points p (input) and Φ(p) (output) are the base points, and the two vectors ⃗u (input) and dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u (output) are the initial vector and its push-forward by Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Notation for the second order Differential Let Φ ∈ C2(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus dΦ ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F)), thus d(dΦ) ∈ C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F)));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So, for p ∈ U and ⃗u ∈ E, we have d(dΦ)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = limh→0 dΦ(p+h⃗u)−dΦ(p) h ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F), and, with ⃗v ∈ E we have (d(dΦ)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 The bilinear map d2Φ(p) ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) is defined by d2Φ(p)(⃗u,⃗v) = (d(dΦ)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v, (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) thanks to the natural canonical isomorphism L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F)) ↔ TL ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) given by TL(⃗u1, ⃗u2) := (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2 for all ⃗u1, ⃗u2 ∈ E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus L =noted TL, thus d(dΦ) =noted d2Φ(p) ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This gives the usual second order Taylor expansion of Φ (supposed C2) near p in the direction ⃗u: Φ(p + h⃗u) = Φ(p) + h dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u + h2 2 d2Φ(p)(⃗u, ⃗u) + o(h2) (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) (=the second order Taylor expansion of f : h → f(h) = Φ(p + h⃗u) near h = 0, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And Schwarz’s theorem tells that d2Φ(p) is symmetric when Φ is C2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' d2Φ(p)(⃗u,⃗v) = d2Φ(p)(⃗v, ⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 A basis and the j-th partial derivative Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 Let Φ ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F), ⃗u ∈ Γ(U) (a vector field), p ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The derivative of Φ at p along ⃗u is defined by ∂⃗uΦ(p) := dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u(p) (= lim h→0 Φ(p + h⃗u(p)) − Φ(p) h ∈ F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) This defines the directional derivative operator along ⃗u: ∂⃗u : � C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) → C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) Φ → ∂⃗u(Φ) := dΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂⃗u(Φ)(p) := dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) (And ∂⃗u(Φ)(p) =noted ⃗u(Φ)(p) in differential geometry thanks to E ≃ E∗∗ which gives ∂⃗u ≃ ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 174 175 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Application 1: Scalar valued functions In particular, if (⃗ei(p)) is a basis at p, then the j-th partial derivative of Φ at p is ∂⃗ejΦ(p) =noted ∂jΦ(p) (the derivative along ⃗ej), and the j-th directional derivative operator is ∂j : � C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) → C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) Φ → ∂jΦ := dΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂j(Φ)(p) := dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) (In differential geometry ∂jΦ =noted ⃗ej(Φ), so ⃗ej(Φ)(p) := dΦ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Application 1: Scalar valued functions S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Differential of a scalar valued function (objective) Here Φ noted = f : � U → R p → f(p) � is a C1 scalar valued function, so df ∈ Ω1(U)∩C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗) (a C0 differential form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So df(p) ∈ E∗ for all p ∈ U, and df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = limh→0 f(p+h⃗u)−f(p) h ∈ R for all ⃗u ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 Prove: If f, g ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) then (derivative of a product) d(fg) = (df)g + f(dg), (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', d(fg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = (df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w)g + f(dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) for all ⃗w ∈ Γ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' limh→0 f(p+h ⃗w)g(p+h ⃗w)−f(p)g(p) h = limh→0 f(p+h ⃗w)g(p+h ⃗w)−f(p)g(p+h ⃗w) h + limh→0 f(p)g(p+h ⃗w)−f(p)g(p) h = limh→0 f(p+h ⃗w)−f(p) h (g(p) + o(1)) + limh→0 f(p) g(p+h ⃗w)−g(p) h ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' calculation that only requires the first order (affine) approximation of f and g: We get the same result as with the affine functions f(x) = a0+a1x and g(x) = b0+b1x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' which give (fg)(x) = a0b0 + (a0b1+a1b0)x + a1b1x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' and then (fg)′(x) = a0b1+a1b0 + 2a1b1x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' which is indeed equal to (f ′g + fg′)(x) = a1(b0+b1x) + (a0+a1x)b1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Quantification Let (⃗ei(p)) be a basis at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So ∂jf(p) =(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej(p) (= limh→0 f(p+h⃗ej(p))−f(p) h ), and we write ∂jf(p) noted = f|j(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) So, with (πei(p)) the dual basis of the basis (⃗ei(p)), and with f|j(p) := πei(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='df(p) (j-th component of df(p) in the basis (πei(p))), we have df = n � j=1 f|jπej, and [df(p)]|⃗e = ( f|1(p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' f|n(p) ) (row matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) So df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = �n j=1f|juj = [df]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗u]|⃗e when ⃗u(p) = � i ui(p)⃗ei(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular with a Cartesian basis, (πei(p)) =noted (dxj), and df = �n j=1 ∂f ∂xj dxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notations: πei = ei, ⃗u = �n j=1uj⃗ej, df = �n j=1f|j ej, df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = �n j=1f|juj, and with a Cartesian basis, πei = dxi and df = �n j=1 ∂f ∂xj dxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12 Prove: (fg)|j = f|j g + f g|j when f, g : U → R are C1 scalar valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Apply (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7): here d(fg) = g df + f dg, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' d(fg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = (df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej) g + f (dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej) for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And df(p) ∈ E∗ satisfies the covariant change of basis formula for linear forms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', if (⃗ai(p)) and (⃗bi(p)) are two bases at p and P(p) is the transition matrix from (⃗ai(p)) to (⃗bi(p)), then [df(p)]|⃗b =(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) [df(p)]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P(p), or in short: [df]|⃗b = [df]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P (covariance formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) 175 176 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Application 2: Coordinate system basis and Christoffel symbols S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Gradients (subjective) associated with a differential through inner dot products Let f ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) (a C1 scalar valued function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Choose (subjective) an inner dot product (·, ·)g in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13 The conjugate gradient ⃗ gradgf(p) of f at p ∈ U relative to (·, ·)g, also called the (·, ·)g-conjugate gradient of f at p, is the (·, ·)g-Riesz representation vector of the linear form df(p) ∈ E∗: ⃗ gradgf(p) := ⃗Rg(df(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', the vector ⃗ gradgf(p) ∈ E is characterized by, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2), ∀⃗u ∈ E, df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = ( ⃗ gradgf(p), ⃗u)g = ⃗ gradgf(p) •g ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) Fundamental: An English observer with his Euclidean dot product (·, ·)a in foot and a French observer with his Euclidean dot product (·, ·)b in metre have the same differential df (defined independently of any unit of measurement);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' But do not have the same gradient: ⃗ gradbf (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) = λ2 ⃗ gradaf with λ2 > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) Quite different vectors isn’t it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The “gradient vector” strongly depends on the chosen inner dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And to forget this fact leads to accidents like the crash of the Mars Climate Orbiter probe, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Subjective first order Taylor expansion: If an inner dot product (·, ·)g exists and is used, then the first order Taylor expansion (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) gives f(p + h⃗u) = f(p) + h ( ⃗ gradgf(p), ⃗u)g + o(h) (= f(p) + h ⃗ gradgf(p) •g ⃗u + o(h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) Fundamental once again (we insist): An inner dot product does not always exist (as a meaningful tool), see § B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 (thermodynamics), thus, for a C1 function, a gradient does not always exists (contrary to a differential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' df(p) is a linear form (covariant) while ⃗ gradgf(p) is a vector (contravariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular the change of basis formulas differ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28): [df]|new = [df]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P, while [ ⃗ gradg]|new = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ ⃗ gradg]|old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22) df cannot be identified ⃗ gradf (with one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') (Recall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' there is no natural canonical isomorphims between E and E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') The differential df is also called the “covariant gradient”, and any of its associated gradient vectors is also called the “contravariant gradient relative to an inner dot product”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Isometric Euclidean framework: If one Euclidean dot product can be imposed to all observers (foot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' metre?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') then ⃗ gradgf =noted ⃗ gradf = ⃗∇f and (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) is written df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = ⃗ gradf • ⃗u = ⃗∇f • ⃗u (isometric framework).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14 Cartesian basis (⃗ei) and (·, ·)g given by [g][⃗e = � 1 0 0 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Give [df]|⃗e and [ ⃗ gradgf]|⃗e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [df]|⃗e = ( ∂f ∂x1 ∂f ∂x2 ) (row matrix) and (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) gives [ ⃗ gradgf]|⃗e = � ∂f ∂x1 1 2 ∂f ∂x2 � (column matrix ̸= [df]T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Application 2: Coordinate system basis and Christoffel symbols (Necessary when dealing with covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Coordinate system, and coordinate system basis Consider a (open) set Upar = {⃗q ∈]a1, b1[×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='×]an, bn[}, called the set of parameters, in the Cartesian space Rn, consider an open set U ⊂ Rn, called the set of geometric positions, and consider a C2- diffeomorphism Ψ : ⃗q ∈ Upar → p ∈ U, called a coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗ai) the canonical basis of the parameter space, let ⃗q = � i qi⃗ai ∈ Upar (the qi are called the parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', see the polar coordinate system at § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 where ⃗q = (q1, q2) = (r, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 176 177 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Application 2: Coordinate system basis and Christoffel symbols Ψ being a diffeomorphism, at any p = Ψ(⃗q) ∈ U the vectors ⃗ai∗(p) := dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) make a basis in E at p, and (⃗ai∗(p)) is called the coordinate system basis at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its dual basis at p is made of the linear forms dqi(p), so where, for all i, j, dqi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj∗(p) = δj i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24) Duality notations: dqi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj∗(p) = δi j for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Parametric expression of the differential of a scalar valued function With a coordinate system Ψ, a scalar valued function f : � U → R p → f(p) � defined in U can be described with the function g = f ◦ Ψ : � Upar → R ⃗q → g(⃗q) := f(p) when p = Ψ(⃗q) � defined in Upar, and g is called the parametric expression of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus dg(⃗q) = df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(⃗q) when p = Ψ(⃗q), (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25) in particular, ∂g ∂qj (⃗q) := dg(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj∗(p) noted = ∂f ∂qj (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) Warning, pay attention: f is a function of p, not a function of ⃗q, and the notations ∂f ∂qj (p) means := ∂(f◦Ψ) ∂qj (⃗q) when p = Ψ(⃗q), and nothing else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus with (dqj(p)) the dual basis of the coordinate basis (⃗ai∗(p)) at p, df(p) (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26) = n � j=1 ∂f ∂qj (p) dqj(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27) Duality notations: df(p) = � j ∂f ∂qj (p) dqj(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15 Pay attention to the notations that could contradict themselves: 1- In Upar the dual basis (πai) of the Cartesian basis (⃗ai) is a uniform basis (independent of ⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' and is (almost) never written (dqi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- Indeed, (dqi(p)) is the name reserved for the dual basis of (⃗ai∗(p)) in the geometric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Mind the notations!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' for polar coordinates (dq1(p), dq2(p)) = (dr(p), dθ(p)) is the dual basis of the polar coordinate system basis (⃗a1∗(p),⃗a2∗(p)) at p, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16 Bases (⃗ai) and (⃗bi) at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A vector ⃗x is expressed as ⃗x = � i xa,i⃗ai = � i xb,i⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Prove: ⃗bi = λ⃗ai, ∀i =⇒ ∂f ∂xb,i = λ ∂f ∂xa,i or ∂f ∂xa,i = ∂f ∂(λxa,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28) (Change of unit formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Duality notations: ∂f ∂xj b = λ ∂f ∂xj a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' df(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj(p) = λdf(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj(p) (linearity of df(p)) reads (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Or [df]|⃗b = [df]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P with P = λI here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17 [df]|⃗b = [df]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂f ∂xj b = �n i=1 ∂f ∂xia P i j is also noted ∂f ∂xj b = n � i=1 ∂f ∂xia ∂xi a ∂xj b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) Why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 177 178 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Application 2: Coordinate system basis and Christoffel symbols Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quick answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗a = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[⃗x]|⃗b, which means [⃗x]|⃗a([⃗x]|⃗b) = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[⃗x]|⃗b, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � � � x1 a(x1 b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', xn b ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' x1 a(x1 b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', xn b ) � � � = � � � �n j=1P 1 j xj b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' �n j=1P n j xj b � � � , thus ∂xi a ∂xj b (x1 b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', xn b ) = P i j , ∀i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30) Thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) means ∂f ∂xj b (p) = n � i=1 ∂f ∂xia (p)∂xi a ∂xj b (x1 b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', xn b ) thus = n � i=1 ∂f ∂xia (p) P i j , (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31) as given in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Detailed answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let O be a point (origin) in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If p ∈ U, let ⃗x = −→ Op = �n i=1xi a⃗ai = �n i=1xi b⃗bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' This define the function [⃗x]|⃗a : [⃗x]|⃗b → [⃗x]|⃗a([⃗x]|⃗b), and we have [⃗x]|⃗a([⃗x]|⃗b) = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[⃗x]|⃗b (change of basis formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then let fa, fb : Rn → R be defined by fa([⃗x]|⃗a) := f(p) and fb([⃗x]|⃗b) := f(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (NB: fa and fb don’t have the same definition domain: They are different).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus fa([⃗x]|⃗a) = fb([⃗x]|⃗b) (= f(p)) when [⃗x([⃗x]|⃗b)]|⃗a = P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='[⃗x]|⃗b, so (fa ◦ [⃗x]|⃗a)([⃗x]|⃗b) = fb([⃗x]|⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ∂(fa◦[⃗x]|⃗a) ∂xi b ([⃗x]|⃗b) = ∂fb ∂xi b ([⃗x]|⃗b), thus the meaning of (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29) is n � j=1 ∂fa ∂xj a ([⃗x]|⃗a)∂xj a ∂xi b ([⃗x]|⃗b) = ∂fb ∂xi b ([⃗x]|⃗b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32) Question: Why did we introduce fa and fb (and not just keep f)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: Because a vector is not just a collection of components (is not just a matrix), and −→ Op cannot be reduced to a matrix of components (which one: [⃗x]|⃗a?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗x]|⃗b?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Here f is a function acting on a point p (independent of a referential), while fa and fb are functions acting on matrices (dependent on the choice of a referential): The domain of definitions are different, so the functions f, fa and fb are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Christoffel symbols We use duality notations for readability and usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18 In a coordinate system basis (⃗ei(p)) in E (previously called (⃗ai∗(p)), the Christoffel symbol γi jk(p) ∈ R are the components of the vector d⃗ek(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej(p), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' d⃗ek(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej(p) = �n k=1γi jk(p)⃗ei(p), so d⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = n � i=1 γi jk⃗ei , or d⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei = n � k=1 γk ij⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) (So, with (ei(p)) the dual basis of (⃗ei(p)), γi jk := ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej, and, for calculations with contractions, d⃗ek = � ij γi jk⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') (The Christoffel symbols vanish in a Cartesian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') (Differential geometry in manifolds: ∇⃗ej⃗ek = �n i=1γi jk⃗ei, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' the γi jk = ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∇⃗ej⃗ek are the component of the connection ∇, the usual connection in a surface in Rn being the Riemannian connection, in which case ∇⃗ej⃗ek is the orthogonal projection of d⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej on the surface relative to a Euclidean dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' for the polar coordinate system, see remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12, d⃗e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗e2 = −r⃗e1, thus γ1 22 = −r and γ2 22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19 Prove: If (⃗ei(p)) is the coordinate system basis of a C2 coordinate system, then: ∀i, j, d⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = d⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei (= ∂2Ψ ∂qi∂qj ), and ∀i, j, k, γk ji = γk ij (symmetry for lower indices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='34) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗ei(p) = (⃗ei ◦ Ψ)(⃗q) =(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23) dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai gives d(⃗ei ◦ Ψ)(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = d(dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj, thus d⃗ei(Ψ(⃗q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dΨ(⃗q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = ∂ ∂Ψ ∂qi ∂qj = ∂ ∂Ψ ∂qj ∂qi (Schwarz theorem since Ψ is C2) = d⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei =noted ∂2Ψ ∂qj∂qi , thus �n k=1γk ij⃗ek = �n k=1γk ji⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20 Consider two coordinate system bases (⃗ai(p)) and (⃗bi(p)) at p, P(p) = [P i j(p)] the tran- sition matrix from (⃗ai(p)) to (⃗bi(p)), and Q = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Using the generic notation d⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1γi jk,e⃗ei, prove the change of basis formula for the Christoffel symbols: γi jk,b = n � λ,µ,ν=1 Qi λP µ j P ν k γλ µν,a+ n � λ,µ=1 Qi λP µ j (dP λ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ) (= n � λ,µ,ν=1 Qi λP µ j P ν k γλ µν,a+ n � λ=1 Qi λ(dP λ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35) (Because of the term � µν Qi λP µ j (dP λ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ), a derivation is not a tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 178 179 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Application 3: Differential of a vector field Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗bk(p) = � ν P ν k (p)⃗aν(p) gives d⃗bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj = � ν(dP ν k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj)⃗aν + � ν P ν k (d⃗aν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj) = � µν P µ j (dP ν k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ)⃗aν + � µν P ν k P µ j (d⃗aν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And bi = � λ Qi λaλ, thus γi jk,b = bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='d⃗bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj = � λµν Qi λP µ j (dP ν k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ)aλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aν + � λµν Qi λP µ j P ν k aλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗aν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ) = � λµ Qi λP µ j (dP λ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ) + � λµν Qi λP µ j P ν k γλ µν,a, thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Application 3: Differential of a vector field Here F = E = ⃗Rn, Φ =noted ⃗w ∈ Γ(U) is a vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus d⃗w(p) ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) and d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u is a vector field in E for all ⃗u ∈ Γ(U), given by (d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u)(p) = d⃗w(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u(p) = limh→0 ⃗w(p+h⃗u(p))− ⃗w(p) h ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification: (⃗ei(p)) is a basis at p in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Call wi(p) ∈ R the components of ⃗w(p), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗w(p) = �n i=1wi(p)⃗ei(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And call wi|j(p) the components of d⃗w(p) (endomorphism in E): ⃗w = n � i=1 wi⃗ei, d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = n � i=1 wi|j⃗ei, [d⃗w]|⃗e = [wi|j] (Jacobian matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) And tensorial notations for calculations with contractions: (πei(p)) being the dual basis, d⃗w = n � i,j=1 wi|j⃗ei ⊗ πej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='37) Duality notations: ⃗w = �n i=1wi⃗ei, d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i,j=1wi |j⃗ei, [d⃗w]|⃗e = [wi |j], and d⃗w = �n i,j=1wi |j⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In a Cartesian basis: Here (⃗ei) is uniform, so ⃗w(p) = �n i=1wi(p)⃗ei gives d⃗w(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1(dwi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej)⃗ei, thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='36) gives wi|j = ∂wi ∂xj (p) noted = wi,j, so [d⃗w]|⃗e = [∂wi ∂xj ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='38) Duality notations: wi |j = ∂wi ∂xj and [d⃗w]|⃗e = [ ∂wi ∂xj ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In a coordinate system basis: With the coordinate system described in § S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 and the duality notations for readability (and usage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗w(p) = �n i=1wi(p)⃗ei(p) gives, for all j, d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = n � i=1 (dwi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej)⃗ei + n � i=1 wi(d⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej) (= n � i=1 wi |j⃗ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='39) (Tensorial notations to be used with contractions: d⃗w = � i ⃗ei ⊗ dwi + � i wi d⃗ei = � ij wi |j⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') And � i wi(d⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej) =(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) � ik wiγk ji⃗ek = � ik wkγi jk⃗ei, thus, for all i, j, wi |j = ∂wi ∂qj + n � k=1 wkγi jk where ∂wi ∂qj := dwi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) ( ∂wi ∂qj := dwi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej is the derivation along the j-th coordinate line of the scalar valued function wi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (In particular, if ⃗w = ⃗eℓ = � i δi ℓ⃗ei, we recover d⃗eℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = � i 0⃗ei + � ik δk ℓ γi jk⃗ei = � i γi jℓ⃗ei, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21 With exercise S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20, and ⃗w = �n i=1ui⃗ai = � n vi⃗bi, check with calculations (d⃗w is an endomorphism defined independently of any basis): [d⃗w]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [d⃗w]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' vi |j = n � k,ℓ=1 Qi kuk |ℓP ℓ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗bj = � ℓ P ℓ j⃗aℓ for all i, Q = P |1, and [⃗w]|⃗b = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [⃗w]|⃗a reads vi = � k Qi kuk for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Cartesian basis: dvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj = d(� k Qi kuk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (� ℓ P ℓ j⃗aℓ) = � kℓ Qi k(duk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aℓ)P ℓ j , qed (here the Qi k are uniform i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' independent of p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 179 180 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Application 4: Differential of a differential form Coordinate system basis: vi = � λ Qi λuλ gives dvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj = � λ(dQi λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj)uλ + � λ Qi λ(duλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus vi |j (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33) = dvi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj + � k vkγi jk,b = � λµ uλP µ j (dQi λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ) + � λµ Qi λP µ j (duλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ) + � kλµν (Qk λuλ)Qi νP µ j (dP ν k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ) + � kωλµν (Qk ωuω)Qi λP µ j P ν k γλ µν,a And Qk ωP λ k = δλ ω gives (dQk ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ)P λ k + Qk ω(dP λ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ) = 0, thus the third term reads � kλµν uλQi νP µ j Qk λ(dP ν k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ) = − � kλµν uλQi νP µ j P ν k (dQk λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ) = − � λµ uλP µ j (dQi λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ), which cancels the first term: Thus vi |j = � λµ Qi λP µ j (duλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aµ) + � λµν uνQi λP µ j γλ µν = � λµ Qi λui |jP µ j , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Application 4: Differential of a differential form Here F = R, Φ =noted ℓ ∈ Ω1(U) (differential form) supposed C1, p ∈ U, so ℓ(p) ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its differential at p in a direction ⃗u is dℓ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = limh→0 ℓ(p+h⃗u)−ℓ(p) h ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (dℓ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v = limh→0 ℓ(p+h⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v−ℓ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v h ∈ R for all ⃗u,⃗v ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification: (πei(p) its the dual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Call ℓi(p) ∈ R the components of ℓ(p), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ℓ(p) = �n i=1ℓi(p)πei(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And call ℓi|j(p) the components of dℓ(p) ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗): ℓ = n � i=1 ℓiπei, dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = n � i=1 ℓi|jπei, [dℓ]|⃗e = [ℓi|j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='42) Tensorial notations, to be used with contractions: dℓ = �n i,j=1ℓi|jπei ⊗ πej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Duality notations: ℓ = � i ℓiei, dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1ℓi|jei, [dℓ]|⃗e = [ℓi|j], and dℓ = �n i,j=1ℓi|jei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In a Cartesian basis: Here (⃗ei) is uniform, so ℓi|j = ∂ℓi ∂xj (p) noted = ℓi,j, so [dℓ]|⃗e = [ ∂ℓi ∂xj ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='43) Duality notations: ℓi|j = dℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = ∂ℓi ∂xj and [dℓ]|⃗e = [ ∂ℓi ∂xj ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In a coordinate system basis: With duality notations and Christoffel symbols: dei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = − n � k=1 γi jkek .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='44) Indeed, ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek = δi k gives (dei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek + ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (d⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej) = 0, thus (dei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek = −ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � ℓ γℓ jk⃗eℓ = −γi jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ℓi|j = ∂ℓi ∂qj − n � k=1 ℓkγk ji where ∂ℓi ∂qj (p) := dℓi(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45) Indeed, ℓ = � i ℓiei gives dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = � i(dℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej)ei + � i ℓi(dei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej) = � i(dℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej)ei − � ik ℓiγi jkek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 Application 5: Differential of a 1 1 tensor Consider a C1 �1 1 � tensor τ : � U → L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) p → τ(p) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Its differential dτ : � U → L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R)) p → dτ(p) � is defined by dτ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = limh→0 τ(p+h⃗u)−τ(p) h ∈ L(E∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), so (dτ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u)(ℓ,⃗v) = limh→0 τ(p+h⃗u)(ℓ,⃗v)−τ(p)(ℓ,⃗v) h (∈ R), for all ⃗u,⃗v ∈ E and ℓ ∈ E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification (duality notations): Basis (⃗ei(p)) in E at p, dual basis (ei(p)), call τ i j(p) the components of τ(p), call τ i j|k(p) the components of dτ(p): τ = � ij τij⃗ei ⊗ ej, dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek = n � i,j=1 τ i j|k⃗ei ⊗ ej .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='46) Tensorial notations, to be used with contractions: dτ = �n i,j,k=1τ i j|k⃗ei ⊗ ej ⊗ ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Classical notations: τ = � ij τij⃗ei ⊗ πej, dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek = � ij τij|k⃗ei ⊗ πej, and dτ = � ijk τij|k⃗ei ⊗ πej ⊗ πek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') 180 181 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Divergence of a vector field: Invariant Cartesian basis: dτ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek = � ij(dτ i j(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek)⃗ei ⊗ ej, so τ i j|k = ∂τ i j ∂xk noted = τ i j,k (:= dτ i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='47) Coordinate system basis: τ(p) = �n i,j=1τ i j(p)⃗ei(p) ⊗ ej(p) gives, for all k, dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek = � ij(dτ i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek)⃗ei ⊗ ej + � ij τ i j(d⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek) ⊗ ej + � ij τ i j⃗ei ⊗ (dej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek) = � ij(dτ i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek)⃗ei ⊗ ej + � ijℓ τ i jγℓ ki⃗eℓ ⊗ ej − � ijℓ τ i jγj kℓ⃗ei ⊗ eℓ = � ij(dτ i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek)⃗ei ⊗ ej + � ijℓ τ ℓ j γi kℓ⃗ei ⊗ ej − � ijℓ τ i ℓγℓ kj⃗ei ⊗ ej (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='48) thus τ i j|k = ∂τ i j ∂qk + n � ℓ=1 τ ℓ j γi kℓ − n � ℓ=1 τ i ℓγℓ kj where ∂τ i j ∂qk := dτ i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='49) (We have the + sign from vector fields, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40), and the − sign from differential forms, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='22 If ⃗u ∈ E, ℓ ∈ E∗ then for the elementary �1 1 � tensor τ = ⃗u ⊗ ℓ prove: d(⃗u ⊗ ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek = (d⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek) ⊗ ℓ + ⃗u ⊗ (dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek), and (⃗u ⊗ ℓ)i j|k = ui |kℓj + uiℓj|k, (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='50) when ⃗u = � i ui⃗ei, ℓ = � j ℓjej, d⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek = � i ui |k⃗ei, dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek = � j ℓj|kej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' τ = ⃗u ⊗ ℓ = � ij τ i j⃗ei ⊗ ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' where τ i j = uiℓj, and dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek = �n i,j=1τ i j|k⃗ei ⊗ ej where τ i j|k = (uiℓj)|k = ui |kℓj + uiℓj|k = (⃗u ⊗ ℓ)i j|k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (similar to the derivation of a product): d(⃗u ⊗ ℓ)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek(p) = lim h→0 (⃗u ⊗ ℓ)(p+h⃗ek(p)) − (⃗u ⊗ ℓ)(p) h = lim h→0 ⃗u(p+h⃗ek(p)) ⊗ ℓ(p+h⃗ek(p)) − ⃗u(p) ⊗ ℓ(p) h = lim h→0 ⃗u(p+h⃗ek(p)) ⊗ ℓ(p+h⃗ek(p)) − ⃗u(p+h⃗ek(p)) ⊗ ℓ(p) h + lim h→0 ⃗u(p+h⃗ek(p)) ⊗ ℓ(p) − ⃗u(p) ⊗ ℓ(p) h = lim h→0(⃗u(p+h⃗ek(p)) ⊗ (ℓ(p+h⃗ek(p)) − ℓ(p) h ) + lim h→0(⃗u(p+h⃗ek(p)) − ⃗u(p) h ) ⊗ ℓ(p) = ⃗u(p) ⊗ (dℓ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek(p)) + (d⃗u(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek(p)) ⊗ ℓ(p), thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='50)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Which gives d(⃗u ⊗ ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek = (� i ui⃗ei) ⊗ (� j ℓj|kej) + (� i ui |k⃗ei) ⊗ (� j ℓjej), thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='50)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 Divergence of a vector field: Invariant Γ(U) is the set of C1 vector fields in U, and Tr : L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) → R is the trace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='23 The divergence operator is div := Tr ◦ d : � Γ(U) → C0(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) ⃗w → div⃗w := Tr(d⃗w), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' div⃗w(p) := Tr(d⃗w(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='51) (So div⃗w(p) = trace of the endomorphism d⃗w(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Tr and d are linear, hence div = Tr ◦ d is R-linear (composed of two R-linear maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='24 The divergence of a vector field is objective (is an invariant): Same value for all observers (objective quantity) intrinsic to ⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The differential and the trace are objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Or computation: ⃗w = � i ui⃗ai = � i vi⃗bi gives vi |j = � kℓ Qi kuk |ℓP ℓ j , see (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='41), thus � i vi |i = � ikℓ P ℓ i Qi kuk |ℓ = � kℓ δℓ kuk |ℓ = � k uk |k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Quantification: ⃗w ∈ Γ(U), (⃗ei) is a basis, ⃗w = �n i=1wi⃗ei with classical notations, and wi|j(p) are the components of the vector d⃗w(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej(p) in the basis (⃗ei(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus div⃗w = n � i=1 wi|i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='52) Duality notations: ⃗w = �n i=1wi⃗ei, d⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = �n i=1wi |j⃗ei, [d⃗w]|⃗e = [wi |j], div⃗w = �n i=1wi |i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 181 182 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Objective divergence for 1 1 tensors Cartesian basis (⃗ei) (classical notations): dwi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej =noted ∂wi ∂xj and wi|j = ∂wi ∂xi , thus div⃗w = n � i=1 ∂wi ∂xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='53) Coordinate system basis (⃗ei) (duality notations): With the Christoffel symbols, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='33), (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='40) gives wi |i = ∂wi ∂qi + n � i=1 wkγi ik, thus div⃗w = n � i=1 ∂wi ∂qi + n � i,k=1 wkγi ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54) Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='25 Prove: div(f ⃗w) = df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + f div⃗w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='55) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' d(f ⃗w) = ⃗w ⊗ df + f d⃗w, thus Tr(d(f ⃗w)) = Tr(⃗w ⊗ df) + Tr(f d⃗w) = df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + f Tr(d⃗w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Use a coordinate system if you prefer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='26 If α is a differential form, if (⃗ei) is a basis and (ei) its dual basis, and if α = �n i=1αiei, then dα = �n i=1αi|jei ⊗ ej, with αi|j := ⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Here it is impossible to define an objective trace of dα like �n i=1αi|i: The result depends on the choice of the basis (the Einstein convention is not satisfied, and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' with a Euclidean basis the result depends on the choice of unit of length: Foot?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Meter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus the objective (or intrinsic) divergence of a differential form is a nonsense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 Objective divergence for 1 1 tensors To create an objective divergence for a second order �1 1 � tensor τ ∈ T 1 1 (U), in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='46) we have to contract an admissible index with the “differential index k”, So, no choice: Contract i and k to get � divτ := �n i,j=1τ i j|iej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let us start with: Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='27 Let ⃗u ∈ Γ(U) and ℓ ∈ Ω1(U) be C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The objective divergence of the elementary �1 1 � tensor ⃗u ⊗ ℓ ∈ T 1 1 (U) is the differential form � div(⃗u ⊗ ℓ) ∈ Ω1(U) defined by � div(⃗u ⊗ ℓ) = (div⃗u)ℓ + dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u, (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='56) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' defined by � div(⃗u ⊗ ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = (div⃗u)(ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) + (dℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w)⃗u for all ⃗w ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the objective divergence operator � div : � T 1 1 (U) → Ω1(U) τ → � divτ � is the linear map defined on elementary tensors with (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Quantification: (⃗ei) is a basis, (ei) its dual basis, ⃗u = � i ui⃗ei, ℓ = � j ℓjej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus ⃗u⊗ℓ = � ij uiℓj⃗ei⊗ej, and (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='50)-(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='56) give � div(⃗u ⊗ ℓ) = n � i,j=1 (ui |iℓj + ℓj|iui)ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='57) So for an elementary tensor τ = ⃗u ⊗ ℓ, τ = � ij τ i j⃗ei ⊗ ej, and dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek = � ijk τ i j|k⃗ei ⊗ ej and � div(τ) = � ij τ i j|iei, with τ i j|k = ui |kℓj + uiℓj|k, here with τ i j = uiℓj, and τ i j|k = ui |kℓj + uiℓj|k, so τ i j|i = ui |iℓj + uiℓj|i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, by linearity of � div, for all tensors τ ∈ T 1 1 (U), we have with (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='46): � divτ = n � i,j=1 τ i j|iej , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ � divτ]|⃗e = � � i τ i 1|i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � i τ i n|i � (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='58) (row matrix since � divτ is a differential form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', we have contracted i and k in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Classical notations: � divτ := �n i,j=1τij|iπej, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [ � divτ]|⃗e = ( � i τi1|i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' � i τin|i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') So Cartesian bases: � divτ = n � i,j=1 ∂τ i j ∂xi ej, Coord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' bases: � divτ = n � i,j=1 �∂τ i j ∂qi + n � k=1 τ k j γi ik − n � k=1 τ k i γi kj � ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='59) Indeed: With τ = � j(� i τ i j⃗ei) ⊗ ej = � j ⃗wj ⊗ ej where ⃗wj = � i τ i j⃗ei, the linearity of � div gives � divτ = � j � div(⃗wj ⊗ ej);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus, with (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='56): 1- Cartesian basis: div⃗wj = � i ∂τ i j ∂xi and dej = 0 give � divτ = 182 183 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Objective divergence for 1 1 tensors � j � i ∂τ i j ∂xi ej = � ij τ i j,iej, thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='59)1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And 2- Coordinate system basis: div⃗wj=(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='54)� i ∂τ i j ∂qi +� ik τ k j γi ik and dej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wj = � k τ k j dej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ek = � k τ k j (− � i γj kiei), thus � j dej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗wj = − � ijk τ k j γj kiei = − � ijk τ k i γi kjej, thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='59)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='28 Prove: If f ∈ C1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) and τ = �n i,j=1τ i j⃗ei ⊗ ej ∈ T 1 1 (U) ∩ C1 then � div(fτ) = df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='τ + f � divτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='60) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' fτ = � ij fτ i j⃗ei ⊗ ej gives d(fτ) = � ijk(fτ i j)|k⃗ei ⊗ ej ⊗ ek = � ijk(f|kτ i j + fτ i j|k)⃗ei ⊗ ej ⊗ ek, thus � div(fτ) = � ij(f|iτ i j + fτ i j|i)ej;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='τ + f � divτ = � ij f|iτ i jej + f � ij τ i j|iej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='29 Prove: If τ ∈ T 1 1 (U) and ⃗w ∈ Γ(U) then div(τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) = � div(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w + τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='. d⃗w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='61) Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' τ = � ij τ i j⃗ei ⊗ ej and ⃗w = � i wi⃗ei give τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w = � ij τ i jwj⃗ei, thus div(τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗w) = � ij τ i j|iwj + τ i jwj |i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Exercice S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='30 If τ ∈ T 1 1 (U) check with component calculations (since � div(τ) is objective): [ � div(τ)]|b = [ � div(τ)]|a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P (covariance formula), (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='62) where P is the transition matrix from a basis (⃗ai) to a basis (⃗bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let τ = � ij σi j⃗ai ⊗ aj = � ij τ i j⃗bi ⊗ bj, so τ i j = � λµ Qi λσλ µP µ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- Cartesian bases: � i τ i j|i = � i dτ i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bi = � i d(� λµ Qi λσλ µP µ j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (� ν P ν i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aν) = � iλµν Qi λP µ j P ν i (dσλ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aν) = � λµν δν λP µ j (dσλ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aν) = � λµ P µ j (dσλ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aλ) = � µ(� λ σλ µ|λ)P µ j as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- Coordinate system bases: � i τ i j|i =(S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='59) � i dτ i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ei + � iℓ τ ℓ j γi iℓ,b − � iℓ τ i ℓγℓ ij,b (with j fixed);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' With � i (dτ i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bi) = � iλµ Qi λ (dσλ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bi) P µ j + � iλµ (dQi λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bi) σλ µ P µ j + � iλµ Qi λ σλ µ (dP µ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bi) = � iλµν Qi λP µ j P ν i (dσλ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aν) + � iλµν σλ µ P µ j P ν i (dQi λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aν) + � iλµν σλ µ Qi λP ν i (dP µ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aν) = � λµ P µ j (dσλ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aλ) − � iλµν σλ µ P µ j Qi λ(dP ν i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aν) + � λµ σλ µ (dP µ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aλ) since P ν i Qi λ = δν λ gives P ν i (dQi λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aν) − Qi λ(dP ν i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And, with (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='35), � iℓ τ ℓ j γi iℓ,b = � iℓ ( � λµ Qℓ λσλ µP µ j )( � αβω Qi αP β i P ω ℓ γα βω,a + � αβ Qi αP β i (dP α ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aβ)) = � λµα σλ µP µ j γα αλ,a + � ℓλµα σλ µQℓ λP µ j (dP α ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aα), (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='63) and − � iℓ τ i ℓγℓ ij,b = − � iℓ ( � λµ Qi λσλ µP µ ℓ )( � αβω P α i P β j Qℓ ωγω αβ,a + � αω P α i Qℓ ω(dP ω j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aα)) = − � λµβ σλ µP β j γµ λβ,a − � λµ σλ µ(dP µ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='64) Thus � i τ i j|i = � λµ P µ j (dσλ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aλ) + � λµα σλ µP µ j γα αλ,a − � λµβ σλ µP β j γµ λβ,a = � λµ P µ j σλ µ|λ as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Divergence of a 2 0 tensor Let τ ∈ T 2 0 (U) and τ = �n i,j=1τ ij⃗ei⊗⃗ej, thus dτ = �n i,j,k=1τ ij |k⃗ei⊗⃗ej⊗ek;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then two objective divergences may be defined: by contracting k with i, or k with j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (The Einstein convention is then satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Divergence of a 0 2 tensor Let τ = �n i,j=1τijei ⊗ ej ∈ T 0 2 (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus dτ = �n i,j,k=1τij|kei ⊗ ej ⊗ ek, and there are no indices to contract to satisfy Einstein convention: There is no objective divergence of 0 2 tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 183 184 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Euclidean framework and “classic divergence” of a tensor (subjective) S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 Euclidean framework and “classic divergence” of a tensor (subjective) Let σ be a C1 tensor of order 2 of any kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' An observer chooses a (Cartesian) Euclidean basis (⃗ei) and call (·, ·)g the associated Euclidean dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And he calls σij the components of σ, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' writes σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗ej = � i σij⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='31 (Usual divergence in classical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') The divergence divσ of σ relative to the basis (⃗ei), is the column matrix (it is not a vector) diveσ = � � � �n j=1 ∂σ1j ∂xj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' �n j=1 ∂σnj ∂xj � � � noted = divσ (a matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='65) (Take the divergences of the “row vectors” of [σ]|e = [σij] to make the “column vector” [diveσ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proposition S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='32 The “so called vector” divσ, in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='67), is not a vector: It does not satisfy the change of basis formula: If (⃗ai) and (⃗bi) are bases, if P is the transition matrix from (⃗ai) to (⃗bi), if [σ]|⃗a = [Aij] and [σ]|⃗b = [Bij], with the divergence of σ relative to (⃗ai) and (⃗bi) called divaσ and divbσ, then neither a contravariant nor a covariant change of basis formula applies in general: neither [divbσ]|⃗b ̸= P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [divaσ]|⃗a nor [divbσ]T |⃗b = [divaσ]T |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='66) (compare with (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='62)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So divσ as given in (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='67) is neither a contravariant vector nor a covariant vector (it is just a matrix which depends on an observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider the simple case ⃗bi = λ⃗ai, for all i, λ > 1: Transition matrix P = λI, and P −1 = 1 λI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' For a �1 1 � tensor: σ = � ij(σb)i j⃗bi ⊗ bj = � ij(σa)i j⃗ai ⊗ aj, [σ]|⃗b = P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [σ]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P = 1 λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [σ]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='λ = [σ]|⃗a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (σa)i j = (σb)i j for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='67) gives divbσ = � ij(d(σb)i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj)⃗bi = � ij(d(σa)i j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (λ⃗aj))(λ⃗ai) = λ2divaσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus [divbσ]|⃗b ̸= P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [divbσ]|⃗a and [divbσ]T |⃗b ̸= [divaσ]T |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' For a �0 2 � tensor: σ = � ij σb,ijbi ⊗ bj = � ij σa,ijai ⊗ aj, and [σ]|⃗b = P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [σ]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P = λ2[σ]|⃗a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' σb,ij = λ2σa,ij for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='67) gives divbσ = � ij(dσb,ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj)⃗bi = λ2 � ij(dσa,ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (λ⃗aj))(λ⃗ai) = λ4divaσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus [divbσ]|⃗b ̸= P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [divbσ]|⃗a and [divbσ]T |⃗b ̸= [divaσ]T |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' For a �2 0 � tensor: σ = � ij σij b ⃗bi ⊗ ⃗bj = � ij σij a ⃗ai ⊗ ⃗aj, and [σ]|⃗b = P −T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [σ]|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P −1 = 1 λ2 [σ]|⃗a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' σij b = 1 λ2 σij a for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='67) gives divbσ = � ij(dσij b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj)⃗bi = 1 λ2 � ij(dσij a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (λ⃗aj))(λ⃗ai) = divaσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus [divbσ]|⃗b ̸= P −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' [divbσ]|⃗a and [divbσ]T |⃗b ̸= [divaσ]T |⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Remark: (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='65) can be written divσ = � ij ∂σij ∂xj ⃗Ei (S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='67) where ( ⃗Ei) is the canonical basis in Mn1 the space of n ∗ 1 column vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' T Natural canonical isomorphisms T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 The adjoint of a linear map Setting of § A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12: E and F are vector spaces, E∗ = L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) and F ∗ = L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) are their dual spaces, and the adjoint of a linear map P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) is the linear map P∗ ∈ L(F ∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗) canonically defined by ∀ℓ ∈ F ∗, P∗(ℓ) := ℓ ◦ P, written P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) (dot notations P∗(ℓ) =noted P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ and ℓ◦P =noted ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P since ℓ and P∗ are linear), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for all (ℓ, ⃗u) ∈ F ∗×E, P∗(ℓ)(⃗u) = ℓ(P(⃗u)), written (P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) Interpretation: If P is the push-forward of vector fields, then P∗ is the pull-back of differential forms, see remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In particular, it will be interpreted with P ∈ Li(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) (linear and invertible = a change of observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 184 185 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' An isomorphism E ≃ E∗ is never natural (never objective) T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 An isomorphism E ≃ E∗ is never natural (never objective) Two observers A and B consider a linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) be the change of observer endomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Willing to work together, A and B (“naturally”) consider the diagram E L −→ E∗ ← considered by observer A P ↓ ↑ P∗ E −→ L E∗ ← considered by observer B (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) Definition T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 (Spivak [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') A linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗) is natural iff the diagram (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) commutes for all P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E): L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗) is natural ⇐⇒ ∀P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E), P∗ ◦ L ◦ P = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) (In that case, if A computes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u with the top line of the diagram, if B computes with the bottom line of the diagram, then they can easily check their results since here L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u = (P∗ ◦ L ◦ P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Question: Does there exist an endomorphism L such that the diagram (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) commutes for all change of observers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' That is, do we have ∃?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E), ∀P ∈ Li(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E), P∗ ◦ L ◦ P = L ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) Answer: Always no (if L ̸= 0): Theorem T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 A (non-zero) linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗) is not natural: If L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗) − {0}, then ∃P ∈ Li(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L ̸= P∗ ◦ L ◦ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Spivak [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') It suffices to prove this proposition for E = ⃗R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let L ∈ L(⃗R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (⃗R)∗), L ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗a1) be a basis in ⃗R (chosen by A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗b1) be a basis in ⃗R (chosen by B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider P ∈ Li(⃗R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗R) defined by P(⃗a1) = ⃗b1 (change of observer), and let λ ∈ R s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗b1 = λ⃗a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) gives P∗(ℓ)(⃗a1) := ℓ(P(⃗a1)) = ℓ(⃗b1) = ℓ(λ⃗a1) = λℓ(⃗a1), thus P∗(ℓ) = λℓ for all ℓ ∈ (⃗R)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus P∗(L(P(⃗a1))) = P∗(L(λ⃗a1)) = λP∗(L(⃗a1)) = λ2L(⃗a1) ̸= L(⃗a1) when λ2 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', P = 2I gives L ̸= P∗ ◦ L ◦ P (= 4L), thus (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6): A (non-zero) linear map E → E∗ cannot be natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Consider E s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' dim E = 1, and consider the linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗) which sends a basis (⃗a1) onto its dual basis (πa1), so L is defined by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1 := πa1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Question: If (⃗b1) is another basis, λ ̸= ±1 and ⃗b1 = λ⃗a1 (change of unit of measurement), does L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗b1 = πb1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' does L also sends (⃗b1) onto its dual basis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Indeed, ⃗b1 = λ⃗a1 gives πb1 = 1 λπa1, thus L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗b1 = λL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1 = λπa1 = λ2πb1 ̸= πb1 since λ2 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In words: L is not natural, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A different presentation: Let LA and LB be defined by LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = πaj and LB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj = πbj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And suppose that ⃗bj = λ⃗aj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then, LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj = λLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = λπaj = λ2πbj = λ2LB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj ̸= LB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj when λ2 ̸= 1, that is, LA ̸= LB when λ2 ̸= 1: An operator that sends a basis onto its dual basis is not natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Let (·, ·)g be an inner dot product in E = ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗Rg ∈ L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) be the Riesz rep- resentation map, that is, defined by ⃗Rg(ℓ) = ⃗ℓg where ⃗ℓg is defined by (⃗ℓg,⃗v)g = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v for all ⃗v ∈ ⃗Rn, cf (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Question: Is ⃗Rg natural?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: No: Consider the diagram � E∗ ⃗Rg −→ E P∗ ↓ ↑ P E∗ −→ ⃗Rg E � with P = λI, λ ̸= ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then P∗ = λI, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ = λ2 ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ ̸= ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ gives P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P∗ ̸= ⃗Rg: So ⃗Rg is not natural, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (You may prefer to consider the diagram (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) with L = ⃗R−1 g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') A different presentation: Consider two distinct Euclidean dot products (·, ·)g and (·, ·)h (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', built with a foot and built with a metre).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So (·, ·)h = λ2(·, ·)g with λ2 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let ⃗Rg, ⃗Rh ∈ L(Rn∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Rn) be the Riesz operators relative to (·, ·)g and (·, ·)h, that is ⃗Rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ = ⃗ℓg and ⃗Rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ = ⃗ℓh are given by ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v = (⃗ℓg,⃗v)g = (⃗ℓh,⃗v)h for all ⃗v ∈ ⃗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have ⃗ℓh = λ2⃗ℓg, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11), thus ⃗Rh = λ2 ⃗Rg ̸= ⃗Rg since λ2 ̸= 1: A Riesz representation operator is not natural (it is observer dependent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 185 186 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Natural canonical isomorphism E ≃ E∗∗ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Natural canonical isomorphism E ≃ E∗∗ Two observers A and B consider the same linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗∗) (where E∗∗ = (E∗)∗ = L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Willing to work together, they (“naturally”) consider the diagram E L −→ E∗∗ ← considered by observer A P ↓ ↓ P∗∗ E −→ L E∗∗ ← considered by observer B (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) where P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E) is a linear diffeomorphism, P∗ ∈ L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗) its adjoint, given by P∗(ℓ) = ℓ ◦ P cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1), and P∗∗ ∈ Li(E∗∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗∗) the adjoint of P∗, thus given by P∗∗(u) = u ◦ P∗ for all u ∈ E∗∗ cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' P∗∗ is given by, for all (ℓ, u) ∈ E∗ × E∗∗, (P∗∗(u))(ℓ) = u(ℓ ◦ P), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (P∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) Question: Does there exist a linear map L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E∗∗) that is natural?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Answer: Yes (particular case of the next proposition): Proposition T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 The canonical isomorphism JE : � E → E∗∗ ⃗u → u = JE(⃗u) defined by JE(⃗u)(ℓ) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u, ∀ℓ ∈ E∗, (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) is natural, that is, F being another finite dimensional vector space, the diagram E JE −→ E∗∗ P ↓ ↓ P∗∗ F −→ JF F ∗∗ written E J −→ E∗∗ P ↓ ↓ P∗∗ F −→ J F ∗∗ (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) commutes for all P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∀P ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F), P∗∗ ◦ JE = JF ◦ P, and we write E ≃ E∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) Thus we can use the unambiguous notation (observer independent) J (⃗u) noted = ⃗u, and J (⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ noted = ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ℓ (= ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) (And u = J (⃗u) is the derivation operator in the direction ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Spivak [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') It is trivial that JE is linear and bijective (E is finite dimensional): It is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (P∗∗ ◦ JE(⃗u))(ℓ) (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) = JE(⃗u)(ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P) (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) = (ℓ ◦ P)(⃗u) = ℓ(P(⃗u)) (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) = JF (P(⃗u))(ℓ), for all ℓ ∈ F ∗ and all ⃗u ∈ E, thus P∗∗ ◦ JE(⃗u) = JF (P(⃗u)), for all ⃗u ∈ E, thus P∗∗ ◦ JE = JF ◦ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 (Characterization of JE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') JE sends any basis (⃗ai) onto its bidual basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (Expected, since JE(⃗u) is the directional derivative in the direction ⃗u, whatever ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (⃗ai) be a basis and (πai) be its dual basis (defined by πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = δij for all i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) gives JE(⃗aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='πai = πai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗aj = δij for all i, j, thus (JE(⃗aj)) is the dual basis of (πai), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', is the bidual basis of (⃗ai);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' True for all basis: JE(⃗bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='πbi = πbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗bj = δij for all i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Natural canonical isomorphisms L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) ≃ L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) ≃ L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F ∗) E, F, A, B are finite dimensional vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider the canonical isomorphism JEF : � L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) → L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) L → �L = JEF (L) where �L(ℓ, ⃗u) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u, ∀(ℓ, ⃗u) ∈ F ∗ × E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) 186 187 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Natural canonical isomorphisms L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F)) ≃ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) ≃ L(F ∗, E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) Let P1 ∈ Li(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A) and P2 ∈ L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B), and consider the diagram L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) JEF −→ L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) IP ↓ ↓ � IP L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B) −→ JAB L(B∗, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) where IP(L) = P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P−1 1 and � IP(�L)(b,⃗a) = �L(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P2, P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a) ∀(b,⃗a) ∈ B∗ × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) (IP and � IP are the push-forwards for linear maps L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) and for bilinear forms �L ∈ L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proposition T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 The canonical isomorphism JEF is natural, that is, the diagram (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) commutes for all P1 ∈ Li(E, A) and all P2 ∈ L(F, B): � IP ◦ JEF = JAB ◦ IP, and L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) natural ≃ L(F ∗, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) Thus L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F ∗) natural ≃ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' JAB(IP(L))(b,⃗a) (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='IP(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P−1 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a = (b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a) (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) = JEF (L)(b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P2, P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a) (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) = � IP(JEF (L))(b,⃗a), true for all L ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F), b ∈ B∗, ⃗a ∈ A, thus (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus L(E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F ∗) (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) ≃ L((F ∗)∗, E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) ≃ L(F, E∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) ≃ L(E∗∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) ≃ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider the canonical isomorphism (defines the transposed of a bilinear map) KEF : � L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) → L(F, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) T → KEF (T) � , KEF (T)(⃗u,⃗v) := T(⃗v, ⃗u), ∀(⃗u,⃗v) ∈ E × F, (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) and ZAB ∈ L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) → L(A, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) defined by ZAB(T)(⃗a,⃗b) := T(P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a, P−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗b) for all (⃗a,⃗b) ∈ A × B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proposition T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 The canonical isomorphism KEF is natural: For all (P1, P2) ∈ Li(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' A)×L(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B), the diagram L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) KEF −→ L(F, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) ZAB ↓ ↓ ZBA L(A, B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) −→ KAB L(B, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) commutes: L(E, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) natural ≃ L(F, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' KEF (ZAB(T))(⃗b,⃗a) = ZAB(T)(⃗a,⃗b) = T(P−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗b, P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a) and ZBA(KEF (T))(⃗a,⃗b) = KEF (T)(P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a, P−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗b) = T(P−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗b, P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a), thus KAB ◦ ZAB = ZBA ◦ KEF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Natural canonical isomorphisms L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F)) ≃ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) ≃ L(F ∗, E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) For application to the second order derivative d(d⃗u) ≃ d2⃗u and, with ⃗u ∈ T 1 0 (U), the notation d⃗u ∈ T 1 1 (U), then d2⃗u ∈ T 1 2 (U), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', dk⃗u ∈ T 1 k (U), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider the canonical isomorphism J12E : � L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F)) → L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) T1 → T2 = J12E(T1) � , J12E(T1)(⃗u1, ⃗u2) := T1(⃗u1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗u2 ∈ F, ∀⃗u1, ⃗u2 ∈ E, (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) and the canonical isomorphism J23E : � L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) → L(F ∗, E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) T2 → J23E(T2) = T3 � , T3(ℓ, ⃗u,⃗v) := ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T2(⃗u1, ⃗u2), ∀⃗u1, ⃗u2 ∈ E, ∀ℓ ∈ F ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) Proposition T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 J12 and J23 are natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Thus J23 ◦ J12 is natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 187 188 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definitions Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 1- We have to prove that the following diagram commutes: L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F)) J12E −→ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) ZAB ↓ ↓ YAB L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B)) J12A −→ L(A, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B) where ZAB(T1)(⃗a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2 := T1(P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2), YAB(T2)(⃗a1,⃗a2) = T2(P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1, P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2), (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='20) (the “push-forwards) for all ⃗a1,⃗a2 ∈ A and LAB ∈ L(A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let T1 ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have J12A(ZAB(T1))(⃗a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2 = ZAB(T1)(⃗a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2 = T1(P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2), and YAB(J12E(T1))(⃗a1,⃗a2) = J12E(T1)(P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1, P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2) = T1(P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2), thus J12A ◦ ZAB = YAB ◦ J12E, thus J12 is natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- We have to prove that the following diagram commutes: L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F) J23E −→ L(F ∗, E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) ZAB ↓ ↓ YAB L(A, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' B) J23A −→ L(B∗, A, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) where ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ZAB(T2)(⃗a1,⃗a2) := (ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T2(P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1, P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2), YAB(T3)(ℓB,⃗a1,⃗a2) = T3(ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P2, P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1, P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2), (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='21) (the “push-forwards) for all ⃗a1,⃗a2 ∈ A and ℓB ∈ B∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let T2 ∈ L(E, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We have J23A(ℓB, ZAB(T2)(⃗a1,⃗a2)) = ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='ZAB(T2)(⃗a1,⃗a2) = (ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T2(P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1, P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2), and YAB(J23A(T2))(ℓB,⃗a1,⃗a2) = J23A(T2)(ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P2, P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1, P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2) = ℓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='T2(P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a1, P−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗a2) thus J23A ◦ ZAB = YAB ◦ J23E, thus J23 is natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' U Distribution in brief: A covariant concept We refer to the books of Laurent Schwartz for a full description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In continuum mechanics, with Ω an open set in Rn and for the space of the finite energy functions L2(Ω) and its sub-spaces, a distribution gives a covariant formulation for the virtual power, as used by Germain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Definitions Usual notations: Let p ∈ [1, ∞[ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' p = 2 for finite energy functions), and let Lp(Ω) := {f : Ω → R : � Ω |f(x)|p dΩ < ∞} and ||f||p = ( � Ω |f(x)|p dΩ) 1 p , (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1) the space of functions such that |f|p is Lebesgue integrable with ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||p its usual norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (Lp(Ω), ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||Lp) is a Banach space (a complete normed space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And let L∞(Ω) := {f : Ω → R : sup x∈Ω (|f(x)|) < ∞}, and ||f||∞ = sup x∈Ω (|f(x)|), (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2) the space of Lebesgue measurable bounded functions with ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||∞ its usual norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then (L∞(Ω), ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||L∞) is a Banach space (a complete normed space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 If f ∈ F(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R), then its support is the set supp(f) := {x ∈ Ω : f(x) ̸= 0} = the closure of {x ∈ Ω : f(x) ̸= 0} (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3) = the set where it is interesting to study f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The closure is required: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', if Ω =]0, 2π[ and f(x) = sin x for all x ∈ ω, then {x ∈ Ω : f(x) ̸= 0} = ]0, π[∪]π, 2π[;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And the point π is a point of interest since sin varies in its vicinity: f ′(π) = cos(π) = −1 ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So it is the closure supp(f) := ]0, π[∪]π, 2π[ = [0, 2π] that is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 (Schwartz notation, D being the letter after C:) Let D(Ω) := C∞ c (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) = {ϕ ∈ C∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' supp(ϕ) is compact in Ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4) 188 189 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Derivation of a distribution E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', Ω = R, ϕ(x) := e− 1 1−x2 if x ∈]−1, 1[ and ϕ(x) := 0 elsewhere: ϕ ∈ D(R) with supp(ϕ) = [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And D(Ω) is a vector space which is dense in (Lp(Ω), ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||Lp) for any p ∈ [1, ∞[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 A distribution in Ω is a linear D(Ω)-continuous3 function T : � D(Ω) → R ϕ → T(ϕ) noted = ⟨T, ϕ⟩ (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) The space of distribution in Ω is named D′(Ω) (the dual of D(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The notation ⟨T, ϕ⟩D′(Ω),D(Ω) = ⟨T, ϕ⟩ is the “duality bracket” = the “covariance–contravariance bracket” between a linear function T ∈ D′(Ω) and a vector ϕ ∈ D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='4 Let f ∈ Lp(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The regular distribution Tf ∈ D′(Ω) associated to f is defined by Tf(ϕ) := � Ω f(x)ϕ(x) dΩ, ∀ϕ ∈ D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6) So Tf is a measuring instrument with density dmf(x) = f(x) dΩ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Tf(ϕ) := � Ω ϕ(x) dmf(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5 Let x0 ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' The Dirac measure at x0 is the distribution T noted = δx0 ∈ D′(R) defined by, for all ϕ ∈ D(R), δx0(ϕ) = ϕ(x0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⟨δx0, ϕ⟩ = ϕ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7) And δx0 is not a regular distribution (δx0 is not a density measure): There is no integrable function f such that Tf = δx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Interpretation: δx0 corresponds to an ideal measuring device: The precision is perfect at x0 (gives the exact value ϕ(x0) at x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' In real life δx0 is the ideal approximation of Tfn where fn is e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' given by fn(x) = n1[x0,x0+ 1 n ] (drawing): For all ϕ ∈ D(Ω), Tfn(ϕ) −→n→∞ δx0(ϕ) = ϕ(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Generalization of the definition: In (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='5) D(Ω) = C∞ c (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is replaced by C∞ c (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So if you consider a basis (⃗ei) then ⃗ϕ ∈ C∞ c (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⃗Rn) reads ⃗ϕ = �n i=1ϕi⃗ei with ϕi ∈ D(Ω) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6 Power: Let α : Ω → T 0 1 (Ω) be a differential form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then P = Tα defined by P(⃗v) = � Ω α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗v dΩ gives the virtual power associated to α relative to the vector field ⃗v (mechanics and thermodynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Derivation of a distribution Let O be a point in Rn (an origin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If p ∈ Rn and if (⃗ei) is a basis in ⃗Rn, let ⃗x = −→ Op = �n i=1xi⃗ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Definition U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='7 The derivative ∂T ∂xi of a distribution T ∈ D′(Ω) is the distribution ∈ D′(Ω) defined by, for all ϕ ∈ D(Ω), ∂T ∂xi (ϕ) := −T( ∂ϕ ∂xi ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ⟨ ∂T ∂xi , ϕ⟩ := −⟨T, ∂ϕ ∂xi ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) ( ∂T ∂xi is indeed a distribution: Easy check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Example U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8 If T = Tf is a regular distribution with f ∈ C1(Ω), then ∂(Tf ) ∂xi = T( ∂f ∂xi ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Indeed, for all ϕ ∈ D(Ω), ∂(Tf ) ∂xi (ϕ) = −Tf( ∂ϕ ∂xi ) = − � Ω f(x) ∂ϕ ∂xi dΩ = + � Ω ∂f ∂xi ϕ(x) dΩ + � Γ 0 dΓ, since ϕ vanishes on Γ = ∂Ω (the support of ϕ is compact in Ω), thus ∂(Tf ) ∂xi (ϕ) = T( ∂f ∂xi )(ϕ) for all ϕ ∈ D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Example U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9 Consider the Heaviside function (the unit step function) H0 := 1R+ and the associated distribution T = TH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then ⟨(TH0)′, ϕ⟩ := −⟨TH0, ϕ′⟩ = − � Ω H0(x)ϕ′(x) dx = − � ∞ 0 ϕ′(x) dx = ϕ(0) = ⟨δ0, ϕ⟩ for any ϕ ∈ D(R), thus (TH0)′ = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Written H0 ′ = δ0 in D′(Ω), which is not in a equality between functions, because H0 is not derivable at 0 as a function, and δ0 is not a function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' It is equality between distributions: The notation H0 ′ can only be used to compute H0 ′(ϕ) (= ⟨H0 ′, ϕ⟩ := −⟨H0, ϕ′⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3The D(Ω)-continuity of T is defined by: 1- A sequence (ϕn)N∗ in D(Ω) converges in D(Ω) towards a function ϕ ∈ D(Ω) iff there exists a compact K ⊂ Ω s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' supp(ϕn) ⊂ K for all n, and || ∂kϕ ∂xi1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂xik − ∂kϕn ∂xi1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='∂xik ||∞ −→n→∞ 0 for all k ∈ N and all ij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- T is continuous at ϕ ∈ D(Ω) iff T(ϕn) −→ n→∞ T(ϕ) for any sequence (ϕn)N ∈ D(Ω)N −→ n→∞ ϕ in D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 189 190 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hilbert space H1(Ω) U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Hilbert space H1(Ω) U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='1 Motivation Consider the hat function Λ(x) � � � � � = x + 1 if x ∈ [−1, 0], = 1 − x if x ∈ [0, 1], = 0 otherwise � � � � � (drawing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' When applying the finite element method, it is well-known that, if you use integrals (if you use the virtual power principle which makes you compute average values), then you can consider the derivative of the hat function Λ as if it was the usual derivative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' at the points where the usual computation of Λ′ is meaningful, that is, Λ′(x) � � � � � = 1 if x ∈] − 1, 0[, = −1 if x ∈]0, 1[, = 0 if x ∈ R − {−1, 0, 1} (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) (drawing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Problem: Λ′ is not defined at −1, 0, 1 (the function Λ is not derivable at −1, 0, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Question: So does the “usual” computation I = � R Λ′(x)ϕ(x) dx with (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='9) gives the good result?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (This is not a trivial question: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', with H0 = 1R+ instead of Λ, we would get the absurd result H′ 0 = 0, absurd since H′ 0 = δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Answer: Yes: 1- Consider TΛ the regular distribution associated to Λ, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 2- Then consider (TΛ)′, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8): We get ⟨(TΛ)′, ϕ⟩ (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='8) = −⟨TΛ, ϕ′⟩ = − � R Λ(x)ϕ′(x) dx = − � 0 −1 Λ(x)ϕ′(x) dx − � 1 0 Λ(x)ϕ′(x) dx = + � 0 −1 1]−1,0[(x)ϕ(x) dx + � 1 0 1]0,1[ϕ(x) dx, for any ϕ ∈ D(R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 3- Thus (TΛ)′ = Tf where f = 1]−1,0[ + 1]0,1[, that is (TΛ)′ is a regular distribution, And its is named f = Λ′ within the distribution framework, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for computations ⟨Λ′, ϕ⟩ := ⟨(TΛ)′, ϕ⟩ with ϕ ∈ D(R) (value = � R f(x)ϕ(x) dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='2 Definition of H1(Ω) The space C1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) is too small in many applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', for the Λ function above);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' We need a larger space where the functions are “derivable is a weaker sense” which is the distribution sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Consider a basis in Rn: Definition U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10 The Sobolev space H1(Ω) is the subspace of L2(Ω) restricted to functions whose gen- eralized derivatives are in L2(Ω): H1(Ω) = {v ∈ L2(Ω) : ∂v ∂xi ∈ L2(Ω), ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='10) Usual shortened notation: H1(Ω) = {v ∈ L2(Ω) : ⃗ gradv ∈ L2(Ω) n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' So to check that v ∈ H1(Ω), even if ∂v ∂xi does not exists in the classic way (see the above hat function Λ), you have to: 1- Consider its associated regular distribution Tv, 2- Compute ∂Tv ∂xi in D′(Ω), 3- And if, for all i, there exists fi ∈ L2(Ω) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' ∂Tv ∂xi = Tfi, then v ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 4- And then fi is noted ∂v ∂xi when used within the Lebesgue integrals � Ω ∂v ∂xi (x)ϕ(x) dx with ϕ ∈ D(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', Λ ∈ H1(R) since (TΛ)′ = Tf with f = 1]−1,0[ + 1]0,1[ ∈ L2(R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (TΛ)′ =noted Λ′ (= f) in the distribution context (integral computations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let (·, ·)L2 and ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||L2 be the usual inner dot product and norm in L2(Ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (u, v)L2 = � Ω u(x)v(x) dΩ, and ||v||L2 = � (v, v)L2 = ( � Ω v(x)2 dΩ) 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11) (L2(Ω), (·, ·)L2) is a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Then define, for all u, v ∈ H1(Ω), (u, v)H1 = (u, v)L2 + n � i=1 ( ∂u ∂xi , ∂v ∂xi )L2, and ||v||H1 = (v, v) 1 2 H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='12) Then (H1(Ω), (·, ·)H1) is a Hilbert space (Riesz–Fisher theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 190 191 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Hilbert space H1(Ω) With a Euclidean dot product (·, ·)g in ⃗Rn and a (·, ·)g-orthonormal basis, (u, v)H1 = (u, v)L2 + ( ⃗ gradu, ⃗ gradv)L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='13) U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='3 Subspace H1 0(Ω) and its dual space H−1(Ω) The boundary Γ = ∂Ω of Ω is supposed to be regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Let H1 0(Ω) := {v ∈ H1(Ω) : v|Γ = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='14) Then (H1 0(Ω), (·, ·)H1) is a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' More generally (without any regularity assumption on Γ), H1 0(Ω) := D(Ω) H1 = the closure of D(Ω) in (H1(Ω), ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='||H1): This closure of D(Ω) in H1(Ω) enables the use of the distribution framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Notation : The dual space of H1 0(Ω) is the space H−1(Ω) = (H1 0(Ω))′ = L(H1 0(Ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' R) (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='15) equipped with the (usual) norm ||T||H−1 := sup ||v||H1 0 =1 |T(v)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' And (duality bracket), if v ∈ H1 0(Ω) and T ∈ H−1(Ω) then T(v) noted = ⟨T, v⟩H−1,H1 0 noted = ⟨T, v⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='16) Theorem U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='11 (Characterization of H−1(Ω) = (H1 0(Ω))′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') A distribution T is in H−1(Ω) iff ∃(f,⃗g) ∈ L2(Ω) × L2(Ω) n s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' T = f − div⃗g (∈ D′(Ω)), (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) that is, for all v ∈ H1 0(Ω), ⟨T, v⟩H−1,H1 0 = � Ω fv dΩ + � Ω dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='⃗g dΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18) And if Ω is bounded then we can choose f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' If moreover ⃗g ∈ H1(Ω) n then ⟨T, v⟩H−1,H1 0 = � Ω f(x)v(x) dx − � Ω div⃗g(x)v(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) (In fact we only need ⃗g ∈ Hdiv(Ω) = {⃗g ∈ L2(Ω) n : div⃗g ∈ L2(Ω)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=', see Brezis [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' For boundary value problems with Neumann boundary conditions, we then need (H1(Ω))′ the dual space of H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' Characterization of (H1(Ω))′: We still have (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='18), but we have to replace (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='17) or (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='19) by, with a Euclidean dot product in ⃗Rn, see Brezis [4], ⟨T, v⟩(H1)′,H1 = � Ω f(x)v(x) dx − � Ω div⃗g(x)v(x) dx + � Γ ⃗g(x) • ⃗n(x) v(x) dx.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content='xml, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} +page_content=' 192' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfH_vd/content/2301.01056v1.pdf'} diff --git a/KdAyT4oBgHgl3EQff_ia/vector_store/index.faiss b/KdAyT4oBgHgl3EQff_ia/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..6c3bb72a138f8ce3fb31e0b07616b6f4bdbea828 --- /dev/null +++ b/KdAyT4oBgHgl3EQff_ia/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e7d514a30f29a4ac3c1c540280d97460c0cd1984d8e1f5debe2fd07c326d9c6b +size 8388653 diff --git a/KdAyT4oBgHgl3EQff_ia/vector_store/index.pkl b/KdAyT4oBgHgl3EQff_ia/vector_store/index.pkl new file mode 100644 index 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This type of +problem led mathematicians to invent solution methods of maxima and +minima, and the genesis of variational calculus as a distinct branch of anal- +ysis. Dido’s problem was inspired by the mythical tale of the foundation +of Carthage (ancient city in North Africa) by a Phoenician princess as told +independently by Roman poet Virgil, and by Latin historian Justinus in +the first two centuries B.C. Historians have debated the facts surrounding +Carthage’s birth; however, contemporary mathematicians have accepted +the vague events described by Virgil in his Aeneid, adding details to Dido’s +story to extrapolate a few verses and use as a basis for the isoperimet- +ric theorem. Was Leonhard Euler or Lord Kelvin who first interpreted +Virgil’s poem as Dido’s problem of variational calculus? In this article I +attempt to resolve a question of historical attribution to identify who first +defined Dido’s problem. +Keywords: Isoperimetrics, variational calculus, Euler, Lord Kelvin +1 +Introduction +In 1937, Karl Menger1 wrote: “The first human being to solve a problem of +calculus of variations seems to have been Queen Dido of Carthage.” Contempo- +rary mathematics books2 go much further than that by adding details to Dido’s +story taken from Virgil’s Aeneid, alleging that Dido established Carthage, an +ancient city in modern Tunisia, by application of the isoperimetric property of +the circle to secure the largest area of land she bought upon arrival to North +Africa. Here are two examples, where I use Italics to highlight details of Dido’s +story that are not in Virgil’s poem: +1Menger (1937) +2See for example, Brunt (2004); Freguglia and Giaquinta (2016); Coppersmith (2017); +Nahin (2004); Rojo and Bloch (2018). +1 +arXiv:2301.02917v1 [math.HO] 7 Jan 2023 + +“Dido was a Carthaginian queen (ca. 850 B.C.?) who came from a dysfunc- +tional family. Her brother, Pygmalion, murdered her husband (who was also +her uncle) and Dido, with the help of various gods, fled to the shores of North +Africa with Pygmalion in pursuit. Upon landing in North Africa, legend has it +that she struck a deal with a local chief to procure as much land as an oxhide +could contain. She then selected an ox and cut its hide into very narrow strips, +which she joined together to form a thread of oxhide more than two and a half +miles long. Dido then used the oxhide thread and the North African sea coast +to define the perimeter of her property ... it is clear that Dido sought to enclose +the maximum area within her ox and the sea. The city of Carthage was then +built within the perimeter defined by the thread and the sea coast. Dido called +the place Byrsa meaning hide of bull.”3 +“Dido ... using the seashore (given as straight) as part of the boundary, she +laid out the hide-strip to enclose the maximum possible area, which she “knew” +would be in the shape of a semicircle.”4 +If these accounts were based on fact, then Dido would be the first woman +in humanity’s history to understand a mathematical principle, much before +the first mathematicians in recorded history. Since Carthage was founded in +814 B.C., Dido was born centuries before Thales of Miletus (c. 624-548 BC), +Pythagoras of Samos (c. 570-490 BC), and Euclid of Alexandria (325-265 BC), +and much earlier than the Greek mathematicians who dealt with isoperimetric +problems, e.g. Zenodorus (c. 200-140 BC) who wrote On Isoperimetric Fig- +ures; this work is lost but details are found in the commentaries by Theon of +Alexandria (335-405 AD), and by Pappus of Alexandria (290-350 AD). In his +Mathematical Collection, Pappus presented results from ancient isoperimetry +studies5 but he did not mention Dido. +Dido’s problem is now taught as the most fundamental isoperimetric prob- +lem: for a fixed perimeter, determine the shape of the closed, planar curve +that encloses the maximum area. The answer is the circle, as any grammar +school child knows, but in variational calculus the solution is determined by an +analytical method introduced by Leonhard Euler and refined by Joseph-Louis +Lagrange. +2 +Dido in Ancient Literature +The story of Dido and the foundation of Carthage was immortalized by Virgil +in his Aeneid, and by third century Roman historian Justinus in his Epitoma +historiarum Philippicarum Pompei Trogi. +However, in these stories there is +absolutely no mention of Dido enclosing a circular shape for the purchased land +3Brunt (2004), pp. 14-15. +4Nahin (2004), p. 45. +5According to Pappus, the first proof of the isoperimetric property of the circle (using +geometric arguments) is due to Zenodorus. +2 + +with the string of hide, or of her using the knowledge that the circle encloses +the largest area. +Virgil wrote in the Aeneid,6 Book 1, lines 365-368, referring to Dido and her +people arriving to Africa: “They came to this place, and bought land, where +you now see the vast walls, and resurgent stronghold, of new Carthage, as much +as they could enclose with the strips of hide from a single bull, and from that +they called it Byrsa.”7 +Justinus, who refers to Dido by her Phoenician name Elissa, wrote in Book +XVIII: “By this means some respite was given to the fugitives; and Elissa, +arriving in a gulf of Africa, attached the inhabitants of the coast, who rejoiced +at the arrival of foreigners, and the opportunity of bartering commodities with +them, to her interest. Having then bargained for a piece of ground, as much +as could be covered with an ox-hide, where she might refresh her companions, +wearied with their long voyage, until she could conveniently resume her progress, +she directed the hide to be cut into the thinnest possible strips, and thus acquired +a greater portion of ground than she had apparently demanded; whence the +place had afterwards the name of Byrsa.”8 +Thus, if neither Virgil nor Justin provided the details given by contemporary +mathematicians about Dido’s problem as we know it, who did? how Dido’s +mythical tale became Dido’s problem? Surely, only a scientist would have made +the connecting leap between “bought land, . . . , as much as they could enclose +with the strips of hide from a single bull . . . ” (Virgil, 1st century BC), and +interpreted these words as “she laid out the hide-strip to enclose the maximum +possible area, which she “knew” would be in the shape of a semicircle” (Nahin, +2004). +3 +Isoperimetry and Calculus of Variations +Isoperimetrics provided the roots for the development of the variational method- +ology, starting with the observation made by ancient scholars that most motion +appears to be in either straight lines or circles. The definition of a straight line as +the shortest path between two points was an early expression of a minimization +principle known to ancient geometers. The isoperimetric problems considered +in antiquity (e.g. the circle in the plane and the sphere in three-dimensional +space were known as the least perimeter figures to enclose a given area and a +given volume, respectively) were solved by geometric means. +Pappus gave credit to Zenodorus (200-140 BC) for solving for the optimal +6Written between 29 and 19 BC, this epic poem in 12 books tells the story of the foundation +of Rome from the ashes of Troy. Virgil describes the foundation of Carthage by Dido in Book +I: 297-371. +7Line 365: Devenere locos, ubi nunc ingentia cernis moenia surgentemque novae Karthagi- +nis arcem, mercatique solum, facti de nomine Byrsam, taurino quantum possent circumdare +tergo. +8Marcus Junianus Justinus, Epitoma historiarum Philippicarum Pompei Trogi (Epitome of +the Philippic History of Pompeius Trogus). Translated by Rev. John Selby Watson. London: +Henry G. Bohn, Convent Garden (1853). +3 + +form of a maximum area surface for a given perimeter. He also expounded the +work of Hero (or Heron) of Alexandria (c. 10-75 AD) who studied the optics of +reflection, finding that reflected light travels in a way that minimizes its travel +time. The law of reflection of light—that the angle of incidence equals the angle +of reflection—was well established since ancient times. In his Catoptrics, Euclid +noted that light travels in straight lines and described the law of reflection (300 +BC). Hero showed by a geometrical method that the actual path taken by a ray +of light reflected from a plane mirror is shorter than any other reflected path +that might be drawn between the source and point of observation. +Ancient Greeks first conceived the idea that Nature selects the shortest, +easiest and most direct path in moving objects between points. In the seven- +teenth and eighteenth centuries, ideas about the economy of Nature continued +preoccupying philosophers and scientists. Finding analytic solutions to more +complicated problems of maxima and minima attracted the greatest mathe- +maticians such as Fermat, Newton, Leibniz, the Bernoulli brothers (Jacob and +Johann I), Euler, Lagrange, and Maupertuis. +Perhaps inspired by Hero’s reflected light minimization problem, Pierre de +Fermat (1601-1665) showed that the time required for a light ray to traverse a +neighboring virtual path differs from the time actually taken by a quantity of +the second order. This is known as Fermat’s principle of least time. +Newton (1643-1727) examined the motion of bodies in a resisting medium, +finding the shape of the body that renders its resistance minimal. +In June of 1696, Johann Bernoulli (1667-1748) posed the following problem +as a challenge to mathematicians: Given two points A and B in a vertical plane, +find the path AMB down which a movable point particle M must, by virtue of its +weight, will traverse in the shortest possible time (assumes that M’s acceleration +is due only to gravity). This is the famous Brachistochrone (from the Greek +brachistos, shortest, and chronos, time) problem, later also called the problem +of least time descent. +The brachistochrone problem does not have a trivial +solution; the Bernoulli brothers (Jacob and Johann I), Newton, Leibniz and +l’Hˆopital solved the problem correctly, each using a different approach.9 +The initial investigations in the maxima and minima principles carried out +by Leonhard Euler began from a study of the work of these mathematicians, +especially motivated by the work of Jacob Bernoulli and prompted by his teacher +Johann Bernoulli. The latter drew his attention to a problem of geodesic lines +in a letter he sent Euler in St. Petersburg in 1728, which led Euler to conceive +in early 1729 an analytical method by which, on any surface, whether convex or +concave, the shortest line can be drawn between two points.10 Euler solved other +isoperimetric problems, obtaining results to help him establish the analytical +foundations of the calculus of variations. +Euler invented variational calculus as a distinct branch of analysis precisely +to systemize the solution methods of maxima and minima, as brilliantly intro- +duced in his 1744 book Methodus inveniendi lineas curvas maximi minimive +9Fregulia and Giaquinta (2016), pp. 3-4. +10Euler (1732) +4 + +proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu +accepti, the first treatise on calculus of variations.11 With Euler’s approach, the +calculus of variations yielded a method for finding an extremum of a quantity +that is expressible as an (variational) integral. +Euler’s Methodus inveniendi represented a substantial break with the then +established tradition set for by his predecessors, including his earlier work in +the subject.12 +In this treatise, Euler formulated the variational principle of +mechanics, which is the principle of least action now attributed to Maupertuis: +For a given projected body, denote its mass by M, half the square of its velocity +by v, the arclength element by ds. Then, among all curves passing through +the same pair of endpoints, the desired curve is the one that minimizes the +integral +� +Mdsv1/2. Details on how Euler formulated the principle are provided +by Goldstine (1980), and by Freguglia and Giaquinta (2016).13 +Euler remarked: “Since the structure of the universe was made most perfect +as designed by the wisest Creator, nothing in the world will occur in which no +maximum or minimum rule is shining forth; wherefore there is absolutely no +doubt that all the effects of the world can be equally successfully determined +from final causes by means of the maximum and least methods, and from the +efficient causes themselves.”14 +Considered as the first variational treatment of mechanics, Euler’s principle +of least action contributed significantly to analytic mechanics and ultimately to +the fundamental underpinnings of twentieth-century physics, including general +relativity and quantum mechanics. +Euler was also known as being able to recite Virgil’s Aeneid by heart. Did +he interpret Dido’s tale as Dido’s isoperimetric problem? +4 +Defining Dido’s Problem in the Calculus of +Variations +A casual survey of the history of mathematics books written in the eighteenth +and nineteenth century yields no clues as to when or how Dido’s mythical story +became part of variational calculus. It required a person with mathematical +brilliance and fertile imagination to connect ancient myth with mathematics. +Two names emerge as potential candidates: Leonhard Euler (1707-1783), the +originator of the calculus of variations, and British mathematician, physicist +and engineer William Thomson, known in physics as Lord Kelvin (1824-1907). +4.1 +Leonhard Euler +In Methodus inveniendi, Euler gives the following example to demonstrate his +analytical method: to find among all admissible curves, enclosing a given area, +11Euler (1744) +12Fraser (1993) +13Goldstine (1980), p. 101; Freguglia and Giaquinta (2016), pp. 181-189. +14Euler (1744), Additamentum I. +5 + +Figure 1: Euler’s sketch from Methodus inveniendi (1744) +the one of least length. Figure 1 is Euler’s sketch to demonstrate that the curved +arc of a circle, BM, is minimum. In his own words:15 +“On the axis AP construct the line BM, so that, when the area ABMP of +a given size is cut off, the curved arc BM corresponding to that area is the +minimum of all.” After solving his variational integral, Euler shows the solution +curve to be an arc of a circle with center somewhere on the line AP, for example, +at C in Fig. 1. +But neither Euler’s Methodus inveniendi nor his other published memoirs in +the field ever mention Dido. +In October 1783, the Marquis de Condorcet16 gave the ´Eloge d’Euler to the +members of the Acad´emie des Sciences in Paris. In this solemn eulogy, Con- +dorcet expounded on Euler’s genius and suggested that a verse from the Aeneid +had given Euler the first idea for a memoir on a question of Mechanics. +In +Condorcet’s own words: +L’´etude de la Litt´erature ancienne et des Langues savantes avait fait partie +de son ´education ; il en conserva le goˆut toute sa vie, et n’oublia rien de ce qu’il +avait appris ; mais il n’eut jamais ni le tems ni le d´esir d’ajouter `a ses premi`eres +´etudes : il n’avait pas lu les Po`etes modernes, et savait par cœur l’Eneide. +Cependant M. Euler ne perdait pas de vue les Math´ematiques, mˆeme lorsqu’il +r´ecitait les vers de Virgile ; tout ´etait propre `a lui rappeler cet objet presque +unique de ses pens´ees, et on trouve dans ses ouvrages un savant M´emoire sur +une question de M´ecanique, dont il racontait qu’un vers de l’Eneide lui avait +donn´e la premi`ere id´ee.17 [The study of ancient literature and scholarly lan- +guages had been part of his education; he retained a taste for it all his life, and +15Euler (1744), Chapter IV, p. +135, Exemplum II: 9. +Super axe AP construere lineam +BM, ita comparatant, ut, abscissa area ABMP datæ magnitudinis, arcus curvæ BM illi areæ +respondens sit omnium minimus. +16Condorcet, Jean-Antoine-Nicolas de Caritat marquis de (1743-1794). +17 ´Eloge d’Euler Prononc´e `a l’Acad´emie, par de Condorcet, Histoire de l’Acad´emie royale +des sciences ... 1783, p. 64. +6 + +Bforgot nothing he had learned; but he never had either the time or the desire to +add to his first studies: he had not read the Modern Poets, and knew the Aeneid +by heart. However, M. Euler did not lose sight of Mathematics, even when he +recited the verses of Virgil; everything was likely to remind him of this almost +unique object of his thoughts, and we find in his works a scholarly Memoir on a +question of Mechanics, of which he said that a verse from the Aeneid had given +him the first idea.] +Euler did take verses from the Aeneid poem to use as mottos for his com- +peting memoirs submitted to the French Academy.18 These are summarized in +Table 1. Was this to what Condorcet referred to? +Table 1. Euler’s Memoirs and Mottos taken from Virgil’s Aeneid. +Year +Memoir Title +Motto +1753 +“On the movement of ships +Tali remigio navis se +(E. 413) +without the wind’s force.” +tarda movebat. +7th winning memoir +Virg. Aeneid Liv. 5 +1759 +“Concernin pitching +Insequitur clamorque virum +(E. 415) +and rolling.” +stridorque rudentum. +9th winning memoir +Virg. Aeneid, Liv. 1 +1770 +“Moon Theory” +Errantem que canit Lunam +(E. 485) +Prize for 1770 +Virg. Aeneid Liv. 1 +10th winning memoir +1772 +“Improved Moon theory” +Hic labor extremus, longarum +(E. 486) +Prize for 1772 +haec meta viarum hinc jam +11th winning memoir +digressi, vestris appellimus oris +Virg. Aeneid, Liv. 3 +However, the mottos were carefully selected by Euler to match the research +topic of the competition.19 In addition to using Virgil’s verses, he also quoted +from other ancient writers such as Marcus Tullius Cicero, Properci, and he +composed other adages, asking Christian Goldbach for suggestions. Ultimately, +Condorcet’s statement “et on trouve dans ses ouvrages un savant M´emoire sur +une question de M´ecanique, dont il racontait qu’un vers de l’Eneide lui avait +donn´e la premi`ere id´ee” does not mean that Euler was inspired by Virgil to +define Dido’s problem. +As a historian, I cannot rely on obituaries to extract factual data, even if +written by an eminent scholar. The much younger Condorcet never met Euler, +and the ´Eloge he wrote, as most eulogies are, was based on hearsay, relaying on +what the French academicians might have recalled about Euler’s life and work. +18Submissions were anonymously and the memoir identified by a motto; the author’s name +enclosed in a sealed envelope was opened only for the winning memoir after the judging of +the contest. +19For the significance of the mottos that Euler selected, see Musielak (2022). +7 + +A contemporary biography (published in 2016) further implies that Euler +solved Dido’s problem. The author refers to a copy of an eight-page manuscript +(preserved in Moscow) that is said to contain Euler’s answer. +Is this the +manuscript that categorically would give Euler credit for connecting Dido’s story +to variational calculus? Unfortunately, the manuscript in question is said to be +“not in Euler’s own handwriting.” Thus, it diminishes its credibility. It is rather +improbable that Euler, a prolific writer, would be the author of a manuscript +inscribed by someone else. Besides, he would have included this solution in a +paper published in 1764, where Euler summarized the results of the Calculus of +Variations in terms of the variational operator. +Joseph-Louis Lagrange (1736-1813) expanded the variational calculus. In his +second letter to Euler dated August 1755, Lagrange outlined his delta-algorithm +(for solving constrained optimization problems), an approach Euler embraced, +prompting him to conceive the term calculus of variations. In the abstract of +a memoir published in 1764, Euler credits Lagrange for enriching the science20 +Their combined work led eventually to the Euler–Lagrange equations, which are +the equilibrium equations for minima of variational integrals.21 +Five years after Euler died, Lagrange published M´ecanique analytique, his +compendium on analytical mechanics, using variational ideas to present me- +chanics from a unified analytic viewpoint. When teaching at the ´Ecole Poly- +technique in 1799, Lagrange published Le¸cons sur le calcul des fonctions and +explained the method of variation. Lagrange provided a brief overview of the +development of problems of maxima and minima, referring only to Greek math- +ematician Apollonius (262-190 BC), which dealt exclusively with the largest +and smallest straight lines which can be drawn from given points to the arcs of +conic sections.22 Dido’s problem is not mentioned here nor in Lagrange’s other +published works. +4.2 +William Thomson, Lord Kelvin +The first instance in which Dido’s name appear in the context of interest is +found in a public lecture delivered by William Thomson in 1893. A great physi- +cist known today as Lord Kelvin, his contributions include a major role in the +development of the second law of thermodynamics; the absolute temperature +scale (measured in kelvins); the dynamical theory of heat; the mathematical +analysis of electricity and magnetism, including the basic ideas for the electro- +magnetic theory of light; and much more. He brought together disparate areas +of physics—heat, thermodynamics, mechanics, hydrodynamics, magnetism, and +20Euler (1764). . . +ex quo Auctori occasio est oblata hanc scientiam novo Calculi genere +locupletandi, quem Calculum variationum appellat et cuis elementa hic tradere ac dilucide +explicare constituit. +21See Freguglia and Giaquinta (2016) for an excellent presentation of the Euler-Lagrange +equations, including a historical perspective. +22Lagrange (1806). +Les questions de maximis et minimis n’ont pas ´et´e incounues aux +anciens g´eom`etres ; car on a un livre entier d’Apollonius, qui traite presqu’uniquement des +plus grandes et des plus petites lignes droites qui peuvent ˆetre men´ees de points donn´es aux +arcs des sections coniques. p. 424. +8 + +Figure 2: Dido’s problem as described by Lord Kelvin in 1893. +electricity. Lord Kelvin played a key role in the final synthesis of 19th-century +science, which viewed all physical change as energy-related phenomena.23 +Lord Kelvin related Dido’s clever approach to bargaining for land as follows, +using the sketch in Fig. 2 to illustrate Dido’s problem: +“. . . +She cut the ox-hide into an exceedingly long strip, and succeeded in +enclosing between it and the sea a very valuable territory on which she built +Carthage. In Dido’s problem the greatest value of land was to be enclosed by a +line of given length. If the land is all of equal value the general solution of the +problem shows that her line of ox-hide should be laid down in a circle. It shows +also that if the sea is to be part of the boundary, starting, let us say, southward +from any given point, A, of the coast, the inland bounding line must at its far +end cut the coast line perpendicularly. Here, then, to complete our solution, we +have a very curious and interesting, but not at all easy, geometrical question to +answer: What must be the radius of a circular arc, ADC, of given length, and +in what direction must it leave the point A, in order that it may cut a given +curve, ABC, perpendicular at some unknown point, C?”24 +Lord Kelvin added that having enough mathematics knowledge, Dido would +determine that the boundary had to be a circle. Of course, as illustrated in Fig. +2, she would have given the boundary a different curvature in different parts to +gain as much as possible of the more valuable parts of the land offered to her, +“even though difference of curvature in different parts would cause the total +area enclosed to be less than it would be with a circular boundary of the same +length.”25 +23Gray (1910) +24Thomson (1894), p. 572-574. +25Ibid., p. 574. +9 + +CARTHAGToday, taught as introduction to calculus of variation, the solution of Dido’s +problem requires an extremization solution under constraint, that is, we max- +imize the area, A = +� +ydx, subject to the condition that the arc, L = +� +ds is +of a given length L. In other words, we wish to maximize the integral A sub- +ject to the condition that another integral L has a given constant value. Note +this is an optimization problem with constraints where we use Lagrange’s strat- +egy for finding the local maxima and minima of a function subject to equality +constraints. +It is clear that, without an original reliable source, I cannot conclude that +Euler defined Dido’s Problem for the first time, inspired by Virgil’s Aeneid, +as Condorcet implied. The evidence points to Lord Kelvin who described the +problem in 1893. And as he stated, whether severe critics will call Dido’s story +mythical or allow it to be historic, it is nevertheless full of scientific interest. +As for me, Dido’s Problem is an excellent example to introduce students to +the calculus of variations, as it expresses a perfectly definite case of isoperimet- +rics, illustrating the fundamental principles introduced by Euler and Lagrange +in the eighteenth century. +5 +Dido and Ancient Mathematics +Nothing is known about Dido’s knowledge. Being a Phoenician princess, it is +highly probable that she was well educated. What we glimpse from Virgil’s +and Justinus’s tales is that Dido was a formidable woman, smart, ambitious, a +foreign leader that left the city of Tyre (on the coastline of modern Lebanon) +with her faithful followers, navigated the waters of the Mediterranean Sea and +landed in the coast of North Africa. There, she established and ruled Carthage +(modern day Tunis), an important port city that rose to the height of its power +in the second century BC, before Rome became supreme and took over that +region. +For the ancient cultures that flourished around the Mediterranean, geom- +etry was fundamental to their development. The Babylonians thriving in the +Mesopotamian River Valley engaged in commerce through the Mediterranean, +and this required considerable mathematical skills. Clay tables preserve records +of what they knew. +For instance, clay tables from Babylon, located in the +southern part of Mesopotamia, about fifty miles south of present-day Baghdad +(Iraq) suggest that the Babylonian had an advanced knowledge of geometry and +arithmetic. +In fact, some scholars believe that the Babylonians knew the Pythagorean +theorem a thousand years before Pythagoras of Samos. At Susa, an ancient +city over two hundred miles from Babylon, a set of tablets were discovered in +1936, which contain the ratios of areas and perimeters of regular polygons to +their respective side lengths. The best known surviving tablet (estimated to be +from between 1900 and 1600 BC) contains a list of Pythagorean triples. This +suggests that the Babylonians had knowledge of the Pythagorean theorem, as +well as certain algebraic identities. +10 + +Moreover, that women knew mathematics in ancient times has been exten- +sively documented. +For example, the Pythagorean society included women, +some of which became famous such as mathematician Theano, who was mar- +ried to Pythagoras. In the dedication of his Introduction to Harmonics, ancient +mathematician and music theorist Nicomachus of Gerasa (c. 60-120 AD) ad- +dresses the lessons to a lady, one of his students.26 In this book, also known as +Manual of Harmonics, Nicomachus dealt with the theory of music, a version of +Pythagorean harmonics, in which he assigned number and numerical ratios to +notes and intervals. And of course, we know about Hypatia of Alexandria (c. +370-415 AD) considered the first woman scholar to attain eminence as mathe- +matician and astronomer.27 +I believe that Dido was educated in mathematics, and so she used the the- +orem of isoperimetry to outsmart the king who sold her the piece of land in +the northern tip of Africa (today’s Tunisia). Therefore, Dido’s Problem should +be viewed not only to illustrate a fundamental problem of variational calculus +but also as a lesson in the history of mathematics and the role ancient women +played in its development. +References +Brunt, B. van (2004). The Calculus of Variations. Published by Springer +New York ISBN: 978-0-387-40247-5 DOI: 10.1007/b97436. +Coppersmith, J. (2017) The Lazy Universe: An Introduction to the Principle +of Least Action. Oxford University Press. +Euler, L. (1732). De linea brevissima in superficie quacunque duo quaelibet +puncta iungente (On the shortest line joining two points on a surface). Com- +mentarii academiae scientiarum Petropolitanae, Volume 3, 1732, pp. 110-124. +(E. 9) +Euler, L. (1738). Problematis isoperimetrici in latissimo sensu accepti so- +lutio generalis (On isoperimetric problems in the widest sense). Commenturii +Academiae Scientiarum Petropolitanae 6 (1732/3), 123-155. Opera Omnia, 125, +13-40. (E. 27) +Euler, L. (1741). Curvarum maximi minimive proprietate gaudentium inven- +tio nova et facilis (New and easy method of finding curves enjoying a maximal +or minimal property). In Commentarii Academiae Scientiarum Petropolitanae +8 (1736). 159-190. Reprinted in Euler, L. Opera Omnia, I 25, 54-80. (E. 56) +Euler, L. (1744). Methodus inveniendi curvas h’neas maximi minimive pro- +prietate gaudentes sive solution problematis isoperimetrici latissimo sensu ac- +cepti. Lausanne, Genf: M.-M. Bousquet. Reprinted in Euler, L. Opera Omnia, +I 24. (E. 65). According to Enestr¨om, Euler completed the manuscript of this +work by April 1743. +Euler, L. (1764). +Elementa calculi variationum (Elements of Calculus of +Variations). Novi commentarii academiae scientiarum Petropolitanea 10 (1764), +26Biographical Note of Nicomachus, in Great Books of the Western World, Robert Maynard +Hutchins, Ed., Vol. 11, p. 807. +27Musielak (2020), pp. 206-207. +11 + +1766, pp. 51-93. This research (E. 296) was presented at the Berlin Academy +in 1756. +Freguglia, P. and Giaquinta, M. (2016). The Early Period of the Calculus +of Variations. Published by Birkh¨auser. +Gelfand, I. M. and Fomin, S. V. (1963). Calculus of Variations. Revised +English Edition Translated and Edited by R. A. Silverman Prentice-Hall, Inc. +Englewood Cliffs, NJ. +Goldstine, H. (1980). A History of the Calculus of Variations from the 17th +through the 19th Century. Springer-Verlag. +Gray, A. (1910). +The Life of William Thomson, Baron Kelvin of Large. +Nature 83, 61–65 (1910). +Lagrange (1806). Le¸cons sur le calcul des fonctions. Nouvelle ´edition, revue, +corrig´ee et augment´ee par l’auteur [J.-L. Lagrange]. Initially published as lecture +notes in 1799 when Lagrange was teaching at the Ecole Polytechnique and +reprinted in 1804. In 1806, Lagrange published a new edition containing two +new lessons. +Menger, K. (1937). What is Calculus of Variations and What Are Its Ap- +plications? The Scientific Monthly 45 (3) (1937), 250-253. +Musielak, D. (2022). Leonhard Euler and the Foundations of Celestial Me- +chanics. Springer History of Physics Series. Springer Nature Switzerland. ISBN +978-3-031-12321-4. +Musielak, D. (2020). Sophie Germain: Revolutionary Mathematician. Springer +Biographies. Springer Nature Switzerland. ISBN 978-3030383770. +Nahin, J.P. (2004). When Least is Best. Princeton University Press, 2004. +Rojo, A. and Bloch, A. (2018). The Principle of Least Action: History and +Physics. Cambridge University Press. doi:10.1017/9781139021029 +Thomson, W. (1894). Popular Lectures and Addresses by Sir William Thom- +son (Baron Kelvin) in Three Volumes. Nature Series, MacMillan and Co. Lon- +don 1894. +Dora Musielak; University of Texas at Arlington, 6 January 2023 +12 + diff --git a/LNE1T4oBgHgl3EQfGwM1/content/tmp_files/load_file.txt b/LNE1T4oBgHgl3EQfGwM1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..520a455e5e7aa54cedcb5b8196996d6dd41ab140 --- /dev/null +++ b/LNE1T4oBgHgl3EQfGwM1/content/tmp_files/load_file.txt @@ -0,0 +1,372 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf,len=371 +page_content='Dido’s Problem: When a myth of ancient literature became a problem of variational calculus Dora Musielak Abstract When introducing the calculus of variations, we may invoke Dido’s problem to illustrate the most fundamental variational problem: to find the curve of given perimeter which bounds the greatest area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' This type of problem led mathematicians to invent solution methods of maxima and minima, and the genesis of variational calculus as a distinct branch of anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Dido’s problem was inspired by the mythical tale of the foundation of Carthage (ancient city in North Africa) by a Phoenician princess as told independently by Roman poet Virgil, and by Latin historian Justinus in the first two centuries B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Historians have debated the facts surrounding Carthage’s birth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' however, contemporary mathematicians have accepted the vague events described by Virgil in his Aeneid, adding details to Dido’s story to extrapolate a few verses and use as a basis for the isoperimet- ric theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Was Leonhard Euler or Lord Kelvin who first interpreted Virgil’s poem as Dido’s problem of variational calculus?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In this article I attempt to resolve a question of historical attribution to identify who first defined Dido’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Keywords: Isoperimetrics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' variational calculus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Euler,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Lord Kelvin 1 Introduction In 1937,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Karl Menger1 wrote: “The first human being to solve a problem of calculus of variations seems to have been Queen Dido of Carthage.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Contempo- rary mathematics books2 go much further than that by adding details to Dido’s story taken from Virgil’s Aeneid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' alleging that Dido established Carthage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' an ancient city in modern Tunisia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' by application of the isoperimetric property of the circle to secure the largest area of land she bought upon arrival to North Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Here are two examples, where I use Italics to highlight details of Dido’s story that are not in Virgil’s poem: 1Menger (1937) 2See for example, Brunt (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Freguglia and Giaquinta (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Coppersmith (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Nahin (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Rojo and Bloch (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='02917v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='HO] 7 Jan 2023 “Dido was a Carthaginian queen (ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 850 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=') who came from a dysfunc- tional family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Her brother, Pygmalion, murdered her husband (who was also her uncle) and Dido, with the help of various gods, fled to the shores of North Africa with Pygmalion in pursuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Upon landing in North Africa, legend has it that she struck a deal with a local chief to procure as much land as an oxhide could contain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' She then selected an ox and cut its hide into very narrow strips, which she joined together to form a thread of oxhide more than two and a half miles long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Dido then used the oxhide thread and the North African sea coast to define the perimeter of her property .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' it is clear that Dido sought to enclose the maximum area within her ox and the sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The city of Carthage was then built within the perimeter defined by the thread and the sea coast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Dido called the place Byrsa meaning hide of bull.”3 “Dido .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' using the seashore (given as straight) as part of the boundary, she laid out the hide-strip to enclose the maximum possible area, which she “knew” would be in the shape of a semicircle.”4 If these accounts were based on fact, then Dido would be the first woman in humanity’s history to understand a mathematical principle, much before the first mathematicians in recorded history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Since Carthage was founded in 814 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=', Dido was born centuries before Thales of Miletus (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 624-548 BC), Pythagoras of Samos (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 570-490 BC), and Euclid of Alexandria (325-265 BC), and much earlier than the Greek mathematicians who dealt with isoperimetric problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Zenodorus (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 200-140 BC) who wrote On Isoperimetric Fig- ures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' this work is lost but details are found in the commentaries by Theon of Alexandria (335-405 AD), and by Pappus of Alexandria (290-350 AD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In his Mathematical Collection, Pappus presented results from ancient isoperimetry studies5 but he did not mention Dido.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Dido’s problem is now taught as the most fundamental isoperimetric prob- lem: for a fixed perimeter, determine the shape of the closed, planar curve that encloses the maximum area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The answer is the circle, as any grammar school child knows, but in variational calculus the solution is determined by an analytical method introduced by Leonhard Euler and refined by Joseph-Louis Lagrange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 2 Dido in Ancient Literature The story of Dido and the foundation of Carthage was immortalized by Virgil in his Aeneid, and by third century Roman historian Justinus in his Epitoma historiarum Philippicarum Pompei Trogi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' However, in these stories there is absolutely no mention of Dido enclosing a circular shape for the purchased land 3Brunt (2004), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 14-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 4Nahin (2004), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 5According to Pappus, the first proof of the isoperimetric property of the circle (using geometric arguments) is due to Zenodorus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 2 with the string of hide, or of her using the knowledge that the circle encloses the largest area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Virgil wrote in the Aeneid,6 Book 1, lines 365-368, referring to Dido and her people arriving to Africa: “They came to this place, and bought land, where you now see the vast walls, and resurgent stronghold, of new Carthage, as much as they could enclose with the strips of hide from a single bull, and from that they called it Byrsa.”7 Justinus, who refers to Dido by her Phoenician name Elissa, wrote in Book XVIII: “By this means some respite was given to the fugitives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' and Elissa, arriving in a gulf of Africa, attached the inhabitants of the coast, who rejoiced at the arrival of foreigners, and the opportunity of bartering commodities with them, to her interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Having then bargained for a piece of ground, as much as could be covered with an ox-hide, where she might refresh her companions, wearied with their long voyage, until she could conveniently resume her progress, she directed the hide to be cut into the thinnest possible strips, and thus acquired a greater portion of ground than she had apparently demanded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' whence the place had afterwards the name of Byrsa.”8 Thus, if neither Virgil nor Justin provided the details given by contemporary mathematicians about Dido’s problem as we know it, who did?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' how Dido’s mythical tale became Dido’s problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Surely, only a scientist would have made the connecting leap between “bought land, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' , as much as they could enclose with the strips of hide from a single bull .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' ” (Virgil, 1st century BC), and interpreted these words as “she laid out the hide-strip to enclose the maximum possible area, which she “knew” would be in the shape of a semicircle” (Nahin, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 3 Isoperimetry and Calculus of Variations Isoperimetrics provided the roots for the development of the variational method- ology, starting with the observation made by ancient scholars that most motion appears to be in either straight lines or circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The definition of a straight line as the shortest path between two points was an early expression of a minimization principle known to ancient geometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The isoperimetric problems considered in antiquity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' the circle in the plane and the sphere in three-dimensional space were known as the least perimeter figures to enclose a given area and a given volume, respectively) were solved by geometric means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Pappus gave credit to Zenodorus (200-140 BC) for solving for the optimal 6Written between 29 and 19 BC, this epic poem in 12 books tells the story of the foundation of Rome from the ashes of Troy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Virgil describes the foundation of Carthage by Dido in Book I: 297-371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 7Line 365: Devenere locos, ubi nunc ingentia cernis moenia surgentemque novae Karthagi- nis arcem, mercatique solum, facti de nomine Byrsam, taurino quantum possent circumdare tergo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 8Marcus Junianus Justinus, Epitoma historiarum Philippicarum Pompei Trogi (Epitome of the Philippic History of Pompeius Trogus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Translated by Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' John Selby Watson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' London: Henry G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Bohn, Convent Garden (1853).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 3 form of a maximum area surface for a given perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' He also expounded the work of Hero (or Heron) of Alexandria (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 10-75 AD) who studied the optics of reflection, finding that reflected light travels in a way that minimizes its travel time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The law of reflection of light—that the angle of incidence equals the angle of reflection—was well established since ancient times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In his Catoptrics, Euclid noted that light travels in straight lines and described the law of reflection (300 BC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Hero showed by a geometrical method that the actual path taken by a ray of light reflected from a plane mirror is shorter than any other reflected path that might be drawn between the source and point of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Ancient Greeks first conceived the idea that Nature selects the shortest, easiest and most direct path in moving objects between points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In the seven- teenth and eighteenth centuries, ideas about the economy of Nature continued preoccupying philosophers and scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Finding analytic solutions to more complicated problems of maxima and minima attracted the greatest mathe- maticians such as Fermat, Newton, Leibniz, the Bernoulli brothers (Jacob and Johann I), Euler, Lagrange, and Maupertuis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Perhaps inspired by Hero’s reflected light minimization problem, Pierre de Fermat (1601-1665) showed that the time required for a light ray to traverse a neighboring virtual path differs from the time actually taken by a quantity of the second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' This is known as Fermat’s principle of least time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Newton (1643-1727) examined the motion of bodies in a resisting medium, finding the shape of the body that renders its resistance minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In June of 1696, Johann Bernoulli (1667-1748) posed the following problem as a challenge to mathematicians: Given two points A and B in a vertical plane, find the path AMB down which a movable point particle M must, by virtue of its weight, will traverse in the shortest possible time (assumes that M’s acceleration is due only to gravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' This is the famous Brachistochrone (from the Greek brachistos, shortest, and chronos, time) problem, later also called the problem of least time descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The brachistochrone problem does not have a trivial solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' the Bernoulli brothers (Jacob and Johann I), Newton, Leibniz and l’Hˆopital solved the problem correctly, each using a different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='9 The initial investigations in the maxima and minima principles carried out by Leonhard Euler began from a study of the work of these mathematicians, especially motivated by the work of Jacob Bernoulli and prompted by his teacher Johann Bernoulli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The latter drew his attention to a problem of geodesic lines in a letter he sent Euler in St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Petersburg in 1728, which led Euler to conceive in early 1729 an analytical method by which, on any surface, whether convex or concave, the shortest line can be drawn between two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='10 Euler solved other isoperimetric problems, obtaining results to help him establish the analytical foundations of the calculus of variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Euler invented variational calculus as a distinct branch of analysis precisely to systemize the solution methods of maxima and minima, as brilliantly intro- duced in his 1744 book Methodus inveniendi lineas curvas maximi minimive 9Fregulia and Giaquinta (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 3-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 10Euler (1732) 4 proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti, the first treatise on calculus of variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='11 With Euler’s approach, the calculus of variations yielded a method for finding an extremum of a quantity that is expressible as an (variational) integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Euler’s Methodus inveniendi represented a substantial break with the then established tradition set for by his predecessors, including his earlier work in the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='12 In this treatise, Euler formulated the variational principle of mechanics, which is the principle of least action now attributed to Maupertuis: For a given projected body, denote its mass by M, half the square of its velocity by v, the arclength element by ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Then, among all curves passing through the same pair of endpoints, the desired curve is the one that minimizes the integral � Mdsv1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Details on how Euler formulated the principle are provided by Goldstine (1980), and by Freguglia and Giaquinta (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='13 Euler remarked: “Since the structure of the universe was made most perfect as designed by the wisest Creator, nothing in the world will occur in which no maximum or minimum rule is shining forth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' wherefore there is absolutely no doubt that all the effects of the world can be equally successfully determined from final causes by means of the maximum and least methods, and from the efficient causes themselves.”14 Considered as the first variational treatment of mechanics, Euler’s principle of least action contributed significantly to analytic mechanics and ultimately to the fundamental underpinnings of twentieth-century physics, including general relativity and quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Euler was also known as being able to recite Virgil’s Aeneid by heart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Did he interpret Dido’s tale as Dido’s isoperimetric problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 4 Defining Dido’s Problem in the Calculus of Variations A casual survey of the history of mathematics books written in the eighteenth and nineteenth century yields no clues as to when or how Dido’s mythical story became part of variational calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' It required a person with mathematical brilliance and fertile imagination to connect ancient myth with mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Two names emerge as potential candidates: Leonhard Euler (1707-1783), the originator of the calculus of variations, and British mathematician, physicist and engineer William Thomson, known in physics as Lord Kelvin (1824-1907).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='1 Leonhard Euler In Methodus inveniendi, Euler gives the following example to demonstrate his analytical method: to find among all admissible curves, enclosing a given area, 11Euler (1744) 12Fraser (1993) 13Goldstine (1980), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 101;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Freguglia and Giaquinta (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 181-189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 14Euler (1744), Additamentum I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 5 Figure 1: Euler’s sketch from Methodus inveniendi (1744) the one of least length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Figure 1 is Euler’s sketch to demonstrate that the curved arc of a circle, BM, is minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In his own words:15 “On the axis AP construct the line BM, so that, when the area ABMP of a given size is cut off, the curved arc BM corresponding to that area is the minimum of all.” After solving his variational integral, Euler shows the solution curve to be an arc of a circle with center somewhere on the line AP, for example, at C in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' But neither Euler’s Methodus inveniendi nor his other published memoirs in the field ever mention Dido.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In October 1783, the Marquis de Condorcet16 gave the ´Eloge d’Euler to the members of the Acad´emie des Sciences in Paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In this solemn eulogy, Con- dorcet expounded on Euler’s genius and suggested that a verse from the Aeneid had given Euler the first idea for a memoir on a question of Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In Condorcet’s own words: L’´etude de la Litt´erature ancienne et des Langues savantes avait fait partie de son ´education ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' il en conserva le goˆut toute sa vie, et n’oublia rien de ce qu’il avait appris ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' mais il n’eut jamais ni le tems ni le d´esir d’ajouter `a ses premi`eres ´etudes : il n’avait pas lu les Po`etes modernes, et savait par cœur l’Eneide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Cependant M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Euler ne perdait pas de vue les Math´ematiques, mˆeme lorsqu’il r´ecitait les vers de Virgile ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' tout ´etait propre `a lui rappeler cet objet presque unique de ses pens´ees, et on trouve dans ses ouvrages un savant M´emoire sur une question de M´ecanique, dont il racontait qu’un vers de l’Eneide lui avait donn´e la premi`ere id´ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='17 [The study of ancient literature and scholarly lan- guages had been part of his education;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' he retained a taste for it all his life, and 15Euler (1744), Chapter IV, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 135, Exemplum II: 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Super axe AP construere lineam BM, ita comparatant, ut, abscissa area ABMP datæ magnitudinis, arcus curvæ BM illi areæ respondens sit omnium minimus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 16Condorcet, Jean-Antoine-Nicolas de Caritat marquis de (1743-1794).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 17 ´Eloge d’Euler Prononc´e `a l’Acad´emie, par de Condorcet, Histoire de l’Acad´emie royale des sciences .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 1783, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 6 Bforgot nothing he had learned;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' but he never had either the time or the desire to add to his first studies: he had not read the Modern Poets, and knew the Aeneid by heart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' However, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Euler did not lose sight of Mathematics, even when he recited the verses of Virgil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' everything was likely to remind him of this almost unique object of his thoughts, and we find in his works a scholarly Memoir on a question of Mechanics, of which he said that a verse from the Aeneid had given him the first idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='] Euler did take verses from the Aeneid poem to use as mottos for his com- peting memoirs submitted to the French Academy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='18 These are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Was this to what Condorcet referred to?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Euler’s Memoirs and Mottos taken from Virgil’s Aeneid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Year Memoir Title Motto 1753 “On the movement of ships Tali remigio navis se (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 413) without the wind’s force.” tarda movebat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 7th winning memoir Virg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Aeneid Liv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 5 1759 “Concernin pitching Insequitur clamorque virum (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 415) and rolling.” stridorque rudentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 9th winning memoir Virg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Aeneid, Liv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 1 1770 “Moon Theory” Errantem que canit Lunam (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 485) Prize for 1770 Virg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Aeneid Liv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 1 10th winning memoir 1772 “Improved Moon theory” Hic labor extremus, longarum (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 486) Prize for 1772 haec meta viarum hinc jam 11th winning memoir digressi, vestris appellimus oris Virg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Aeneid, Liv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 3 However, the mottos were carefully selected by Euler to match the research topic of the competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='19 In addition to using Virgil’s verses, he also quoted from other ancient writers such as Marcus Tullius Cicero, Properci, and he composed other adages, asking Christian Goldbach for suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Ultimately, Condorcet’s statement “et on trouve dans ses ouvrages un savant M´emoire sur une question de M´ecanique, dont il racontait qu’un vers de l’Eneide lui avait donn´e la premi`ere id´ee” does not mean that Euler was inspired by Virgil to define Dido’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' As a historian, I cannot rely on obituaries to extract factual data, even if written by an eminent scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The much younger Condorcet never met Euler, and the ´Eloge he wrote, as most eulogies are, was based on hearsay, relaying on what the French academicians might have recalled about Euler’s life and work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 18Submissions were anonymously and the memoir identified by a motto;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' the author’s name enclosed in a sealed envelope was opened only for the winning memoir after the judging of the contest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 19For the significance of the mottos that Euler selected, see Musielak (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 7 A contemporary biography (published in 2016) further implies that Euler solved Dido’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The author refers to a copy of an eight-page manuscript (preserved in Moscow) that is said to contain Euler’s answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Is this the manuscript that categorically would give Euler credit for connecting Dido’s story to variational calculus?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Unfortunately, the manuscript in question is said to be “not in Euler’s own handwriting.” Thus, it diminishes its credibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' It is rather improbable that Euler, a prolific writer, would be the author of a manuscript inscribed by someone else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Besides, he would have included this solution in a paper published in 1764, where Euler summarized the results of the Calculus of Variations in terms of the variational operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Joseph-Louis Lagrange (1736-1813) expanded the variational calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In his second letter to Euler dated August 1755, Lagrange outlined his delta-algorithm (for solving constrained optimization problems), an approach Euler embraced, prompting him to conceive the term calculus of variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In the abstract of a memoir published in 1764, Euler credits Lagrange for enriching the science20 Their combined work led eventually to the Euler–Lagrange equations, which are the equilibrium equations for minima of variational integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='21 Five years after Euler died, Lagrange published M´ecanique analytique, his compendium on analytical mechanics, using variational ideas to present me- chanics from a unified analytic viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' When teaching at the ´Ecole Poly- technique in 1799, Lagrange published Le¸cons sur le calcul des fonctions and explained the method of variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Lagrange provided a brief overview of the development of problems of maxima and minima, referring only to Greek math- ematician Apollonius (262-190 BC), which dealt exclusively with the largest and smallest straight lines which can be drawn from given points to the arcs of conic sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='22 Dido’s problem is not mentioned here nor in Lagrange’s other published works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='2 William Thomson, Lord Kelvin The first instance in which Dido’s name appear in the context of interest is found in a public lecture delivered by William Thomson in 1893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' A great physi- cist known today as Lord Kelvin, his contributions include a major role in the development of the second law of thermodynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' the absolute temperature scale (measured in kelvins);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' the dynamical theory of heat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' the mathematical analysis of electricity and magnetism, including the basic ideas for the electro- magnetic theory of light;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' and much more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' He brought together disparate areas of physics—heat, thermodynamics, mechanics, hydrodynamics, magnetism, and 20Euler (1764).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' ex quo Auctori occasio est oblata hanc scientiam novo Calculi genere locupletandi, quem Calculum variationum appellat et cuis elementa hic tradere ac dilucide explicare constituit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 21See Freguglia and Giaquinta (2016) for an excellent presentation of the Euler-Lagrange equations, including a historical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 22Lagrange (1806).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Les questions de maximis et minimis n’ont pas ´et´e incounues aux anciens g´eom`etres ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' car on a un livre entier d’Apollonius, qui traite presqu’uniquement des plus grandes et des plus petites lignes droites qui peuvent ˆetre men´ees de points donn´es aux arcs des sections coniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 8 Figure 2: Dido’s problem as described by Lord Kelvin in 1893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' electricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Lord Kelvin played a key role in the final synthesis of 19th-century science, which viewed all physical change as energy-related phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='23 Lord Kelvin related Dido’s clever approach to bargaining for land as follows, using the sketch in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 2 to illustrate Dido’s problem: “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' She cut the ox-hide into an exceedingly long strip, and succeeded in enclosing between it and the sea a very valuable territory on which she built Carthage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In Dido’s problem the greatest value of land was to be enclosed by a line of given length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' If the land is all of equal value the general solution of the problem shows that her line of ox-hide should be laid down in a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' It shows also that if the sea is to be part of the boundary, starting, let us say, southward from any given point, A, of the coast, the inland bounding line must at its far end cut the coast line perpendicularly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Here, then, to complete our solution, we have a very curious and interesting, but not at all easy, geometrical question to answer: What must be the radius of a circular arc, ADC, of given length, and in what direction must it leave the point A, in order that it may cut a given curve, ABC, perpendicular at some unknown point, C?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='24 Lord Kelvin added that having enough mathematics knowledge, Dido would determine that the boundary had to be a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Of course, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 2, she would have given the boundary a different curvature in different parts to gain as much as possible of the more valuable parts of the land offered to her, “even though difference of curvature in different parts would cause the total area enclosed to be less than it would be with a circular boundary of the same length.”25 23Gray (1910) 24Thomson (1894), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 572-574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 25Ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=', p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 9 CARTHAGToday, taught as introduction to calculus of variation, the solution of Dido’s problem requires an extremization solution under constraint, that is, we max- imize the area, A = � ydx, subject to the condition that the arc, L = � ds is of a given length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In other words, we wish to maximize the integral A sub- ject to the condition that another integral L has a given constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Note this is an optimization problem with constraints where we use Lagrange’s strat- egy for finding the local maxima and minima of a function subject to equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' It is clear that, without an original reliable source, I cannot conclude that Euler defined Dido’s Problem for the first time, inspired by Virgil’s Aeneid, as Condorcet implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The evidence points to Lord Kelvin who described the problem in 1893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' And as he stated, whether severe critics will call Dido’s story mythical or allow it to be historic, it is nevertheless full of scientific interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' As for me, Dido’s Problem is an excellent example to introduce students to the calculus of variations, as it expresses a perfectly definite case of isoperimet- rics, illustrating the fundamental principles introduced by Euler and Lagrange in the eighteenth century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 5 Dido and Ancient Mathematics Nothing is known about Dido’s knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Being a Phoenician princess, it is highly probable that she was well educated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' What we glimpse from Virgil’s and Justinus’s tales is that Dido was a formidable woman, smart, ambitious, a foreign leader that left the city of Tyre (on the coastline of modern Lebanon) with her faithful followers, navigated the waters of the Mediterranean Sea and landed in the coast of North Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' There, she established and ruled Carthage (modern day Tunis), an important port city that rose to the height of its power in the second century BC, before Rome became supreme and took over that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' For the ancient cultures that flourished around the Mediterranean, geom- etry was fundamental to their development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The Babylonians thriving in the Mesopotamian River Valley engaged in commerce through the Mediterranean, and this required considerable mathematical skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Clay tables preserve records of what they knew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' For instance, clay tables from Babylon, located in the southern part of Mesopotamia, about fifty miles south of present-day Baghdad (Iraq) suggest that the Babylonian had an advanced knowledge of geometry and arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In fact, some scholars believe that the Babylonians knew the Pythagorean theorem a thousand years before Pythagoras of Samos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' At Susa, an ancient city over two hundred miles from Babylon, a set of tablets were discovered in 1936, which contain the ratios of areas and perimeters of regular polygons to their respective side lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The best known surviving tablet (estimated to be from between 1900 and 1600 BC) contains a list of Pythagorean triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' This suggests that the Babylonians had knowledge of the Pythagorean theorem, as well as certain algebraic identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 10 Moreover, that women knew mathematics in ancient times has been exten- sively documented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' For example, the Pythagorean society included women, some of which became famous such as mathematician Theano, who was mar- ried to Pythagoras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In the dedication of his Introduction to Harmonics, ancient mathematician and music theorist Nicomachus of Gerasa (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 60-120 AD) ad- dresses the lessons to a lady, one of his students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='26 In this book, also known as Manual of Harmonics, Nicomachus dealt with the theory of music, a version of Pythagorean harmonics, in which he assigned number and numerical ratios to notes and intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' And of course, we know about Hypatia of Alexandria (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 370-415 AD) considered the first woman scholar to attain eminence as mathe- matician and astronomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='27 I believe that Dido was educated in mathematics, and so she used the the- orem of isoperimetry to outsmart the king who sold her the piece of land in the northern tip of Africa (today’s Tunisia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Therefore, Dido’s Problem should be viewed not only to illustrate a fundamental problem of variational calculus but also as a lesson in the history of mathematics and the role ancient women played in its development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' References Brunt, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' van (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The Calculus of Variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Published by Springer New York ISBN: 978-0-387-40247-5 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='1007/b97436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Coppersmith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (2017) The Lazy Universe: An Introduction to the Principle of Least Action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Oxford University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Euler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (1732).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' De linea brevissima in superficie quacunque duo quaelibet puncta iungente (On the shortest line joining two points on a surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Com- mentarii academiae scientiarum Petropolitanae, Volume 3, 1732, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 110-124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 9) Euler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (1738).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Problematis isoperimetrici in latissimo sensu accepti so- lutio generalis (On isoperimetric problems in the widest sense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Commenturii Academiae Scientiarum Petropolitanae 6 (1732/3), 123-155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Opera Omnia, 125, 13-40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 27) Euler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (1741).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Curvarum maximi minimive proprietate gaudentium inven- tio nova et facilis (New and easy method of finding curves enjoying a maximal or minimal property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In Commentarii Academiae Scientiarum Petropolitanae 8 (1736).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 159-190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Reprinted in Euler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Opera Omnia, I 25, 54-80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 56) Euler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (1744).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Methodus inveniendi curvas h’neas maximi minimive pro- prietate gaudentes sive solution problematis isoperimetrici latissimo sensu ac- cepti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Lausanne, Genf: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Bousquet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Reprinted in Euler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Opera Omnia, I 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' According to Enestr¨om, Euler completed the manuscript of this work by April 1743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Euler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (1764).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Elementa calculi variationum (Elements of Calculus of Variations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Novi commentarii academiae scientiarum Petropolitanea 10 (1764), 26Biographical Note of Nicomachus, in Great Books of the Western World, Robert Maynard Hutchins, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 27Musielak (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 206-207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 11 1766, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 51-93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' This research (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' 296) was presented at the Berlin Academy in 1756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Freguglia, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' and Giaquinta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The Early Period of the Calculus of Variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Published by Birkh¨auser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Gelfand, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' and Fomin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Calculus of Variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Revised English Edition Translated and Edited by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Silverman Prentice-Hall, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Englewood Cliffs, NJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Goldstine, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' A History of the Calculus of Variations from the 17th through the 19th Century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Springer-Verlag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Gray, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (1910).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The Life of William Thomson, Baron Kelvin of Large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Nature 83, 61–65 (1910).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Lagrange (1806).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Le¸cons sur le calcul des fonctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Nouvelle ´edition, revue, corrig´ee et augment´ee par l’auteur [J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Lagrange].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Initially published as lecture notes in 1799 when Lagrange was teaching at the Ecole Polytechnique and reprinted in 1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' In 1806, Lagrange published a new edition containing two new lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Menger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (1937).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' What is Calculus of Variations and What Are Its Ap- plications?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The Scientific Monthly 45 (3) (1937), 250-253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Musielak, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Leonhard Euler and the Foundations of Celestial Me- chanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Springer History of Physics Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Springer Nature Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' ISBN 978-3-031-12321-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Musielak, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Sophie Germain: Revolutionary Mathematician.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Springer Biographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Springer Nature Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' ISBN 978-3030383770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Nahin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' When Least is Best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Princeton University Press, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Rojo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' and Bloch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' The Principle of Least Action: History and Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content='1017/9781139021029 Thomson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' (1894).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Popular Lectures and Addresses by Sir William Thom- son (Baron Kelvin) in Three Volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Nature Series, MacMillan and Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Lon- don 1894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' Dora Musielak;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} +page_content=' University of Texas at Arlington, 6 January 2023 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE1T4oBgHgl3EQfGwM1/content/2301.02917v1.pdf'} diff --git a/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf b/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..197ee8de3fdf88fc7d8788b21d2bc5100685c343 --- /dev/null +++ b/LNFAT4oBgHgl3EQfwR78/content/2301.08681v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8a77b5f95172694e5085a5f80a1fca30c5e0be9e260fe63f319e13db64b00838 +size 612596 diff --git a/LNFAT4oBgHgl3EQfwR78/vector_store/index.pkl b/LNFAT4oBgHgl3EQfwR78/vector_store/index.pkl new file 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+++ b/MNE0T4oBgHgl3EQf0AKB/content/tmp_files/2301.02680v1.pdf.txt @@ -0,0 +1,2809 @@ +Goldstone bosons and fluctuating hydrodynamics with dipole and momentum +conservation +Paolo Glorioso,1, ∗ Xiaoyang Huang,2, † Jinkang Guo,2 Joaquin Rodriguez-Nieva,1 and Andrew Lucas2, ‡ +1Department of Physics, Stanford University, Stanford CA 94305, USA +2Department of Physics and Center for Theory of Quantum Matter, +University of Colorado, Boulder, CO 80309, USA +(Dated: January 10, 2023) +We develop a Schwinger-Keldysh effective field theory describing the hydrodynamics of a fluid +with conserved charge and dipole moments, together with conserved momentum. +The resulting +hydrodynamic modes are highly unusual, including sound waves with quadratic (magnon-like) dis- +persion relation and subdiffusive decay rate. Hydrodynamics itself is unstable below four spatial +dimensions. We show that the momentum density is, at leading order, the Goldstone boson for +a dipole symmetry which appears spontaneously broken at finite charge density. Unlike an ordi- +nary fluid, the presence or absence of energy conservation qualitatively changes the decay rates of +the hydrodynamic modes. This effective field theory naturally couples to curved spacetime and +background gauge fields; in the flat spacetime limit, we reproduce the “mixed rank tensor fields” +previously coupled to fracton matter. +CONTENTS +1. Introduction +2 +2. Effective field theory of hydrodynamics +3 +2.1. General setup +3 +2.2. Classical limit and hydrodynamic effective theory +5 +2.3. Ideal hydrodynamics +7 +2.4. Dissipative hydrodynamics and higher-order terms +8 +2.5. Relevant perturbations in low dimensions +12 +3. Spontaneous symmetry breaking +12 +3.1. Mermin-Wagner Theorem +12 +3.2. Goldstone’s Theorem +13 +3.3. Existence of a symmetry-breaking state +14 +4. Hydrodynamics with energy conservation +15 +5. From Galilean symmetry to dipole symmetry +17 +6. Conclusions +19 +Acknowledgements +19 +A. Memory matrix methods +19 +1. Momentum susceptibility +20 +2. Dynamics without energy +21 +B. Dipole fluids with momentum in a curved spacetime +21 +C. Consistency of the symmetry algebra +24 +References +27 +∗ paolog@stanford.edu +† xiaoyang.huang@colorado.edu +‡ andrew.j.lucas@colorado.edu +arXiv:2301.02680v1 [hep-th] 6 Jan 2023 + +2 +1. +INTRODUCTION +One of the oldest and most successful theories in physics is hydrodynamics. While hydrodynamics was first under- +stood as a phenomenological set of equations that govern liquids and gases [1], over the past century we have instead +recognized that hydrodynamics is best understood as the universal effective field theory that governs thermalization +in a chaotic many-body system [2–6]. In the simplest scenarios, the degrees of freedom of a hydrodynamic theory +correspond to locally conserved quantities; the way that these modes interact with each other and decay is constrained +only by basic symmetries of the theory. Using modern effective field theory methods, sophisticated nonlinear theories +of fluctuating hydrodynamics have been developed and applied to increasingly sophisticated systems. +One family of novel phases of matter which has interesting dynamics arises when the microscopic degrees of freedom +are fractons – excitations which are individually immobile, and can only move in tandem [7–21].1 As a simple example, +we can consider a phase of matter in which charge/mass is conserved together with dipole moment/center of mass +– in this case, a single particle cannot move without violating the dipole conservation law! +The past few years +have seen an intense study of the fracton phases of matter that can result by combining many of these interacting +fractons. And over the past two years, it has been understood that when such fracton phases thermalize [22, 23], the +resulting hydrodynamics is non-trivial [24–27]: Fick’s law of diffusion, for example, becomes replaced by subdiffusive +equations, with the dynamical critical exponent dependent on how many multipole moments are conserved. Many +further “fracton hydrodynamics” universality classes have since been discovered [28–34]. +In this paper, we detail a qualitatively new universality class of hydrodynamics that emerges when fracton-like +multipole conservation laws are combined with canonical energy and momentum conservation, which was first pre- +sented by four of us in a shorter paper [35]; see also [36, 37]. We focus on the case where dipole moment is the +only additional conserved quantity, and where the theory has parity and time-reversal symmetry; in the absence of +momentum conservation, the consequences of breaking these symmetries were recently discussed in [34]. Without +dipole conservation, such a theory is essentially described by textbook Navier-Stokes equations with incoherent con- +ductivities [2]. With dipole conservation, the Navier-Stokes equations are completely changed [35]. At finite charge +density, the conventional propagating sound modes are replaced by magnon-like propagating modes. The decay rates +of these magnon-like modes is diffusive if energy is conserved, but subdiffusive if energy is not conserved. And at zero +density, the character of the hydrodynamic modes completely changes; the naive derivative expansion of hydrodynam- +ics at finite density is singular as low density is approached. At zero density, assuming particle-hole symmetry, the +momentum dynamics decouples from that of charge within linear response. The character of collective modes in this +case is completely different, where momentum and charge display diffusive and subdiffusive damping, respectively. +The subtle nature of this emergent hydrodynamics is intricately related to the fact that (in quantum mechanics) +the dipole moment operator D, and net momentum operator P, do not commute [38]: +[D, P] = iQ, +(1.1) +where Q represents total charge. An analogous classical statement holds for Poisson brackets. One might expect +that such a commutation relation is similar to angular momentum commutation relations in an isotropic fluid – such +commutation relations lead not to new propagating degrees of freedom, but rather constraints on the currents of other +modes (the stress tensor, in this case). However, at finite density, (1.1) implies that momentum susceptibility (the +generalization of mass density) is singular! This means that a naive hydrodynamic degree of freedom – fluid velocity – +is non-local. One of the main results of this paper is that we can nevertheless construct a local hydrodynamic theory, +using unconventional degrees of freedom. In Section 2, we will describe how to construct this EFT following the +constructions of [2–6]. In the process, we comment on the coupling of this theory to background geometry (vielbein), +although much of this technical work is relegated to appendices. Following [39–42], we hope this can further stimulate +work on understanding how and when fractons can be coupled to gravity. +As reported in [35], these hydrodynamic theories can be unstable below four dimensions. This is true both without +energy conservation, and with energy conservation at infinite temperature (under mild assumptions). This result +generalizes the well-known Kardar-Parisi-Zhang instability of an equilibrium fluid (without dipole conservation) in +one dimension [43], and implies the existence of a non-equilibrium fixed point in three dimensions, in an undriven +system. +From many perspectives, we will show that the commutation relation (1.1) implies there is spontaneous symmetry +breaking. In a finite density state, D and P do not commute, so they cannot be diagonalized simultaneously – in a state +with fixed momentum, there are large fluctuations in dipole moment. Unlike more conventional compact non-Abelian +1 While it can be desirable to strengthen this definition to demand that fractons cannot move under the action of any local operator, +following the “fracton hydrodynamics” literature, we will take a looser definition. +(In our theory, a local operator that inserts a +quadrupole can move an isolated charge.) + +3 +symmetry groups such as SU(2), here one cannot find any physical “singlet” states in the Hilbert space which lie in +trivial representations. (This is analogous to textbook quantum mechanics: one cannot find simultaneous eigenstates +of x and p.) So it seems that trivially, dipole and/or momentum will be spontaneously broken, in agreement with +previous literature on low-dimensional SSB with non-compact symmetry groups [44]. In this paper, we focus on +ensembles with fixed momentum density (which seems more physical to us). A consequence is that the propagating +momentum density is (at leading order in the hydrodynamic expansion) proportional to the Goldstone boson for +broken dipole symmetry. In one and two space dimensions, the fluctuations of this Goldstone boson are very large, +and for this reason a recent work [45] proved that there is no SSB within the context of the Mermin-Wagner theorem. +On the other hand, we will see that the hydrodynamics in low dimension a single hydrodynamic mode still contains +all of the spectral weight required to saturate the Goldstone theorem. The presence of this Goldstone boson in the +hydrodynamic theory suggests an unusual paradigm for possible “spontaneous symmetry breaking.” Following recent +discussions on the spontaneous breaking of boost symmetry in fluids [46, 47], we will discuss at some length the nature +of the apparent spontaneous symmetry breaking in Section 3. +In Section 4, we extend the discussion of the EFT to models with energy conservation. The observation of interest +is that energy conservation changes the dynamical universality class of the dipole-momentum conserving fixed point.2 +Intuitively, this can be understood as follows: energy can diffuse, while charge must subdiffuse. Due to thermody- +namics at finite charge and energy density, however, charge and energy modes generically “mix” (e.g. the propagating +sound wave would involve both charge and energy fluctuations). As a consequence, the dominant decay channel is +always through energy diffusion, which leads to z = 2, rather than z = 4, at the hydrodynamic (Gaussian) fixed point. +Lastly, in Section 5, we discuss how a dipole-conserving theory can arise in the infinite mass limit of a theory with +Galilean invariance. This may suggest one way to look for this physics in experimental systems, e.g. in (nearly) flat +bands [52–56] in condensed matter systems. +Several complementary discussions and technical computations are included in the appendices. In Appendix A, +the memory matrix formalism is applied to derive the normal modes and diverging momentum susceptibility. Ap- +pendix B consists of a detailed derivation of dipole-conserving hydrodynamics in curved spacetime. In Appendix C, +the consistency between dipole symmetry and geometry is verified. +2. +EFFECTIVE FIELD THEORY OF HYDRODYNAMICS +One main result is that hydrodynamics with dipole conservation possesses anomalous scaling, which is due to the +interplay between the nonlinear hydrodynamic interactions and hydrodynamic fluctuations [35]. To derive this we shall +use a recently formulated effective field theory (EFT) of hydrodynamics, which systematically describes fluctuations +by encoding hydrodynamics into an effective action [2–6]. +2.1. +General setup +The aim of the EFT approach is to systematically encode the correlation functions of hydrodynamic densities and +currents. Such correlation functions have the general form +Tr(T (J1J2 · · · )ρ0 ˜T (J3J4 · · · · · · )) = +� +ρ0 +Dψ1Dψ2 eiS0[ψ1]−iS0[ψ2] J1[ψ1]J2[ψ1]J3[ψ2]J4[ψ2] · · · , +(2.1) +where, in the first expression, T and ˜T denote time- and anti time-ordering, ρ0 is the initial state, which we take to be +thermal ρ0 = e−βH/ tr(e−βH), with H the microscopic Hamiltonian of the system, and J1, J2, . . . are operators inserted +at (t1, ⃗x1), (t2, ⃗x2), . . . . On the right-hand side, we formally rewrote the correlator as a path-integral, where S0 is the +action of the microscopic dynamics, and ψ1, ψ2 are a doubled copy of the degrees of freedom of the system. Since on +the left-hand side we have a forward (backward) time evolution given by the time-ordered (anti-time ordered) product, +the path integral contains two exponentials of the action S0, with a relative minus sign, as the first one corresponds +to forward evolution, while the second one to backward evolution. In other words, the doubling of degrees of freedom +comes from that the evolution of the density matrix ρ0 → U(t)ρ0U †(t) contains two factors of the evolution, one +forward and one backward. Computing hydrodynamic correlation functions from the microscopic dynamics is very +2 Interestingly, there was a debate in past literature about the role of energy conservation in flowing from the Navier-Stokes to the KPZ +fixed point [48–50]. The consensus is now that energy conservation indeed does not disturb the KPZ point [51]. What was missing in +the past was just to incorporate dipole conservation! + +4 +hard. We thus want to introduce an EFT approach that substitutes the right-hand side of (2.1) with a simpler action: +Tr(T (J1J2 · · · )ρ0 ˜T (J3J4 · · · · · · )) = +� +Dχ1Dχ2 eiS[χ1,χ2] J1[χ1]J2[χ1]J3[χ2]J4[χ2] · · · , +(2.2) +where S is the effective action for hydrodynamics, and χ1, χ2 denote the doubled hydrodynamic degrees of freedom. +The action S will encode the effects of fluctuation and dissipation and, in particular, will allow us to predict the +existence of anomalous scaling. +We shall now introduce the degrees of freedom of this EFT. These should be fields that nonlinearly realize the +symmetries associated to conservation of charge, dipole and momentum. For momentum conservation, we introduce +a set of coordinate fields Xi = Xi(σt, σI) which nonlinearly realize translations P i, i.e. +Xi(σt, σI) → Xi(σt, σI) + ξi , +(2.3) +where ξi is a constant vector. We are using σI to denote an auxiliary coordinate system which can be thought of as +labeling the fluid parcels at a fixed value of time σt.3 The coordinates Xi(σt, σI) describe the trajectory of the fluid +parcel labeled by σI as a function of time σt. The coordinates (σt, Xi) are the “physical” ones, in the sense that they +label the time and space in the lab reference frame.4 +Next, we also have a vector degree of freedom ϕi(σt, σI) that nonlinearly realizes the dipole shift symmetry Di: +ϕi(σt, σI) → ϕi(σt, σI) + ci , +(2.4) +where ci is a constant vector. +Finally, for charge Q, the associated degree of freedom is a scalar ϕ(σt, σI), and +transforms as +ϕ(σt, σI) → ϕ(σt, σI) + a − ciXi , +(2.5) +where a is a constant denoting the parameter of transformations associated to Q. +The fields ϕ and ϕi can be +heuristically viewed as describing the “local phase” of the fluid ei(ϕ+Xiϕi), where this particular form is motivated +from the fact that, for dipole-conserving field theories, U(1) global transformations can have a linear dependence in +spatial coordinates [11]. Note that ϕ transforms also under dipole shifts. This particular transformation rule is implied +by the commutator (1.1). Indeed, writing infinitesimal translation and dipole shift as δξϕ = ξi∂iϕ, δcϕ = −ciXi, we +have +(δcδξ − δξδc)ϕ = ciξi , +(2.6) +i.e. the commutator is an infinitesimal shift of ϕ, as required by (1.1). It can also be verified that the last term in (2.5) +is the most general transormation consistent with (1.1). The effective action will be invariant under transformations +(2.3), (2.4), and (2.5) which, as a consequence of Noether’s theorem, correspond to the statement of conservation of +momentum, dipole and charge, respectively. Moreover, we provide a thorough and careful consistency check of the +symmetry algebra in Appendix C. +Now recall from above that all the degrees of freedom have to be doubled, so we will have Xi +1, Xi +2, ϕi +1, ϕi +2, ϕ1 and +ϕ2. The symmetries (2.3), (2.4), and (2.5) will also be doubled, which in turn correspond to the conservation of +the corresponding hydrodynamic currents defined in the forward and backward time contours. Unlike in the path +integral (2.1), the effective action appearing in (2.2) does not have a factorized form. This is because, as a result of +the coarse-graining, where the fast-moving degrees of freedom have been integrated out, new couplings that are local +in the “folded” time have been generated. These cross-couplings are responsible for dissipations and fluctuations. +While the effective action loses factorization, it still satisfies several properties that come from the unitarity of the +underlying microscopic evolution [2, 5]: +S[χ, χ] = 0, +S[χ2, χ1] = −S∗[χ1, χ2], +Im S[χ1, χ2] ≥ 0 , +(2.7) +where χ1, χ2 collectively denote the two copies of Xi, ϕi, ϕ. Note in particular that the action can (and will) be +complex-valued; as we will see this is a basic consequence of having thermal fluctuations. Additionally, since the +initial state ρ0 is thermal, and assuming that the microscopic Hamiltonian H is invariant under time-reversal, the +effective action satisfies a discrete Z2 symmetry called “dynamical KMS symmetry”: +S[χ1, χ2] = S[˜χ1, ˜χ2], +˜χ1(σt, σI) = (−1)ηχ1(−σt, σI), +˜χ2(σt, σ) = (−1)ηχ2(−σt − iβ, σI) , +(2.8) +3 In older literature, these are the so-called “Lagrangian specification” of the fluid [57]. +4 It is convenient to denote time by σt as, in what follows, we will often need to take derivatives with respect to time at fixed σI, not at +fixed Xi. + +5 +where (−1)η = ±1 denotes the time-reversal eigenvalue of χ. This symmetry is equivalent to the Euclidean time +periodicity of correlation functions on a thermal state with inverse temperature β. In our effective action, it will +relate couplings responsible for dissipation with those describing fluctuations, and it will ensure consistency with +the second law of thermodynamics, Onsager relations, and existence of equilibrium. Eq. (2.8) can be extended to +situations where the microscopic Hamiltonian is invariant under a more general discrete symmetry, so long as such +symmetry contains time-reversal. A proof of (2.8) is given in [2]. +To complete our effective field theory, we need an additional set of symmetries that characterize the fact that the +late-time behavior of the system is that of a fluid. Recall that σI should be interpreted as labels of fluid elements at a +fixed value of σt. Adiabatically reshuffling fluid elements has a vanishing cost in energy, since, in contrast to a solid, +fluid parcels are not pinned to a particular spatial location. This means that a specific way to label fluid elements at +a given time is not physical, and thus the effective action should be invariant under time-independent redefinitions of +σI: +σI → σ +′I(σJ) . +(2.9) +Had we not considered this symmetry, the action could depend on arbitrary derivatives ∂IXi, and we would describe +a solid instead of a liquid. Analogously, in the charge sector, we have the freedom to relabel the local phase ei(ϕ+Xiϕi) +at a fixed time. This amounts to requiring the symmetry +ϕ1(σt, σI) → ϕ1(σt, σI) + λ(σI), +ϕ2(σt, σI) → ϕ2(σt, σI) + λ(σI) , +(2.10) +where λ(σI) is a time-independent redefinition of the phase and can be arbitrarily assigned on each fluid element σI. +The symmetry (2.10), dubbed diagonal shift symmetry in [2, 5], states the absence of spontaneous symmetry breaking +of the global U(1) symmetry. Indeed, in the occurrence of spontaneous symmetry breaking, the full information about +the phase would be a physical (of course, up to constant shifts of the phase), which would give rise to a superfluid. +Instead, in the present context, we are merely interested in the conservation of charge (and dipole) in the absence of +spontaneous symmetry breaking of charge. +Unlike ϕ1,2, ϕi +1,2 does not have a diagonal shift symmetry. Indeed, one can show that imposing the symmetry +ϕi +1,2 → ϕi +1,2 + λi(σI) leads to the incorrect Ward identities, as we will show below (2.18c) in the following Section. +We will later see in Sec. 3 that this can be understood as a consequence of spontaneous symmetry breaking — ϕi is +a Goldstone boson. +2.2. +Classical limit and hydrodynamic effective theory +The formalism we have introduced above is based on quantum mechanics. +In the present paper, however, we +are interested in the emergent classical, high-temperature hydrodynamic behavior of many-body systems. There is a +simple way to take the classical limit of this framework which retains the physics we are interested in and has the benefit +of considerably simplifying various technical aspects. To this aim, we restore factors of ℏ and write χ1 = χ + 1 +2ℏχa, +χ2 = χ − 1 +2ℏχa, where aggain χ collectively denotes the hydrodynamic fields, for example: Xi +1 = Xi + 1 +2ℏXi +a, etc. +The fact that χ1 − χ2 is linear in ℏ can be heuristically understood from the fact that the forward and backward time +evolutions are located a distance ℏβ from each other, and thus, as ℏ → 0, χ1 − χ2 should vanish linearly in ℏ. In this +limit, the dynamical KMS symmetry becomes +˜χ(σt, σI) = (−1)ηχ(−σt, σI), +˜χa(σt, σI) = (−1)η{χa(−σt, σI) + iβ∂tχ(−σt, σI)} , +(2.11) +where the dependence on ℏ has factorized out, and the nonlocal time shift in (2.8) reduced to an exact time derivative, +allowing for a more straightforward implementation. +We now proceed to writing down the invariant blocks that will be used to write the effective action. We assume +rotational invariance, but a generalization to discrete rotational symmetry is straightforward [58]. We recall that the +energy conservation is not assumed, so we take β0 as a constant inverse temperature.5 We will come back to include +energy conservation in Section 4. For completeness, we summarize the notation here; see Appendix B for more details. +We denote µ, ν = 0, 1, 2, 3 for the physical spacetime, and A, B = t, x, y, z for the fluid spacetime, and use i, j, I, J to +indicate their spatial subspace, respectively. We introduce the internal spacetime indices α, β and b, c for its spatial +subspace (we reserve a to describe a-fields in the Keldysh contour!). +5 The recent formalism of [34], which can build effective field theories for non-thermal systems (i.e. those whose steady state is not of the +form exp[−βH], may allow us to put this construction on a firmer footing. However, it is not known how to incorporate the non-Abelian +multipole algebra or spontaneous symmetry breaking into this formalism. Revisiting this question would be interesting in future work. + +6 +To incorporate the gauge invariance, we introduce the background gauge field eb +µ, Aµ and Ab +µ, such that the invariant +building-blocks in the fluid spacetime are defined as (s = 1, 2) +eb +s,A(σ) = ∂Xµ +s (σ) +∂σA +eb +s,µ(σ), +(2.12a) +Bs,A(σ) = ∂Xµ +s (σ) +∂σA +� +As,µ(σ) + eb +s,µ(σ)ϕs,b(σ) +� ++ ∂ϕs(σ) +∂σA , +(2.12b) +Kb +s,A(σ) = ∂Xµ +s (σ) +∂σA +� +Ab +s,µ + ωb +s,µcϕc +s +� ++ ∂ϕb +s(σ) +∂σA . +(2.12c) +See a derivation in Appendix B. In the main text, we will be particularly interested in the geometry where eb +1µ +eb +2µ = +2δb +µ and ωb +s,µc = 0. In the classical limit, this corresponds to working with a flat spacetime and allowing eb +a,µ = eb +1µ−eb +2µ +to source the stress tensor. Throughout this section, we take e0 +sµ = δ0 +sµ, but will consider a more general background +when evaluating the energy fluctuations in Section 4. We denote the r, a-fields as follows +Λr = Λ1 + Λ2 +2 +, +(2.13a) +Λa = Λ1 − Λ2, +(2.13b) +where Λr,a denote collectively the background and dynamical fields. The r-fields of the blocks are +eb +r,A = ∂AXµeb +µ, +(2.14a) +Br,A = ∂Aϕ + ∂AXµAµ + ∂AXµeb +µϕb, +(2.14b) +Kb +r,A = ∂AXµKb +r,µ = ∂AXµ(∂µϕb + Ab +µ), +(2.14c) +While a-fields are always invariant under the relabeling symmetries (2.9) and (2.10), the r-fields are not, and the +invariant r-fields without derivatives are +eb +r,t = ∂tXµeb +µ ≡ β0uµeb +µ = β0ub, +Br,t = β0uµBr,µ ≡ β0µ, +Kb +r,µ, +(2.15) +from which we defined the thermodynamic variables uµ, µ and Kb +r,µ. Below, we omit the index r for simplicity. In the +classical limit and physical spacetime, the a-fields of the invariant blocks can be written as +eb +a,A = ∂AXµEb +a,µ, +Eb +a,µ = eb +a,µ + LXaeb +µ, +(2.16a) +Ba,A = ∂AXµCa,µ, +Ca,µ = Aa,µ + ∂µϕa + LXaAµ + eb +µϕa,b + Eb +a,µϕb, +(2.16b) +Kb +a,A = ∂AXµKb +a,µ, +Kb +a,µ = ∂µϕb +a + Ab +a,µ + LXaAb +µ. +(2.16c) +To the leading order in a-fields, the effective Lagrangian is given by +L = ˆT µ +b Eb +a,µ + JµCa,µ + Jµ +b Kb +a,µ + . . . . +(2.17) +Note that in this case the stress tensor is not equal to the coefficient T µ +b ̸= ˆT µ +b . Indeed, the Ward identities in the +absence of stochastic fluctuations are obtained by varying L with respect to Xi +a, ϕa and ϕb +a and then setting a-fields +to zero. This leads to +∂µ( ˆT µ +b + Jµϕb)eb +i + Ab +ieµbJµ − F b +iµJµ +b − FiµJµ = 0, +(2.18a) +∂µJµ = 0, +(2.18b) +∂µJµ +b − Jµeµb = 0, +(2.18c) +where Fµν = ∂µAν − ∂νAµ and F b +µν = ∂µAb +ν − ∂νAb +µ are the U(1) and dipole field strength, respectively. In the +following, we will construct the effective field theory to determine the stress tensor and currents. We will see that the +stress tensor is indeed given by the term in the bracket in (2.18a). +Note that, had we imposed diagonal shift symmetry on the dipole field ϕb → ϕb + λb(σI), this would affect the +structure of the Ward identities. Indeed, neglecting background fields, Ca,µ = ∂µϕa + eb +µϕa,b + ∂µXb +aϕb, and we see +that Ca,µ would transform nontrivially under λb. This in turn would affect the structure of the action (2.17) and +thus alter the Ward identities. Ward identities should only be determined in terms of the global symmetries of the + +7 +system (or their gauged version), i.e. eqs. (2.3)-(2.5), therefore it is necessary that ϕb does not possess a diagonal +shift symmetry, unlike ϕ. +Additionally, we note that, unlike most studies about dipole field theory (e.g. [59]), the dipole gauge field Ab +µ as +well as the dipole current Jµ +b by no means need to be symmetric in their spatial indices. In dipole hydrodynamics +without momentum, Jij ∼ ∂i∂jµ, which automatically decouples the antisymmetric part at this derivative order. +The momentum density, on the other hand, can contribute to the antisymmetric part of Jij. As we will see later, +such antisymmetric term contributes to the momentum subdiffusion mode in (2.59) through the coefficient a3 that is +defined in (2.29), and thus has physical consequences. +At last, let us consider a system that preserves the symmetry Θ = PT , where P acts as flipping all the spatial +coordinates. The KMS transformation of dynamical fields in fluid spacetime is given by +� +Xµ +a (−σ) = −Xµ +a (σ) − iβ0∂tXµ(σ) + iβ0δµ +0 , +�ϕa(−σ) = −ϕa(σ) − iβ0∂tϕ(σ), +�ϕb +a(−σ) = ϕb +a(σ) + iβ0∂tϕb(σ), (2.19) +and that of external gauge fields is given by +�eb +a,µ(−σ) = eb +a,µ(σ) + iβ0∂teb +µ(σ), +�Aa,µ(−σ) = Aa,µ(σ) + iβ0∂tAµ(σ), +�Ab +a,µ(−σ) = −Ab +a,µ(σ) − iβ0∂tAb +µ(σ). +(2.20) +In the classical limit and physical spacetime, we thus have +�eb +µ(−x) = eb +µ(x), +(2.21a) +�Eb +a,µ(−x) = Eb +a,µ(x) + iLβeb +µ(x), +(2.21b) +�Bµ(−x) = Bµ(x), +(2.21c) +�Ca,µ(−x) = Ca,µ(x) + iLβBµ(x), +(2.21d) +�Kb +µ(−x) = −Kb +µ(x), +(2.21e) +�Kb +a,µ(−x) = −Kb +a,µ(x) − iLβKb +µ(x), +(2.21f) +where βµ ≡ β0uµ, and +Lβeb +µ = β0∂µub, +(2.22a) +LβBµ = β0∂µµ + βν(Fνµ + 2eb +[µ∂ν]ϕb), +(2.22b) +LβKb +µ = ∂µ +� +βν(∂νϕb + Ab +ν) +� ++ βνF b +νµ. +(2.22c) +2.3. +Ideal hydrodynamics +To describe ideal hydrodynamics, it is convenient to first introduce the single-time equilibrium action [60]: +S0 = +� +dd+1xP(eb +t, Bt, Kb +t , f IJKb +IKc +J). +(2.23) +Then, the factorizability condition leads to +IEFT,eq = +� +dd+1xLeq = S0[Λ1] − S0[Λ2], +(2.24a) +Leq = peµ +b Eb +a,µ + nuµCa,µ + ˆπbuµEb +a,µ + ψµ +b +� +Kb +a,µ − Ec +a,µeν +cKb +ν +� +, +(2.24b) +where we used eµ +a,α = −eν +αeµ +βeβ +a,ν. The coefficients in Leq define the equation of state, +p ≡ P, +n = β0 +∂P +∂Bt +, +ˆπb = β0 +∂P +∂eb +t +, +ψµ +b = ∂P +∂Kbµ +, +(2.25) +where the partial derivatives of thermodynamic pressure P are taken with other arguments being fixed. It can be +verified that Leq satisfies the KMS condition. Now, we can read off the equilibrium stress tensor and currents from +varying the action with respect to eb +a,µ, Aa,µ and Ab +a,µ: +T µ +(0)b = peµ +b + πbuµ − ψµ +c Kc +νeν +b, +(2.26a) + +8 +Jµ +(0) = nuµ, +(2.26b) +Jµ +(0)b = ψµ +b , +(2.26c) +where we defined the momentum density as +πb ≡ ˆπb + nϕb ≡ ˆρub + nϕb, +(2.27) +with ˆρ ≥ 0 an O(1) coefficients. We can further express the dipole currents by expanding the pressure up to quadratic +terms in field amplitude +P ∼ 1 +2bK0bK0b − 1 +2aibjcKibKjc + . . . , +(2.28) +where the invariant tensor is given by +aijkl = a1δijδkl + a2δij + 2a3δi[kδl]j, +(2.29) +with A = Aij + Aji − 2 +dδijAkk, A[ij] = 1 +2(Aij − Aji). Thermodynamic stability requires that b, a1,2,3 ≥ 0. Thus, +we have +J0 +(0)b = ψ0 +b = bK0b, +(2.30a) +Ji +(0)b = ψi +b = −aibjcKjc. +(2.30b) +Let us turn off the gauge field temporarily and plugin (2.26) into (2.18). Then, we find exactly the momentum and +charge conservations, but with an additional dipole constraint: +nub = −aibkc∂i∂kϕc. +(2.31) +Here, we have neglected the terms with time derivatives; in fact, as we will see in Section 2.4, there is a relaxation +time that relaxes non-hydrodynamic modes, which makes (2.31) exact (up to a fluid frame change). As a result, (2.27) +reduces to +πb = nϕb + · · · , +(2.32) +up to higher order corrections. The low energy excitations for the dipole-conserving fluid are not single particle (frac- +ton) excitations, but rather propagating dipole Goldstone/density waves. In particular, the momentum susceptibility +ρ, defined as πb ≡ ρub (different from ˆρ), will be non-local: +ρ ∼ n2 +k2 , +(2.33) +and diverges at large distance k → 0; see Appendix A for another derivation of non-local ρ using the memory matrix +formalism. +At leading order in amplitude expansion, the conservation equations for δn ≡ n − n0 and πb are +∂0πb + n0 +χ ∂iδnδi +b = 0, +∂0δn − a +n0 +∂2 +i ∂jπb = 0 , +(2.34) +where we only retained terms at leading order in derivatives and a = a1 +2a2(d−1)/d. Upon Fourier transformation, +the normal modes are given by +ω = ± +� a +χk2. +(2.35) +We therefore find that the dipole “sound” modes are magnon-like. +2.4. +Dissipative hydrodynamics and higher-order terms +We are now ready to write down the most general (leading order) dissipative part of the effective field theory. We +expand the Lagrangian to containing at most two factors of a-fields +L = Leq + L(1) + L +(1) + L(2), +(2.36) + +9 +where the superscript (n) represents the number of a-fields, and we will always keep the leading derivative orders. +As mentioned around (2.21), we will restrict to systems whose microscopic dynamics is PT -even dynamics. The +dynamical KMS symmetry then implies that L(2) is PT -even and relates it to L(1); moreover it allows the presence +of an additional L +(1) that is KMS-invariant by itself (and is thus also PT -even). It is helpful to first introduce a +combined field +∆Ua,ib ≡ ∂iCa,jej +b − Ka,ib += ∂i∂jϕaej +b + ∂i(Aa,j + LXaAj)ej +b + ∂i(Ec +a,jϕc)ej +b − (Aa,ib + LXaAib), +(2.37) +whose KMS transformation is given by +� +∆U a,ib(−x) = −∆Ua,ib(x) − iLβ∆Uib(x), +(2.38) +where +LβUib ≡ β0ej +b∂i∂jµ + ∂i(βνFνj)ej +b + ∂i(βc∂jϕc)ej +b − ∂i(βνAb +ν) − βνF b +νi +≈ β0ej +b∂i∂jµ + β0∂iF0jej +b − β0∂0Ab +i, +(2.39) +with the second line being the approximation of linear response. With the benefit of hindsight, we have constructed +this field to be independent of ϕb +a. Using this field, we can write the most general PT -even L(2) as +−iβ0L(2) = σijCa,iCa,j + sibjcEa,ibEa,jc + tibjc∆Ua,ib∆Ua,jc + 2rijkb +1 +∂iCa,j∆Ua,kb + rijkl +2 +∂iCa,j∂kCa,l, +(2.40) +where σij = σδij, σ ≥ 0. +In the above action, we only considered leading derivative contributions from a-type +fields, and we additionally included first derivative terms in Ca,i as they are of the same order as ∆Ua,ib. +As +usual, the terms proportional to Eb +a,0, Ca,0 can be eliminated by field redefinition as shown below, and, at the +same time, Kb +a,0 ∼ ∂tϕb is neglected as it is subleading to the last term which contains first spatial derivatives of +ϕb, and we are keeping into account the scaling ω ∼ k2 found in (2.35). Under KMS transformations, we require +[L(2) + L′(1) − ( ˜L(2) + ˜L′(1))]O(a) = 0, where O(a) indicates the (first) order of a-fields. Since L(2)|O(a) = 0, the +constraint reduces to L′(1) − ˜L′(1)|O(a) = ˜L(2)|O(a). Next, we redefine L(1) ≡ 1 +2(L′(1) − ˜L′(1))O(a) = 1 +2 ˜L(2)|O(a), thus, +by construction, L(1) is PT -odd. This leads to +β0L(1) = − σijLβBiCa,j − sibjcLβeibEa,jc − tibjcLβ∆Uib∆Ua,jc − rijkb +1 +∂iLβBj∆Ua,kb − rkbij +1 +Lβ∆Ukb∂iCa,j +(2.41) +− rijkl +2 +∂iLβBj∂kCa,l. +From the above discussion, the effective Lagrangian allows a PT -even L +(1) that itself remains invariant under KMS +transformation. This is given by +β0L +(1) = dibjc (Lβeib∆Ua,jc − Ea,ibLβ∆Ujc) + f ijkb (Lβekb∂iCa,j − Ea,kb∂iLβBj) , +(2.42) +which describes non-dissipative dynamics. So far, the effective field theory is general and complete, but we will see +below that simplifications can be made by ignoring certain higher order corrections. +From (2.31), we see that the dipole Ward identity is not a conservation law but a force balance equation. We +now show that the associated field ϕb +a can be eliminated and still preserve locality of the effective action. Indeed, by +integrating out Xi, we obtain +0 = ∂0 +� +Ca,i + ˆρ +n0 +Ea,0beb +i + · · · +� += ∂0 +� +∂iϕa + Aa,i + LXaAi + ϕa,beb +i + Eb +a,iϕb + ˆρ +n0 +Ea,0beb +i + · · · +� +, +(2.43) +where the dots include higher derivative orders. As all the fields are set to be zero at spacetime infinity, the expression +in the bracket is also zero, and since ϕb +a appears without derivatives we can eliminate it from the effective action +without generating non-local terms. Importantly, the combined field ∆Ua,ij does not contain ϕb +a, so we simply need +to replace Ca,i: +Ca,i = − ˆρ +n0 +Ea,0beb +i + · · · . +(2.44) + +10 +After such replacement, the possible additional effective Lagrangian can be added is +β0L(1) +addition ∼ −ALβB0Ca,0 − BLβe0,bEa,0b. +(2.45) +Now, suppose that we are able to shift the r-fields through6 +µ → µ + δµ, +ϕb → ϕb + δϕb, +(2.46) +then the correction to L(1) from Leq is given by +δrLeq ∼ δn0Ca,0 + n0δϕbEa,0b, +(2.47) +where δn0 = δµ∂µn0 + δϕb∂ϕbn0, and we have neglected the contribution to the bulk viscosity. We find that if we +choose the field redefinition as +δn0 = β−1 +0 ALβB0, +δϕb = (n0β0)−1BLβe0,b, +(2.48) +then the additional Lagrangian (2.45) can be eliminated. This indicates that terms proportional to Ca,i, ∂iCa,j can +be safely ignored as a change of frame, and the effective Lagrangian becomes +−iβ0L(2) = sibjcEa,ibEa,jc + tibjc∆Ua,ib∆Ua,jc, +(2.49a) +β0L(1) = −sibjcLβeibEa,jc − tibjcLβ∆Uib∆Ua,jc, +(2.49b) +β0L +(1) = dibjc (Lβeib∆Ua,jc − Ea,ibLβ∆Ujc) . +(2.49c) +In parallel, if we do not integrate out Xi, we find that the coefficient σ associated with Ca,i in (2.40) gives the +relaxation of a non-hydrodynamic mode. By varying (2.41) with respect to ϕb +a, we obtain the leading-order dipole +Ward identity +b∂2 +0ϕb − aibjc∂i∂jϕc = nub − σ +� +ei +b∂iµ + ∂0ϕb +� +. +(2.50) +Clearly, ∂0ϕb acquires a relaxation time τ: +τ ≡ b +σ . +(2.51) +Therefore, on a finite time scale τ, ∂0ϕb relaxes to ub (schematically) – thus they are not independent degrees of +freedom in our hydrodynamic limit (t → ∞): ϕb is the hydrodynamic mode corresponding to momentum density. We +thus understand that σ is not the usual transport coefficient but determines the relaxation rate for the dipole Ward +identity to become a force balance equation. This allows us to ignore it in the hydrodynamic limit. +Hence, we see that Xi is not necessarily a physical degree of freedom (at finite density). What is non-trivial is that +the momentum density, which ordinarily would be ∂0Xi, is approximately proportional to the dipole Goldstone. It +seems non-trivial to uncover the ultimate structure we have found without introducing these extra degrees of freedom, +but it may be possible to achieve this in future work. In particular, we find it most instructive to couple this theory +to geometry (see Appendix B) in the presence of such additional degrees of freedom. +The derivative expansion of the stress tensor and currents are obtained from variation of L(1) + L +(1), which leads +to (neglecting nonlinear terms, which will not be relevant for the remainder of this section) +T ib +(1) = −sibjc∂juc − dibkl (∂k∂lµ + ∂kF0l − Akcδc +l ) , +(2.52a) +Jib +(1) = tibkl (∂k∂lµ + ∂kF0l − Akcδc +l ) − djcib∂juc, +(2.52b) +where ub is fixed by (2.31). In a rotationally invariant theory, we have +sijkl = ζδijδkl + ηδij, +(2.53a) +tijkl = t1δijδkl + t2δij +(2.53b) +dijkl = d1δijδkl + d2δij. +(2.53c) +6 Since Xi has been integrated out, uµ is not a low-energy degree of freedom to which the field redefinition can be applied. + +11 +From the unitarity of the effective action, ImL(2) ≥ 0, we find that the dissipative coefficients satisfy the following +positivity constraint, +ζ, η, t1, t2 ≥ 0, +(2.54) +while d1,2 are unconstrained and non-dissipative. +Let us now analyze the normal modes around a homogeneous background charge density n0 > 0. We also turned +off the background fields for simplicity. Treating the deviation δn = n − n0 and ϕb as small, we obtain the derivative +expansion of the pressure as +p ≈ p0 + ∂P +∂Bt +|Kbµ,ei +tδBt + 1 +2 +∂P +∂Kb +i +|Bt,ei +t∂iϕb + . . . , +(2.55) +≈ p0 + +� +χ−1n0δn + 1 +2 +∂2p0 +∂n2 +0 +(δn)2 +� +− 1 +2 +� +aibjc + ∂aibjc +∂n0 +δn +� +∂iϕb∂jϕc + . . . , +where χ = ∂n +∂µ is the normal charge susceptibility. Hence, the stress tensor and currents are given by +T ib ≈ +� +p0 + χ−1n0δn + 1 +2 +∂2p0 +∂n2 +0 +(δn)2 − 1 +2akdjc∂kϕd∂jϕc +� +δib +(2.56a) +− ajikcϕb∂j∂kϕc + aicjd∂jϕd∂kϕcδkb + T ib +(1) + τ ib, +Jib ≈ − +� +aibjc + ∂aibjc +∂n0 +δn +� +∂jϕc + Jib +(1) + ξib, +(2.56b) +where τ ib, ξib are the noise whose variance satisfy the fluctuation-dissipation theorem: +⟨τ ib(x)τ jc(0)⟩ = 2T0sibjcδ(d+1)(x), +(2.57a) +⟨ξib(x)ξjc(0)⟩ = 2T0tibjcδ(d+1)(x). +(2.57b) +The expressions for T ib +(1) and Jib +(1) are given by (2.52) with all the coefficients taking their equilibrium values; we can +neglect the non-dissipative terms proportional to d1,2 since they are sub-leading corrections to the ideal hydrodynamics. +So far, we have included the leading order derivatives and only kept non-linear terms in the ideal hydrodynamics +because the non-linearity in dissipative coefficients are irrelevant [35]. Plugging (2.56) in (2.18) and using (2.31) and +(2.32), we obtain equation of motions: +n0∂0ϕb + χ−1n0∂iδnδib + λn0δn∂iδnδib + 2aicjd∂i∂jϕd∂[kϕc]δkb + n−1 +0 sibjcakcld∂i∂j∂k∂lϕd + ∂iτ ib = 0, +(2.58a) +∂0δn − aijkb∂i∂j∂kϕb − ¯λijkb∂i∂j(δn∂kϕb) + χ−1tijkl∂i∂j∂k∂lδn + ∂i∂jξibδj +b = 0, +(2.58b) +where we denoted λ = n−1 +0 ∂2 +n0p0, ¯λijkb = ∂n0aijkb as the non-linear coefficients. In the above equation, we have +neglected the higher order time derivatives and assumed that all the coefficients upon expansion do not depend on +δn and ϕi. The normal modes are defined as non-vanishing solutions to the equation of motion. By neglecting the +non-linear terms, we obtain +ω = ± +� a +χk2 − i +� t +χ + 1 +n2 +0 +(Γ1 + Γ2) +� +k4, +(2.59a) +ω = −iΓ1 +n2 +0 +k4, +(2.59b) +where +a = a1 + 2d − 1 +d +a2, +t = t1 + 2d − 1 +d +t2, +Γ1 = (a2 + a3)η, +Γ2 = ζa1 + 2d − 1 +d +(ζa2 + ηa1) + +� +3 − 8 +d + 4 +d2 +� +ηa2 − ηa3. +(2.60) +Therefore, we find two longitudinal propagating mode with magnon-like dispersion relation ∼ ±k2 and attenuation +∼ −ik4 in (2.59a), and transverse subdiffusive modes ∼ −ik4 with multiplicity d − 1 in (2.59b). + +12 +2.5. +Relevant perturbations in low dimensions +Following [35], we give a zeroth-order scaling analysis on the nonlinear effect. The dissipative scaling ω ∼ k4 causes +the noise to scale as τ ib, ξib ∼ k(d+4)/2 based on (2.57). To match the scaling to the dynamical terms with time +derivatives, we find ϕb ∼ kd/2−1 and δn ∼ kd/2. As per the usual renormalization group analysis, the nonlinear +coefficients would scale as λ, ¯λijkb ∼ k(4−d)/2, making them relevant when d < 4. As a consequence, we expect the +true IR fixed point to have anomalous dissipative scaling: ω ∼ ±k2 − ikz, with z < 4. We crudely estimate z by +assuming that the thermodynamic field does not renormalize due to its Gaussian fluctuations at long wavelength. +Then, requiring λ, ¯λijkb ̸= 0 to not depend on k at fixed point, we obtain z = d/2 + 2. We emphasize that the +critical exponent z has been testified numerically in [35] with excellent agreement in d = 1, 2, suggesting a breakdown +of hydrodynamics. +Meanwhile, naive scaling analysis also implies that the transverse subdiffusive modes would +acquire an anomalous scaling ω ∼ −ikz′ with z′ = z. We hope to report on a 1-loop analysis of the corrections to +hydrodynamics in the near future in order to further investigate these claims. +Interestingly, there was previous work on a stochastic molecular-beam-epitaxy (MBE) process [61] which shares +some similarity with our model. In [61], the authors study +∂th + ∇2 (∇h)2 + ∇4h + noise = 0. +(2.61) +This equation, like our theory, has z = 4 subdiffusion at the linear level. However, the authors of [61] did not demand +invariance of the renormalized theory under dipole shift h → h + c0 + c1x, and as such they argued that while (like +us) the critical dimension d = 4, they instead found a distinct fixed point with z = (8 + d)/3, which we understand +as coming from fixing the scaling dimension of noise, rather than of h. It may be possible to interpret the results +of [61], in light of our work, as highlighting the possibly “accidental” appearance of the same dynamical fixed point +(KPZ) in both a hydrodynamic setting (Navier-Stokes equations in 1d), and as a model of surface growth. As our +symmetries and anticipated scaling exponents differ from [61], the analogy between surface growth and hydrodynamics +with momentum conservation may break down in multipole-conserving theories. +3. +SPONTANEOUS SYMMETRY BREAKING +In this section, we discuss a subtle yet very interesting feature of the dipole and momentum conserving fluid: the +identification of ϕb as a Goldstone boson of the dipole field, and the corresponding intuition that dipole symmetry is +(by many reasonable definitions) spontaneously broken. +Let us first briefly justify the identification of ϕb as a Goldstone boson. Under dipole shift symmetry, ϕb → ϕb +cb, +just as a conventional Goldstone. +Moreover, if we wanted to consider (even in the absence of momentum) the +spontaneous symmetry breaking of the dipole charge [62], this could be achieved by adding ∇ϕb∇ϕa,b terms to the +Lagrangian (this will be discussed further elsewhere). Recent papers on dipole symmetry breaking in the absence of +momentum conservation include [63, 64], while [45] discusses the role of dipole symmetry breaking in a translation +invariant state. Lastly, as we will see, the dynamics of the collective and long-lived ϕb mode saturate Goldstone’s +Theorem. For these three reasons, we will call ϕb the Goldstone boson associated with dipole symmetry. +There are three notions through which prior authors have justified the presence of spontaneous symmetry breaking +that we are aware of. Briefly, and we will elaborate more on each in subsequent sections: (1) The physical state +of interest is not invariant under the action of the global symmetry group; (2) the existence of a Goldstone boson +which saturates Goldstone’s Theorem; (3) the presence of long-range order in expectation values of operators that +shift under the symmetry (Mermin-Wagner Theorem). In a nutshell, the theory of interest here is compatible with all +three definitions in spatial dimensions d > 2, but only with the first two when d ≤ 2. In low dimensions, our theory +thus seems to represent an unusual paradigm not encountered before, where many (but not all) of the usual features +of SSB exist. We believe that it is appropriate to consider dipole symmetry as spontaneously broken, but leave it to +future authors to more firmly settle the possibly semantic question. We conclude this section with a discussion of a +quantized version of Model A from [35], which will give a concrete example of the more abstract ideas discussed. +3.1. +Mermin-Wagner Theorem +First, we focus on a physically transparent test for spontaneous symmetry breaking: correlation functions of the +order parameter πi(t, x). A microscopic argument along these lines was presented in [45], and previously in [62] in +the absence of momentum conservation. While this operator is charged under dipole transformations, it transforms + +13 +nonlinearly: πi → πi + ci, where ci is an arbitrary vector parameter.7 To conclude that dipole symmetry is sponta- +neously broken, one often asks whether πi is a well-defined order parameter: any finite value (including zero) will be +sufficient to conclude spontaneous breaking. From the discussion around (2.33), the equal-time two-point function of +πi diverges at low momenta: +⟨πi(k)πj(−k)⟩ ∼ 1 +k2 . +(3.1) +Let us consider the average momentum density on a region of linear size L. The fluctuations of the average momentum +density on this region scale as +� +(πi − ⟨πi⟩)2� +L ∼ +� +� +� +L +d = 1 +log L +d = 2 +constant +d > 2 +. +(3.2) +This means that the momentum density is well-defined as a thermodynamic variable only for d > 2, in which case +(3.4) implies spontaneous symmetry breaking of dipole transformations. For d ≤ 2, fluctuations are too large to make +sense of the expectation value of πi, and therefore we cannot conclude that there is spontaneous symmetry breaking +from this perspective. +3.2. +Goldstone’s Theorem +Let us now discuss how the classic Goldstone’s Theorem can be generalized to our theory. Consistency with the +algebra (1.1) demands the following commutation relation between dipole and momentum density: +[Di, πj] = inδj +i . +(3.3) +On a thermal state ρ0 at finite background charge density n0, we have +tr(ρ0[Di, πj]) = in0δij ̸= 0 . +(3.4) +For concreteness, we shall take ρ0 to be microcanonical with respect to momentum and charge, and canonical with +respect to energy (if the latter is conserved); see an explicit example in Section 3.3. As we discussed in the previous +section, πi has large fluctuations in low dimensions; nevertheless the expression on the left-hand side of (3.4) can +still be well-defined as we demonstrate in the explicit example of Section 3.3. We now show how this relation (3.4), +entirely dictated by symmetry, will imply the existence of a Goldstone mode [46]. Let us start by writing the total +dipole charge in terms of the charge density operator Di = +� +ddx xin,8 which allows us to express the above as +� +ddx xiGR +nπj(t, x) = −n0θ(t)δij, +GR +nπi(t, x) = −iθ(t) tr(ρ0[n(t, x), πi(0, 0)]) , +(3.5) +where GR +nπi is the retarded two-point function of charge and momentum densities. Doing a Fourier transform leads to +lim +⃗p→0 +∂ +∂pi ImGR +nπi(ω, p) = −n0πδ(ω)δij . +(3.6) +We see that (3.6) implies a zero-frequency contribution to the spectral density Jnπi as a direct consequence of (3.4). +Using the approach of Kadanoff-Martin and (2.34), we now show that this spectral weight is entirely captured by +a single hydrodynamic mode: the “magnon-like” sound mode. We can obtain the retarded two-point function GR +nπi +(without missing any important counterterm). Doing a Laplace transform +� −izδij +n0 +χ iki +ikjk2 a +n0 +−iz +� � +πj(z, p) +δn(z, p) +� += +� +π(0) +i +(p) +δn(0)(p) +� +, +(3.7) +7 In this subsection we will be agnostic of the global structure of the dipole group and of possible subtleties related to boundary conditions. +We will describe these in more detail for a specific model in the next subsection. +8 The total dipole moment Di could in principle receive further contributions from “bond dipoles”, i.e. degrees of freedom with an internal +dipole charge. This is entirely analogous to the contribution of orbital angular momentum and spin to the total angular momentum. +We shall not investigate this situation here, but note that these bound dipole degrees of freedom are not conserved and have a finite +lifetime. Therefore it is unlikely they would contribute to the singular spectral weight we calculate below. + +14 +where the right-hand side represents densities configurations at the initial time. The initial density configuration δn(0) +is related to a perturbation in the chemical potential at initial time through δn(0) = χδµ, so we find +∂πi(z, p) +∂δµ(p) = +−n0iki +−z2 + a +χk4 . +(3.8) +From linear response we know that, for a density δna conjugated to a chemical potential δµa, izδna(z, k) = (GR +ab(z, k)− +GR +ab(0, k))δµb(k) and GR +ab(0, k → 0) = −χab. We then find +GR +nπi(ω, p) = +−n0ωki +−ω2 + a +χk4 . +(3.9) +At leading order in ki, we have +Im GR +nπ(ω, p) → −πn0kiδ(ω) +(3.10) +which can be obtained by taking into account that dissipative corrections in the retarded two-point fuction contribute +through ω → ω + iε, where ε → 0 as k → 0. We then see that we recovered eq. (3.6). This contribution comes from +the pressure term, and is thus present in any fluid which conserves momentum and density. +Hence we call ϕb a Goldstone mode: this IR mode by itself saturates the requisite Goldstone’s Theorem for dipole +symmetry. The fact that in the EFT, ϕb did not have a diagonal shift symmetry like ϕ and Xi, further suggests that +we should interpret ϕb as a Goldstone mode. In previous hydrodynamic effective field theories such as [2, 5], when +this diagonal shift symmetry is not present, the conclusion has been that the corresponding continuous symmetry is +spontaneously broken. +At charge neutrality n0 = 0, the Goldstone’s theorem become trivial, and we will not have such dipole Goldstones. +3.3. +Existence of a symmetry-breaking state +Finally, we now show that a system with dipole and momentum symmetry can possess a symmetry-breaking ground +state even in one and two space dimensions. +The dipole and momentum algebra is isomorphic to the Heisenberg algebra of position and momentum (in a +microcanonical ensemble where Q is fixed). In textbook quantum mechanics (and in models interest here, as discussed +below), the physical Hilbert space contains no trivial representation of this algebra. Therefore it is not possible to +construct a state which is invariant under the symmetry group, and one can say that the symmetry must be broken, +either explicitly or spontaneously. +A physical example of another system that has a (sub)algebra isomorphic to dipole and momentum is a Galilean- +invariant fluid. +This is slightly different because the commutator of [H, D] ̸= 0 in the Galilean-invariant fluid. +Nevertheless, recent papers [46, 47] have argued that Galilean boosts are spontaneously broken – in any dimension – +by the fluid rest frame. We believe this is most succinctly justified by the argument in the previous paragraph. We +discuss an interesting relation between our dipole fluids and the Galilean-invariant fluids in Section 5. +One challenge in showing whether the ground state is invariant or not, is that historically this was done by a careful +consideration of finite volume regularization [65]. When compactifying space onto a toroidal lattice with finite length +in each direction, the dipole conservation law is isomorphic to Z, not R.9 This regularization is not acceptable for +us here since it qualitatively changes the nature of the symmetry group; we seek an alternative regularization that +manifestly preserves the Heisenberg algebra. +Model A of [35] provides us with a concrete model which we can suitably regularize. The Hamiltonian for Model A +is +H = +N−1 +� +i=1 +�(pi − pi+1)2 +2 ++ V (xi+1 − xi) +� +(3.11) +with [xi, pj] = iδij conventional position and momentum operators. We can view this as a chain where each site i +possesses an infinite-dimensional Hilbert space. Alternatively, we can interpret this as a generalization of an atomistic +Hamiltonian modelling a solid in one dimension, where xi and pi describe displacements and momenta. Similar models +9 We thank Nathan Seiberg for emphasizing the points in this paragraph. + +15 +(without dipole conservation) are known to capture hydrodynamics (and its breakdown) in one dimension [51]. The +details of V are not important for our discussion, though we would likely consider a function Taylor expanded about +a minimum at argument x = 1. One can easily see that H commutes with the dipole and momentum operators of +the theory: +P = +N +� +i=1 +pi, +D = +N +� +i=1 +xi. +(3.12) +Note that this theory is in a microcanonical ensemble for charge: Q = N is a fixed c-number. +To see that the +thermodynamics is well defined, we take ρ0 = e−βH restricted to superselection sectors satisfying � +i pi = P and +� xi = D fixed; we also take lattice constant a = 1. Upon changing variables to si = pi − pi+1, ri = xi − xi+1, the +Hamiltonian becomes H = �N−1 +i=1 +1 +2s2 +i + V (ri). The classical partition function is then (up to an overall constant) +� �N−1 +i=1 dsidrie−βH. The partition function factorizes into manifestly convergent integrals, and is therefore finite. In +quantum mechanics, the partition function will also be finite; this is most easily seen by working in a plane wave +basis. We additionally notice that due to the commutator [D, pi] = i, tr(ρ0[D, pi]) is finite, thus showing that the +atomistic analogue of (3.4) is well-defined in the present model, despite the large fluctuations of momentum density +discussed in Section 3.1. +Let us now look in more detail at the quantum ground state of Model A. This will be a wave function of the form +ψgs(x1, . . . , xN) = ψ0(x1 − x2, . . . , xN−1 − xN) · eik(x1+···+xN) e−(x1+···+xN)2/2Na +(2πa)1/4 +. +(3.13) +This wave function has an eigenvalue independent of k and a. If we take a = ∞ above, the wave function is a +non-normalizable eigenstate of P, but not D. Clearly no pure state can be found that is an eigenstate of both P and +D. By taking a < ∞, the above state is normalizable and physical, with controllably small fluctuations in the value +of P. We don’t think the situation improves much for mixed states: the only possibility invariant under both P and +D is the “identity matrix” in the center of mass coordinate, but it is dubious whether (even representing just one +coordinate in the N → ∞ limit) such a highly non-normalizable state could be considered. This paragraph simply +re-states, in a concrete model, what we already noted – it is not possible to simultaneously diagonalize D and P. +Is this a physical effect? Since the center of mass coordinates completely decouple from H, they can arguably be +removed from Hilbert space entirely without loss of generality. What remains in the Hilbert space are the long-lived +fluctuations of πi(k), which are subject to the same Mermin-Wagner fluctuations mentioned above. There is therefore +no notion of long range order measurable by correlations of local operators that are not group-invariant: in d = 1, +⟨(p1 − pN/2)2⟩ ∼ L. +To throw one more wrench into the mix, however, there is a critical difference between SSB of compact U(1) and +non-compact dipole symmetry (alone, isomorphic to R). In the U(1) case, the Goldstone ϕ is not singly-valued: +one must look at the well-posed operators Am = eimϕ for m ̸= 0. These operators have ⟨Am⟩ = 0 implying no +SSB for d ≤ 2 in any physical ensemble. However for dipole symmetry, the global momentum P and its average +πi(k = 0) = +1 +N P are perfectly well-defined operators. We have constructed normalizable (ground!) states in which +the physical operator ⟨π⟩ takes on a finite value, which shifts under dipole transformations. In this sense, the large +fluctuations assured by the Mermin-Wagner Theorem seem less dangerous as in a conventional U(1) superfluid, and +in fact the Mermin-Wagner theorem is argued to only apply to compact symmetry groups [66]. However, like in the +superfluid, there can be no long-range coherence in π, which exhibits large local fluctuations. We leave open the +question of whether this is a semantic or a crucial physical difference. +4. +HYDRODYNAMICS WITH ENERGY CONSERVATION +We now discuss how to incorporate energy conservation into our hydrodynamic theory. +We will need another +hydrodynamic degree of freedom X0(σt, σI) that non-linearly realizes the time translation P 0, where now σt is the +proper time in the fluid’s local rest frame. This leads to an additional invariant building-block in the fluid spacetime +defined as +e0 +s,A(σ) = ∂Xµ +s (σ) +∂σA +e0 +s,µ(σ) +(4.1) +In order to distinguish from solid phases, the effective theory must be invariant under time-independent reparametriza- +tions of the proper time σt, +σt → σt + f(σI), +(4.2) + +16 +which implies that the invariant r-field without derivatives can only be e0 +t. We then introduce the proper temperature +β ≡ e0 +t, +(4.3) +as the thermodynamic temperature10. In the classical limit and physical spacetime, the corresponding a-field can be +written as +e0 +a,A = ∂AXµE0 +a,µ, +E0 +a,µ = e0 +a,µ + LXae0 +µ. +(4.4) +Then, we can include a term T µ +0 E0 +a,µ into the leading effective Lagrangian (2.17), and by variation with respect to +X0 +a, we get a new Ward identity, i.e. the energy conservation equation, +∂µT µ +0 + Ab +0eµbJµ − F b +0µJµ +b − F0µJµ = 0. +(4.5) +The dynamical KMS transformation for the energy variables is given by +˜e0 +µ(−x) = e0 +µ(x), +(4.6a) +˜E0 +a,µ(−x) = E0 +a,µ(x) + iLβe0 +µ(x), +(4.6b) +with Lβe0 +µ = ∂µβ. +We now generalize the fluid Lagrangian to include energy conservation, focusing on quadratic order in perturbations +(i.e., linear response), and switching off background fields. Keeping into account dynamical KMS symmetry, the new +terms in the total Lagrangian that depend on E0 +a,µ are Lε = Leq,ε + L(1) +ε ++ L(2) +ε , with +Leq,ε = −(ε + p)uµE0 +a,µ + peµ +0E0 +a,µ, +(4.7a) +−iβL(2) +ε += κijE0 +a,iE0 +a,j + 2αijCa,iE0 +a,j, +(4.7b) +βL(1) +ε += −κijLβe0 +i E0 +a,j − αij(Ca,iLβe0 +j + LβBiE0 +a,j), +(4.7c) +where κij = κδij is the isotropic thermal conductivity, αij = αδij, α, κ ≥ 0, and where we defined the energy density +through ε + p = −β ∂P +∂β . In the above, we are taking ε = ε0 + δε and p = p0 + χ−1 +ε (ε0 + p0)δε + χ−1n0δn, where +χε = −β ∂ε +∂β is the specific heat, and ε0, p0 are background values of energy density and pressure. The other possible +O(a2) contributions in L(2) +ε +come with a time derivative, so we can eliminate them through a frame redefinition as +explained below (2.45). At linear order, the ideal charge and dipole currents in (2.26) are not modified, and thus we +can still use eq. (2.31) to eliminate ui in favor of the dipole Goldstone ϕb. On the other hand, eq. (2.43) is updated +to +0 = ∂0 +� +Ca,i + ˆρ +n0 +Ei +a,0 − ε0 + p0 +n0 +E0 +a,i +� +(4.8) +where the last term above comes from the first term in Leq,ε. Eliminating Ca,i will produce terms proportional either +to Ei +a,0, or to E0 +a,i. The former contributions can be eliminated through a frame redefinition, following similar steps to +the discussion below (2.43). The latter will renormalize the value of the thermal conductivity κ (preserving dynamical +KMS invariance) and can also be eliminated.11 We can thus set α = 0. By varying with respect to the e0 +a,µ, we find +the equilibrium energy-4-current as +T µ +(0)0 = −εuµ − p(uµ − eµ +0) . +(4.9) +Unlike the momentum density, the energy density is still governed by the fluid elements; on the algebraic level, this +is because the time translation and the dipole symmetry commute. The dissipative part is the usual temperature +gradient: +T i +(1)0 = −κijβ−1∂jβ , +(4.10) +10 All the β0 appeared in the previous sections must be taken to be β(σ) as a function of spacetime. We automatically take that into +account in the following derivation without lengthy repetitions. +11 To see this, note that LβBi, appearing in the last term in (4.7c), can be replaced using n0LβBi = (ε0 + p0)Lβe0 +i , which can be inferred +from the ideal part of the momentum conservation equation. + +17 +and the noise contribution to the energy current is τ i, with ⟨τ i(x)τ j(0)⟩ = 2T0κijδ(d+1)(x). +We now analyze the normal modes. Keeping into account corrections to (2.56), T ib +(0),ε ≈ χ−1 +ε (ε0 + p0)δε δib, and +substituting the expressions found above into the Ward identities, we obtain equation of motions for the hydrodynamic +modes and the dipole Goldstone. The complete expression of the normal modes is complicated due to the additional +coupling to the energy sector. Thus, we do not present the full solutions here for the sake of conciseness but emphasize +on few consequences after including the energy sector. First, the d − 1 subdiffusive modes in (2.59b) continue to exist +but with a different subdiffusion constant. Second, the energy diffusion will generically mix with the magnon-like +sound mode since they both scale as ∼ k2. To see it, we keep the dissipative hydrodynamics in the energy sector but +ideal hydrodynamics in the momentum and charge sectors. The resulting equation of motion is given by, in the linear +response, +∂0δε + n−1 +0 (ε0 + p0)aijkc∂i∂j∂kϕc − χ−1 +ε κij∂i∂jδε = 0 +(4.11a) +n0∂0ϕb + χ−1n0∂iδnδib + χ−1 +ε (ε0 + p0)∂iδεδib = 0 +(4.11b) +∂0δn − aijkb∂i∂j∂kϕb = 0 . +(4.11c) +After doing a Fourier transform, the normal modes are the solutions of the following cubic equation +ω3 + i κ +χε +ω2k2 + +� +a +χε +�ε0 + p0 +n0 +�2 +− a +χ +� +ωk4 − i aκ +χχε +k6 = 0 +(4.12) +The resulting modes are 3-fold: two propagating modes ω = ±ck2 − iγk2 and one diffusion mode ω = −iγ′k2, where +explicit expressions for c, γ, γ′ are not illuminating. The propagating modes still have the magnon-like dispersion but +the leading dissipative contribution is now ∼ −ik2. The diffusion mode is reminiscent of the energy diffusion in a +normal fluid. +We conclude by noting that, because of the diffusive nature of the three longitudinal modes above, it is not clear +a priori whether the hydrodynamic instability and the associated non-Gaussian universality class emergent at long +wavelengths found in [35] will survive at long times, or whether it will be replaced by a different universality class. +This is an interesting question that we leave for future work. +5. +FROM GALILEAN SYMMETRY TO DIPOLE SYMMETRY +Lastly, let us point out an interesting connection between the physics described above, and the m → ∞ (infinite +mass) limit of the Galilean symmetry algebra. This was described in a different formalism in the recent work [45]. +Consider a system preserving both time H and space Pi translations, U(1) charge Q, and Galilean boost Ki. Similar +to the dipole symmetry, Galilean boosts are also broken spontaneously in hydrodynamics [46, 47] due to the following +algebra: +[Ki, Pj] = imQδij, +(5.1a) +[Ki, H] = iPi, +(5.1b) +where m is the mass of underlying particles. Due to the second commutator above, we note that the resulting algebra +is distinct from the dipole-momentum algebra we discuss in this paper. Still, observe that in conventional liquids +and gases there are no obvious “Goldstone bosons”: the hydrodynamics are described by simply charge, energy and +momentum conservation.12 +Following general constructions in Appendix B, we obtain the invariant blocks in the flat spacetime as +eµ +A = ∂AXµ − ∂AX0δµ +i ηi, +(5.2a) +V i +A = ∂Aηi, +(5.2b) +BA = ∂Aϕ + m∂AXiηb − 1 +2m∂AX0ηiηi, +(5.2c) +12 In the literature it is remarked that the Goldstone of broken boosts is the conventional sound wave [47], but we remark that even +without any boost or rotational symmetries, such sound waves still exist [58]. So the sound wave is not crucially reliant on the broken +continuous boost symmetry, while the “sound mode” of the dipole-momentum theory cannot be disentangled from symmetry breaking, +as far as we can tell. + +18 +where we associated the Goldstone ηi to the Galilean boost symmetry Ki. Following the construction described in +Section 2, the r-fields invariant under relabeling symmetries are eµ +t , Bt and V i +µ (upon index contraction). We can also +derive the a-fields in the classical limit. For our purpose, we will ignore nonlinear terms associated with Xµ +a but keep +relevant terms for ηi +a. The resulting a-fields are +eµ +a,A = ∂AXνEµ +a,ν, +Eµ +a,ν ≈ ∂νXµ +a − δ0 +νδµ +i ηi +a, +(5.3a) +V i +a,A = ∂AXµV i +a,µ, +V i +a,µ = ∂µηi +a, +(5.3b) +Ba,A = ∂AXµCa,µ, +Ca,µ ≈ ∂µϕa + mδi +µηa,i − mδ0 +µηiηa,i, +(5.3c) +To the leading order in a-fields, the effective Lagrangian is given by +L = ˆT µ +ν Eν +a,µ + JµCa,µ + W µ +i V i +a,µ + . . . . +(5.4) +Varying with respect to ηa,i, we obtain the boost Ward identity +− ˆT 0 +i + mJi − mJ0ηi − ∂µW µ +i = 0. +(5.5) +Writing the stress tensor and currents in terms of thermodynamic variables, ˆT 0 +i ∼ ui + ηi, Ji ∼ n0(ui + ηi), J0 ∼ n0 +and ∂µW µ +i ∼ O(∂2ηi), we find that +ηi ∼ ui + O(∂2). +(5.6) +This immediately tells that the boost Goldstones are redundant degree of freedoms. More explicitly, if including the +first-order dissipative effect +βL(1) ∼ −ΓLβei +0Ei +a,0 ⊃ −Γβ∂0ηiηa,i, +(5.7) +we see that the boost equation of motion (5.5) will be damped at the timescale ∼ Γ. Therefore, the broken boost +does not provide additional hydrodynamic modes, and the long-time dynamics is equivalent to an ordinary (boost- +invariant) charge fluid. As a bonus, by setting ηi = 0 (as it will decay to this value), (5.5) implies that the momentum +density is equal to the mass current: +T 0 +i |ηi=0 ≡ +� +ˆT 0 +i + mJ0ηi +� +|ηi=0 = mJi|ηi=0. +(5.8) +Note however that, as in the dipole fluid, the Galilean boost with parameter ci causes a shift to the momentum +density: +T 0 +i → T 0 +i + mn0ci. +(5.9) +This is well known from textbook fluid mechanics – it is simply the statement that hydrodynamics is the same in all +inertial reference frames, and that momentum transforms from one such frame to another. The fact that [Ki, Pi] ∼ Q is +mathematically analogous to [Di, Pi] ∝ Q explains why the dipole shift symmetry causes such a similar transformation. +Unlike the dipole symmetry however, [Ki, H] ̸= 0 and this causes the ηi “dipole Goldstone” to also show up in Eµ +a,0 +as well as its quadratic form in Ca,0, which causes ηi to relax. +Remarkably, there exists an well-defined limit for the Galilean boost symmetry – taking the infinite mass limit +m → ∞ and keeping Di ≡ Ki/m (but not Ki itself) a good symmetry generator. To keep everything in the same +order, the boost Goldstone needs to be scaled as ϕi ≡ mηi since ηi ∼ m−1 → 0. In this limit, the boost algebra +reduces exactly to the dipole algebra by identifying Di and ϕi as the dipole generator and Goldstone correspondingly. +Moreover, by defining ∂µJµ +i ≡ ∂µW µ +i /m as the divergence of dipole current and upon charge conjugation, the boost +Ward identity (5.5) becomes the dipole Ward identity (2.18c). Consequently, the relation (5.6) is incomplete at the +leading order since ηi → 0, and going to the next leading order, we find it reproduces the dipole constraint (2.31). So +long as Γ remains finite, the dissipation term in (5.7) vanishes since ηa → m−1ϕa → 0. The (dipole) Goldstone can +no longer be integrated out; in fact, as seen before, it is the velocity which becomes redundant13. This reveals how +the algebraic observation that the Galilean algebra becomes the multipole algebra at m = ∞ is realized in the EFT. +We are not sure whether or not existing methods [71–75] used to study “non-relativistic conformal field theories” +(with Galilean symmetry) can be neatly used in the infinite mass limit; this could be a fruitful direction for future +research. +The physics discussed above might be useful to study physics in strongly correlated flat bands, in the regime where +the infinite mass limit is exact [76]. +13 The number of degrees of freedom is nevertheless the same in both cases, and it is smaller than that to start with – we are losing +d redundant modes in spatial dimension d. The reduction of the true degrees of freedom is a common feature (not yet proven) for +spacetime symmetry breaking based upon the inverse Higgs mechanism [67–70], therefore, it is interesting to understand to what extend +can we generalize this statement. + +19 +6. +CONCLUSIONS +In this paper, we have developed an effective field theory for fluids with dipole and momentum conservation. Our +construction highlights a few subtle aspects of this problem: in particular, it appears necessary to write down a local +action in terms of more degrees of freedom than are actually present in the effective theory, despite the absence of +an obvious need for Lagrange multipliers. A crucial observation that arises out of this construction is that the dipole +symmetry is generally spontaneously broken, and that the Goldstone boson for this broken symmetry is essentially +the momentum density. +The fact that dipole symmetry is spontaneously broken is also found in [77]. Unlike in that reference however, we +find this conclusion is deeply related to having momentum conservation; without momentum, there is no need to have +spontaneously broken the dipole symmetry. That this effect appears common in the large N models of [77] may be an +artifact of the solution method in the large N limit. At the very least, there is no evidence for spontaneous symmetry +breaking of dipole symmetry in any classical (Markov chain) models of dipole-conserving hydrodynamics studied thus +far; in contrast, the numerical simulations of [35] did find compelling evidence for the universality class whose field +theory was derived in the present paper. It would be interesting to understand this issue further in future work. +An important lesson from this effective field theory construction was the generalization from flat to curved back- +ground. Here, our approach perhaps differs conceptually from other attempts in the recent literature [59]. In our +approach, we followed recent work [58] which emphasized the importance of using vielbein indices (not spatial indices) +to encode conservation laws in curved space: this construction made it possible to couple anisotropic fluids to geom- +etry. Since in principle it may be desirable to study anisotropic dipole- and momentum-conserving fluids, we expect +that such a vielbein construction is also preferable here. More importantly, by starting with a first-order formulation, +it was natural to assert that in curved space one should relax the requirement that the gauge field is a mixed rank +object (At, Aij), and to instead simply require a pair of gauge fields Aµ and Ab +µ corresponding to charge and dipole. +The mixed-rank gauge field can then only emerge in a flat space limit. Moreover, the vielbein formalism helps us to +understand two key physical consequences: the existence of the dipole Goldstones and the asymmetric part of the +dipole fields. +Looking forward, we anticipate that our methods can be generalized to discover infinitely many new hydrodynamic +universality classes that arise in fracton-like classical or quantum matter. It may be straightforward conceptually, if +tedious in practice, to generalize this construction to include higher multipole conservation laws. A more important +and interesting direction will be to understand how to generalize the geometrically inspired construction presented +here to non-thermal matter – after all, the highlight of this work is the dynamics without energy conservation! Recent +work [34] along these lines has begun, but the consequences or diagnosis of spontaneous symmetry breaking in this +new approach have not yet been understood. +It will also be fascinating to look for experimental realizations of the dipole and momentum conserving hydrody- +namics developed here, whether in high quality solid-state devices in very large electric fields [78], or in low density +interacting ultracold atoms in tilted trapped optical lattices [79]. We also hope that progress along these lines will +be made in the next few years. It may also be possible to explore similar (though it appears distinct) fixed points +which arise from the symmetry group of volume-preserving diffeomorphisms (which can arise in lowest Landau level +physics) [80, 81]; this algebra is equivalent to the dipole-momentum algebra at the linearized level. +ACKNOWLEDGEMENTS +We acknowledge useful discussions with Anton Kapustin, Rahul Nandkishore, Shu-Heng Shao, Dam Thanh Son, Lev +Spodyneiko, and especially Kristan Jensen and Nathan Seiberg. PG and AL thank the Simons Center for Geometry +and Physics for hospitality. +This work was supported by the Department of Energy through Award DE-SC0019380 (PG), the Simons Foundation +through Award No. 620869 (PG), the National Science Foundation under CAREER Award DMR-2145544 (XH, JG, +AL), the Gordon and Betty Moore Foundation’s EPiQS Initiative via Grants GBMF4302 and GBMF8686 (JFRN), +and GBMF10279 (XH, JG, AL), and by Research Fellowships from the Alfred P. Sloan Foundation under Grant +FG-2020-13615 (PG) and FG-2020-13795 (AL). +Appendix A: Memory matrix methods +In this appendix, we use the memory matrix formalism [82] to derive the linearized hydrodynamics of the main text, +both near and away from charge neutrality. This approach provides an independent check on many of the non-trivial +properties of hydrodynamics that we found above and can give some interesting perspectives on the results. + +20 +The memory matrix formalism is an old set of formal manipulations, used to isolate the contributions to linear +response theory (two-point functions) which arise from parametrically slow dynamics. Since long wavelength hydro- +dynamic modes are arbitrarily long lived, this method can be well-suited for calculations of their properties. We +now tersely summarize the main results of this method: for details see [82]. Consider a many-body system with +Hamiltonian H, at temperature T. One can construct a vector space consisting of all operators A acting on this +system: to emphasize the vector nature, we can write |A). An inner product on this space is +(A|B) := T +β +� +0 +dλ⟨A†(iλ)B⟩T +(A.1) +with T = 1/β and ⟨· · · ⟩T = +1 +tr(e−βH)tr(e−βH · · · ) the thermal expectation value. Note that the susceptibility matrix +is +(A|B) = TχAB. +(A.2) +Suppose that we have a designated set of “slow” operators |OA). For us, these are naturally taken to be n(k) and +πi(k) (the Fourier wave number is k, and is held fixed). We may define the projectors +p = +� +slowA,B +|A)(Tχ)−1 +AB(B|, +q = 1 − p, +(A.3) +which project degrees of freedom onto slow (p) and fast (q) modes. By noticing that (A|(z−L)−1|B) is linearly related +to the retarded Green’s function GR +AB(z), one can show that there are hydrodynamic quasinormal modes whenever [] +det(M + N − iωχ) = 0. +(A.4) +Here M (the memory matrix) and N are given by +MAB = ( ˙A|qi(z − qLq)−1q| ˙B), +(A.5a) +NAB = −NBA = χ ˙AB. +(A.5b) +Here L = i[H, ·] denotes the Liouvillian, and ˙A = i[H, A], with H the overall Hamiltonian. +In this paper, we aim to use this framework to gain further insight (and justification) for the non-trivial hydro- +dynamics discovered in Section 2. Strictly speaking, one can object to this on the grounds that energy conservation +is explicit in any theory satisfying the above postulates. Ultimately, we will use this approach to discern what hap- +pens when energy is conserved along with dipole and momentum; however, we believe that this approach remains +instructive even if we “ignore” energy conservation as an unjustified assumption. As we will see, some of the confusing +features of this fluid are consequences of very general, and even semi-microscopic, arguments. +1. +Momentum susceptibility +Let us begin by determining the momentum susceptibility; in the memory matrix language, this is (π|π) = Tχππ +(we’ll leave the Fourier index implicit for the remainder of this section). While a microscopic computation is not +possible (nor important for hydrodynamic considerations), we can easily bound susceptibility using the Cauchy- +Schwarz inequality: +(πx|πx) ≥ (πx|Jx)2 +(Jx|Jx) . +(A.6) +Here Jx is the x-component of the charge current operator; for simplicity, we’ll also take k = kˆx. Now, observe two +key properties. Firstly, in a generic many-body system, +(πx|Jx) = Tn0, +(A.7) +with n0 the equilibrium charge density: n0 = ⟨n⟩T . A heuristic and fully general argument for this fact, which we +have not seen in the literature, is as follows: +(πx|Jx)k→0 +T += +� ∞ +0 +dt +� ddx +V +i⟨[Jx(x, t), Px]⟩ = +� ∞ +0 +dt +� ddx +V +⟨∂xJx⟩ = − +� ∞ +0 +dt +� ddx +V +⟨∂tn⟩ = Tn0 +(A.8) + +21 +where in the last step we have used integration by parts. This argument is not rigorous because if the k → 0 limit is +taken too quickly the integral trivially vanishes. Secondly, using (2.18c), +(Jx|Jx) = k2(Jxx|Jxx). +(A.9) +Since Jxx is the local current operator which is well-defined with local dipole conservation, we conclude that (Jxx|Jxx) +is k-independent as k → 0, and should remain finite as n0 → 0. Combining these 3 equations, we find that for some +constant c > 0, which does not vanish as n0 → 0, +χππ = cn2 +0 +k2 , +(A.10) +in agreement with the result (2.33) of the main text. +2. +Dynamics without energy +Now, let us consider the dynamics without energy conservation. In this case, we’ll include πi and n as degrees of +freedom in the memory matrix formalism. The non-zero matrix elements of the N matrix are +Nπin = Nnπi = (πi| ˙n) = ikj(πi|Jj) = ikjδijTn0 = T × ikin0. +(A.11) +The most important non-zero matrix elements of the M matrix are +Mnn = ( ˙n|qi(ω − qLq)−1q| ˙n) = kikjkkkl(Jij|i(ω − qLq)−1|Jkl) = Tαk4, +(A.12a) +Mπiπj = ( ˙πi|qi(ω − qLq)−1| ˙πj) = kkkl(Tik|q|Tjl) = T +� +βkikj + γk2δij +� +. +(A.12b) +for some constants α, β, γ. The hydrodynamic normal modes thus come from solving +0 = det(M + N − iωχ), +(A.13) +which in the transverse sector gives +ωχππ = γk2, +(A.14a) +ω = −iγk4 +cn2 +0 +, +(A.14b) +and in the longitudinal sector gives +det +� αk4 − iωχnn +ikn0 +ikn0 +(β + γ)k2 − iωcn2 +0 +k2 +− iωA +� +(A.15) +Here A > 0 is a finite constant that persists even as n0 → 0, arising from χππ Indeed, if n0 = 0, then we see that both +longitudinal and transverse momentum get k2 decay rates, and charge has k4 subdiffusion, while if n0 > 0, we have +0 = k4n2 +0 + (αk4 − iωχnn)((β + γ)k4 − iωcn2 +0) +(A.16) +which is solved by +ω = ± +k2 +√cχnn +− iΓk4 + · · · +(A.17) +in agreement with the prediction (2.59) of the main text. +Appendix B: Dipole fluids with momentum in a curved spacetime +In this section, we will discuss the change of dipole hydrodynamics by coupling to a curved spacetime. Such analysis +also tells us how to source various currents in the flat spacetime limit as discussed in Section 2. It is interesting to +notice that because of the non-commutativity between the dipole moment and the momentum operator, the momentum + +22 +density is not invariant under dipole shifts. However, since the energy operator commutes with the dipole moment, +the energy density is invariant. Consequently, our theory forbids any type of boost symmetries between space and +time, and a natural choice of spacetime to describe such theory is the Aristotelian background [40, 59, 83, 84]. It +is most natural to use vielbein formalism to describe the geometry, thus we introduce the internal (flat) spacetime +indices α, β and b, c for its spatial subspace (we reserve a to describe a-fields in the Keldysh contour!). The dipole +and spacetime algebra is given by +[Pb, Dc] = −iQδbc, +(B.1a) +[Pd, Lbc] = i(δdcPb − δdbPc), +(B.1b) +[Dd, Lbc] = i(δdcDb − δdbDc), +(B.1c) +[Lbc, Lde] = i(δbdLce − δbeLcd − δcdLbe + δceLbd), +(B.1d) +where Lbc generates the SO(d) rotational symmetry. The time translation P0 commutes with every other generator. +Before constructing the field theory, it is instructive to define the dipole moment operationally as +Db ≡ +� +ddxe yb(xi)n(x), +(B.2) +for some arbitrary function yb(xi) in terms of spatial coordinates. In particular, let us assume for now that the +coordinates {yb} form the “natural” coordinates where the vielbein would be constant: eµ +b = δµ +b . Nevertheless, we will +not for now invoke any such constraint. We also emphasize that on a general curved background, yb is not globally +well-defined, with the 2-sphere the simplest example. This means that one should only understand the resulting theory +as a ‘covariant’ way of coupling a dipole and momentum conserving theory to curved space – the explicit coupling to +the metric ends up breaking the conservation laws. This phenomenon is not unusual, and is well-known to happen +already with momentum conservation on a generic curved background. In what follows, we will treat derivatives on +the vielbein at the same order as derivatives on the hydrodynamic fields; thus e.g. the curvature scalar R ∼ O(k2). +The time derivative of the dipole moment implies +∂0Db = +� +ddxe yb(xi)∂0n(x) = − +� +ddx yb(xi)∂k(eJk) = +� +ddxe ∂k(yb(xi))Jk ≜ +� +ddxe ekbJk, +(B.3) +where we have used the covariant charge conservation in (B.20b). The last equation defines the vielbein eb +k ≡ ∂kyb +that give us the transformation from yb to the uglier coordinates xi, such that the dipole moment will be conserved +according to the dipole Ward identity in (B.20c). We emphasize that the existence of such a choice of viebein is not +generic: in particular, the Ricci curvature tensor vanishes. Therefore, only on a flat space can the dipole moment be +defined in terms of operators in (B.2). +To build the invariant blocks and to include the spontaneous dipole symmetry breaking, we apply the coset construc- +tion for the spacetime symmetries [85, 86] (for an introduction to the coset construction, see the references therein). +Let us focus on dipole fluids without energy conservation. The unbroken generators are the translations Pα, and the +charge Q; the broken generator is the dipole moment Db. Thus, the most general group element is parametrized as +g(σ) = eiβ0σtP0eiyb(X(σ))Pbeiϕb(σ)Dbe− i +2 θbc(σ)Lbceiϕ(σ)Q, +(B.4) +where σA represent the fluid spacetime, and we associate with each symmetry generator a dynamical field. P0 is +an effective time-translation supporting a fixed temperature T0 = β−1 +0 +determined by the noise. We introduce the +gauge fields for translations (ˆeα +µ), rotations (ωb +µc), charges ( ˆAµ), and dipole moments (Ab +µ) , thus the gauge invariant +Maurer-Cartan one-form is given by +g−1 +� +∂A + i∂AXµˆeα +µPb + i +2∂AXµωbc +µ Lbc + i∂AXµ ˆAµQ + i∂AXµAb +µDb +� +g = ieα +APα+iKb +ADb+iBAQ+i1 +2Θbc +A Lbc. (B.5) +Hence, the useful invariant blocks are given by +eb +A = ∂AXµec +µR b +c , +(B.6a) +BA = ∂Aϕ + ∂AXµAµ + ∂AXµeb +µϕb, +(B.6b) +Kb +A = +� +∂Aϕc + ∂AXµωc +µdϕd + ∂AXµAc +µ +� +R b +c , +(B.6c) +Θb +Ac = ∂AXµ � +(R−1)bd∂µRdc − (R−1)bdωe +µdRec +� +, +(B.6d) + +23 +where to linear order in θbc the rotation matrix reads Rbc = δbc − θbc, and we defined +eb +µ ≡ ∂µyb(X) + ˆeb +µ + ωb +µcyc(X), +(B.7) +Aµ ≡ ˆAµ − Ab +µyb(X). +(B.8) +Clearly, we should regard eb +µ as the vielbein, ωb +µc = −ωc +µb as the spin connection, and Aµ as the U(1) gauge field. The +vielbein formalism requires also the Christoffel connection Γρ +µν, and the covariant derivative is defined as +∇µe0 +ν = ∂µe0 +ν − Γρ +µνe0 +ρ, +(B.9a) +∇µeb +ν = ∂µeb +ν + ωb +µcec +ν − Γρ +µνeb +ρ. +(B.9b) +The spin connection and Christoffel connection are not independent, so we impose the metric compatibility +∇µeα +ν = 0, +(B.10) +and treat the vielbeins and the spin connections as the independent background fields with respect to which the action +would vary. We note a useful relation +Γµ +νµ = e−1∂νe, +(B.11) +where e ≡ det eα +µ. +To incorporate the blocks into the Schwinger-Keldysh formalism, we introduce the two-time copies (s = 1, 2): +eb +s,A(σ) = ∂Xµ +s (σ) +∂σA +ec +s,µ(σ)R +b +s,c (σ), +(B.12a) +Bs,A(σ) = ∂Xµ +s (σ) +∂σA +� +As,µ(σ) + eb +s,µ(σ)ϕs,b(σ) +� ++ ∂ϕs(σ) +∂σA , +(B.12b) +Kb +s,A(σ) = ∂Xµ +s (σ) +∂σA +� +Ac +s,µ(σ) + ωc +s,µd(σ)ϕd +s(σ) +� +R +b +s,c (σ) + ∂ϕc +s(σ) +∂σA R +b +s,c (σ), +(B.12c) +Θb +s,Ac(σ) = ∂Xµ +s (σ) +∂σA +� +(R−1 +s )bd∂s,µRs,dc − (R−1 +s )bdωe +s,µdRs,ec +� +(σ), +(B.12d) +Restricting to θbc +r = 0, the r-fields are +eb +r,A = ∂AXµeb +µ, +(B.13a) +Br,A = ∂Aϕ + ∂AXµAµ + ∂AXµeb +µϕb, +(B.13b) +Kb +r,A = ∂Aϕb + ∂AXµωb +µdϕd + ∂AXµAb +µ, +(B.13c) +Θb +r,Ac = ∂AXµωb +µc. +(B.13d) +In the classical limit and physical spacetime, the a-fields are given by +eb +a,A = ∂AXµEb +a,µ, +Eb +a,µ = eb +a,µ + LXaeb +µ + ec +µθb +a,c, +(B.14a) +Ba,A = ∂AXµCa,µ, +Ca,µ = Aa,µ + ∂µϕa + LXaAµ + eb +µϕa,b + (eb +a,µ + LXaeb +µ)ϕb, +(B.14b) +Kb +a,A = ∂AXµKb +a,µ, +Kb +a,µ = ∇µϕb +a + ωb +a,µcϕc + ϕcLXaωb +µc + Ab +a,µ + LXaAb +µ + Kc +µθb +a,c, +(B.14c) +Θb +a,Ac = ∂AXµΩb +a,µc, +Ωb +a,µc = −∇µθb +a,c + ωb +a,µc + LXaωb +µc , +(B.14d) +where ∇µθb +a,c = ∂µθb +a,c + ωb +µdθd +a,c − ωd +µcθb +a,d. It is straightforward to check that the invariant blocks defined above +reflect the consistency of symmetry algebra in Appendix C. To the leading order in a-fields, the effective Lagrangian +can be written as +L = ˆT µ +b Eb +a,µ + JµCa,µ + Jµ +b Kb +a,µ + ˆSµc +bΩb +a,µc + . . . . +(B.15) +To derive the momentum Ward identity, let us consider the variation with respect to Xν +a. Denoting compactly the +total stress tensor T µ +b ≡ ˆT µ +b + Jµϕb and the total spin current Sµc +b ≡ ˆSµc +b + Jµ +[bϕc], we obtain +δS +δXνa += −e−1∂µ[e(T µ +b eb +ν + Sµc +bωb +νc + Jµ +b Ab +ν + JµAν)] + T µ +b ∂νeb +µ + Sµc +b∂νωb +µc + Jµ +b ∂νAb +µ + Jµ∂νAµ + +24 += −e−1∂µ(eT µ +b )eb +ν − e−1∂µ(eSµc +b)ωb +νc − e−1∂µ(eJµ +b )Ab +ν − e−1∂µ(eJµ)Aν ++ 2T µ +b ∂[νeb +µ] + 2Sµc +b∂[νωb +µ]c + 2Jµ +b ∂[νAb +µ] + 2Jµ∂[νAµ] += −∇′ +µ(T µ +b )eb +ν − ∇′ +µ(Jµ +b )Ab +ν − ∇′ +µJµAν + T µ +b Gb +νµ + Sµc +bRb +cνµ + Jµ +b F b +νµ + JµFνµ, +(B.16) +where we defined the modified covariant derivative ∇′ +µ ≡ ∇µ + Gρ +µρ with Gλ +µν ≡ 2Γλ +[µν], and the field strength +Fµν ≡ ∂µAν − ∂νAµ, +(B.17a) +F b +µν ≡ ∂µAb +ν − ∂νAb +µ + ωb +µcAc +ν − ωb +νcAc +µ, +(B.17b) +Gb +µν ≡ ∂µeb +ν − ∂νeb +µ + ωb +µcec +ν − ωb +νcec +µ, +(B.17c) +Rb +cµν ≡ ∂µωb +νc − ∂νωb +µc + ωb +µdωd +νc − ωb +νdωd +µc. +(B.17d) +In the last step of (B.16), we used the spin current Ward identity obtained by varying the action with respect to θbd +a : +T µ +[bed] +µ + Jµ +[bAd] +µ + ∇′ +µSµd +b = 0 , +(B.18) +as well as the dipole Ward identity (B.20c). It is known, however, that the “intrinsic” spin current ˆSµc +b is a non- +hydrodynamic mode [87]. This can be seen through −∂0 ˆS0c +b ∼ ˆS0c +b ⊂ ˆT µ +[bec] +µ which means that ˆS0c +b will relax to +zero at long time.14 Since the spatial part ˆSic +b is proportional to (gradient of) ˆS0c +b, we are allowed to set ˆSµc +b = 0 in +the following.15 On the other hand, the “non-intrinsic” spin current Sµc +b = Jµ +[bϕd] is not relaxed. Last, the variation +with respect to ϕa and ϕb +a will give charge and dipole Ward identities. Combining them, we obtain, with ordinary +derivatives, +e−1∂µ +� +e( ˆT µ +b + Jµϕb) +� ++ e−1∂µ (eJµ +d ϕc) ωd +νceν +b + e−1∂µ(eJµ +c )Ac +νeν +b +−2( ˆT µ +c + Jµϕc)∂[νec +µ]eν +b − 2Jµ +d ϕc∂[νωd +µ]ceν +b − 2Jµ∂[νAµ]eν +b − 2Jµ +c ∂[νAc +µ]eν +b = 0, +(B.19a) +e−1∂µ(eJµ) = 0, +(B.19b) +e−1∂µ(eJµ +b ) − ωc +µbJµ +c − Jµeµb = 0, +(B.19c) +and with covariant derivatives, +∇′ +µ +� +ˆT µ +b + Jµϕb +� ++ Ac +νeµcJµeν +b − Rd +cνµϕcJµ +d eν +b − F d +νµJµ +d eν +b − FνµJµeν +b − Gc +νµ +� +ˆT µ +c + Jµϕc +� +eν +b = 0, +(B.20a) +∇′ +µJµ = 0, +(B.20b) +∇′ +µJµ +b − Jµeµb = 0. +(B.20c) +These are the momentum, charge and dipole equations of motions in an generic curved spacetime. Remarkably, the +dipole Goldstone will couple to the curvature to give a force in the momentum equation, and we have showed that it +is best understood as the Joule heating due to spin currents [84]. +If we set ϕb = 0, our result is consistent with [59] in that the structure of their Eq.(4.13) is recovered: ∇′ +µ ˆT µ +b = +fµeµ +b −e0 +µ ˆT µ +c ec +ρeν +b∇νeρ +0, where we denoted collectively the Joule heating term fµ and used Gb +µν = 2eb +λe0 +[µ∇ν]eλ +0. However, +unlike [59], our formalism includes the dipole Goldstone ϕb, which leads to the coupling between spin current and +background curvature. Note that this feature is ultimately related to the breakdown of dipole gauge theory in curved +spacetime [39, 88], but here the dipole symmetry is valid so long as we count background curvature as derivative +corrections. There is another conceptual difference from [59] that our Ab +µ needs not to be symmetric. This is possible +if we include the dipole Goldstone ϕb. Unlike [59], the antisymmetric part of Ab +µ is not fixed by Fµν, and does have +physical consequences as emphasized in the main text. +Appendix C: Consistency of the symmetry algebra +The analysis of this section follows largely [59] but contains several generalizations of it. +14 ˆS0c +b is identified as the local angular momentum density in [87]. +15 Note that in an ordinary fluid without dipole symmetry, imposing ˆSµc +b = 0 to (B.18) would lead to a symmetric stress tensor ˆT µ +[bed] +µ = 0. + +25 +Let χµ be an infinitesimal diffeomorphism, Ωb +c an infinitesimal rotation, Λ a U(1) gauge transformation, and ξb a +dipole shift parameter. We start by defining the action of transformations Ξ = (χµ, Ωb +c, Λ, ξb) on the dynamical fields +Xµ(σ), ϕ(σ), ϕb(σ). We have +eδΞXµ = (1 + δΞ + · · · )Xµ = Xµ − χµ(X) + · · · , +(C.1) +thus, in leading order, +[δΞ′, δΞ] = [eδΞ′ , eδΞ] . +(C.2) +We have, up to second order, +eδΞ′ eδΞXµ = Xµ − χµ − χ′µ + χ′ν∂νχµ, +(C.3) +so that16 +[eδΞ′ , eδΞ]Xµ = Lχ′χµ = −δ[Ξ′,Ξ]Xµ, +(C.4) +where we carefully kept into account zeroth, first, and second-order terms and verified that zeroth and first-order +terms cancel out. We then find +χµ +[Ξ′,Ξ] ≡ δΞ′χµ = Lχ′χµ . +(C.5) +Next, let’s look at ϕb. We have +eδΞϕb = ϕb + ξb − Ωb +cϕc. +(C.6) +Note that we are not including Lχϕb in the above as we view ϕb as a function of σ, which is a singlet under physical +spacetime diffeomorphisms χµ. Then, +eδΞ′ eδΞϕb = ϕb + ξb + ξ′b − (Ω′b +c + Ωb +c)ϕc − Ωb +cξ′c + Ωb +dΩ′d +cϕc − Lχ′ξb + Lχ′Ωb +cϕc, +(C.7) +and +[eδΞ′ , eδΞ]ϕb = −Lχ′ξb + Lχξ′b + Ω′b +cξc − Ωb +cξ′c + (Lχ′Ωb +c − LχΩ′b +c)ϕc + (Ωb +dΩ′d +c − Ω′b +dΩd +c)ϕc, +(C.8) +thus giving +ξb +[Ξ′,Ξ] ≡ δΞ′ξb = Lχ′ξb − Lχξ′b + Ωb +cξ′c − Ω′b +cξc , +(C.9a) +Ωb +c,[Ξ′,Ξ] ≡ δΞ′Ωb +c = Lχ′Ωb +c − LχΩ′b +c + Ωb +dΩ′d +c − Ω′b +dΩd +c. +(C.9b) +Now it’s ϕ’s turn: +eδΞϕ = ϕ + Λ ≡ ϕ + λ + M bξb + Nµχµ , +(C.10) +where M b and Nµ are expressions in terms of Xµ, ϕ and ϕb. Up to a normalization of the transformation parameters, +the most general choice consistent with charge conjugation invariance is +M b = (1 − c)eb +µXµ, +Nµ = ceb +µϕb . +(C.11) +We will determine c = 0 later from consistency requirements, but now it can be any value. Again we view ϕ as a +function of σ and thus we do not have the term Lχϕ. Composing two transformations, +eδΞ′ eδΞϕ = ϕ + Λ + Λ′ − Lχ′Λ + ceb +µχµξ′ +b , +(C.12) +gives +[eδΞ′ , eδΞ]ϕ = −Lχ′Λ + LχΛ′ + ceb +µχµξ′ +b − ceb +µχ′µξb +(C.13) +16 We take passive transformations on dynamical fields, i.e. [δΞ′, δΞ] = −δ[Ξ′,Ξ]. + +26 +To summarize this part, the algebra of local transformations is +χµ +[Ξ′,Ξ] ≡δΞ′χµ = Lχ′χµ, +(C.14a) +Ωb +c,[Ξ′,Ξ] ≡δΞ′Ωb +c = Lχ′Ωb +c − LχΩ′b +c + Ωb +dΩ′d +c − Ω′b +dΩd +c, +(C.14b) +ξb +[Ξ′,Ξ] ≡δΞ′ξb = Lχ′ξb − Lχξ′b + Ωb +cξ′c − Ω′b +cξc, +(C.14c) +Λ[Ξ′,Ξ] ≡δΞ′Λ = Lχ′Λ − LχΛ′ − ceb +µχµξ′ +b + ceb +µχ′µξb. +(C.14d) +This is a generalization of Eq.(4.11) in [59]. To see the consistency with the dipole algebra, we decompose the field +variation into +δΞ = −iχµPµ + iλQ + iξbDb, +(C.15) +where Pµ, Q and Db are symmetry generators, and we have ignored the rotation symmetry generator for simplicity. +The background field is also turned off. As a consequence of (C.10), we have +iQϕ = 1, +iDbϕ = Mb, +−iPµϕ = Nµ. +(C.16) +Now, let us take Ξ′ = ξ′ +b and Ξ = χµ, then +−[δΞ′, δΞ]ϕ = −χµδb +µξ′ +c[Db, Pc]ϕ − (1 − c)Xµδb +µχρ∂ρξ′ +b. +(C.17) +At the same time, we have +−[δΞ′, δΞ]ϕ = Λ[ξ′ +b,χµ] = −χµδb +µξ′ +b − (1 − c)Xµδb +µχρ∂ρξ′ +b. +(C.18) +Thus we conclude that +[Pb, Dc] = −iQδbc +(C.19) +is the desired dipole algebra. +There is an unwanted free parameter c appearing in the algebra. Here, we show that in order for the transformation +Ξ itself to also form a closed algebra, one must set c = 0. First, as a warm-up, let us consider the variation of +χµ, ξb, Ωb +c. We have +[δΞ′′, δΞ′]χµ = Lχ′′Lχ′χµ − Lχ′Lχ′′χµ = Lχ[Ξ′′,Ξ′]χµ = δ[Ξ′′,Ξ′]χµ. +(C.20) +Next, using product rules, e.g. δΞ′′Lχ′ξb = LδΞ′′χ′ξb + Lχ′δΞ′′ξb , we find +δΞ′′δΞ′ξb =Ω′′b +cLχξ′c + Ωb +cLχ′′ξ′c − Lχ′′Ω′b +cξc − LχΩ′b +cξ′′c + Ωb +cΩ′c +dξ′′d + Ω′′b +dΩ′d +cξc ++ +� +Lχ′′Ωb +cξ′c + Lχ′Ωb +cξ′′c − Ω′′b +cLχ′ξc − Ω′b +cLχ′′ξc − Ω′′b +dΩd +cξ′c − Ω′b +dΩd +cξ′′c� +, +(C.21) +thus +[δΞ′′, δΞ′]ξb = Lχ[Ξ′′,Ξ′]ξb − Lχξb +[Ξ′′,Ξ′] + Ωb +cξc +[Ξ′′,Ξ′] − Ωb +c,[Ξ′′,Ξ′]ξc = δ[Ξ′′,Ξ′]ξb. +(C.22) +Similar calculation leads to +[δΞ′′, δΞ′]Ωb +c = Lχ[Ξ′′,Ξ′]Ωb +c − LχΩb +c,[Ξ′′,Ξ′] + Ωb +dΩd +c,[Ξ′′,Ξ′] − Ωb +d,[Ξ′′,Ξ′]Ωd +c = δ[Ξ′′,Ξ′]Ωb +c. +(C.23) +Now, we look at Λ. For brevity, let us take eb +µ = δb +µ and temporarily turn off Ωb +c. Note, this is merely a simplification +for manipulations and the final result should still hold in arbitrarily curved spacetime. We have +δΞ′′δΞ′Λ = δΞ′′(Lχ′Λ − LχΛ′ − cδb +µχµξ′ +b + cδb +µχ′µξb) += LLχ′′χ′Λ + Lχ′(Lχ′′Λ − LχΛ′′ − cδb +µ(χµξ′′ +b − χ′′µξb)) − LLχ′′χΛ′ − Lχ(Lχ′′Λ′ − Lχ′Λ′′ − cδb +µ(χ′µξ′′ +b − χ′′µξ′ +b)) +− cδb +µLχ′′χµξ′ +b − cδb +µχµ(Lχ′′ξ′ +b − Lχ′ξ′′ +b ) + cδb +µLχ′′χ′µξb + cδb +µχ′µ(Lχ′′ξb − Lχξ′′ +b ) += Lχ′′Lχ′Λ + LχLχ′Λ′′ − (Lχ′′LχΛ′ + Lχ′LχΛ′′) + cδb +µ [Lχχ′µξ′′ +b − Lχ(χ′′µξ′ +b) − χµLχ′′ξ′ +b] ++ cδb +µ [−(Lχ′χµξ′′ +b + Lχ′′χµξ′ +b) + (Lχ′χ′′µξb + Lχ′′χ′µξb) + (χ′′µLχ′ξb + χ′µLχ′′ξb)] , +(C.24) + +27 +thus +[δΞ′′, δΞ′]Λ = Lχ[Ξ′′,Ξ′]Λ − Lχ(Lχ′′Λ′ − Lχ′Λ′′) ++ cδb +µ [−Lχ(χ′′µξ′ +b − χ′µξ′′ +b ) − χµ(Lχ′′ξ′ +b − Lχ′ξ′′ +b ) + Lχχ′µξ′′ +b − Lχχ′′µξ′ +b] . +(C.25) +We note that the r.h.s. cannot be written as Lχ[Ξ′′,Ξ′]Λ−LχΛ[Ξ′′,Ξ′]−cδµbχµξb +[Ξ′′,Ξ′]+cδµbχµ +[Ξ′′,Ξ′]ξb, hence, the algebra +is not closed. However, if we set c = 0, we have +[δΞ′′, δΞ′]Λ = Lχ[Ξ′′,Ξ′]Λ − LχΛ[Ξ′′,Ξ′] = δ[Ξ′′,Ξ′]Λ. +(C.26) +To summarize, when c = 0, Ξ itself forms a closed algebra +[δΞ′′, δΞ′]Ξ = δ[Ξ′′,Ξ′]Ξ. +(C.27) +Lastly, let us turn to the background gauge fields. The variation is defined as +δΞeb +µ = Lχeb +µ − Ωb +cec +µ, +(C.28a) +δΞωb +µc = Lχωb +µc + ∇µΩb +c, +(C.28b) +δΞAµ = LχAµ − ∂µΛ − eb +µξb, +(C.28c) +δΞAb +µ = LχAb +µ − ∇µξb − Ωb +cAc +µ. +(C.28d) +Composing two transformations, we have +δΞ′δΞeb +µ = Lχ′Lχeb +µ + (LχΩ′b +c + Ω′b +dΩd +c)ec +µ − +� +Lχ(Ω′b +cec +µ) + Lχ′(Ωb +cec +µ) +� +, +(C.29a) +δΞ′δΞAµ = Lχ′LχAµ + ∂µLχΛ′ + eb +µLχξ′ +b − eb +µΩbcξ′c − +� +Lχ∂µΛ′ + Lχ′∂µΛ + Lχ′(eb +µξb) + Lχ(eb +µξ′ +b) +� +, +(C.29b) +δΞ′δΞωb +µc = Lχ′Lχωb +µc − ∇µ(LχΩ′b +c + Ω′b +dΩd +c) + (Lχ∇µΩ′b +c + Lχ′∇µΩb +c + ∂µΩb +dΩ′d +c + ∂µΩ′b +dΩd +c ++ ωb +µeΩe +dΩ′d +c + ωb +µeΩ′e +dΩd +c − Ω′b +eωe +µdΩd +c − Ωb +eωe +µdΩ′d +c). +(C.29c) +The variation of Ab +µ is a bit tedious, but taking lessons from previous manipulations, we can immediately see that the +infinitesimal rotation should be closed. Hence, we are allowed to focus on the piece involving ξb only. 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Lett. 122, 076403 (2019). + diff --git a/MNE0T4oBgHgl3EQf0AKB/content/tmp_files/load_file.txt b/MNE0T4oBgHgl3EQf0AKB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..84f4cfdf9d668e0a42e55934611ced14fd5e365f --- /dev/null +++ b/MNE0T4oBgHgl3EQf0AKB/content/tmp_files/load_file.txt @@ -0,0 +1,1430 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf,len=1429 +page_content='Goldstone bosons and fluctuating hydrodynamics with dipole and momentum conservation Paolo Glorioso,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' ∗ Xiaoyang Huang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' † Jinkang Guo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2 Joaquin Rodriguez-Nieva,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1 and Andrew Lucas2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' ‡ 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Stanford University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Stanford CA 94305,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' USA 2Department of Physics and Center for Theory of Quantum Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' University of Colorado,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' CO 80309,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' USA (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 2023) We develop a Schwinger-Keldysh effective field theory describing the hydrodynamics of a fluid with conserved charge and dipole moments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' together with conserved momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The resulting hydrodynamic modes are highly unusual, including sound waves with quadratic (magnon-like) dis- persion relation and subdiffusive decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Hydrodynamics itself is unstable below four spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We show that the momentum density is, at leading order, the Goldstone boson for a dipole symmetry which appears spontaneously broken at finite charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Unlike an ordi- nary fluid, the presence or absence of energy conservation qualitatively changes the decay rates of the hydrodynamic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This effective field theory naturally couples to curved spacetime and background gauge fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' in the flat spacetime limit, we reproduce the “mixed rank tensor fields” previously coupled to fracton matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' CONTENTS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Effective field theory of hydrodynamics 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' General setup 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Classical limit and hydrodynamic effective theory 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Ideal hydrodynamics 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Dissipative hydrodynamics and higher-order terms 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Relevant perturbations in low dimensions 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Spontaneous symmetry breaking 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Mermin-Wagner Theorem 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Goldstone’s Theorem 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Existence of a symmetry-breaking state 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Hydrodynamics with energy conservation 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' From Galilean symmetry to dipole symmetry 17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Conclusions 19 Acknowledgements 19 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Memory matrix methods 19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Momentum susceptibility 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Dynamics without energy 21 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Dipole fluids with momentum in a curved spacetime 21 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Consistency of the symmetry algebra 24 References 27 ∗ paolog@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='edu † xiaoyang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='huang@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='edu ‡ andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='lucas@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='02680v1 [hep-th] 6 Jan 2023 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' INTRODUCTION One of the oldest and most successful theories in physics is hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' While hydrodynamics was first under- stood as a phenomenological set of equations that govern liquids and gases [1], over the past century we have instead recognized that hydrodynamics is best understood as the universal effective field theory that governs thermalization in a chaotic many-body system [2–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In the simplest scenarios, the degrees of freedom of a hydrodynamic theory correspond to locally conserved quantities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' the way that these modes interact with each other and decay is constrained only by basic symmetries of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Using modern effective field theory methods, sophisticated nonlinear theories of fluctuating hydrodynamics have been developed and applied to increasingly sophisticated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' One family of novel phases of matter which has interesting dynamics arises when the microscopic degrees of freedom are fractons – excitations which are individually immobile, and can only move in tandem [7–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1 As a simple example, we can consider a phase of matter in which charge/mass is conserved together with dipole moment/center of mass – in this case, a single particle cannot move without violating the dipole conservation law!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The past few years have seen an intense study of the fracton phases of matter that can result by combining many of these interacting fractons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' And over the past two years, it has been understood that when such fracton phases thermalize [22, 23], the resulting hydrodynamics is non-trivial [24–27]: Fick’s law of diffusion, for example, becomes replaced by subdiffusive equations, with the dynamical critical exponent dependent on how many multipole moments are conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Many further “fracton hydrodynamics” universality classes have since been discovered [28–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In this paper, we detail a qualitatively new universality class of hydrodynamics that emerges when fracton-like multipole conservation laws are combined with canonical energy and momentum conservation, which was first pre- sented by four of us in a shorter paper [35];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' see also [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We focus on the case where dipole moment is the only additional conserved quantity, and where the theory has parity and time-reversal symmetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' in the absence of momentum conservation, the consequences of breaking these symmetries were recently discussed in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Without dipole conservation, such a theory is essentially described by textbook Navier-Stokes equations with incoherent con- ductivities [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' With dipole conservation, the Navier-Stokes equations are completely changed [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' At finite charge density, the conventional propagating sound modes are replaced by magnon-like propagating modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The decay rates of these magnon-like modes is diffusive if energy is conserved, but subdiffusive if energy is not conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' And at zero density, the character of the hydrodynamic modes completely changes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' the naive derivative expansion of hydrodynam- ics at finite density is singular as low density is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' At zero density, assuming particle-hole symmetry, the momentum dynamics decouples from that of charge within linear response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The character of collective modes in this case is completely different, where momentum and charge display diffusive and subdiffusive damping, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The subtle nature of this emergent hydrodynamics is intricately related to the fact that (in quantum mechanics) the dipole moment operator D, and net momentum operator P, do not commute [38]: [D, P] = iQ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1) where Q represents total charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' An analogous classical statement holds for Poisson brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' One might expect that such a commutation relation is similar to angular momentum commutation relations in an isotropic fluid – such commutation relations lead not to new propagating degrees of freedom, but rather constraints on the currents of other modes (the stress tensor, in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' However, at finite density, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1) implies that momentum susceptibility (the generalization of mass density) is singular!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This means that a naive hydrodynamic degree of freedom – fluid velocity – is non-local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' One of the main results of this paper is that we can nevertheless construct a local hydrodynamic theory, using unconventional degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In Section 2, we will describe how to construct this EFT following the constructions of [2–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In the process, we comment on the coupling of this theory to background geometry (vielbein), although much of this technical work is relegated to appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Following [39–42], we hope this can further stimulate work on understanding how and when fractons can be coupled to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' As reported in [35], these hydrodynamic theories can be unstable below four dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This is true both without energy conservation, and with energy conservation at infinite temperature (under mild assumptions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This result generalizes the well-known Kardar-Parisi-Zhang instability of an equilibrium fluid (without dipole conservation) in one dimension [43], and implies the existence of a non-equilibrium fixed point in three dimensions, in an undriven system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' From many perspectives, we will show that the commutation relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1) implies there is spontaneous symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In a finite density state, D and P do not commute, so they cannot be diagonalized simultaneously – in a state with fixed momentum, there are large fluctuations in dipole moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Unlike more conventional compact non-Abelian 1 While it can be desirable to strengthen this definition to demand that fractons cannot move under the action of any local operator, following the “fracton hydrodynamics” literature, we will take a looser definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (In our theory, a local operator that inserts a quadrupole can move an isolated charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=') 3 symmetry groups such as SU(2), here one cannot find any physical “singlet” states in the Hilbert space which lie in trivial representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (This is analogous to textbook quantum mechanics: one cannot find simultaneous eigenstates of x and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=') So it seems that trivially, dipole and/or momentum will be spontaneously broken, in agreement with previous literature on low-dimensional SSB with non-compact symmetry groups [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In this paper, we focus on ensembles with fixed momentum density (which seems more physical to us).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' A consequence is that the propagating momentum density is (at leading order in the hydrodynamic expansion) proportional to the Goldstone boson for broken dipole symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In one and two space dimensions, the fluctuations of this Goldstone boson are very large, and for this reason a recent work [45] proved that there is no SSB within the context of the Mermin-Wagner theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' On the other hand, we will see that the hydrodynamics in low dimension a single hydrodynamic mode still contains all of the spectral weight required to saturate the Goldstone theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The presence of this Goldstone boson in the hydrodynamic theory suggests an unusual paradigm for possible “spontaneous symmetry breaking.” Following recent discussions on the spontaneous breaking of boost symmetry in fluids [46, 47], we will discuss at some length the nature of the apparent spontaneous symmetry breaking in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In Section 4, we extend the discussion of the EFT to models with energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The observation of interest is that energy conservation changes the dynamical universality class of the dipole-momentum conserving fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2 Intuitively, this can be understood as follows: energy can diffuse, while charge must subdiffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Due to thermody- namics at finite charge and energy density, however, charge and energy modes generically “mix” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' the propagating sound wave would involve both charge and energy fluctuations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' As a consequence, the dominant decay channel is always through energy diffusion, which leads to z = 2, rather than z = 4, at the hydrodynamic (Gaussian) fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Lastly, in Section 5, we discuss how a dipole-conserving theory can arise in the infinite mass limit of a theory with Galilean invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This may suggest one way to look for this physics in experimental systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' in (nearly) flat bands [52–56] in condensed matter systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Several complementary discussions and technical computations are included in the appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In Appendix A, the memory matrix formalism is applied to derive the normal modes and diverging momentum susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Ap- pendix B consists of a detailed derivation of dipole-conserving hydrodynamics in curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In Appendix C, the consistency between dipole symmetry and geometry is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' EFFECTIVE FIELD THEORY OF HYDRODYNAMICS One main result is that hydrodynamics with dipole conservation possesses anomalous scaling, which is due to the interplay between the nonlinear hydrodynamic interactions and hydrodynamic fluctuations [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' To derive this we shall use a recently formulated effective field theory (EFT) of hydrodynamics, which systematically describes fluctuations by encoding hydrodynamics into an effective action [2–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' General setup The aim of the EFT approach is to systematically encode the correlation functions of hydrodynamic densities and currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Such correlation functions have the general form Tr(T (J1J2 · · · )ρ0 ˜T (J3J4 · · · · · · )) = � ρ0 Dψ1Dψ2 eiS0[ψ1]−iS0[ψ2] J1[ψ1]J2[ψ1]J3[ψ2]J4[ψ2] · · · , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1) where, in the first expression, T and ˜T denote time- and anti time-ordering, ρ0 is the initial state, which we take to be thermal ρ0 = e−βH/ tr(e−βH), with H the microscopic Hamiltonian of the system, and J1, J2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' are operators inserted at (t1, ⃗x1), (t2, ⃗x2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' On the right-hand side, we formally rewrote the correlator as a path-integral, where S0 is the action of the microscopic dynamics, and ψ1, ψ2 are a doubled copy of the degrees of freedom of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Since on the left-hand side we have a forward (backward) time evolution given by the time-ordered (anti-time ordered) product, the path integral contains two exponentials of the action S0, with a relative minus sign, as the first one corresponds to forward evolution, while the second one to backward evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In other words, the doubling of degrees of freedom comes from that the evolution of the density matrix ρ0 → U(t)ρ0U †(t) contains two factors of the evolution, one forward and one backward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Computing hydrodynamic correlation functions from the microscopic dynamics is very 2 Interestingly, there was a debate in past literature about the role of energy conservation in flowing from the Navier-Stokes to the KPZ fixed point [48–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The consensus is now that energy conservation indeed does not disturb the KPZ point [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' What was missing in the past was just to incorporate dipole conservation!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 4 hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We thus want to introduce an EFT approach that substitutes the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1) with a simpler action: Tr(T (J1J2 · · · )ρ0 ˜T (J3J4 · · · · · · )) = � Dχ1Dχ2 eiS[χ1,χ2] J1[χ1]J2[χ1]J3[χ2]J4[χ2] · · · , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2) where S is the effective action for hydrodynamics, and χ1, χ2 denote the doubled hydrodynamic degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The action S will encode the effects of fluctuation and dissipation and, in particular, will allow us to predict the existence of anomalous scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We shall now introduce the degrees of freedom of this EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' These should be fields that nonlinearly realize the symmetries associated to conservation of charge, dipole and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' For momentum conservation, we introduce a set of coordinate fields Xi = Xi(σt, σI) which nonlinearly realize translations P i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Xi(σt, σI) → Xi(σt, σI) + ξi , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3) where ξi is a constant vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We are using σI to denote an auxiliary coordinate system which can be thought of as labeling the fluid parcels at a fixed value of time σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3 The coordinates Xi(σt, σI) describe the trajectory of the fluid parcel labeled by σI as a function of time σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The coordinates (σt, Xi) are the “physical” ones, in the sense that they label the time and space in the lab reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4 Next, we also have a vector degree of freedom ϕi(σt, σI) that nonlinearly realizes the dipole shift symmetry Di: ϕi(σt, σI) → ϕi(σt, σI) + ci , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4) where ci is a constant vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Finally, for charge Q, the associated degree of freedom is a scalar ϕ(σt, σI), and transforms as ϕ(σt, σI) → ϕ(σt, σI) + a − ciXi , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5) where a is a constant denoting the parameter of transformations associated to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The fields ϕ and ϕi can be heuristically viewed as describing the “local phase” of the fluid ei(ϕ+Xiϕi), where this particular form is motivated from the fact that, for dipole-conserving field theories, U(1) global transformations can have a linear dependence in spatial coordinates [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Note that ϕ transforms also under dipole shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This particular transformation rule is implied by the commutator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Indeed, writing infinitesimal translation and dipole shift as δξϕ = ξi∂iϕ, δcϕ = −ciXi, we have (δcδξ − δξδc)ϕ = ciξi , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='6) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' the commutator is an infinitesimal shift of ϕ, as required by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' It can also be verified that the last term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5) is the most general transormation consistent with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The effective action will be invariant under transformations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5) which, as a consequence of Noether’s theorem, correspond to the statement of conservation of momentum, dipole and charge, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Moreover, we provide a thorough and careful consistency check of the symmetry algebra in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Now recall from above that all the degrees of freedom have to be doubled, so we will have Xi 1, Xi 2, ϕi 1, ϕi 2, ϕ1 and ϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The symmetries (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5) will also be doubled, which in turn correspond to the conservation of the corresponding hydrodynamic currents defined in the forward and backward time contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Unlike in the path integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1), the effective action appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2) does not have a factorized form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This is because, as a result of the coarse-graining, where the fast-moving degrees of freedom have been integrated out, new couplings that are local in the “folded” time have been generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' These cross-couplings are responsible for dissipations and fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' While the effective action loses factorization, it still satisfies several properties that come from the unitarity of the underlying microscopic evolution [2, 5]: S[χ, χ] = 0, S[χ2, χ1] = −S∗[χ1, χ2], Im S[χ1, χ2] ≥ 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='7) where χ1, χ2 collectively denote the two copies of Xi, ϕi, ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Note in particular that the action can (and will) be complex-valued;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' as we will see this is a basic consequence of having thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Additionally, since the initial state ρ0 is thermal, and assuming that the microscopic Hamiltonian H is invariant under time-reversal, the effective action satisfies a discrete Z2 symmetry called “dynamical KMS symmetry”: S[χ1, χ2] = S[˜χ1, ˜χ2], ˜χ1(σt, σI) = (−1)ηχ1(−σt, σI), ˜χ2(σt, σ) = (−1)ηχ2(−σt − iβ, σI) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='8) 3 In older literature, these are the so-called “Lagrangian specification” of the fluid [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 4 It is convenient to denote time by σt as, in what follows, we will often need to take derivatives with respect to time at fixed σI, not at fixed Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 5 where (−1)η = ±1 denotes the time-reversal eigenvalue of χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This symmetry is equivalent to the Euclidean time periodicity of correlation functions on a thermal state with inverse temperature β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In our effective action, it will relate couplings responsible for dissipation with those describing fluctuations, and it will ensure consistency with the second law of thermodynamics, Onsager relations, and existence of equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='8) can be extended to situations where the microscopic Hamiltonian is invariant under a more general discrete symmetry, so long as such symmetry contains time-reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' A proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='8) is given in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' To complete our effective field theory, we need an additional set of symmetries that characterize the fact that the late-time behavior of the system is that of a fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Recall that σI should be interpreted as labels of fluid elements at a fixed value of σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Adiabatically reshuffling fluid elements has a vanishing cost in energy, since, in contrast to a solid, fluid parcels are not pinned to a particular spatial location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This means that a specific way to label fluid elements at a given time is not physical, and thus the effective action should be invariant under time-independent redefinitions of σI: σI → σ ′I(σJ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='9) Had we not considered this symmetry, the action could depend on arbitrary derivatives ∂IXi, and we would describe a solid instead of a liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Analogously, in the charge sector, we have the freedom to relabel the local phase ei(ϕ+Xiϕi) at a fixed time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This amounts to requiring the symmetry ϕ1(σt, σI) → ϕ1(σt, σI) + λ(σI), ϕ2(σt, σI) → ϕ2(σt, σI) + λ(σI) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='10) where λ(σI) is a time-independent redefinition of the phase and can be arbitrarily assigned on each fluid element σI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The symmetry (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='10), dubbed diagonal shift symmetry in [2, 5], states the absence of spontaneous symmetry breaking of the global U(1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Indeed, in the occurrence of spontaneous symmetry breaking, the full information about the phase would be a physical (of course, up to constant shifts of the phase), which would give rise to a superfluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Instead, in the present context, we are merely interested in the conservation of charge (and dipole) in the absence of spontaneous symmetry breaking of charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Unlike ϕ1,2, ϕi 1,2 does not have a diagonal shift symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Indeed, one can show that imposing the symmetry ϕi 1,2 → ϕi 1,2 + λi(σI) leads to the incorrect Ward identities, as we will show below (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='18c) in the following Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We will later see in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 3 that this can be understood as a consequence of spontaneous symmetry breaking — ϕi is a Goldstone boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Classical limit and hydrodynamic effective theory The formalism we have introduced above is based on quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In the present paper, however, we are interested in the emergent classical, high-temperature hydrodynamic behavior of many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' There is a simple way to take the classical limit of this framework which retains the physics we are interested in and has the benefit of considerably simplifying various technical aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' To this aim, we restore factors of ℏ and write χ1 = χ + 1 2ℏχa, χ2 = χ − 1 2ℏχa, where aggain χ collectively denotes the hydrodynamic fields, for example: Xi 1 = Xi + 1 2ℏXi a, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The fact that χ1 − χ2 is linear in ℏ can be heuristically understood from the fact that the forward and backward time evolutions are located a distance ℏβ from each other, and thus, as ℏ → 0, χ1 − χ2 should vanish linearly in ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In this limit, the dynamical KMS symmetry becomes ˜χ(σt, σI) = (−1)ηχ(−σt, σI), ˜χa(σt, σI) = (−1)η{χa(−σt, σI) + iβ∂tχ(−σt, σI)} , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='11) where the dependence on ℏ has factorized out, and the nonlocal time shift in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='8) reduced to an exact time derivative, allowing for a more straightforward implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We now proceed to writing down the invariant blocks that will be used to write the effective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We assume rotational invariance, but a generalization to discrete rotational symmetry is straightforward [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We recall that the energy conservation is not assumed, so we take β0 as a constant inverse temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5 We will come back to include energy conservation in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' For completeness, we summarize the notation here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' see Appendix B for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We denote µ, ν = 0, 1, 2, 3 for the physical spacetime, and A, B = t, x, y, z for the fluid spacetime, and use i, j, I, J to indicate their spatial subspace, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We introduce the internal spacetime indices α, β and b, c for its spatial subspace (we reserve a to describe a-fields in the Keldysh contour!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 5 The recent formalism of [34], which can build effective field theories for non-thermal systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' those whose steady state is not of the form exp[−βH], may allow us to put this construction on a firmer footing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' However, it is not known how to incorporate the non-Abelian multipole algebra or spontaneous symmetry breaking into this formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Revisiting this question would be interesting in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 6 To incorporate the gauge invariance, we introduce the background gauge field eb µ, Aµ and Ab µ, such that the invariant building-blocks in the fluid spacetime are defined as (s = 1, 2) eb s,A(σ) = ∂Xµ s (σ) ∂σA eb s,µ(σ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='12a) Bs,A(σ) = ∂Xµ s (σ) ∂σA � As,µ(σ) + eb s,µ(σ)ϕs,b(σ) � + ∂ϕs(σ) ∂σA , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='12b) Kb s,A(σ) = ∂Xµ s (σ) ∂σA � Ab s,µ + ωb s,µcϕc s � + ∂ϕb s(σ) ∂σA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='12c) See a derivation in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In the main text, we will be particularly interested in the geometry where eb 1µ +eb 2µ = 2δb µ and ωb s,µc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In the classical limit, this corresponds to working with a flat spacetime and allowing eb a,µ = eb 1µ−eb 2µ to source the stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Throughout this section, we take e0 sµ = δ0 sµ, but will consider a more general background when evaluating the energy fluctuations in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We denote the r, a-fields as follows Λr = Λ1 + Λ2 2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='13a) Λa = Λ1 − Λ2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='13b) where Λr,a denote collectively the background and dynamical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The r-fields of the blocks are eb r,A = ∂AXµeb µ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='14a) Br,A = ∂Aϕ + ∂AXµAµ + ∂AXµeb µϕb, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='14b) Kb r,A = ∂AXµKb r,µ = ∂AXµ(∂µϕb + Ab µ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='14c) While a-fields are always invariant under the relabeling symmetries (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='10), the r-fields are not, and the invariant r-fields without derivatives are eb r,t = ∂tXµeb µ ≡ β0uµeb µ = β0ub, Br,t = β0uµBr,µ ≡ β0µ, Kb r,µ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='15) from which we defined the thermodynamic variables uµ, µ and Kb r,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Below, we omit the index r for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In the classical limit and physical spacetime, the a-fields of the invariant blocks can be written as eb a,A = ∂AXµEb a,µ, Eb a,µ = eb a,µ + LXaeb µ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='16a) Ba,A = ∂AXµCa,µ, Ca,µ = Aa,µ + ∂µϕa + LXaAµ + eb µϕa,b + Eb a,µϕb, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='16b) Kb a,A = ∂AXµKb a,µ, Kb a,µ = ∂µϕb a + Ab a,µ + LXaAb µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='16c) To the leading order in a-fields, the effective Lagrangian is given by L = ˆT µ b Eb a,µ + JµCa,µ + Jµ b Kb a,µ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='17) Note that in this case the stress tensor is not equal to the coefficient T µ b ̸= ˆT µ b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Indeed, the Ward identities in the absence of stochastic fluctuations are obtained by varying L with respect to Xi a, ϕa and ϕb a and then setting a-fields to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This leads to ∂µ( ˆT µ b + Jµϕb)eb i + Ab ieµbJµ − F b iµJµ b − FiµJµ = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='18a) ∂µJµ = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='18b) ∂µJµ b − Jµeµb = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='18c) where Fµν = ∂µAν − ∂νAµ and F b µν = ∂µAb ν − ∂νAb µ are the U(1) and dipole field strength, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In the following, we will construct the effective field theory to determine the stress tensor and currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We will see that the stress tensor is indeed given by the term in the bracket in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='18a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Note that, had we imposed diagonal shift symmetry on the dipole field ϕb → ϕb + λb(σI), this would affect the structure of the Ward identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Indeed, neglecting background fields, Ca,µ = ∂µϕa + eb µϕa,b + ∂µXb aϕb, and we see that Ca,µ would transform nontrivially under λb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This in turn would affect the structure of the action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='17) and thus alter the Ward identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Ward identities should only be determined in terms of the global symmetries of the 7 system (or their gauged version), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5), therefore it is necessary that ϕb does not possess a diagonal shift symmetry, unlike ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Additionally, we note that, unlike most studies about dipole field theory (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' [59]), the dipole gauge field Ab µ as well as the dipole current Jµ b by no means need to be symmetric in their spatial indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In dipole hydrodynamics without momentum, Jij ∼ ∂i∂jµ, which automatically decouples the antisymmetric part at this derivative order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The momentum density, on the other hand, can contribute to the antisymmetric part of Jij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' As we will see later, such antisymmetric term contributes to the momentum subdiffusion mode in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='59) through the coefficient a3 that is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='29), and thus has physical consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' At last, let us consider a system that preserves the symmetry Θ = PT , where P acts as flipping all the spatial coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The KMS transformation of dynamical fields in fluid spacetime is given by � Xµ a (−σ) = −Xµ a (σ) − iβ0∂tXµ(σ) + iβ0δµ 0 , �ϕa(−σ) = −ϕa(σ) − iβ0∂tϕ(σ), �ϕb a(−σ) = ϕb a(σ) + iβ0∂tϕb(σ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='19) and that of external gauge fields is given by �eb a,µ(−σ) = eb a,µ(σ) + iβ0∂teb µ(σ), �Aa,µ(−σ) = Aa,µ(σ) + iβ0∂tAµ(σ), �Ab a,µ(−σ) = −Ab a,µ(σ) − iβ0∂tAb µ(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='20) In the classical limit and physical spacetime, we thus have �eb µ(−x) = eb µ(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='21a) �Eb a,µ(−x) = Eb a,µ(x) + iLβeb µ(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='21b) �Bµ(−x) = Bµ(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='21c) �Ca,µ(−x) = Ca,µ(x) + iLβBµ(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='21d) �Kb µ(−x) = −Kb µ(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='21e) �Kb a,µ(−x) = −Kb a,µ(x) − iLβKb µ(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='21f) where βµ ≡ β0uµ, and Lβeb µ = β0∂µub, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='22a) LβBµ = β0∂µµ + βν(Fνµ + 2eb [µ∂ν]ϕb), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='22b) LβKb µ = ∂µ � βν(∂νϕb + Ab ν) � + βνF b νµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='22c) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Ideal hydrodynamics To describe ideal hydrodynamics, it is convenient to first introduce the single-time equilibrium action [60]: S0 = � dd+1xP(eb t, Bt, Kb t , f IJKb IKc J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='23) Then, the factorizability condition leads to IEFT,eq = � dd+1xLeq = S0[Λ1] − S0[Λ2], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='24a) Leq = peµ b Eb a,µ + nuµCa,µ + ˆπbuµEb a,µ + ψµ b � Kb a,µ − Ec a,µeν cKb ν � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='24b) where we used eµ a,α = −eν αeµ βeβ a,ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The coefficients in Leq define the equation of state, p ≡ P, n = β0 ∂P ∂Bt , ˆπb = β0 ∂P ∂eb t , ψµ b = ∂P ∂Kbµ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='25) where the partial derivatives of thermodynamic pressure P are taken with other arguments being fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' It can be verified that Leq satisfies the KMS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Now, we can read off the equilibrium stress tensor and currents from varying the action with respect to eb a,µ, Aa,µ and Ab a,µ: T µ (0)b = peµ b + πbuµ − ψµ c Kc νeν b, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='26a) 8 Jµ (0) = nuµ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='26b) Jµ (0)b = ψµ b , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='26c) where we defined the momentum density as πb ≡ ˆπb + nϕb ≡ ˆρub + nϕb, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='27) with ˆρ ≥ 0 an O(1) coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We can further express the dipole currents by expanding the pressure up to quadratic terms in field amplitude P ∼ 1 2bK0bK0b − 1 2aibjcKibKjc + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='28) where the invariant tensor is given by aijkl = a1δijδkl + a2δij + 2a3δi[kδl]j, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='29) with A = Aij + Aji − 2 dδijAkk, A[ij] = 1 2(Aij − Aji).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Thermodynamic stability requires that b, a1,2,3 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Thus, we have J0 (0)b = ψ0 b = bK0b, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='30a) Ji (0)b = ψi b = −aibjcKjc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='30b) Let us turn off the gauge field temporarily and plugin (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='26) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Then, we find exactly the momentum and charge conservations, but with an additional dipole constraint: nub = −aibkc∂i∂kϕc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='31) Here, we have neglected the terms with time derivatives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' in fact, as we will see in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4, there is a relaxation time that relaxes non-hydrodynamic modes, which makes (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='31) exact (up to a fluid frame change).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' As a result, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='27) reduces to πb = nϕb + · · · , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='32) up to higher order corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The low energy excitations for the dipole-conserving fluid are not single particle (frac- ton) excitations, but rather propagating dipole Goldstone/density waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In particular, the momentum susceptibility ρ, defined as πb ≡ ρub (different from ˆρ), will be non-local: ρ ∼ n2 k2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='33) and diverges at large distance k → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' see Appendix A for another derivation of non-local ρ using the memory matrix formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' At leading order in amplitude expansion, the conservation equations for δn ≡ n − n0 and πb are ∂0πb + n0 χ ∂iδnδi b = 0, ∂0δn − a n0 ∂2 i ∂jπb = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='34) where we only retained terms at leading order in derivatives and a = a1 +2a2(d−1)/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Upon Fourier transformation, the normal modes are given by ω = ± � a χk2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='35) We therefore find that the dipole “sound” modes are magnon-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Dissipative hydrodynamics and higher-order terms We are now ready to write down the most general (leading order) dissipative part of the effective field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We expand the Lagrangian to containing at most two factors of a-fields L = Leq + L(1) + L (1) + L(2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='36) 9 where the superscript (n) represents the number of a-fields, and we will always keep the leading derivative orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' As mentioned around (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='21), we will restrict to systems whose microscopic dynamics is PT -even dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The dynamical KMS symmetry then implies that L(2) is PT -even and relates it to L(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' moreover it allows the presence of an additional L (1) that is KMS-invariant by itself (and is thus also PT -even).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' It is helpful to first introduce a combined field ∆Ua,ib ≡ ∂iCa,jej b − Ka,ib = ∂i∂jϕaej b + ∂i(Aa,j + LXaAj)ej b + ∂i(Ec a,jϕc)ej b − (Aa,ib + LXaAib), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='37) whose KMS transformation is given by � ∆U a,ib(−x) = −∆Ua,ib(x) − iLβ∆Uib(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='38) where LβUib ≡ β0ej b∂i∂jµ + ∂i(βνFνj)ej b + ∂i(βc∂jϕc)ej b − ∂i(βνAb ν) − βνF b νi ≈ β0ej b∂i∂jµ + β0∂iF0jej b − β0∂0Ab i, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='39) with the second line being the approximation of linear response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' With the benefit of hindsight, we have constructed this field to be independent of ϕb a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Using this field, we can write the most general PT -even L(2) as −iβ0L(2) = σijCa,iCa,j + sibjcEa,ibEa,jc + tibjc∆Ua,ib∆Ua,jc + 2rijkb 1 ∂iCa,j∆Ua,kb + rijkl 2 ∂iCa,j∂kCa,l, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='40) where σij = σδij, σ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In the above action, we only considered leading derivative contributions from a-type fields, and we additionally included first derivative terms in Ca,i as they are of the same order as ∆Ua,ib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' As usual, the terms proportional to Eb a,0, Ca,0 can be eliminated by field redefinition as shown below, and, at the same time, Kb a,0 ∼ ∂tϕb is neglected as it is subleading to the last term which contains first spatial derivatives of ϕb, and we are keeping into account the scaling ω ∼ k2 found in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Under KMS transformations, we require [L(2) + L′(1) − ( ˜L(2) + ˜L′(1))]O(a) = 0, where O(a) indicates the (first) order of a-fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Since L(2)|O(a) = 0, the constraint reduces to L′(1) − ˜L′(1)|O(a) = ˜L(2)|O(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Next, we redefine L(1) ≡ 1 2(L′(1) − ˜L′(1))O(a) = 1 2 ˜L(2)|O(a), thus, by construction, L(1) is PT -odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This leads to β0L(1) = − σijLβBiCa,j − sibjcLβeibEa,jc − tibjcLβ∆Uib∆Ua,jc − rijkb 1 ∂iLβBj∆Ua,kb − rkbij 1 Lβ∆Ukb∂iCa,j (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='41) − rijkl 2 ∂iLβBj∂kCa,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' From the above discussion, the effective Lagrangian allows a PT -even L (1) that itself remains invariant under KMS transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This is given by β0L (1) = dibjc (Lβeib∆Ua,jc − Ea,ibLβ∆Ujc) + f ijkb (Lβekb∂iCa,j − Ea,kb∂iLβBj) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='42) which describes non-dissipative dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' So far, the effective field theory is general and complete, but we will see below that simplifications can be made by ignoring certain higher order corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='31), we see that the dipole Ward identity is not a conservation law but a force balance equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We now show that the associated field ϕb a can be eliminated and still preserve locality of the effective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Indeed, by integrating out Xi, we obtain 0 = ∂0 � Ca,i + ˆρ n0 Ea,0beb i + · · · � = ∂0 � ∂iϕa + Aa,i + LXaAi + ϕa,beb i + Eb a,iϕb + ˆρ n0 Ea,0beb i + · · · � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='43) where the dots include higher derivative orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' As all the fields are set to be zero at spacetime infinity, the expression in the bracket is also zero, and since ϕb a appears without derivatives we can eliminate it from the effective action without generating non-local terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Importantly, the combined field ∆Ua,ij does not contain ϕb a, so we simply need to replace Ca,i: Ca,i = − ˆρ n0 Ea,0beb i + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='44) 10 After such replacement, the possible additional effective Lagrangian can be added is β0L(1) addition ∼ −ALβB0Ca,0 − BLβe0,bEa,0b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='45) Now, suppose that we are able to shift the r-fields through6 µ → µ + δµ, ϕb → ϕb + δϕb, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='46) then the correction to L(1) from Leq is given by δrLeq ∼ δn0Ca,0 + n0δϕbEa,0b, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='47) where δn0 = δµ∂µn0 + δϕb∂ϕbn0, and we have neglected the contribution to the bulk viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We find that if we choose the field redefinition as δn0 = β−1 0 ALβB0, δϕb = (n0β0)−1BLβe0,b, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='48) then the additional Lagrangian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='45) can be eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This indicates that terms proportional to Ca,i, ∂iCa,j can be safely ignored as a change of frame, and the effective Lagrangian becomes −iβ0L(2) = sibjcEa,ibEa,jc + tibjc∆Ua,ib∆Ua,jc, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='49a) β0L(1) = −sibjcLβeibEa,jc − tibjcLβ∆Uib∆Ua,jc, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='49b) β0L (1) = dibjc (Lβeib∆Ua,jc − Ea,ibLβ∆Ujc) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='49c) In parallel, if we do not integrate out Xi, we find that the coefficient σ associated with Ca,i in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='40) gives the relaxation of a non-hydrodynamic mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' By varying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='41) with respect to ϕb a, we obtain the leading-order dipole Ward identity b∂2 0ϕb − aibjc∂i∂jϕc = nub − σ � ei b∂iµ + ∂0ϕb � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='50) Clearly, ∂0ϕb acquires a relaxation time τ: τ ≡ b σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='51) Therefore, on a finite time scale τ, ∂0ϕb relaxes to ub (schematically) – thus they are not independent degrees of freedom in our hydrodynamic limit (t → ∞): ϕb is the hydrodynamic mode corresponding to momentum density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We thus understand that σ is not the usual transport coefficient but determines the relaxation rate for the dipole Ward identity to become a force balance equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This allows us to ignore it in the hydrodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Hence, we see that Xi is not necessarily a physical degree of freedom (at finite density).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' What is non-trivial is that the momentum density, which ordinarily would be ∂0Xi, is approximately proportional to the dipole Goldstone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' It seems non-trivial to uncover the ultimate structure we have found without introducing these extra degrees of freedom, but it may be possible to achieve this in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In particular, we find it most instructive to couple this theory to geometry (see Appendix B) in the presence of such additional degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The derivative expansion of the stress tensor and currents are obtained from variation of L(1) + L (1), which leads to (neglecting nonlinear terms, which will not be relevant for the remainder of this section) T ib (1) = −sibjc∂juc − dibkl (∂k∂lµ + ∂kF0l − Akcδc l ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='52a) Jib (1) = tibkl (∂k∂lµ + ∂kF0l − Akcδc l ) − djcib∂juc, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='52b) where ub is fixed by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In a rotationally invariant theory, we have sijkl = ζδijδkl + ηδij, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='53a) tijkl = t1δijδkl + t2δij (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='53b) dijkl = d1δijδkl + d2δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='53c) 6 Since Xi has been integrated out, uµ is not a low-energy degree of freedom to which the field redefinition can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 11 From the unitarity of the effective action, ImL(2) ≥ 0, we find that the dissipative coefficients satisfy the following positivity constraint, ζ, η, t1, t2 ≥ 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='54) while d1,2 are unconstrained and non-dissipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Let us now analyze the normal modes around a homogeneous background charge density n0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We also turned off the background fields for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Treating the deviation δn = n − n0 and ϕb as small, we obtain the derivative expansion of the pressure as p ≈ p0 + ∂P ∂Bt |Kbµ,ei tδBt + 1 2 ∂P ∂Kb i |Bt,ei t∂iϕb + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='55) ≈ p0 + � χ−1n0δn + 1 2 ∂2p0 ∂n2 0 (δn)2 � − 1 2 � aibjc + ∂aibjc ∂n0 δn � ∂iϕb∂jϕc + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' , where χ = ∂n ∂µ is the normal charge susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Hence, the stress tensor and currents are given by T ib ≈ � p0 + χ−1n0δn + 1 2 ∂2p0 ∂n2 0 (δn)2 − 1 2akdjc∂kϕd∂jϕc � δib (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='56a) − ajikcϕb∂j∂kϕc + aicjd∂jϕd∂kϕcδkb + T ib (1) + τ ib, Jib ≈ − � aibjc + ∂aibjc ∂n0 δn � ∂jϕc + Jib (1) + ξib, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='56b) where τ ib, ξib are the noise whose variance satisfy the fluctuation-dissipation theorem: ⟨τ ib(x)τ jc(0)⟩ = 2T0sibjcδ(d+1)(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='57a) ⟨ξib(x)ξjc(0)⟩ = 2T0tibjcδ(d+1)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='57b) The expressions for T ib (1) and Jib (1) are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='52) with all the coefficients taking their equilibrium values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' we can neglect the non-dissipative terms proportional to d1,2 since they are sub-leading corrections to the ideal hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' So far, we have included the leading order derivatives and only kept non-linear terms in the ideal hydrodynamics because the non-linearity in dissipative coefficients are irrelevant [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Plugging (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='56) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='18) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='31) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='32), we obtain equation of motions: n0∂0ϕb + χ−1n0∂iδnδib + λn0δn∂iδnδib + 2aicjd∂i∂jϕd∂[kϕc]δkb + n−1 0 sibjcakcld∂i∂j∂k∂lϕd + ∂iτ ib = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='58a) ∂0δn − aijkb∂i∂j∂kϕb − ¯λijkb∂i∂j(δn∂kϕb) + χ−1tijkl∂i∂j∂k∂lδn + ∂i∂jξibδj b = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='58b) where we denoted λ = n−1 0 ∂2 n0p0, ¯λijkb = ∂n0aijkb as the non-linear coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In the above equation, we have neglected the higher order time derivatives and assumed that all the coefficients upon expansion do not depend on δn and ϕi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The normal modes are defined as non-vanishing solutions to the equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' By neglecting the non-linear terms, we obtain ω = ± � a χk2 − i � t χ + 1 n2 0 (Γ1 + Γ2) � k4, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='59a) ω = −iΓ1 n2 0 k4, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='59b) where a = a1 + 2d − 1 d a2, t = t1 + 2d − 1 d t2, Γ1 = (a2 + a3)η, Γ2 = ζa1 + 2d − 1 d (ζa2 + ηa1) + � 3 − 8 d + 4 d2 � ηa2 − ηa3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='60) Therefore, we find two longitudinal propagating mode with magnon-like dispersion relation ∼ ±k2 and attenuation ∼ −ik4 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='59a), and transverse subdiffusive modes ∼ −ik4 with multiplicity d − 1 in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='59b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Relevant perturbations in low dimensions Following [35], we give a zeroth-order scaling analysis on the nonlinear effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The dissipative scaling ω ∼ k4 causes the noise to scale as τ ib, ξib ∼ k(d+4)/2 based on (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' To match the scaling to the dynamical terms with time derivatives, we find ϕb ∼ kd/2−1 and δn ∼ kd/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' As per the usual renormalization group analysis, the nonlinear coefficients would scale as λ, ¯λijkb ∼ k(4−d)/2, making them relevant when d < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' As a consequence, we expect the true IR fixed point to have anomalous dissipative scaling: ω ∼ ±k2 − ikz, with z < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We crudely estimate z by assuming that the thermodynamic field does not renormalize due to its Gaussian fluctuations at long wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Then, requiring λ, ¯λijkb ̸= 0 to not depend on k at fixed point, we obtain z = d/2 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We emphasize that the critical exponent z has been testified numerically in [35] with excellent agreement in d = 1, 2, suggesting a breakdown of hydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Meanwhile, naive scaling analysis also implies that the transverse subdiffusive modes would acquire an anomalous scaling ω ∼ −ikz′ with z′ = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We hope to report on a 1-loop analysis of the corrections to hydrodynamics in the near future in order to further investigate these claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Interestingly, there was previous work on a stochastic molecular-beam-epitaxy (MBE) process [61] which shares some similarity with our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In [61], the authors study ∂th + ∇2 (∇h)2 + ∇4h + noise = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='61) This equation, like our theory, has z = 4 subdiffusion at the linear level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' However, the authors of [61] did not demand invariance of the renormalized theory under dipole shift h → h + c0 + c1x, and as such they argued that while (like us) the critical dimension d = 4, they instead found a distinct fixed point with z = (8 + d)/3, which we understand as coming from fixing the scaling dimension of noise, rather than of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' It may be possible to interpret the results of [61], in light of our work, as highlighting the possibly “accidental” appearance of the same dynamical fixed point (KPZ) in both a hydrodynamic setting (Navier-Stokes equations in 1d), and as a model of surface growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' As our symmetries and anticipated scaling exponents differ from [61], the analogy between surface growth and hydrodynamics with momentum conservation may break down in multipole-conserving theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' SPONTANEOUS SYMMETRY BREAKING In this section, we discuss a subtle yet very interesting feature of the dipole and momentum conserving fluid: the identification of ϕb as a Goldstone boson of the dipole field, and the corresponding intuition that dipole symmetry is (by many reasonable definitions) spontaneously broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Let us first briefly justify the identification of ϕb as a Goldstone boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Under dipole shift symmetry, ϕb → ϕb +cb, just as a conventional Goldstone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Moreover, if we wanted to consider (even in the absence of momentum) the spontaneous symmetry breaking of the dipole charge [62], this could be achieved by adding ∇ϕb∇ϕa,b terms to the Lagrangian (this will be discussed further elsewhere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Recent papers on dipole symmetry breaking in the absence of momentum conservation include [63, 64], while [45] discusses the role of dipole symmetry breaking in a translation invariant state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Lastly, as we will see, the dynamics of the collective and long-lived ϕb mode saturate Goldstone’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' For these three reasons, we will call ϕb the Goldstone boson associated with dipole symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' There are three notions through which prior authors have justified the presence of spontaneous symmetry breaking that we are aware of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Briefly, and we will elaborate more on each in subsequent sections: (1) The physical state of interest is not invariant under the action of the global symmetry group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2) the existence of a Goldstone boson which saturates Goldstone’s Theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (3) the presence of long-range order in expectation values of operators that shift under the symmetry (Mermin-Wagner Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In a nutshell, the theory of interest here is compatible with all three definitions in spatial dimensions d > 2, but only with the first two when d ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In low dimensions, our theory thus seems to represent an unusual paradigm not encountered before, where many (but not all) of the usual features of SSB exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We believe that it is appropriate to consider dipole symmetry as spontaneously broken, but leave it to future authors to more firmly settle the possibly semantic question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We conclude this section with a discussion of a quantized version of Model A from [35], which will give a concrete example of the more abstract ideas discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Mermin-Wagner Theorem First, we focus on a physically transparent test for spontaneous symmetry breaking: correlation functions of the order parameter πi(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' A microscopic argument along these lines was presented in [45], and previously in [62] in the absence of momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' While this operator is charged under dipole transformations, it transforms 13 nonlinearly: πi → πi + ci, where ci is an arbitrary vector parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='7 To conclude that dipole symmetry is sponta- neously broken, one often asks whether πi is a well-defined order parameter: any finite value (including zero) will be sufficient to conclude spontaneous breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' From the discussion around (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='33), the equal-time two-point function of πi diverges at low momenta: ⟨πi(k)πj(−k)⟩ ∼ 1 k2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1) Let us consider the average momentum density on a region of linear size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The fluctuations of the average momentum density on this region scale as � (πi − ⟨πi⟩)2� L ∼ � � � L d = 1 log L d = 2 constant d > 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2) This means that the momentum density is well-defined as a thermodynamic variable only for d > 2, in which case (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4) implies spontaneous symmetry breaking of dipole transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' For d ≤ 2, fluctuations are too large to make sense of the expectation value of πi, and therefore we cannot conclude that there is spontaneous symmetry breaking from this perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Goldstone’s Theorem Let us now discuss how the classic Goldstone’s Theorem can be generalized to our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Consistency with the algebra (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1) demands the following commutation relation between dipole and momentum density: [Di, πj] = inδj i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3) On a thermal state ρ0 at finite background charge density n0, we have tr(ρ0[Di, πj]) = in0δij ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4) For concreteness, we shall take ρ0 to be microcanonical with respect to momentum and charge, and canonical with respect to energy (if the latter is conserved);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' see an explicit example in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' As we discussed in the previous section, πi has large fluctuations in low dimensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' nevertheless the expression on the left-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4) can still be well-defined as we demonstrate in the explicit example of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We now show how this relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4), entirely dictated by symmetry, will imply the existence of a Goldstone mode [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Let us start by writing the total dipole charge in terms of the charge density operator Di = � ddx xin,8 which allows us to express the above as � ddx xiGR nπj(t, x) = −n0θ(t)δij, GR nπi(t, x) = −iθ(t) tr(ρ0[n(t, x), πi(0, 0)]) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5) where GR nπi is the retarded two-point function of charge and momentum densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Doing a Fourier transform leads to lim ⃗p→0 ∂ ∂pi ImGR nπi(ω, p) = −n0πδ(ω)δij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='6) We see that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='6) implies a zero-frequency contribution to the spectral density Jnπi as a direct consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Using the approach of Kadanoff-Martin and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='34), we now show that this spectral weight is entirely captured by a single hydrodynamic mode: the “magnon-like” sound mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We can obtain the retarded two-point function GR nπi (without missing any important counterterm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Doing a Laplace transform � −izδij n0 χ iki ikjk2 a n0 −iz � � πj(z, p) δn(z, p) � = � π(0) i (p) δn(0)(p) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='7) 7 In this subsection we will be agnostic of the global structure of the dipole group and of possible subtleties related to boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We will describe these in more detail for a specific model in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 8 The total dipole moment Di could in principle receive further contributions from “bond dipoles”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' degrees of freedom with an internal dipole charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This is entirely analogous to the contribution of orbital angular momentum and spin to the total angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We shall not investigate this situation here, but note that these bound dipole degrees of freedom are not conserved and have a finite lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Therefore it is unlikely they would contribute to the singular spectral weight we calculate below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 14 where the right-hand side represents densities configurations at the initial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The initial density configuration δn(0) is related to a perturbation in the chemical potential at initial time through δn(0) = χδµ, so we find ∂πi(z, p) ∂δµ(p) = −n0iki −z2 + a χk4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='8) From linear response we know that, for a density δna conjugated to a chemical potential δµa, izδna(z, k) = (GR ab(z, k)− GR ab(0, k))δµb(k) and GR ab(0, k → 0) = −χab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We then find GR nπi(ω, p) = −n0ωki −ω2 + a χk4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='9) At leading order in ki, we have Im GR nπ(ω, p) → −πn0kiδ(ω) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='10) which can be obtained by taking into account that dissipative corrections in the retarded two-point fuction contribute through ω → ω + iε, where ε → 0 as k → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We then see that we recovered eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This contribution comes from the pressure term, and is thus present in any fluid which conserves momentum and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Hence we call ϕb a Goldstone mode: this IR mode by itself saturates the requisite Goldstone’s Theorem for dipole symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The fact that in the EFT, ϕb did not have a diagonal shift symmetry like ϕ and Xi, further suggests that we should interpret ϕb as a Goldstone mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In previous hydrodynamic effective field theories such as [2, 5], when this diagonal shift symmetry is not present, the conclusion has been that the corresponding continuous symmetry is spontaneously broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' At charge neutrality n0 = 0, the Goldstone’s theorem become trivial, and we will not have such dipole Goldstones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Existence of a symmetry-breaking state Finally, we now show that a system with dipole and momentum symmetry can possess a symmetry-breaking ground state even in one and two space dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The dipole and momentum algebra is isomorphic to the Heisenberg algebra of position and momentum (in a microcanonical ensemble where Q is fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In textbook quantum mechanics (and in models interest here, as discussed below), the physical Hilbert space contains no trivial representation of this algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Therefore it is not possible to construct a state which is invariant under the symmetry group, and one can say that the symmetry must be broken, either explicitly or spontaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' A physical example of another system that has a (sub)algebra isomorphic to dipole and momentum is a Galilean- invariant fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This is slightly different because the commutator of [H, D] ̸= 0 in the Galilean-invariant fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Nevertheless, recent papers [46, 47] have argued that Galilean boosts are spontaneously broken – in any dimension – by the fluid rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We believe this is most succinctly justified by the argument in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We discuss an interesting relation between our dipole fluids and the Galilean-invariant fluids in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' One challenge in showing whether the ground state is invariant or not, is that historically this was done by a careful consideration of finite volume regularization [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' When compactifying space onto a toroidal lattice with finite length in each direction, the dipole conservation law is isomorphic to Z, not R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='9 This regularization is not acceptable for us here since it qualitatively changes the nature of the symmetry group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' we seek an alternative regularization that manifestly preserves the Heisenberg algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Model A of [35] provides us with a concrete model which we can suitably regularize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The Hamiltonian for Model A is H = N−1 � i=1 �(pi − pi+1)2 2 + V (xi+1 − xi) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='11) with [xi, pj] = iδij conventional position and momentum operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We can view this as a chain where each site i possesses an infinite-dimensional Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Alternatively, we can interpret this as a generalization of an atomistic Hamiltonian modelling a solid in one dimension, where xi and pi describe displacements and momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Similar models 9 We thank Nathan Seiberg for emphasizing the points in this paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 15 (without dipole conservation) are known to capture hydrodynamics (and its breakdown) in one dimension [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The details of V are not important for our discussion, though we would likely consider a function Taylor expanded about a minimum at argument x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' One can easily see that H commutes with the dipole and momentum operators of the theory: P = N � i=1 pi, D = N � i=1 xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='12) Note that this theory is in a microcanonical ensemble for charge: Q = N is a fixed c-number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' To see that the thermodynamics is well defined, we take ρ0 = e−βH restricted to superselection sectors satisfying � i pi = P and � xi = D fixed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' we also take lattice constant a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Upon changing variables to si = pi − pi+1, ri = xi − xi+1, the Hamiltonian becomes H = �N−1 i=1 1 2s2 i + V (ri).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The classical partition function is then (up to an overall constant) � �N−1 i=1 dsidrie−βH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The partition function factorizes into manifestly convergent integrals, and is therefore finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In quantum mechanics, the partition function will also be finite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' this is most easily seen by working in a plane wave basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We additionally notice that due to the commutator [D, pi] = i, tr(ρ0[D, pi]) is finite, thus showing that the atomistic analogue of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4) is well-defined in the present model, despite the large fluctuations of momentum density discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Let us now look in more detail at the quantum ground state of Model A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This will be a wave function of the form ψgs(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' , xN) = ψ0(x1 − x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' , xN−1 − xN) · eik(x1+···+xN) e−(x1+···+xN)2/2Na (2πa)1/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='13) This wave function has an eigenvalue independent of k and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' If we take a = ∞ above, the wave function is a non-normalizable eigenstate of P, but not D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Clearly no pure state can be found that is an eigenstate of both P and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' By taking a < ∞, the above state is normalizable and physical, with controllably small fluctuations in the value of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We don’t think the situation improves much for mixed states: the only possibility invariant under both P and D is the “identity matrix” in the center of mass coordinate, but it is dubious whether (even representing just one coordinate in the N → ∞ limit) such a highly non-normalizable state could be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This paragraph simply re-states, in a concrete model, what we already noted – it is not possible to simultaneously diagonalize D and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Is this a physical effect?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Since the center of mass coordinates completely decouple from H, they can arguably be removed from Hilbert space entirely without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' What remains in the Hilbert space are the long-lived fluctuations of πi(k), which are subject to the same Mermin-Wagner fluctuations mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' There is therefore no notion of long range order measurable by correlations of local operators that are not group-invariant: in d = 1, ⟨(p1 − pN/2)2⟩ ∼ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' To throw one more wrench into the mix, however, there is a critical difference between SSB of compact U(1) and non-compact dipole symmetry (alone, isomorphic to R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In the U(1) case, the Goldstone ϕ is not singly-valued: one must look at the well-posed operators Am = eimϕ for m ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' These operators have ⟨Am⟩ = 0 implying no SSB for d ≤ 2 in any physical ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' However for dipole symmetry, the global momentum P and its average πi(k = 0) = 1 N P are perfectly well-defined operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We have constructed normalizable (ground!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=') states in which the physical operator ⟨π⟩ takes on a finite value, which shifts under dipole transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In this sense, the large fluctuations assured by the Mermin-Wagner Theorem seem less dangerous as in a conventional U(1) superfluid, and in fact the Mermin-Wagner theorem is argued to only apply to compact symmetry groups [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' However, like in the superfluid, there can be no long-range coherence in π, which exhibits large local fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We leave open the question of whether this is a semantic or a crucial physical difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' HYDRODYNAMICS WITH ENERGY CONSERVATION We now discuss how to incorporate energy conservation into our hydrodynamic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We will need another hydrodynamic degree of freedom X0(σt, σI) that non-linearly realizes the time translation P 0, where now σt is the proper time in the fluid’s local rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This leads to an additional invariant building-block in the fluid spacetime defined as e0 s,A(σ) = ∂Xµ s (σ) ∂σA e0 s,µ(σ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1) In order to distinguish from solid phases, the effective theory must be invariant under time-independent reparametriza- tions of the proper time σt, σt → σt + f(σI), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2) 16 which implies that the invariant r-field without derivatives can only be e0 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We then introduce the proper temperature β ≡ e0 t, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3) as the thermodynamic temperature10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In the classical limit and physical spacetime, the corresponding a-field can be written as e0 a,A = ∂AXµE0 a,µ, E0 a,µ = e0 a,µ + LXae0 µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4) Then, we can include a term T µ 0 E0 a,µ into the leading effective Lagrangian (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='17), and by variation with respect to X0 a, we get a new Ward identity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' the energy conservation equation, ∂µT µ 0 + Ab 0eµbJµ − F b 0µJµ b − F0µJµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5) The dynamical KMS transformation for the energy variables is given by ˜e0 µ(−x) = e0 µ(x), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='6a) ˜E0 a,µ(−x) = E0 a,µ(x) + iLβe0 µ(x), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='6b) with Lβe0 µ = ∂µβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We now generalize the fluid Lagrangian to include energy conservation, focusing on quadratic order in perturbations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=', linear response), and switching off background fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Keeping into account dynamical KMS symmetry, the new terms in the total Lagrangian that depend on E0 a,µ are Lε = Leq,ε + L(1) ε + L(2) ε , with Leq,ε = −(ε + p)uµE0 a,µ + peµ 0E0 a,µ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='7a) −iβL(2) ε = κijE0 a,iE0 a,j + 2αijCa,iE0 a,j, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='7b) βL(1) ε = −κijLβe0 i E0 a,j − αij(Ca,iLβe0 j + LβBiE0 a,j), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='7c) where κij = κδij is the isotropic thermal conductivity, αij = αδij, α, κ ≥ 0, and where we defined the energy density through ε + p = −β ∂P ∂β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In the above, we are taking ε = ε0 + δε and p = p0 + χ−1 ε (ε0 + p0)δε + χ−1n0δn, where χε = −β ∂ε ∂β is the specific heat, and ε0, p0 are background values of energy density and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The other possible O(a2) contributions in L(2) ε come with a time derivative, so we can eliminate them through a frame redefinition as explained below (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' At linear order, the ideal charge and dipole currents in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='26) are not modified, and thus we can still use eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='31) to eliminate ui in favor of the dipole Goldstone ϕb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' On the other hand, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='43) is updated to 0 = ∂0 � Ca,i + ˆρ n0 Ei a,0 − ε0 + p0 n0 E0 a,i � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='8) where the last term above comes from the first term in Leq,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Eliminating Ca,i will produce terms proportional either to Ei a,0, or to E0 a,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The former contributions can be eliminated through a frame redefinition, following similar steps to the discussion below (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The latter will renormalize the value of the thermal conductivity κ (preserving dynamical KMS invariance) and can also be eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='11 We can thus set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' By varying with respect to the e0 a,µ, we find the equilibrium energy-4-current as T µ (0)0 = −εuµ − p(uµ − eµ 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='9) Unlike the momentum density, the energy density is still governed by the fluid elements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' on the algebraic level, this is because the time translation and the dipole symmetry commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The dissipative part is the usual temperature gradient: T i (1)0 = −κijβ−1∂jβ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='10) 10 All the β0 appeared in the previous sections must be taken to be β(σ) as a function of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We automatically take that into account in the following derivation without lengthy repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 11 To see this, note that LβBi, appearing in the last term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='7c), can be replaced using n0LβBi = (ε0 + p0)Lβe0 i , which can be inferred from the ideal part of the momentum conservation equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 17 and the noise contribution to the energy current is τ i, with ⟨τ i(x)τ j(0)⟩ = 2T0κijδ(d+1)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We now analyze the normal modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Keeping into account corrections to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='56), T ib (0),ε ≈ χ−1 ε (ε0 + p0)δε δib, and substituting the expressions found above into the Ward identities, we obtain equation of motions for the hydrodynamic modes and the dipole Goldstone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The complete expression of the normal modes is complicated due to the additional coupling to the energy sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Thus, we do not present the full solutions here for the sake of conciseness but emphasize on few consequences after including the energy sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' First, the d − 1 subdiffusive modes in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='59b) continue to exist but with a different subdiffusion constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Second, the energy diffusion will generically mix with the magnon-like sound mode since they both scale as ∼ k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' To see it, we keep the dissipative hydrodynamics in the energy sector but ideal hydrodynamics in the momentum and charge sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The resulting equation of motion is given by, in the linear response, ∂0δε + n−1 0 (ε0 + p0)aijkc∂i∂j∂kϕc − χ−1 ε κij∂i∂jδε = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='11a) n0∂0ϕb + χ−1n0∂iδnδib + χ−1 ε (ε0 + p0)∂iδεδib = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='11b) ∂0δn − aijkb∂i∂j∂kϕb = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='11c) After doing a Fourier transform, the normal modes are the solutions of the following cubic equation ω3 + i κ χε ω2k2 + � a χε �ε0 + p0 n0 �2 − a χ � ωk4 − i aκ χχε k6 = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='12) The resulting modes are 3-fold: two propagating modes ω = ±ck2 − iγk2 and one diffusion mode ω = −iγ′k2, where explicit expressions for c, γ, γ′ are not illuminating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The propagating modes still have the magnon-like dispersion but the leading dissipative contribution is now ∼ −ik2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The diffusion mode is reminiscent of the energy diffusion in a normal fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We conclude by noting that, because of the diffusive nature of the three longitudinal modes above, it is not clear a priori whether the hydrodynamic instability and the associated non-Gaussian universality class emergent at long wavelengths found in [35] will survive at long times, or whether it will be replaced by a different universality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This is an interesting question that we leave for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' FROM GALILEAN SYMMETRY TO DIPOLE SYMMETRY Lastly, let us point out an interesting connection between the physics described above, and the m → ∞ (infinite mass) limit of the Galilean symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This was described in a different formalism in the recent work [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Consider a system preserving both time H and space Pi translations, U(1) charge Q, and Galilean boost Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Similar to the dipole symmetry, Galilean boosts are also broken spontaneously in hydrodynamics [46, 47] due to the following algebra: [Ki, Pj] = imQδij, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1a) [Ki, H] = iPi, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1b) where m is the mass of underlying particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Due to the second commutator above, we note that the resulting algebra is distinct from the dipole-momentum algebra we discuss in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Still, observe that in conventional liquids and gases there are no obvious “Goldstone bosons”: the hydrodynamics are described by simply charge, energy and momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='12 Following general constructions in Appendix B, we obtain the invariant blocks in the flat spacetime as eµ A = ∂AXµ − ∂AX0δµ i ηi, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2a) V i A = ∂Aηi, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2b) BA = ∂Aϕ + m∂AXiηb − 1 2m∂AX0ηiηi, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2c) 12 In the literature it is remarked that the Goldstone of broken boosts is the conventional sound wave [47], but we remark that even without any boost or rotational symmetries, such sound waves still exist [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' So the sound wave is not crucially reliant on the broken continuous boost symmetry, while the “sound mode” of the dipole-momentum theory cannot be disentangled from symmetry breaking, as far as we can tell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 18 where we associated the Goldstone ηi to the Galilean boost symmetry Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Following the construction described in Section 2, the r-fields invariant under relabeling symmetries are eµ t , Bt and V i µ (upon index contraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We can also derive the a-fields in the classical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' For our purpose, we will ignore nonlinear terms associated with Xµ a but keep relevant terms for ηi a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The resulting a-fields are eµ a,A = ∂AXνEµ a,ν, Eµ a,ν ≈ ∂νXµ a − δ0 νδµ i ηi a, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3a) V i a,A = ∂AXµV i a,µ, V i a,µ = ∂µηi a, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3b) Ba,A = ∂AXµCa,µ, Ca,µ ≈ ∂µϕa + mδi µηa,i − mδ0 µηiηa,i, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3c) To the leading order in a-fields, the effective Lagrangian is given by L = ˆT µ ν Eν a,µ + JµCa,µ + W µ i V i a,µ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4) Varying with respect to ηa,i, we obtain the boost Ward identity − ˆT 0 i + mJi − mJ0ηi − ∂µW µ i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5) Writing the stress tensor and currents in terms of thermodynamic variables, ˆT 0 i ∼ ui + ηi, Ji ∼ n0(ui + ηi), J0 ∼ n0 and ∂µW µ i ∼ O(∂2ηi), we find that ηi ∼ ui + O(∂2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='6) This immediately tells that the boost Goldstones are redundant degree of freedoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' More explicitly, if including the first-order dissipative effect βL(1) ∼ −ΓLβei 0Ei a,0 ⊃ −Γβ∂0ηiηa,i, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='7) we see that the boost equation of motion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5) will be damped at the timescale ∼ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Therefore, the broken boost does not provide additional hydrodynamic modes, and the long-time dynamics is equivalent to an ordinary (boost- invariant) charge fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' As a bonus, by setting ηi = 0 (as it will decay to this value), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5) implies that the momentum density is equal to the mass current: T 0 i |ηi=0 ≡ � ˆT 0 i + mJ0ηi � |ηi=0 = mJi|ηi=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='8) Note however that, as in the dipole fluid, the Galilean boost with parameter ci causes a shift to the momentum density: T 0 i → T 0 i + mn0ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='9) This is well known from textbook fluid mechanics – it is simply the statement that hydrodynamics is the same in all inertial reference frames, and that momentum transforms from one such frame to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The fact that [Ki, Pi] ∼ Q is mathematically analogous to [Di, Pi] ∝ Q explains why the dipole shift symmetry causes such a similar transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Unlike the dipole symmetry however, [Ki, H] ̸= 0 and this causes the ηi “dipole Goldstone” to also show up in Eµ a,0 as well as its quadratic form in Ca,0, which causes ηi to relax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Remarkably, there exists an well-defined limit for the Galilean boost symmetry – taking the infinite mass limit m → ∞ and keeping Di ≡ Ki/m (but not Ki itself) a good symmetry generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' To keep everything in the same order, the boost Goldstone needs to be scaled as ϕi ≡ mηi since ηi ∼ m−1 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In this limit, the boost algebra reduces exactly to the dipole algebra by identifying Di and ϕi as the dipole generator and Goldstone correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Moreover, by defining ∂µJµ i ≡ ∂µW µ i /m as the divergence of dipole current and upon charge conjugation, the boost Ward identity (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5) becomes the dipole Ward identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='18c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Consequently, the relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='6) is incomplete at the leading order since ηi → 0, and going to the next leading order, we find it reproduces the dipole constraint (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' So long as Γ remains finite, the dissipation term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='7) vanishes since ηa → m−1ϕa → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The (dipole) Goldstone can no longer be integrated out;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' in fact, as seen before, it is the velocity which becomes redundant13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This reveals how the algebraic observation that the Galilean algebra becomes the multipole algebra at m = ∞ is realized in the EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We are not sure whether or not existing methods [71–75] used to study “non-relativistic conformal field theories” (with Galilean symmetry) can be neatly used in the infinite mass limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' this could be a fruitful direction for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The physics discussed above might be useful to study physics in strongly correlated flat bands, in the regime where the infinite mass limit is exact [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 13 The number of degrees of freedom is nevertheless the same in both cases, and it is smaller than that to start with – we are losing d redundant modes in spatial dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The reduction of the true degrees of freedom is a common feature (not yet proven) for spacetime symmetry breaking based upon the inverse Higgs mechanism [67–70], therefore, it is interesting to understand to what extend can we generalize this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' CONCLUSIONS In this paper, we have developed an effective field theory for fluids with dipole and momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Our construction highlights a few subtle aspects of this problem: in particular, it appears necessary to write down a local action in terms of more degrees of freedom than are actually present in the effective theory, despite the absence of an obvious need for Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' A crucial observation that arises out of this construction is that the dipole symmetry is generally spontaneously broken, and that the Goldstone boson for this broken symmetry is essentially the momentum density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The fact that dipole symmetry is spontaneously broken is also found in [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Unlike in that reference however, we find this conclusion is deeply related to having momentum conservation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' without momentum, there is no need to have spontaneously broken the dipole symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' That this effect appears common in the large N models of [77] may be an artifact of the solution method in the large N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' At the very least, there is no evidence for spontaneous symmetry breaking of dipole symmetry in any classical (Markov chain) models of dipole-conserving hydrodynamics studied thus far;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' in contrast, the numerical simulations of [35] did find compelling evidence for the universality class whose field theory was derived in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' It would be interesting to understand this issue further in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' An important lesson from this effective field theory construction was the generalization from flat to curved back- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Here, our approach perhaps differs conceptually from other attempts in the recent literature [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In our approach, we followed recent work [58] which emphasized the importance of using vielbein indices (not spatial indices) to encode conservation laws in curved space: this construction made it possible to couple anisotropic fluids to geom- etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Since in principle it may be desirable to study anisotropic dipole- and momentum-conserving fluids, we expect that such a vielbein construction is also preferable here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' More importantly, by starting with a first-order formulation, it was natural to assert that in curved space one should relax the requirement that the gauge field is a mixed rank object (At, Aij), and to instead simply require a pair of gauge fields Aµ and Ab µ corresponding to charge and dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The mixed-rank gauge field can then only emerge in a flat space limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Moreover, the vielbein formalism helps us to understand two key physical consequences: the existence of the dipole Goldstones and the asymmetric part of the dipole fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Looking forward, we anticipate that our methods can be generalized to discover infinitely many new hydrodynamic universality classes that arise in fracton-like classical or quantum matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' It may be straightforward conceptually, if tedious in practice, to generalize this construction to include higher multipole conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' A more important and interesting direction will be to understand how to generalize the geometrically inspired construction presented here to non-thermal matter – after all, the highlight of this work is the dynamics without energy conservation!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Recent work [34] along these lines has begun, but the consequences or diagnosis of spontaneous symmetry breaking in this new approach have not yet been understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' It will also be fascinating to look for experimental realizations of the dipole and momentum conserving hydrody- namics developed here, whether in high quality solid-state devices in very large electric fields [78], or in low density interacting ultracold atoms in tilted trapped optical lattices [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We also hope that progress along these lines will be made in the next few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' It may also be possible to explore similar (though it appears distinct) fixed points which arise from the symmetry group of volume-preserving diffeomorphisms (which can arise in lowest Landau level physics) [80, 81];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' this algebra is equivalent to the dipole-momentum algebra at the linearized level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We acknowledge useful discussions with Anton Kapustin, Rahul Nandkishore, Shu-Heng Shao, Dam Thanh Son, Lev Spodyneiko, and especially Kristan Jensen and Nathan Seiberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' PG and AL thank the Simons Center for Geometry and Physics for hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This work was supported by the Department of Energy through Award DE-SC0019380 (PG), the Simons Foundation through Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 620869 (PG), the National Science Foundation under CAREER Award DMR-2145544 (XH, JG, AL), the Gordon and Betty Moore Foundation’s EPiQS Initiative via Grants GBMF4302 and GBMF8686 (JFRN), and GBMF10279 (XH, JG, AL), and by Research Fellowships from the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Sloan Foundation under Grant FG-2020-13615 (PG) and FG-2020-13795 (AL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Appendix A: Memory matrix methods In this appendix, we use the memory matrix formalism [82] to derive the linearized hydrodynamics of the main text, both near and away from charge neutrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This approach provides an independent check on many of the non-trivial properties of hydrodynamics that we found above and can give some interesting perspectives on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 20 The memory matrix formalism is an old set of formal manipulations, used to isolate the contributions to linear response theory (two-point functions) which arise from parametrically slow dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Since long wavelength hydro- dynamic modes are arbitrarily long lived, this method can be well-suited for calculations of their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We now tersely summarize the main results of this method: for details see [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Consider a many-body system with Hamiltonian H, at temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' One can construct a vector space consisting of all operators A acting on this system: to emphasize the vector nature, we can write |A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' An inner product on this space is (A|B) := T β � 0 dλ⟨A†(iλ)B⟩T (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1) with T = 1/β and ⟨· · · ⟩T = 1 tr(e−βH)tr(e−βH · · · ) the thermal expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Note that the susceptibility matrix is (A|B) = TχAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2) Suppose that we have a designated set of “slow” operators |OA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' For us, these are naturally taken to be n(k) and πi(k) (the Fourier wave number is k, and is held fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We may define the projectors p = � slowA,B |A)(Tχ)−1 AB(B|, q = 1 − p, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3) which project degrees of freedom onto slow (p) and fast (q) modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' By noticing that (A|(z−L)−1|B) is linearly related to the retarded Green’s function GR AB(z), one can show that there are hydrodynamic quasinormal modes whenever [] det(M + N − iωχ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4) Here M (the memory matrix) and N are given by MAB = ( ˙A|qi(z − qLq)−1q| ˙B), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5a) NAB = −NBA = χ ˙AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5b) Here L = i[H, ·] denotes the Liouvillian, and ˙A = i[H, A], with H the overall Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In this paper, we aim to use this framework to gain further insight (and justification) for the non-trivial hydro- dynamics discovered in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Strictly speaking, one can object to this on the grounds that energy conservation is explicit in any theory satisfying the above postulates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Ultimately, we will use this approach to discern what hap- pens when energy is conserved along with dipole and momentum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' however, we believe that this approach remains instructive even if we “ignore” energy conservation as an unjustified assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' As we will see, some of the confusing features of this fluid are consequences of very general, and even semi-microscopic, arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Momentum susceptibility Let us begin by determining the momentum susceptibility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' in the memory matrix language, this is (π|π) = Tχππ (we’ll leave the Fourier index implicit for the remainder of this section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' While a microscopic computation is not possible (nor important for hydrodynamic considerations), we can easily bound susceptibility using the Cauchy- Schwarz inequality: (πx|πx) ≥ (πx|Jx)2 (Jx|Jx) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='6) Here Jx is the x-component of the charge current operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' for simplicity, we’ll also take k = kˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Now, observe two key properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Firstly, in a generic many-body system, (πx|Jx) = Tn0, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='7) with n0 the equilibrium charge density: n0 = ⟨n⟩T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' A heuristic and fully general argument for this fact, which we have not seen in the literature, is as follows: (πx|Jx)k→0 T = � ∞ 0 dt � ddx V i⟨[Jx(x, t), Px]⟩ = � ∞ 0 dt � ddx V ⟨∂xJx⟩ = − � ∞ 0 dt � ddx V ⟨∂tn⟩ = Tn0 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='8) 21 where in the last step we have used integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This argument is not rigorous because if the k → 0 limit is taken too quickly the integral trivially vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Secondly, using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='18c), (Jx|Jx) = k2(Jxx|Jxx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='9) Since Jxx is the local current operator which is well-defined with local dipole conservation, we conclude that (Jxx|Jxx) is k-independent as k → 0, and should remain finite as n0 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Combining these 3 equations, we find that for some constant c > 0, which does not vanish as n0 → 0, χππ = cn2 0 k2 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='10) in agreement with the result (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='33) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Dynamics without energy Now, let us consider the dynamics without energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In this case, we’ll include πi and n as degrees of freedom in the memory matrix formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The non-zero matrix elements of the N matrix are Nπin = Nnπi = (πi| ˙n) = ikj(πi|Jj) = ikjδijTn0 = T × ikin0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='11) The most important non-zero matrix elements of the M matrix are Mnn = ( ˙n|qi(ω − qLq)−1q| ˙n) = kikjkkkl(Jij|i(ω − qLq)−1|Jkl) = Tαk4, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='12a) Mπiπj = ( ˙πi|qi(ω − qLq)−1| ˙πj) = kkkl(Tik|q|Tjl) = T � βkikj + γk2δij � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='12b) for some constants α, β, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The hydrodynamic normal modes thus come from solving 0 = det(M + N − iωχ), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='13) which in the transverse sector gives ωχππ = γk2, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='14a) ω = −iγk4 cn2 0 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='14b) and in the longitudinal sector gives det � αk4 − iωχnn ikn0 ikn0 (β + γ)k2 − iωcn2 0 k2 − iωA � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='15) Here A > 0 is a finite constant that persists even as n0 → 0, arising from χππ Indeed, if n0 = 0, then we see that both longitudinal and transverse momentum get k2 decay rates, and charge has k4 subdiffusion, while if n0 > 0, we have 0 = k4n2 0 + (αk4 − iωχnn)((β + γ)k4 − iωcn2 0) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='16) which is solved by ω = ± k2 √cχnn − iΓk4 + · · · (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='17) in agreement with the prediction (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='59) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Appendix B: Dipole fluids with momentum in a curved spacetime In this section, we will discuss the change of dipole hydrodynamics by coupling to a curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Such analysis also tells us how to source various currents in the flat spacetime limit as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' It is interesting to notice that because of the non-commutativity between the dipole moment and the momentum operator, the momentum 22 density is not invariant under dipole shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' However, since the energy operator commutes with the dipole moment, the energy density is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Consequently, our theory forbids any type of boost symmetries between space and time, and a natural choice of spacetime to describe such theory is the Aristotelian background [40, 59, 83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' It is most natural to use vielbein formalism to describe the geometry, thus we introduce the internal (flat) spacetime indices α, β and b, c for its spatial subspace (we reserve a to describe a-fields in the Keldysh contour!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The dipole and spacetime algebra is given by [Pb, Dc] = −iQδbc, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1a) [Pd, Lbc] = i(δdcPb − δdbPc), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1b) [Dd, Lbc] = i(δdcDb − δdbDc), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1c) [Lbc, Lde] = i(δbdLce − δbeLcd − δcdLbe + δceLbd), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1d) where Lbc generates the SO(d) rotational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The time translation P0 commutes with every other generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Before constructing the field theory, it is instructive to define the dipole moment operationally as Db ≡ � ddxe yb(xi)n(x), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2) for some arbitrary function yb(xi) in terms of spatial coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In particular, let us assume for now that the coordinates {yb} form the “natural” coordinates where the vielbein would be constant: eµ b = δµ b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Nevertheless, we will not for now invoke any such constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We also emphasize that on a general curved background, yb is not globally well-defined, with the 2-sphere the simplest example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This means that one should only understand the resulting theory as a ‘covariant’ way of coupling a dipole and momentum conserving theory to curved space – the explicit coupling to the metric ends up breaking the conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This phenomenon is not unusual, and is well-known to happen already with momentum conservation on a generic curved background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In what follows, we will treat derivatives on the vielbein at the same order as derivatives on the hydrodynamic fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' thus e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' the curvature scalar R ∼ O(k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The time derivative of the dipole moment implies ∂0Db = � ddxe yb(xi)∂0n(x) = − � ddx yb(xi)∂k(eJk) = � ddxe ∂k(yb(xi))Jk ≜ � ddxe ekbJk, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3) where we have used the covariant charge conservation in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='20b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The last equation defines the vielbein eb k ≡ ∂kyb that give us the transformation from yb to the uglier coordinates xi, such that the dipole moment will be conserved according to the dipole Ward identity in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='20c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We emphasize that the existence of such a choice of viebein is not generic: in particular, the Ricci curvature tensor vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Therefore, only on a flat space can the dipole moment be defined in terms of operators in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' To build the invariant blocks and to include the spontaneous dipole symmetry breaking, we apply the coset construc- tion for the spacetime symmetries [85, 86] (for an introduction to the coset construction, see the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Let us focus on dipole fluids without energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The unbroken generators are the translations Pα, and the charge Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' the broken generator is the dipole moment Db.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Thus, the most general group element is parametrized as g(σ) = eiβ0σtP0eiyb(X(σ))Pbeiϕb(σ)Dbe− i 2 θbc(σ)Lbceiϕ(σ)Q, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4) where σA represent the fluid spacetime, and we associate with each symmetry generator a dynamical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' P0 is an effective time-translation supporting a fixed temperature T0 = β−1 0 determined by the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We introduce the gauge fields for translations (ˆeα µ), rotations (ωb µc), charges ( ˆAµ), and dipole moments (Ab µ) , thus the gauge invariant Maurer-Cartan one-form is given by g−1 � ∂A + i∂AXµˆeα µPb + i 2∂AXµωbc µ Lbc + i∂AXµ ˆAµQ + i∂AXµAb µDb � g = ieα APα+iKb ADb+iBAQ+i1 2Θbc A Lbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5) Hence, the useful invariant blocks are given by eb A = ∂AXµec µR b c , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='6a) BA = ∂Aϕ + ∂AXµAµ + ∂AXµeb µϕb, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='6b) Kb A = � ∂Aϕc + ∂AXµωc µdϕd + ∂AXµAc µ � R b c , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='6c) Θb Ac = ∂AXµ � (R−1)bd∂µRdc − (R−1)bdωe µdRec � , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='6d) 23 where to linear order in θbc the rotation matrix reads Rbc = δbc − θbc, and we defined eb µ ≡ ∂µyb(X) + ˆeb µ + ωb µcyc(X), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='7) Aµ ≡ ˆAµ − Ab µyb(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='8) Clearly, we should regard eb µ as the vielbein, ωb µc = −ωc µb as the spin connection, and Aµ as the U(1) gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The vielbein formalism requires also the Christoffel connection Γρ µν, and the covariant derivative is defined as ∇µe0 ν = ∂µe0 ν − Γρ µνe0 ρ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='9a) ∇µeb ν = ∂µeb ν + ωb µcec ν − Γρ µνeb ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='9b) The spin connection and Christoffel connection are not independent, so we impose the metric compatibility ∇µeα ν = 0, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='10) and treat the vielbeins and the spin connections as the independent background fields with respect to which the action would vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We note a useful relation Γµ νµ = e−1∂νe, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='11) where e ≡ det eα µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' To incorporate the blocks into the Schwinger-Keldysh formalism, we introduce the two-time copies (s = 1, 2): eb s,A(σ) = ∂Xµ s (σ) ∂σA ec s,µ(σ)R b s,c (σ), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='12a) Bs,A(σ) = ∂Xµ s (σ) ∂σA � As,µ(σ) + eb s,µ(σ)ϕs,b(σ) � + ∂ϕs(σ) ∂σA , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='12b) Kb s,A(σ) = ∂Xµ s (σ) ∂σA � Ac s,µ(σ) + ωc s,µd(σ)ϕd s(σ) � R b s,c (σ) + ∂ϕc s(σ) ∂σA R b s,c (σ), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='12c) Θb s,Ac(σ) = ∂Xµ s (σ) ∂σA � (R−1 s )bd∂s,µRs,dc − (R−1 s )bdωe s,µdRs,ec � (σ), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='12d) Restricting to θbc r = 0, the r-fields are eb r,A = ∂AXµeb µ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='13a) Br,A = ∂Aϕ + ∂AXµAµ + ∂AXµeb µϕb, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='13b) Kb r,A = ∂Aϕb + ∂AXµωb µdϕd + ∂AXµAb µ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='13c) Θb r,Ac = ∂AXµωb µc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='13d) In the classical limit and physical spacetime, the a-fields are given by eb a,A = ∂AXµEb a,µ, Eb a,µ = eb a,µ + LXaeb µ + ec µθb a,c, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='14a) Ba,A = ∂AXµCa,µ, Ca,µ = Aa,µ + ∂µϕa + LXaAµ + eb µϕa,b + (eb a,µ + LXaeb µ)ϕb, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='14b) Kb a,A = ∂AXµKb a,µ, Kb a,µ = ∇µϕb a + ωb a,µcϕc + ϕcLXaωb µc + Ab a,µ + LXaAb µ + Kc µθb a,c, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='14c) Θb a,Ac = ∂AXµΩb a,µc, Ωb a,µc = −∇µθb a,c + ωb a,µc + LXaωb µc , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='14d) where ∇µθb a,c = ∂µθb a,c + ωb µdθd a,c − ωd µcθb a,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' It is straightforward to check that the invariant blocks defined above reflect the consistency of symmetry algebra in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' To the leading order in a-fields, the effective Lagrangian can be written as L = ˆT µ b Eb a,µ + JµCa,µ + Jµ b Kb a,µ + ˆSµc bΩb a,µc + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='15) To derive the momentum Ward identity, let us consider the variation with respect to Xν a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Denoting compactly the total stress tensor T µ b ≡ ˆT µ b + Jµϕb and the total spin current Sµc b ≡ ˆSµc b + Jµ [bϕc],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' we obtain δS δXνa = −e−1∂µ[e(T µ b eb ν + Sµc bωb νc + Jµ b Ab ν + JµAν)] + T µ b ∂νeb µ + Sµc b∂νωb µc + Jµ b ∂νAb µ + Jµ∂νAµ 24 = −e−1∂µ(eT µ b )eb ν − e−1∂µ(eSµc b)ωb νc − e−1∂µ(eJµ b )Ab ν − e−1∂µ(eJµ)Aν + 2T µ b ∂[νeb µ] + 2Sµc b∂[νωb µ]c + 2Jµ b ∂[νAb µ] + 2Jµ∂[νAµ] = −∇′ µ(T µ b )eb ν − ∇′ µ(Jµ b )Ab ν − ∇′ µJµAν + T µ b Gb νµ + Sµc bRb cνµ + Jµ b F b νµ + JµFνµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='16) where we defined the modified covariant derivative ∇′ µ ≡ ∇µ + Gρ µρ with Gλ µν ≡ 2Γλ [µν], and the field strength Fµν ≡ ∂µAν − ∂νAµ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='17a) F b µν ≡ ∂µAb ν − ∂νAb µ + ωb µcAc ν − ωb νcAc µ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='17b) Gb µν ≡ ∂µeb ν − ∂νeb µ + ωb µcec ν − ωb νcec µ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='17c) Rb cµν ≡ ∂µωb νc − ∂νωb µc + ωb µdωd νc − ωb νdωd µc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='17d) In the last step of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='16), we used the spin current Ward identity obtained by varying the action with respect to θbd a : T µ [bed] µ + Jµ [bAd] µ + ∇′ µSµd b = 0 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='18) as well as the dipole Ward identity (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='20c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' It is known, however, that the “intrinsic” spin current ˆSµc b is a non- hydrodynamic mode [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This can be seen through −∂0 ˆS0c b ∼ ˆS0c b ⊂ ˆT µ [bec] µ which means that ˆS0c b will relax to zero at long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='14 Since the spatial part ˆSic b is proportional to (gradient of) ˆS0c b, we are allowed to set ˆSµc b = 0 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='15 On the other hand, the “non-intrinsic” spin current Sµc b = Jµ [bϕd] is not relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Last, the variation with respect to ϕa and ϕb a will give charge and dipole Ward identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Combining them, we obtain, with ordinary derivatives, e−1∂µ � e( ˆT µ b + Jµϕb) � + e−1∂µ (eJµ d ϕc) ωd νceν b + e−1∂µ(eJµ c )Ac νeν b −2( ˆT µ c + Jµϕc)∂[νec µ]eν b − 2Jµ d ϕc∂[νωd µ]ceν b − 2Jµ∂[νAµ]eν b − 2Jµ c ∂[νAc µ]eν b = 0, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='19a) e−1∂µ(eJµ) = 0, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='19b) e−1∂µ(eJµ b ) − ωc µbJµ c − Jµeµb = 0, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='19c) and with covariant derivatives, ∇′ µ � ˆT µ b + Jµϕb � + Ac νeµcJµeν b − Rd cνµϕcJµ d eν b − F d νµJµ d eν b − FνµJµeν b − Gc νµ � ˆT µ c + Jµϕc � eν b = 0, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='20a) ∇′ µJµ = 0, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='20b) ∇′ µJµ b − Jµeµb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='20c) These are the momentum, charge and dipole equations of motions in an generic curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Remarkably, the dipole Goldstone will couple to the curvature to give a force in the momentum equation, and we have showed that it is best understood as the Joule heating due to spin currents [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' If we set ϕb = 0, our result is consistent with [59] in that the structure of their Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='13) is recovered: ∇′ µ ˆT µ b = fµeµ b −e0 µ ˆT µ c ec ρeν b∇νeρ 0, where we denoted collectively the Joule heating term fµ and used Gb µν = 2eb λe0 [µ∇ν]eλ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' However, unlike [59], our formalism includes the dipole Goldstone ϕb, which leads to the coupling between spin current and background curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Note that this feature is ultimately related to the breakdown of dipole gauge theory in curved spacetime [39, 88], but here the dipole symmetry is valid so long as we count background curvature as derivative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' There is another conceptual difference from [59] that our Ab µ needs not to be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' This is possible if we include the dipole Goldstone ϕb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Unlike [59], the antisymmetric part of Ab µ is not fixed by Fµν, and does have physical consequences as emphasized in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Appendix C: Consistency of the symmetry algebra The analysis of this section follows largely [59] but contains several generalizations of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 14 ˆS0c b is identified as the local angular momentum density in [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 15 Note that in an ordinary fluid without dipole symmetry, imposing ˆSµc b = 0 to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='18) would lead to a symmetric stress tensor ˆT µ [bed] µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 25 Let χµ be an infinitesimal diffeomorphism, Ωb c an infinitesimal rotation, Λ a U(1) gauge transformation, and ξb a dipole shift parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We start by defining the action of transformations Ξ = (χµ, Ωb c, Λ, ξb) on the dynamical fields Xµ(σ), ϕ(σ), ϕb(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We have eδΞXµ = (1 + δΞ + · · · )Xµ = Xµ − χµ(X) + · · · , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='1) thus, in leading order, [δΞ′, δΞ] = [eδΞ′ , eδΞ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='2) We have, up to second order, eδΞ′ eδΞXµ = Xµ − χµ − χ′µ + χ′ν∂νχµ, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='3) so that16 [eδΞ′ , eδΞ]Xµ = Lχ′χµ = −δ[Ξ′,Ξ]Xµ, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='4) where we carefully kept into account zeroth, first, and second-order terms and verified that zeroth and first-order terms cancel out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We then find χµ [Ξ′,Ξ] ≡ δΞ′χµ = Lχ′χµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='5) Next, let’s look at ϕb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We have eδΞϕb = ϕb + ξb − Ωb cϕc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='6) Note that we are not including Lχϕb in the above as we view ϕb as a function of σ, which is a singlet under physical spacetime diffeomorphisms χµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Then, eδΞ′ eδΞϕb = ϕb + ξb + ξ′b − (Ω′b c + Ωb c)ϕc − Ωb cξ′c + Ωb dΩ′d cϕc − Lχ′ξb + Lχ′Ωb cϕc, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='7) and [eδΞ′ , eδΞ]ϕb = −Lχ′ξb + Lχξ′b + Ω′b cξc − Ωb cξ′c + (Lχ′Ωb c − LχΩ′b c)ϕc + (Ωb dΩ′d c − Ω′b dΩd c)ϕc, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='8) thus giving ξb [Ξ′,Ξ] ≡ δΞ′ξb = Lχ′ξb − Lχξ′b + Ωb cξ′c − Ω′b cξc , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='9a) Ωb c,[Ξ′,Ξ] ≡ δΞ′Ωb c = Lχ′Ωb c − LχΩ′b c + Ωb dΩ′d c − Ω′b dΩd c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='9b) Now it’s ϕ’s turn: eδΞϕ = ϕ + Λ ≡ ϕ + λ + M bξb + Nµχµ , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='10) where M b and Nµ are expressions in terms of Xµ, ϕ and ϕb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Up to a normalization of the transformation parameters, the most general choice consistent with charge conjugation invariance is M b = (1 − c)eb µXµ, Nµ = ceb µϕb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='11) We will determine c = 0 later from consistency requirements, but now it can be any value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Again we view ϕ as a function of σ and thus we do not have the term Lχϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Composing two transformations, eδΞ′ eδΞϕ = ϕ + Λ + Λ′ − Lχ′Λ + ceb µχµξ′ b , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='12) gives [eδΞ′ , eδΞ]ϕ = −Lχ′Λ + LχΛ′ + ceb µχµξ′ b − ceb µχ′µξb (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='13) 16 We take passive transformations on dynamical fields, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' [δΞ′, δΞ] = −δ[Ξ′,Ξ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' 26 To summarize this part, the algebra of local transformations is χµ [Ξ′,Ξ] ≡δΞ′χµ = Lχ′χµ, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='14a) Ωb c,[Ξ′,Ξ] ≡δΞ′Ωb c = Lχ′Ωb c − LχΩ′b c + Ωb dΩ′d c − Ω′b dΩd c, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='14b) ξb [Ξ′,Ξ] ≡δΞ′ξb = Lχ′ξb − Lχξ′b + Ωb cξ′c − Ω′b cξc, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='14c) Λ[Ξ′,Ξ] ≡δΞ′Λ = Lχ′Λ − LχΛ′ − ceb µχµξ′ b + ceb µχ′µξb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='14d) This is a generalization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='11) in [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' To see the consistency with the dipole algebra, we decompose the field variation into δΞ = −iχµPµ + iλQ + iξbDb, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='15) where Pµ, Q and Db are symmetry generators, and we have ignored the rotation symmetry generator for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The background field is also turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' As a consequence of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='10), we have iQϕ = 1, iDbϕ = Mb, −iPµϕ = Nµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='16) Now, let us take Ξ′ = ξ′ b and Ξ = χµ, then −[δΞ′, δΞ]ϕ = −χµδb µξ′ c[Db, Pc]ϕ − (1 − c)Xµδb µχρ∂ρξ′ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='17) At the same time, we have −[δΞ′, δΞ]ϕ = Λ[ξ′ b,χµ] = −χµδb µξ′ b − (1 − c)Xµδb µχρ∂ρξ′ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='18) Thus we conclude that [Pb, Dc] = −iQδbc (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='19) is the desired dipole algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' There is an unwanted free parameter c appearing in the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Here, we show that in order for the transformation Ξ itself to also form a closed algebra, one must set c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' First, as a warm-up, let us consider the variation of χµ, ξb, Ωb c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We have [δΞ′′, δΞ′]χµ = Lχ′′Lχ′χµ − Lχ′Lχ′′χµ = Lχ[Ξ′′,Ξ′]χµ = δ[Ξ′′,Ξ′]χµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='20) Next, using product rules, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' δΞ′′Lχ′ξb = LδΞ′′χ′ξb + Lχ′δΞ′′ξb , we find δΞ′′δΞ′ξb =Ω′′b cLχξ′c + Ωb cLχ′′ξ′c − Lχ′′Ω′b cξc − LχΩ′b cξ′′c + Ωb cΩ′c dξ′′d + Ω′′b dΩ′d cξc + � Lχ′′Ωb cξ′c + Lχ′Ωb cξ′′c − Ω′′b cLχ′ξc − Ω′b cLχ′′ξc − Ω′′b dΩd cξ′c − Ω′b dΩd cξ′′c� , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='21) thus [δΞ′′, δΞ′]ξb = Lχ[Ξ′′,Ξ′]ξb − Lχξb [Ξ′′,Ξ′] + Ωb cξc [Ξ′′,Ξ′] − Ωb c,[Ξ′′,Ξ′]ξc = δ[Ξ′′,Ξ′]ξb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='22) Similar calculation leads to [δΞ′′, δΞ′]Ωb c = Lχ[Ξ′′,Ξ′]Ωb c − LχΩb c,[Ξ′′,Ξ′] + Ωb dΩd c,[Ξ′′,Ξ′] − Ωb d,[Ξ′′,Ξ′]Ωd c = δ[Ξ′′,Ξ′]Ωb c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='23) Now, we look at Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' For brevity, let us take eb µ = δb µ and temporarily turn off Ωb c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Note, this is merely a simplification for manipulations and the final result should still hold in arbitrarily curved spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' We have δΞ′′δΞ′Λ = δΞ′′(Lχ′Λ − LχΛ′ − cδb µχµξ′ b + cδb µχ′µξb) = LLχ′′χ′Λ + Lχ′(Lχ′′Λ − LχΛ′′ − cδb µ(χµξ′′ b − χ′′µξb)) − LLχ′′χΛ′ − Lχ(Lχ′′Λ′ − Lχ′Λ′′ − cδb µ(χ′µξ′′ b − χ′′µξ′ b)) − cδb µLχ′′χµξ′ b − cδb µχµ(Lχ′′ξ′ b − Lχ′ξ′′ b ) + cδb µLχ′′χ′µξb + cδb µχ′µ(Lχ′′ξb − Lχξ′′ b ) = Lχ′′Lχ′Λ + LχLχ′Λ′′ − (Lχ′′LχΛ′ + Lχ′LχΛ′′) + cδb µ [Lχχ′µξ′′ b − Lχ(χ′′µξ′ b) − χµLχ′′ξ′ b] + cδb µ [−(Lχ′χµξ′′ b + Lχ′′χµξ′ b) + (Lχ′χ′′µξb + Lχ′′χ′µξb) + (χ′′µLχ′ξb + χ′µLχ′′ξb)] , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='24) 27 thus [δΞ′′, δΞ′]Λ = Lχ[Ξ′′,Ξ′]Λ − Lχ(Lχ′′Λ′ − Lχ′Λ′′) + cδb µ [−Lχ(χ′′µξ′ b − χ′µξ′′ b ) − χµ(Lχ′′ξ′ b − Lχ′ξ′′ b ) + Lχχ′µξ′′ b − Lχχ′′µξ′ b] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='25) We note that the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' cannot be written as Lχ[Ξ′′,Ξ′]Λ−LχΛ[Ξ′′,Ξ′]−cδµbχµξb [Ξ′′,Ξ′]+cδµbχµ [Ξ′′,Ξ′]ξb, hence, the algebra is not closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' However, if we set c = 0, we have [δΞ′′, δΞ′]Λ = Lχ[Ξ′′,Ξ′]Λ − LχΛ[Ξ′′,Ξ′] = δ[Ξ′′,Ξ′]Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='26) To summarize, when c = 0, Ξ itself forms a closed algebra [δΞ′′, δΞ′]Ξ = δ[Ξ′′,Ξ′]Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='27) Lastly, let us turn to the background gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' The variation is defined as δΞeb µ = Lχeb µ − Ωb cec µ, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='28a) δΞωb µc = Lχωb µc + ∇µΩb c, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='28b) δΞAµ = LχAµ − ∂µΛ − eb µξb, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='28c) δΞAb µ = LχAb µ − ∇µξb − Ωb cAc µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='28d) Composing two transformations, we have δΞ′δΞeb µ = Lχ′Lχeb µ + (LχΩ′b c + Ω′b dΩd c)ec µ − � Lχ(Ω′b cec µ) + Lχ′(Ωb cec µ) � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='29a) δΞ′δΞAµ = Lχ′LχAµ + ∂µLχΛ′ + eb µLχξ′ b − eb µΩbcξ′c − � Lχ∂µΛ′ + Lχ′∂µΛ + Lχ′(eb µξb) + Lχ(eb µξ′ b) � , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='29b) δΞ′δΞωb µc = Lχ′Lχωb µc − ∇µ(LχΩ′b c + Ω′b dΩd c) + (Lχ∇µΩ′b c + Lχ′∇µΩb c + ∂µΩb dΩ′d c + ∂µΩ′b dΩd c + ωb µeΩe dΩ′d c + ωb µeΩ′e dΩd c − Ω′b eωe µdΩd c − Ωb eωe µdΩ′d c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='29c) The variation of Ab µ is a bit tedious, but taking lessons from previous manipulations, we can immediately see that the infinitesimal rotation should be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Hence, we are allowed to focus on the piece involving ξb only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' In particular, δΞ′δΞAb µ ⊃ ∇µLχξ′b + ∂µ(Ω′b cξc) − ωb µdΩd cξ′c − � Lχ(∇µξ′b) + Lχ′(∇µξb) + ∂µΩb cξ′c + ∂µΩ′b cξc + Ωb cωc µdξ′d + Ω′b cωc µdξd� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='30a) Then, it is straightforward to check that [δΞ′, δΞ] = δ[Ξ′,Ξ] holds for the background fields as well: [δΞ′, δΞ]eb µ = Lχ[Ξ′,Ξ]eb µ − Ωb c,[Ξ′,Ξ]ec µ = δ[Ξ′,Ξ]eb µ, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='31a) [δΞ′, δΞ]Aµ = Lχ[Ξ′,Ξ]Aµ − ∂µΛ[Ξ′,Ξ] − eµbξb [Ξ′,Ξ] = δ[Ξ′,Ξ]Aµ, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='31b) [δΞ′, δΞ]Ab µ = Lχ[Ξ′,Ξ]Ab µ − ∇µξb [Ξ′,Ξ] − Ωb c,[Ξ′,Ξ]Ac µ = δ[Ξ′,Ξ]Ab µ, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='31c) [δΞ′, δΞ]ωb µc = Lχ[Ξ′,Ξ]ωb µc + ∇µΩb c,[Ξ′,Ξ] = δ[Ξ′,Ξ]ωb µc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='31d) [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Landau and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Lifshitz, Fluid Mechanics, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' (Butterworth Heinemann, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' [2] Michael Crossley, Paolo Glorioso, and Hong Liu, “Effective field theory of dissipative fluids,” JHEP 09, 095 (2017), arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content='03646 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' [3] Felix M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Haehl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNE0T4oBgHgl3EQf0AKB/content/2301.02680v1.pdf'} +page_content=' Loganayagam, 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a/MtFOT4oBgHgl3EQf1jQj/content/tmp_files/2301.12939v1.pdf.txt b/MtFOT4oBgHgl3EQf1jQj/content/tmp_files/2301.12939v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b9fcdd926991ad899b54250ad72720b93d7d22d --- /dev/null +++ b/MtFOT4oBgHgl3EQf1jQj/content/tmp_files/2301.12939v1.pdf.txt @@ -0,0 +1,666 @@ +Data-driven soiling detection in PV modules +Alexandros Kalimeris1, Ioannis Psarros1, Giorgos Giannopoulos1, Manolis Terrovitis1, +George Papastefanatos1, and Gregory Kotsis2 +1Athena RC +2INACCESS Networks +January 31, 2023 +Abstract +Soiling is the accumulation of dirt in solar panels which leads to a decreasing trend in solar energy +yield and may be the cause of vast revenue losses. The effect of soiling can be reduced by washing the +panels, which is, however, a procedure of non-negligible cost. Moreover, soiling monitoring systems are +often unreliable or very costly. We study the problem of estimating the soiling ratio in photo-voltaic (PV) +modules, i.e., the ratio of the real power output to the power output that would be produced if solar panels +were clean. A key advantage of our algorithms is that they estimate soiling, without needing to train on +labelled data, i.e., periods of explicitly monitoring the soiling in each park, and without relying on generic +analytical formulas which do not take into account the peculiarities of each installation. We consider +as input a time series comprising a minimum set of measurements, that are available to most PV park +operators. Our experimental evaluation shows that we significantly outperform current state-of-the-art +methods for estimating soiling ratio. +Keywords: Solar energy, solar panels, soiling, performance loss, time series analysis +1 +Introduction +Soiling is the accumulation of dirt on the surfaces of photo-voltaic (PV) modules, which leads to a loss in +the power output. Soiling is typically caused by airborne particles, including for example dust, pollen and +soot. Depending on the location, soiling may also be caused by heavier material such as ice, bird droppings, +or falling leaves. +One standard way to quantify soiling is by the soiling ratio SR [iec21], which is defined as the ratio of the +real power output to the power output that would be produced if solar panels were clean. Soiling loss is then +defined as 1 − SR, and soiling rate is defined as the (daily) rate of change of the soiling loss. Other metrics +have been also proposed, e.g., the insolation-weighted soiling ratio [DMM18], aiming to better capture the +loss induced by soiling. +To reduce the effect of soiling, PV modules must be cleaned on strategically chosen dates to reduce +the cost induced by energy loss while taking into account cleaning costs. Detection of time periods during +which soiling severely affects power output is therefore significant for the efficient scheduling of cleanings. +What makes the problem challenging is the shortage of labelled data which is caused by the fact that +soiling monitoring systems are often considered unreliable or costly. For example, soiling stations which +are the most common commercially available soiling monitoring solution [BMAF21], still require regular +cleanings and maintenance, which can be expensive, especially in remote locations, and imperfect cleanings +can result in significant measurement uncertainty [MMMM17]. Therefore, soiling periods must be deduced +from measurements of a number of reliable variables, e.g., power output, irradiance, temperature. +Existing methods that detect soiling follow two alternative strategies: a) they train a model on labelled +data, i.e., data where the soiling of the panels has been logged using specialized sensors and cleaning events +1 +arXiv:2301.12939v1 [eess.SP] 30 Jan 2023 + +have been explicitly recorded (e.g., [MMD11, MMDK13]) and b) by using an analytical formula for optimal +energy output based on environmental readings (e.g., [KMNW06, DMM18, MTL+21]). The former strategy +is more accurate but requires significant resources to produce the labelled data, which must be produced for +each different installation. The latter strategy does not take into account the peculiarities of each installation +and leads to less accurate results (as we demonstrate in Section 4). +The main advantage of our method, +is that it is purely data-driven, in the sense that it does not require a generic analytical formula for the +relation between power output and the commonly used environmental readings, but it learns this relation +in a self-supervised manner (without the need for labelled data). This way we achieve better results than +methods that rely on analytical formulas without the cost of methods that need explicitly labelled data. +We consider as input the monitoring data from the park operation, i.e., a time series with measurements of +power output, irradiance, and module temperature for a certain array or string of PV modules, precipitation, +and dates on which the solar panels were manually cleaned for maintenance (if such information exists). The +soiling ratio over a sequence of timestamps t1, . . . , tn is defined as SR = Pt1 +P ∗ +t1 , . . . , Ptn +P ∗ +tn , where each Pti is +the actual power output corresponding to timestamp ti, and P ∗ +ti is the expected power output assuming +that the solar panels are clean, corresponding to the same timestamp. Our framework trains a regression +model M which accurately predicts P ∗ +t1, . . . , P ∗ +tn (which are not given as input). This yields an estimate +for the soiling ratio as SRM = Pt1 +˜ +Pt1 , . . . , Ptn +˜ +Ptn , where each ˜Pti is the value predicted by M for timestamp +ti. We aim for M such that SRM ≈ SR. Raining periods (extracted from precipitation measurements), +and manual cleanings, are used in the “learning” phase of our proposed model. One of our methods can +run exclusively on rain information, in case manual cleanings are not performed or logged. Our approach is +robust to misinformation about manual cleanings because it checks each potential cleaning to determine its +effect on power output. Manual cleanings that are not logged, have a negligible effect; they can only affect +the quality of the training set positively. +The main advantages of our method are that they do not require measurements of soiling from specialized +equipment which can be costly or inaccurate, they do not rely on the accuracy of an analytical formula for +the optimal energy output of the park, and they agnostic to the type of PV modules employed. +As a +purely data-driven approach, it solely depends on the availability of data, and in particular a minimal set +of generally available variables. Our approach is robust to misinformation about manual cleanings because +it checks each potential cleaning to determine its effect on power output. Moreover, manual cleanings that +are not logged, have a negligible effect on our approach; their existence can only affect the quality of the +training set positively. +In Section 2, we discuss related work, in Section 3.1 we provide necessary background, in Section 3.2 we +present a detailed description of our methods, and in Section 4 we present our experimental findings. +2 +Related work +PVUSA introduced a method for rating PV systems based on a simple regression model [DG95] which +employs the simplified assumption that array current depends only on irradiance and that array voltage +depends only on module temperature. Massi Pavan et al. [MMD11] compare the standard test conditions +(STC) (irradiance: 1000W/m2, module temperature: 25◦C) performance of a PV park before and after its +cleaning. In order to determine the performance at STC conditions they use a regression model, suggested +in [MWPP08], that accepts as input the two main climate features, i.e. the in-plane global irradiance and +the photo voltaic module temperature. However, their work requires as input labelled data, i.e. time series +extracted from both clean and soiled PV modules. Massi Pavan et al. [MMDK13] developed four Bayesian +Neural Network (BNN) models with the aim to calculate the STC performance of two plants before and +after a complete clean-up of their modules. The idea is that differences between the STC power before and +after the clean-up represent the losses due to the soiling effect. However, their work also requires as input +labelled data, i.e. time series extracted from both clean and soiled PV modules. +Closer to our work are methods which estimate soiling losses based on PV system data. The Fixed Rate +Precipitation (FRP) method [KMNW06] calculates the daily soiling loss. The method requires as input: +2 + +the slope of the performance metric/index during the longest dry period, a cleaning threshold for rains, +i.e., the minimum amount of daily precipitation required to have a cleaning effect on PV modules, and a +number of days after a raining period for which no soiling occurs. The method implicitly assumes that the +soiling rate remains the same throughout time. This requirement can be very restrictive, because of the +different types of soiling that may occur, depending also on the location or the season. For the same reason, +it is unrealistic to assume that there is a certain minimum value classifying rains as effective. More recently, +Deceglie, Micheli, and Muller [DMM18] developed a new method for quantifying soiling loss, which compares +favourably to FRP. The new method is termed the stochastic rate and recovery (SRR) method. It uses an +analytical formula, calculated over values for irradiance and module temperature, to compute the expected +power output, which is then used to compute a performance metric. The method first detects soiling intervals +in a dataset, and then, based on the observed characteristics of each interval, estimates the total loss. Notice +that SRR provides an aggregate estimate of soiling loss, calculated for the whole input period, while our +focus lies on determining soiling loss even on shorter periods of time. Skomedal and Deceglie [SD20] proposed +the combined degradation and soiling method for further analyzing a performance metric signal. Finally, +Micheli et al. [MTL+21] consider non-linear degradation in soiling intervals, and they apply various methods +for changepoints detection to obtain a refined soiling profile. All methods studied there are based on finding +changepoints on the performance metric curve, as calculated by SRR. On the contrary, our approach detects +changepoints as an intermediate step towards computing a performance metric. It is apparent from recent +work that improvements on estimating the expected power output directly translate to improvements on +various tasks in PV data analysis. +3 +Methodology +3.1 +Preliminaries +3.1.1 +Basic assumptions and definitions +Our input consists of a multi-variate time series containing measurements for: i) power output, ii) irradiance, +iii) module temperature, iv) precipitation. Our methods can be further enhanced if we are also given as +input the dates on which the PV modules were manually cleaned. +Let R be the set of all rains, defined as follows: [t, t′] ∈ R if and only if there is a rain starting at t +and ending at t′. Rains are extracted from input as maximal time intervals containing positive precipitation +values. Similarly, if manual cleanings are provided let C be the set of all such intervals, defined as follows: +[t, t′] ∈ C if and only if we know that the PV modules were being cleaned between timestamps t and t′. We +denote by Wp the set of all potential cleaning events, defined as Wp = C ∪ R. We assume that precipitation +measurements are sufficiently frequent, so that we can accurately detect rains. +3.1.2 +Regression models +A basic component of our methods is regression. We fit regression models to represent power output during +“dirty” or “clean” periods and we use prediction errors to detect performance changes. We consider as feature +variables the irradiance and the module temperature, and the target outcome corresponds to the power +output. We apply Ridge Regression with polynomial features, which is parameterized by the degree of the +regression polynomial, and a regularization strength parameter for the linear least squares function (the loss +function) where regularization is given by the ℓ2-norm. The parameters were selected during the initial stages +of the algorithm development process, where we experimented with cross-validation and hyper-parameter +tuning techniques. The exact values used in our experiments are discussed in Section 4. Our model selection +was a consequence of preliminary experiments with various (simple) regression models such as Ordinary Least +Squares, Support Vector Regression, etc., that we executed in a CPU with maximum processor frequency +at 3.7GHz, and available RAM at 256Gb. In the experiment that we conducted, we randomly choose 100 +time intervals of maximum duration of one month from the time series provided in [MAD+14], which are +also discussed in Section 4, and we randomly split them into training and testing subsets containing 80% +3 + +and 20% of the points respectively. Our choice satisfies a bifold objective: i) good accuracy and ii) fast +fitting time. The latter is vital in our method which fits one model for each potential cleaning. Table 3 +contains MAPE values and fitting times for four different models. Polynomial features and the polynomial +kernel used in Support Vector Regression (SVR) are of degree 3. The highest accuracy is achieved by SVR +with linear kernel and polynomial features, being roughly 0, 4% better than Ridge Regression which is the +second best. However, the fitting time of SVR is at least one order of magnitude higher than that of Ridge +Regression. +Ridge Regression is a simple model that adds only one extra tunable parameter to our learning +pipeline, and the regularization it provides acts as a measure to prevent overfitting. We also emphasize the +fact that one can easily plug-in any regression model in our approach. +Table 1: Evaluation of regression models. +Model +MAPE +Fitting time (s) +Linear Regression +with polynomial features +0.0812 +0.0015 +Ridge Regression +with polynomial features +0.0807 +0.0012 +Support Vector Regression +with polynomial kernel +1.0648 +0.0177 +Support Vector Regression +with linear kernel and polynomial features +0.0770 +0.0666 +Several steps in our approach rely on computing measures for the prediction accuracy of our model. Let +Y = Yt1, . . . , Ytn, ˜Y = ˜Yt′ +1, . . . , ˜Yt′n be two univariate time series, and let T = {t1, . . . , tn}, T ′ = {t′ +1, . . . , t′ +n}. +We use a variant of the mean absolute percentage error (MAPE) which is defined over time intervals as +follows: for any [t, t′] ⊆ T ∩ T ′, +mape0(Y, ˜Y, [t, t′]) = mean({|Yj − ˜Yj| | j ∈ [t, t′]} +mean({|Yj| | j ∈ [t, t′]}) +. +Note that mape0 is robust to zero true values (as long as not all of them are zeroes) since it uses as +denominator the mean of the values, as opposed to standard MAPE where all actual values appear as +denominators leading to singularities even if there is only one zero true value. When Y and ˜Y are clear from +the context, we omit them from our notation and we simply write mape0([t, t′]). We also use the median +multiplicative error defined as mede(Y, ˜Y) = median +�� +Yi +˜Yj | i ∈ T, j ∈ T ′�� +. +3.2 +Soiling detection +In this section, we formally describe our methods, which are composed of two main steps. The first step is +that of detecting cleaning events. Then, using these cleaning events we define training periods for regression +models aiming to capture the optimal performance of the PV modules. +In all our methods, we fit regression +models which capture the dependence of power output on the values of irradiance and module temperature, +i.e., power output is the dependent variable, while irradiance and module temperature are the feature vari- +ables. Measurements are scaled to [0, 1] by subtracting the minimum value and dividing by the range of +values. Figure 1 summarizes the main steps of our methods. +3.2.1 +Baseline soiling estimator +We first present our baseline approach for estimating the soiling ratio. Our baseline algorithm is based on +the following assumption: manual cleanings alone define points in time where the PV modules are clean. +While these points are not sufficiently many to define a training set, we can extend them to short intervals of +a user-defined length wtrain. This is the amount of time during which we can safely assume that the panels +remain clean. +4 + +PV data +precipitation data +(manual cleaning dates) +potential cleanings +cleaning event detection +fit regression +model before +potential +cleaning +compare prediction +errors before/after +potential cleaning +cleaning events +fit regression model +on periods following +the cleaning events +predict optimal per- +formance and esti- +mate soiling ratio +Forward Checking Soiling Estimator (FCSE) +PV data +precipitation data +manual cleaning dates +potential cleanings +cleaning event detection +fit one regression +model on periods +following manual +cleanings +compare prediction +errors before/after +potential cleaning +cleaning events fit regression model +on periods following +the cleaning events +predict optimal per- +formance and esti- +mate soiling ratio +Backward Checking Soiling Estimator (BCSE) +Baseline estimator +Figure 1: Basic steps of our methods. Manual cleanings are optional for FCSE. To detect cleaning events, +FCSE fits one regression model before each potential cleaning event, while BCSE fits one regression model +using manual cleaning dates and uses it in classifying all cleaning events. +We fit a regression model that aims to capture the power output when PV modules are clean. To this +purpose, we fit a regression model M on the set of input points with timestamps from � +[t,t′]∈C[t′, t′ +wtrain]. +We define SRM = Pt1 +˜ +Pt1 , . . . , Ptn +˜ +Ptn as the modelled soiling ratio where each Pti is an input power output value, +and ˜Pti is the value predicted M. +3.2.2 +Forward checking soiling estimator (FCSE) +Our first method examines each potential cleaning event independently and assigns scores which represent +the significance of the detected change of behavior. Five input parameters are required: the length of the +training period w1, the length of the validation period w2, the length of the test period w3, a parameter +q defining the quantile of the scores which classifies events as cleanings, and the length wtrain defining the +training set for the final regression model used to estimate soiling. For each interval [t, t′] ∈ Wp, we fit a +regression model in the time interval [t − w1 − w2, t − w2), we validate it in the time interval [t − w2, t) and +we test it in the time interval (t′, t′ + w3]. We compute the function mape0 on the validation interval and +if the returned value is greater than 5% then we consider this event invalid and we discard it from further +consideration. This threshold aims to discard events that we are unable to classify with certainty. +The +reasons behind choosing 5% as our threshold are the following. First, due to the nature of our task, the +regression model is required to make very accurate predictions and detect power deviations at a very small +scale. This requires high accuracy of our regression models; therefore a tight threshold. On the other hand, +this threshold must be pragmatic: having an extremely small value as a threshold will lead to unrealistic +outputs where no cleaning events are detected and, consequently, no soiling estimation can be derived. +We +experimentally validate our choice of 5% in Section 4.2.2. +The intuition is that if the PV modules under-perform due to soiling, for a time period preceding t, then +the regression model captures this under-performing behaviour and if [t, t′] is a cleaning event then the model +should underestimate the power output in (t′, t′ + w3]. To compute the score of the potential cleaning event +[t, t′], we first compute PIval as the sequence of actual power output values divided by the predicted power +output values for the time interval [t − w2, t), and PItest as the sequence of actual power output values +divided by the predicted power output values for the time interval (t′, t′ + w3]. Then, the score assigned +to [t, t′] is mede(PIval, PItest). We define as cleaning events all intervals [t, t′] ∈ Wp with score above the +qth-quantile of all scores. Let W1 be the set of detected cleaning events. We fit a regression model M on +the input points with timestamps from � +[t,t′]∈W1[t′, t′ + wtrain]. The intuition is that cleaning events define +points in time where the PV modules are clean. Obviously, these points are not sufficiently many to define +a proper training set. By extending these points to (short) intervals, of length wtrain, we increase the size of +5 + +the training set without (significantly) affecting its quality. We define SRM = +Pt1 +˜ +Pt1 , . . . , Ptn +˜ +Ptn as the estimated +soiling ratio where each Pti is an input power output value, and ˜Pti is the value predicted by the regression +model M. +Notice that FCSE does not require having the cleaning dates C as input, and we could simply have +Wp = R. +3.2.3 +Backward checking soiling estimator (BCSE) +Our second method builds upon the baseline approach. This method requires five input parameters w1, +w2, w3, q, wtrain. Parameters w1 and w2 denote the length of the testing period preceding the potential +cleaning event and the length of the validation period following the potential cleaning event respectively. +Parameter w3 denotes the length of the time period following each [t, t′] ∈ C such that the modules remain +clean. Parameter q defines the quantile of the scores which classifies events as cleanings. Parameter wtrain +is used to define the training set of the final regression model for estimating the soiling ratio. We train +one regression model on the set of points defined by timestamps in � +[t,t′]∈C[t′, t′ + w3]. This model aims +to capture modules’ “clean” performance. For each [t, t′] ∈ Wp, we use our model to make predictions on +[t − w1, t) and (t′, t′ + w2]. If mape0((t′, t′ + w2]) is greater than 5% then we consider this interval invalid +and we discard if from further consideration. As in FCSE, this filtering step is to avoid considering events +that our models fail to classify with a good amount of certainty. +The intuition is that if [t, t′] is a cleaning event, then the PV modules’ performance during [t′, t′ + w2] +must resemble the “clean” performance as predicted by our regression model. +Similarly, if the modules +under-perform during [t − w1, t), then the induced ratio of the actual power output over the predicted power +output must be significantly smaller than 1. To compute the score of the potential cleaning event [t, t′], we +first compute PIbefore as the sequence of actual power output values divided by the predicted power output +values for the time interval [t − w1, t), and PIafter as the sequence of actual power output values divided +by the predicted power output values for the time interval (t′, t′ + w2]. Then, the score assigned to [t, t′] +is mede(PIbefore, PIafter). We define as our threshold parameter thrsh the qth-quantile of all scores. We +define as cleaning events all intervals [t, t′] ∈ Wp with score above the qth-quantile of all scores. Let W2, +be the set of detected cleaning events. We fit a regression model M on the input points with timestamps +from � +[t,t′]∈W2[t′, t′ + wtrain]. As in FCSE, the intuition is that cleaning events define points in time where +the PV modules are clean. Obviously, these points are not sufficiently many to define a training set. By +extending these points to (short) intervals, of length wtrain, we increase the size of the training set without +(significantly) affecting its quality. We define SRM = Pt1 +˜ +Pt1 , . . . , Ptn +˜ +Ptn as the estimated soiling ratio where each +Pti is an input power output value, and ˜Pti is the value predicted by M. +4 +Experiments +4.1 +Datasets +State-of-the-art dataset +To evaluate our methods, we use a dataset provided in [MAD+14], which +contains a set of current-voltage (I-V) curves and associated meteorological data for PV modules representing +all flat-plate PV technologies and for three different locations and climates for approximately one-year +periods. +For each location, we are given values for a normalized metric, called soiling derate which is +computed using measurements for short-circuit current and irradiance from two identical PV modules; one +that is cleaned during daily maintenance, and one that is not. +Soiling derate is the result of dividing +daily values of ampere-hours per kilowatt-hours per square meter Plane of Array (POA) irradiance for the +not-cleaned PV module, by the corresponding values of the cleaned PV module [MAD+14]. The soiling +derate aims to provide a performance index analogous to soiling ratio, estimated on real measurements. We +emphasize that soiling derate is only used for the evaluation of our methods and are not utilized as input (nor +in SRR). The time granularity is 5 minutes, and measurements are provided for all hours of daylight. The +6 + +three locations are Cocoa, Florida, USA; Eugene, Oregon, USA; and Golden, Colorado, USA. PV modules +in Cocoa and Eugene were cleaned when this was necessary in order to ensure that levels of soiling loss were +maintained at a reasonable level. PV modules in Golden were not cleaned because frequent rains helped +maintaining a reasonable level of soiling loss. Cocoa has a minimum soiling derate of 0.985, Eugene has a +minimum soiling derate of 0.964, and Golden has a minimum soiling derate of 0.977. +In our methods, we use measurements for the maximum power of the PV module in watts, the amount of +solar irradiance in watts per square meter received on the PV module surface, the PV module back-surface +temperature and the accumulated daily total precipitation. The dataset also provides dates on which all PV +modules were cleaned. We apply our methods on PV modules that were used in estimating the soiling derate, +and in particular on those that were not cleaned every day. As discussed in Section 3.1.2 our methods utilize +Ridge Regression models. For those models, we use polynomial features of the 3rd degree and a regularization +strength parameter alpha = 10−4 during the fitting stages. +Real-world dataset +We also consider a real-world scenario, where no ground truth is available. +We +test our methods on a dataset from a very different location and of different climate conditions, comprising +measurements from a solar park located in Greece. We are given values for power output, irradiance, module +temperature and precipitation on a time granularity of 15 min for a period of approximately 7 years, and 15 +dates of manual cleanings. +4.2 +Method evaluation and discussion +4.2.1 +Soiling estimation +We evaluate our methods, by comparing them to the analogous model used in SRR. To show robustness of our +methods in different parameter settings, we try various lengths for the periods used in changepoint detection. +Table 2 lists the respecting values (in days) for parameters w1, w2, w3 in FCSE and w1, w2 in BCSE. The +rest of the parameters are set as follows: we apply FCSE with parameters q = 0.9, and wtrain = 30 days and +BCSE with parameters q = 0.9, w3 = 30 days, and wtrain = 30 days. The baseline soiling estimator is applied +with wtrain = 30 days. Since our methods are unsupervised, classic automated methods fail to optimize the +above parameters. Essentially, domain expertise is the main lead for selecting parameters appropriately, also +depending on the properties of each location that affect the rate at which soiling progresses. However, as +Table 2 indicates, the methods are robust within a range of reasonable values for the parameters. The fixed +parameters wtrain (and w3 in BCSE) define time periods during which a clean solar panel is likely to remain +clean. While smaller values for wtrain (resp. w3) seem to provide safer conclusions, larger values provide a +bigger size and diversity of the induced training set. The parameter q defines a threshold on how important +a changepoint should be to be considered as a cleaning event. Setting q = 0.9 implies that the top-scored +10% of potential cleanings will be considered as cleaning events. Factors that must be taken into account +when setting this parameter include the total number of potential changepoints, parameters w3, wtrain, and +the size of the dataset. While larger values of q tend to lead to safer conclusions about cleaning events, this +may lead to a decreased size of the training set, negatively affecting the final regression model. +We juxtapose our estimated soiling ratio with the ground-truth soiling derate and the performance metric +used in SRR. We have three different ways of estimating the soiling ratio: our baseline approach, FCSE and +BCSE, which are +described in Section 3.2. In our estimates, we map negative values and values greater +than one to zero and one, respectively. Then, we apply a rolling median with windows of one day. +For computing the performance metric as in SRR, we rely again on the publicly available RdTools +package [DNS+22]. We use as input aggregate daily values calculated on measurements taken between 12:00 +and 14:00, with irradiance greater than 500W/m2. We first compute the performance metric as the ratio of +realized to modelled PV energy yield, where modelled PV energy yield is derived from a standard formula +which is implemented in pvlib package [HHM22]. Then, we perform a few processing steps as suggested in +RdTools’ tutorials1. We first normalize the time series with the expected power, we then apply default filters +to remove clipping effects and outliers, and finally, we resample to one-day values. +1https://rdtools.readthedocs.io/en/stable/examples/degradation_and_soiling_example_pvdaq_4.html +7 + +Figure 2: Soiling ratio predicted by our models, and the performance metric used in SRR, for the Eugene +dataset. FCSE with parameters w1 = 10, w2 = 5, w3 = 10 and BCSE with parameters w1 = 5, w2 = 10. +Let SD be the soiling derate time series. We denote by PM the performance metric used in SRR. In +Figure 2, we plot our estimated soiling ratio, for all three models discussed in Section 3.2, the soiling derate +and the performance metric used in SRR, for the site of Eugene. Compared to the other datasets, Eugene +has periods of declining performance which are more apparent. PV modules at the Eugene site were cleaned +on March 11, July 10, August 14, August 21, and August 26. No significant precipitation is observed during +July and August, which leads to a rapid drop in the performance. +We also calculate the root-mean-square error (RMSE) comparing the soiling derate with each modelled +ratio, for all three sites. Since no manual cleanings were performed in Golden, the baseline algorithm and +BCSE cannot be executed. We list these results in Table 2. It becomes evident, both from the RMSE values +and from the visual inspection of the figure, that a better estimation of the soiling ratio can be derived by +our models, compared to the model based on an analytical formula which is employed by SRR, in a setting +where a soiling tendency needs to be detected, nearly real-time, on newly incoming data. Further, BCSE +compares favourably to FCSE, and improves upon the baseline algorithm in the Eugene dataset. On the other +hand, both the baseline algorithm and BCSE cannot be executed in the Golden dataset, due to the lack of +manual cleanings. FCSE and BCSE present slightly diverse behaviors, rendering each potentially preferable +in diverse real-world settings, depending on the exact objective of a solar park operator. Specifically, BCSE +provides the most accurate method in approximating soiling ratio, thus preferable when small to medium +soiling events are tolerable by the operator, as long as “false alarms” are minimised. On the other hand, +FCSE, while slightly missing in accuracy, it is more sensitive in the detection of smaller (potential) soiling +events, making it ideal in cases when even small soiling events need to be handled. Finally, we can see that +the formula used in SRR essentially predicts the majority of the considered period as soiling; a behavior +with small practical use in a real-world deployment scenario. +8 + +Baseline +1DO +0.98 +0.96 +soiling derate +estimated soiling ratio +FCSE +0.98 +0.96 +soiling derate +estimated soiling ratio +EC5E +1DO +0.98 +soiling derate +0.96 +estimated soiling ratio +LDO +FeormanceMetc[SRR +0.98 +0.96 +soiling derate +Perf. metric (SRR) +imestampTable 2: Evaluation. +Model +RMSE against SD +Eugene +Cocoa +Golden +Baseline +0.006 +0.006 +- +FCSE (w1 = 2, w2 = 1, w3 = 2) +0.010 +0.006 +0.008 +FCSE (w1 = 10, w2 = 5, w3 = 10) +0.007 +0.008 +0.008 +FCSE (w1 = 30, w2 = 10, w3 = 30) +0.009 +0.007 +0.008 +BCSE (w1 = 1, w2 = 2) +0.008 +0.006 +- +BCSE (w1 = 5, w2 = 10) +0.005 +0.007 +- +BCSE (w1 = 10, w2 = 30) +0.007 +0.007 +- +PM used in SRR +0.019 +0.020 +0.028 +Figure 3: Segmentation and estimated soiling ratio obtained by FCSE. +4.2.2 +Required accuracy of regression models +We experimentally justify our choice of 5% as a threshold for validation MAPE of our models in methods +FCSE (w1 = 10, w2 = 5, w3 = 10), BCSE (w1 = 5, w2 = 10), as discussed in Section 3.2. To be able to +execute both methods for various thresholds, we employ them on the two datasets that are accompanied by +manual cleaning information, i.e., Eugene and Cocoa. For both methods, we calculate the mean (over the +two sites) RMSE against SD. Experiments in Table 3 indicate that the best result is obtained for 5% (or +above), for BCSE. +Table 3: Choice of validation MAPE threshold. +MAPE threshold +mean RMSE against SD +FCSE +BCSE +3% +0.007 +0.009 +4% +0.008 +0.007 +5%, 10%, 15%, 20% +0.008 +0.006 +9 + +1DO +0.95 +0.90 +0.B5 +estimated soiling ratio (smoothed) +0.:0] +cleaning events +imestampFigure 4: Segmentation and estimated soiling ratio obtained by BCSE. +4.2.3 +Industrial use-case (absence of ground-truth) +In this section, we test our methods on the dataset described in Section 4.1. First, we apply FCSE for the +detection of cleaning events. We filter out rains with maximum precipitation of at most 0.1 to remove noise. +Figure 3 (resp. Figure 4) illustrates the cleaning events detected by FCSE(resp. BCSE) and our modelled +soiling ratio. +Within each interval defined by two consecutive changepoints, we compute a line using the +Theil–Sen method [The92, Sen68] on the estimated soiling ratio (on a 15min granularity). The Theil-Sen +method is a way of fitting a line to a set of points, which is robust to outliers. The line is chosen by selecting +the median slope over all lines defined by pairs of points. We plot the lines with negative slope as red dotted +line segments lying in the corresponding intervals, over the course of 5 years. We also plot a smoothed +version of our estimated soiling ratio, where we have applied a rolling median of 5 days. +In both figures, in almost all time periods defined by two consecutive changepoints, we observe that there +is a decreasing trend in the time series for the detected period, as dictated by the slope of the line fitted by +the Theil-Sen regression (red-dotted line segments). This decreasing trend ends with rain or manual cleaning, +illustrated by a blue vertical line, which is detected by our method as a cleaning event. This example is an +indication of the effectiveness and generalizability of the proposed method. Despite the lack of labels to be +able to explicitly verify the result, the trend identified is consistent with soiling and it is verifiable through +the effect of washing. +5 +Conclusion +We have described a method for estimating the soiling ratio, which uses a set of easily accessed measurements +from sensors that are commonly deployed in PV parks. Our method is data-driven, in the sense that it models +the optimal performance by efficiently learning it from the data, without relying on generic formulas that +fail to capture the peculiarities of the site. +Estimating the soiling ratio is useful for PV park administrators since it allows them to schedule cleaning +procedures more effectively by taking into account the rate of soil accumulation and the effectiveness of past +cleaning efforts without the need for frequent visual inspections or installing specialized equipment which +induces extra cost and maintenance efforts. +Our method effectively estimates the soiling ratio in historical data. Future possible directions include +extending our method to forecasting soiling losses in the future, which would assist in deciding cleaning +10 + +LDO0 +5L60 +S60 +0.925 +0.900 +0.B75 +0.B50 +0.B25 +estimated soiling ratio (smoothed) +0.B0.0 +cleaning events +101:2036actions at a short notice. +6 +Acknowledgements +The authors were partially supported by the EU’s Horizon 2020 Research and Innovation programme, under +the grant agreement No. 957345: “MORE”. +References +[BMAF21] +João Gabriel Bessa, Leonardo Micheli, Florencia Almonacid, and Eduardo F. Fernández. Mon- +itoring photovoltaic soiling: assessment, challenges, and perspectives of current and potential +strategies. iScience, 24(3):102165, 2021. +[DG95] +R N Dows and E J Gough. PVUSA procurement, acceptance, and rating practices for photo- +voltaic power plants. 9 1995. +[DMM18] +Michael G. Deceglie, Leonardo Micheli, and Matthew Muller. Quantifying soiling loss directly +from PV yield. 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Springer Netherlands, Dordrecht, 1992. +12 + diff --git a/MtFOT4oBgHgl3EQf1jQj/content/tmp_files/load_file.txt b/MtFOT4oBgHgl3EQf1jQj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9123b67e4ed86649b1dc1914db77b35996c8b7c --- /dev/null +++ b/MtFOT4oBgHgl3EQf1jQj/content/tmp_files/load_file.txt @@ -0,0 +1,543 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf,len=542 +page_content='Data-driven soiling detection in PV modules Alexandros Kalimeris1, Ioannis Psarros1, Giorgos Giannopoulos1, Manolis Terrovitis1, George Papastefanatos1, and Gregory Kotsis2 1Athena RC 2INACCESS Networks January 31, 2023 Abstract Soiling is the accumulation of dirt in solar panels which leads to a decreasing trend in solar energy yield and may be the cause of vast revenue losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The effect of soiling can be reduced by washing the panels, which is, however, a procedure of non-negligible cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Moreover, soiling monitoring systems are often unreliable or very costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We study the problem of estimating the soiling ratio in photo-voltaic (PV) modules, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=', the ratio of the real power output to the power output that would be produced if solar panels were clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' A key advantage of our algorithms is that they estimate soiling, without needing to train on labelled data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=', periods of explicitly monitoring the soiling in each park, and without relying on generic analytical formulas which do not take into account the peculiarities of each installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We consider as input a time series comprising a minimum set of measurements, that are available to most PV park operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Our experimental evaluation shows that we significantly outperform current state-of-the-art methods for estimating soiling ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Keywords: Solar energy, solar panels, soiling, performance loss, time series analysis 1 Introduction Soiling is the accumulation of dirt on the surfaces of photo-voltaic (PV) modules, which leads to a loss in the power output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Soiling is typically caused by airborne particles, including for example dust, pollen and soot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Depending on the location, soiling may also be caused by heavier material such as ice, bird droppings, or falling leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' One standard way to quantify soiling is by the soiling ratio SR [iec21], which is defined as the ratio of the real power output to the power output that would be produced if solar panels were clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Soiling loss is then defined as 1 − SR, and soiling rate is defined as the (daily) rate of change of the soiling loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Other metrics have been also proposed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=', the insolation-weighted soiling ratio [DMM18], aiming to better capture the loss induced by soiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' To reduce the effect of soiling, PV modules must be cleaned on strategically chosen dates to reduce the cost induced by energy loss while taking into account cleaning costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Detection of time periods during which soiling severely affects power output is therefore significant for the efficient scheduling of cleanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' What makes the problem challenging is the shortage of labelled data which is caused by the fact that soiling monitoring systems are often considered unreliable or costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' For example, soiling stations which are the most common commercially available soiling monitoring solution [BMAF21], still require regular cleanings and maintenance, which can be expensive, especially in remote locations, and imperfect cleanings can result in significant measurement uncertainty [MMMM17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Therefore, soiling periods must be deduced from measurements of a number of reliable variables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=', power output, irradiance, temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Existing methods that detect soiling follow two alternative strategies: a) they train a model on labelled data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=', data where the soiling of the panels has been logged using specialized sensors and cleaning events 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='12939v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='SP] 30 Jan 2023 have been explicitly recorded (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=', [MMD11, MMDK13]) and b) by using an analytical formula for optimal energy output based on environmental readings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=', [KMNW06, DMM18, MTL+21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The former strategy is more accurate but requires significant resources to produce the labelled data, which must be produced for each different installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The latter strategy does not take into account the peculiarities of each installation and leads to less accurate results (as we demonstrate in Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The main advantage of our method, is that it is purely data-driven, in the sense that it does not require a generic analytical formula for the relation between power output and the commonly used environmental readings, but it learns this relation in a self-supervised manner (without the need for labelled data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' This way we achieve better results than methods that rely on analytical formulas without the cost of methods that need explicitly labelled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We consider as input the monitoring data from the park operation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=', a time series with measurements of power output, irradiance, and module temperature for a certain array or string of PV modules, precipitation, and dates on which the solar panels were manually cleaned for maintenance (if such information exists).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The soiling ratio over a sequence of timestamps t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' , tn is defined as SR = Pt1 P ∗ t1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' , Ptn P ∗ tn , where each Pti is the actual power output corresponding to timestamp ti, and P ∗ ti is the expected power output assuming that the solar panels are clean, corresponding to the same timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Our framework trains a regression model M which accurately predicts P ∗ t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' , P ∗ tn (which are not given as input).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' This yields an estimate for the soiling ratio as SRM = Pt1 ˜ Pt1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' , Ptn ˜ Ptn , where each ˜Pti is the value predicted by M for timestamp ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We aim for M such that SRM ≈ SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Raining periods (extracted from precipitation measurements), and manual cleanings, are used in the “learning” phase of our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' One of our methods can run exclusively on rain information, in case manual cleanings are not performed or logged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Our approach is robust to misinformation about manual cleanings because it checks each potential cleaning to determine its effect on power output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Manual cleanings that are not logged, have a negligible effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' they can only affect the quality of the training set positively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The main advantages of our method are that they do not require measurements of soiling from specialized equipment which can be costly or inaccurate, they do not rely on the accuracy of an analytical formula for the optimal energy output of the park, and they agnostic to the type of PV modules employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' As a purely data-driven approach, it solely depends on the availability of data, and in particular a minimal set of generally available variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Our approach is robust to misinformation about manual cleanings because it checks each potential cleaning to determine its effect on power output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Moreover, manual cleanings that are not logged, have a negligible effect on our approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' their existence can only affect the quality of the training set positively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' In Section 2, we discuss related work, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='1 we provide necessary background, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2 we present a detailed description of our methods, and in Section 4 we present our experimental findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 2 Related work PVUSA introduced a method for rating PV systems based on a simple regression model [DG95] which employs the simplified assumption that array current depends only on irradiance and that array voltage depends only on module temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Massi Pavan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' [MMD11] compare the standard test conditions (STC) (irradiance: 1000W/m2, module temperature: 25◦C) performance of a PV park before and after its cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' In order to determine the performance at STC conditions they use a regression model, suggested in [MWPP08], that accepts as input the two main climate features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' the in-plane global irradiance and the photo voltaic module temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' However, their work requires as input labelled data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' time series extracted from both clean and soiled PV modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Massi Pavan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' [MMDK13] developed four Bayesian Neural Network (BNN) models with the aim to calculate the STC performance of two plants before and after a complete clean-up of their modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The idea is that differences between the STC power before and after the clean-up represent the losses due to the soiling effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' However, their work also requires as input labelled data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' time series extracted from both clean and soiled PV modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Closer to our work are methods which estimate soiling losses based on PV system data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The Fixed Rate Precipitation (FRP) method [KMNW06] calculates the daily soiling loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The method requires as input: 2 the slope of the performance metric/index during the longest dry period, a cleaning threshold for rains, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=', the minimum amount of daily precipitation required to have a cleaning effect on PV modules, and a number of days after a raining period for which no soiling occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The method implicitly assumes that the soiling rate remains the same throughout time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' This requirement can be very restrictive, because of the different types of soiling that may occur, depending also on the location or the season.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' For the same reason, it is unrealistic to assume that there is a certain minimum value classifying rains as effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' More recently, Deceglie, Micheli, and Muller [DMM18] developed a new method for quantifying soiling loss, which compares favourably to FRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The new method is termed the stochastic rate and recovery (SRR) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' It uses an analytical formula, calculated over values for irradiance and module temperature, to compute the expected power output, which is then used to compute a performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The method first detects soiling intervals in a dataset, and then, based on the observed characteristics of each interval, estimates the total loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Notice that SRR provides an aggregate estimate of soiling loss, calculated for the whole input period, while our focus lies on determining soiling loss even on shorter periods of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Skomedal and Deceglie [SD20] proposed the combined degradation and soiling method for further analyzing a performance metric signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Finally, Micheli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' [MTL+21] consider non-linear degradation in soiling intervals, and they apply various methods for changepoints detection to obtain a refined soiling profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' All methods studied there are based on finding changepoints on the performance metric curve, as calculated by SRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' On the contrary, our approach detects changepoints as an intermediate step towards computing a performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' It is apparent from recent work that improvements on estimating the expected power output directly translate to improvements on various tasks in PV data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 3 Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='1 Preliminaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='1 Basic assumptions and definitions Our input consists of a multi-variate time series containing measurements for: i) power output, ii) irradiance, iii) module temperature, iv) precipitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Our methods can be further enhanced if we are also given as input the dates on which the PV modules were manually cleaned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Let R be the set of all rains, defined as follows: [t, t′] ∈ R if and only if there is a rain starting at t and ending at t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Rains are extracted from input as maximal time intervals containing positive precipitation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Similarly, if manual cleanings are provided let C be the set of all such intervals, defined as follows: [t, t′] ∈ C if and only if we know that the PV modules were being cleaned between timestamps t and t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We denote by Wp the set of all potential cleaning events, defined as Wp = C ∪ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We assume that precipitation measurements are sufficiently frequent, so that we can accurately detect rains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2 Regression models A basic component of our methods is regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We fit regression models to represent power output during “dirty” or “clean” periods and we use prediction errors to detect performance changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We consider as feature variables the irradiance and the module temperature, and the target outcome corresponds to the power output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We apply Ridge Regression with polynomial features, which is parameterized by the degree of the regression polynomial, and a regularization strength parameter for the linear least squares function (the loss function) where regularization is given by the ℓ2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The parameters were selected during the initial stages of the algorithm development process, where we experimented with cross-validation and hyper-parameter tuning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The exact values used in our experiments are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Our model selection was a consequence of preliminary experiments with various (simple) regression models such as Ordinary Least Squares, Support Vector Regression, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=', that we executed in a CPU with maximum processor frequency at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='7GHz, and available RAM at 256Gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' In the experiment that we conducted, we randomly choose 100 time intervals of maximum duration of one month from the time series provided in [MAD+14], which are also discussed in Section 4, and we randomly split them into training and testing subsets containing 80% 3 and 20% of the points respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Our choice satisfies a bifold objective: i) good accuracy and ii) fast fitting time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The latter is vital in our method which fits one model for each potential cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Table 3 contains MAPE values and fitting times for four different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Polynomial features and the polynomial kernel used in Support Vector Regression (SVR) are of degree 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The highest accuracy is achieved by SVR with linear kernel and polynomial features, being roughly 0, 4% better than Ridge Regression which is the second best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' However, the fitting time of SVR is at least one order of magnitude higher than that of Ridge Regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Ridge Regression is a simple model that adds only one extra tunable parameter to our learning pipeline, and the regularization it provides acts as a measure to prevent overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We also emphasize the fact that one can easily plug-in any regression model in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Table 1: Evaluation of regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Model MAPE Fitting time (s) Linear Regression with polynomial features 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='0812 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='0015 Ridge Regression with polynomial features 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='0807 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='0012 Support Vector Regression with polynomial kernel 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='0648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='0177 Support Vector Regression with linear kernel and polynomial features 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='0770 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='0666 Several steps in our approach rely on computing measures for the prediction accuracy of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Let Y = Yt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' , Ytn, ˜Y = ˜Yt′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' , ˜Yt′n be two univariate time series, and let T = {t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' , tn}, T ′ = {t′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' , t′ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We use a variant of the mean absolute percentage error (MAPE) which is defined over time intervals as follows: for any [t, t′] ⊆ T ∩ T ′, mape0(Y, ˜Y, [t, t′]) = mean({|Yj − ˜Yj| | j ∈ [t, t′]} mean({|Yj| | j ∈ [t, t′]}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Note that mape0 is robust to zero true values (as long as not all of them are zeroes) since it uses as denominator the mean of the values, as opposed to standard MAPE where all actual values appear as denominators leading to singularities even if there is only one zero true value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' When Y and ˜Y are clear from the context, we omit them from our notation and we simply write mape0([t, t′]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We also use the median multiplicative error defined as mede(Y, ˜Y) = median �� Yi ˜Yj | i ∈ T, j ∈ T ′�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2 Soiling detection In this section, we formally describe our methods, which are composed of two main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The first step is that of detecting cleaning events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Then, using these cleaning events we define training periods for regression models aiming to capture the optimal performance of the PV modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' In all our methods, we fit regression models which capture the dependence of power output on the values of irradiance and module temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=', power output is the dependent variable, while irradiance and module temperature are the feature vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Measurements are scaled to [0, 1] by subtracting the minimum value and dividing by the range of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Figure 1 summarizes the main steps of our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='1 Baseline soiling estimator We first present our baseline approach for estimating the soiling ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Our baseline algorithm is based on the following assumption: manual cleanings alone define points in time where the PV modules are clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' While these points are not sufficiently many to define a training set, we can extend them to short intervals of a user-defined length wtrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' This is the amount of time during which we can safely assume that the panels remain clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='PV data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='precipitation data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='(manual cleaning dates) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='potential cleanings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='cleaning event detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='fit regression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='model before ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='potential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='cleaning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='compare prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='errors before/after ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='potential cleaning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='cleaning events ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='fit regression model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='on periods following ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='the cleaning events ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='predict optimal per- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='formance and esti- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='mate soiling ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='Forward Checking Soiling Estimator (FCSE) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='PV data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='precipitation data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='manual cleaning dates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='potential cleanings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='cleaning event detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='fit one regression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='model on periods ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='following manual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='cleanings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='compare prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='errors before/after ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='potential cleaning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='cleaning events fit regression model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='on periods following ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='the cleaning events ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='predict optimal per- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='formance and esti- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='mate soiling ratio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='Backward Checking Soiling Estimator (BCSE) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='Baseline estimator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='Figure 1: Basic steps of our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Manual cleanings are optional for FCSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' To detect cleaning events, FCSE fits one regression model before each potential cleaning event, while BCSE fits one regression model using manual cleaning dates and uses it in classifying all cleaning events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We fit a regression model that aims to capture the power output when PV modules are clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' To this purpose, we fit a regression model M on the set of input points with timestamps from � [t,t′]∈C[t′, t′ +wtrain].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We define SRM = Pt1 ˜ Pt1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' , Ptn ˜ Ptn as the modelled soiling ratio where each Pti is an input power output value, and ˜Pti is the value predicted M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2 Forward checking soiling estimator (FCSE) Our first method examines each potential cleaning event independently and assigns scores which represent the significance of the detected change of behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Five input parameters are required: the length of the training period w1, the length of the validation period w2, the length of the test period w3, a parameter q defining the quantile of the scores which classifies events as cleanings, and the length wtrain defining the training set for the final regression model used to estimate soiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' For each interval [t, t′] ∈ Wp, we fit a regression model in the time interval [t − w1 − w2, t − w2), we validate it in the time interval [t − w2, t) and we test it in the time interval (t′, t′ + w3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We compute the function mape0 on the validation interval and if the returned value is greater than 5% then we consider this event invalid and we discard it from further consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' This threshold aims to discard events that we are unable to classify with certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The reasons behind choosing 5% as our threshold are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' First, due to the nature of our task, the regression model is required to make very accurate predictions and detect power deviations at a very small scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' This requires high accuracy of our regression models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' therefore a tight threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' On the other hand, this threshold must be pragmatic: having an extremely small value as a threshold will lead to unrealistic outputs where no cleaning events are detected and, consequently, no soiling estimation can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We experimentally validate our choice of 5% in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The intuition is that if the PV modules under-perform due to soiling, for a time period preceding t, then the regression model captures this under-performing behaviour and if [t, t′] is a cleaning event then the model should underestimate the power output in (t′, t′ + w3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' To compute the score of the potential cleaning event [t, t′], we first compute PIval as the sequence of actual power output values divided by the predicted power output values for the time interval [t − w2, t), and PItest as the sequence of actual power output values divided by the predicted power output values for the time interval (t′, t′ + w3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Then, the score assigned to [t, t′] is mede(PIval, PItest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We define as cleaning events all intervals [t, t′] ∈ Wp with score above the qth-quantile of all scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Let W1 be the set of detected cleaning events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We fit a regression model M on the input points with timestamps from � [t,t′]∈W1[t′, t′ + wtrain].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The intuition is that cleaning events define points in time where the PV modules are clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Obviously, these points are not sufficiently many to define a proper training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' By extending these points to (short) intervals, of length wtrain, we increase the size of 5 the training set without (significantly) affecting its quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We define SRM = Pt1 ˜ Pt1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' , Ptn ˜ Ptn as the estimated soiling ratio where each Pti is an input power output value, and ˜Pti is the value predicted by the regression model M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Notice that FCSE does not require having the cleaning dates C as input, and we could simply have Wp = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='3 Backward checking soiling estimator (BCSE) Our second method builds upon the baseline approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' This method requires five input parameters w1, w2, w3, q, wtrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Parameters w1 and w2 denote the length of the testing period preceding the potential cleaning event and the length of the validation period following the potential cleaning event respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Parameter w3 denotes the length of the time period following each [t, t′] ∈ C such that the modules remain clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Parameter q defines the quantile of the scores which classifies events as cleanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Parameter wtrain is used to define the training set of the final regression model for estimating the soiling ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We train one regression model on the set of points defined by timestamps in � [t,t′]∈C[t′, t′ + w3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' This model aims to capture modules’ “clean” performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' For each [t, t′] ∈ Wp, we use our model to make predictions on [t − w1, t) and (t′, t′ + w2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' If mape0((t′, t′ + w2]) is greater than 5% then we consider this interval invalid and we discard if from further consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' As in FCSE, this filtering step is to avoid considering events that our models fail to classify with a good amount of certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The intuition is that if [t, t′] is a cleaning event, then the PV modules’ performance during [t′, t′ + w2] must resemble the “clean” performance as predicted by our regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Similarly, if the modules under-perform during [t − w1, t), then the induced ratio of the actual power output over the predicted power output must be significantly smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' To compute the score of the potential cleaning event [t, t′], we first compute PIbefore as the sequence of actual power output values divided by the predicted power output values for the time interval [t − w1, t), and PIafter as the sequence of actual power output values divided by the predicted power output values for the time interval (t′, t′ + w2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Then, the score assigned to [t, t′] is mede(PIbefore, PIafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We define as our threshold parameter thrsh the qth-quantile of all scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We define as cleaning events all intervals [t, t′] ∈ Wp with score above the qth-quantile of all scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Let W2, be the set of detected cleaning events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We fit a regression model M on the input points with timestamps from � [t,t′]∈W2[t′, t′ + wtrain].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' As in FCSE, the intuition is that cleaning events define points in time where the PV modules are clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Obviously, these points are not sufficiently many to define a training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' By extending these points to (short) intervals, of length wtrain, we increase the size of the training set without (significantly) affecting its quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We define SRM = Pt1 ˜ Pt1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' , Ptn ˜ Ptn as the estimated soiling ratio where each Pti is an input power output value, and ˜Pti is the value predicted by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='1 Datasets State-of-the-art dataset To evaluate our methods, we use a dataset provided in [MAD+14], which contains a set of current-voltage (I-V) curves and associated meteorological data for PV modules representing all flat-plate PV technologies and for three different locations and climates for approximately one-year periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' For each location, we are given values for a normalized metric, called soiling derate which is computed using measurements for short-circuit current and irradiance from two identical PV modules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' one that is cleaned during daily maintenance, and one that is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Soiling derate is the result of dividing daily values of ampere-hours per kilowatt-hours per square meter Plane of Array (POA) irradiance for the not-cleaned PV module, by the corresponding values of the cleaned PV module [MAD+14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The soiling derate aims to provide a performance index analogous to soiling ratio, estimated on real measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We emphasize that soiling derate is only used for the evaluation of our methods and are not utilized as input (nor in SRR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The time granularity is 5 minutes, and measurements are provided for all hours of daylight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The 6 three locations are Cocoa, Florida, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Eugene, Oregon, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' and Golden, Colorado, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' PV modules in Cocoa and Eugene were cleaned when this was necessary in order to ensure that levels of soiling loss were maintained at a reasonable level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' PV modules in Golden were not cleaned because frequent rains helped maintaining a reasonable level of soiling loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Cocoa has a minimum soiling derate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='985, Eugene has a minimum soiling derate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='964, and Golden has a minimum soiling derate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' In our methods, we use measurements for the maximum power of the PV module in watts, the amount of solar irradiance in watts per square meter received on the PV module surface, the PV module back-surface temperature and the accumulated daily total precipitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The dataset also provides dates on which all PV modules were cleaned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We apply our methods on PV modules that were used in estimating the soiling derate, and in particular on those that were not cleaned every day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' As discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2 our methods utilize Ridge Regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' For those models, we use polynomial features of the 3rd degree and a regularization strength parameter alpha = 10−4 during the fitting stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Real-world dataset We also consider a real-world scenario, where no ground truth is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We test our methods on a dataset from a very different location and of different climate conditions, comprising measurements from a solar park located in Greece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We are given values for power output, irradiance, module temperature and precipitation on a time granularity of 15 min for a period of approximately 7 years, and 15 dates of manual cleanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2 Method evaluation and discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='1 Soiling estimation We evaluate our methods, by comparing them to the analogous model used in SRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' To show robustness of our methods in different parameter settings, we try various lengths for the periods used in changepoint detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Table 2 lists the respecting values (in days) for parameters w1, w2, w3 in FCSE and w1, w2 in BCSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The rest of the parameters are set as follows: we apply FCSE with parameters q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='9, and wtrain = 30 days and BCSE with parameters q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='9, w3 = 30 days, and wtrain = 30 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The baseline soiling estimator is applied with wtrain = 30 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Since our methods are unsupervised, classic automated methods fail to optimize the above parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Essentially, domain expertise is the main lead for selecting parameters appropriately, also depending on the properties of each location that affect the rate at which soiling progresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' However, as Table 2 indicates, the methods are robust within a range of reasonable values for the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The fixed parameters wtrain (and w3 in BCSE) define time periods during which a clean solar panel is likely to remain clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' While smaller values for wtrain (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' w3) seem to provide safer conclusions, larger values provide a bigger size and diversity of the induced training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The parameter q defines a threshold on how important a changepoint should be to be considered as a cleaning event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Setting q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='9 implies that the top-scored 10% of potential cleanings will be considered as cleaning events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Factors that must be taken into account when setting this parameter include the total number of potential changepoints, parameters w3, wtrain, and the size of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' While larger values of q tend to lead to safer conclusions about cleaning events, this may lead to a decreased size of the training set, negatively affecting the final regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We juxtapose our estimated soiling ratio with the ground-truth soiling derate and the performance metric used in SRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We have three different ways of estimating the soiling ratio: our baseline approach, FCSE and BCSE, which are described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' In our estimates, we map negative values and values greater than one to zero and one, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Then, we apply a rolling median with windows of one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' For computing the performance metric as in SRR, we rely again on the publicly available RdTools package [DNS+22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We use as input aggregate daily values calculated on measurements taken between 12:00 and 14:00, with irradiance greater than 500W/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We first compute the performance metric as the ratio of realized to modelled PV energy yield, where modelled PV energy yield is derived from a standard formula which is implemented in pvlib package [HHM22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Then, we perform a few processing steps as suggested in RdTools’ tutorials1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We first normalize the time series with the expected power, we then apply default filters to remove clipping effects and outliers, and finally, we resample to one-day values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 1https://rdtools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='io/en/stable/examples/degradation_and_soiling_example_pvdaq_4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='html 7 Figure 2: Soiling ratio predicted by our models, and the performance metric used in SRR, for the Eugene dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' FCSE with parameters w1 = 10, w2 = 5, w3 = 10 and BCSE with parameters w1 = 5, w2 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Let SD be the soiling derate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We denote by PM the performance metric used in SRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' In Figure 2, we plot our estimated soiling ratio, for all three models discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2, the soiling derate and the performance metric used in SRR, for the site of Eugene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Compared to the other datasets, Eugene has periods of declining performance which are more apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' PV modules at the Eugene site were cleaned on March 11, July 10, August 14, August 21, and August 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' No significant precipitation is observed during July and August, which leads to a rapid drop in the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We also calculate the root-mean-square error (RMSE) comparing the soiling derate with each modelled ratio, for all three sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Since no manual cleanings were performed in Golden, the baseline algorithm and BCSE cannot be executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We list these results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' It becomes evident, both from the RMSE values and from the visual inspection of the figure, that a better estimation of the soiling ratio can be derived by our models, compared to the model based on an analytical formula which is employed by SRR, in a setting where a soiling tendency needs to be detected, nearly real-time, on newly incoming data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Further, BCSE compares favourably to FCSE, and improves upon the baseline algorithm in the Eugene dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' On the other hand, both the baseline algorithm and BCSE cannot be executed in the Golden dataset, due to the lack of manual cleanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' FCSE and BCSE present slightly diverse behaviors, rendering each potentially preferable in diverse real-world settings, depending on the exact objective of a solar park operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Specifically, BCSE provides the most accurate method in approximating soiling ratio, thus preferable when small to medium soiling events are tolerable by the operator, as long as “false alarms” are minimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' On the other hand, FCSE, while slightly missing in accuracy, it is more sensitive in the detection of smaller (potential) soiling events, making it ideal in cases when even small soiling events need to be handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Finally, we can see that the formula used in SRR essentially predicts the majority of the considered period as soiling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' a behavior with small practical use in a real-world deployment scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 8 Baseline 1DO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='96 soiling derate estimated soiling ratio FCSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='96 soiling derate estimated soiling ratio EC5E 1DO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='98 soiling derate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='96 estimated soiling ratio LDO FeormanceMetc[SRR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='96 soiling derate Perf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' metric (SRR) imestampTable 2: Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Model RMSE against SD Eugene Cocoa Golden Baseline 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='006 FCSE (w1 = 2, w2 = 1, w3 = 2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='008 FCSE (w1 = 10, w2 = 5, w3 = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='008 FCSE (w1 = 30, w2 = 10, w3 = 30) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='008 BCSE (w1 = 1, w2 = 2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='006 BCSE (w1 = 5, w2 = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='007 BCSE (w1 = 10, w2 = 30) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='007 PM used in SRR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='028 Figure 3: Segmentation and estimated soiling ratio obtained by FCSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2 Required accuracy of regression models We experimentally justify our choice of 5% as a threshold for validation MAPE of our models in methods FCSE (w1 = 10, w2 = 5, w3 = 10), BCSE (w1 = 5, w2 = 10), as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' To be able to execute both methods for various thresholds, we employ them on the two datasets that are accompanied by manual cleaning information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=', Eugene and Cocoa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' For both methods, we calculate the mean (over the two sites) RMSE against SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Experiments in Table 3 indicate that the best result is obtained for 5% (or above), for BCSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Table 3: Choice of validation MAPE threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' MAPE threshold mean RMSE against SD FCSE BCSE 3% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='009 4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='007 5%, 10%, 15%, 20% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='006 9 1DO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='B5 estimated soiling ratio (smoothed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=':0] cleaning events imestampFigure 4: Segmentation and estimated soiling ratio obtained by BCSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='3 Industrial use-case (absence of ground-truth) In this section, we test our methods on the dataset described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' First, we apply FCSE for the detection of cleaning events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We filter out rains with maximum precipitation of at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='1 to remove noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Figure 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Figure 4) illustrates the cleaning events detected by FCSE(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' BCSE) and our modelled soiling ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Within each interval defined by two consecutive changepoints, we compute a line using the Theil–Sen method [The92, Sen68] on the estimated soiling ratio (on a 15min granularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The Theil-Sen method is a way of fitting a line to a set of points, which is robust to outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' The line is chosen by selecting the median slope over all lines defined by pairs of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We plot the lines with negative slope as red dotted line segments lying in the corresponding intervals, over the course of 5 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' We also plot a smoothed version of our estimated soiling ratio, where we have applied a rolling median of 5 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' In both figures, in almost all time periods defined by two consecutive changepoints, we observe that there is a decreasing trend in the time series for the detected period, as dictated by the slope of the line fitted by the Theil-Sen regression (red-dotted line segments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' This decreasing trend ends with rain or manual cleaning, illustrated by a blue vertical line, which is detected by our method as a cleaning event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' This example is an indication of the effectiveness and generalizability of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Despite the lack of labels to be able to explicitly verify the result, the trend identified is consistent with soiling and it is verifiable through the effect of washing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 5 Conclusion We have described a method for estimating the soiling ratio, which uses a set of easily accessed measurements from sensors that are commonly deployed in PV parks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Our method is data-driven, in the sense that it models the optimal performance by efficiently learning it from the data, without relying on generic formulas that fail to capture the peculiarities of the site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Estimating the soiling ratio is useful for PV park administrators since it allows them to schedule cleaning procedures more effectively by taking into account the rate of soil accumulation and the effectiveness of past cleaning efforts without the need for frequent visual inspections or installing specialized equipment which induces extra cost and maintenance efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Our method effectively estimates the soiling ratio in historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Future possible directions include extending our method to forecasting soiling losses in the future, which would assist in deciding cleaning 10 LDO0 5L60 S60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='B75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='B50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='B25 estimated soiling ratio (smoothed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content='0 cleaning events 101:2036actions at a short notice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 6 Acknowledgements The authors were partially supported by the EU’s Horizon 2020 Research and Innovation programme, under the grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 957345: “MORE”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' References [BMAF21] João Gabriel Bessa, Leonardo Micheli, Florencia Almonacid, and Eduardo F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Fernández.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' Mon- itoring photovoltaic soiling: assessment, challenges, and perspectives of current and potential strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' iScience, 24(3):102165, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' [DG95] R N Dows and E J Gough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' PVUSA procurement, acceptance, and rating practices for photo- voltaic power plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' 9 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFOT4oBgHgl3EQf1jQj/content/2301.12939v1.pdf'} +page_content=' [DMM18] Michael G.' metadata={'source': 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SUPERVISED SPEECH +REPRESENTATION FOR SPOKEN LANGUAGE MODELING +Amitay Sicherman and Yossi Adi +School of Engineering and Computer Science +The Hebrew University of Jerusalem, Israel +ABSTRACT +This work profoundly analyzes discrete self-supervised speech +representations through the eyes of Generative Spoken Lan- +guage Modeling (GSLM). Following the findings of such an +analysis, we propose practical improvements to the discrete +unit for the GSLM. First, we start comprehending these units +by analyzing them in three axes: interpretation, visualization, +and resynthesis. Our analysis finds a high correlation between +the speech units to phonemes and phoneme families, while +their correlation with speaker or gender is weaker. Addition- +ally, we found redundancies in the extracted units and claim +that one reason may be the units’ context. Following this +analysis, we propose a new, unsupervised metric to measure +unit redundancies. +Finally, we use this metric to develop +new methods that improve the robustness of units clustering +and show significant improvement considering zero-resource +speech metrics such as ABX. Code and analysis tools are +available under the following link. +Index Terms— self supervised learning, generative spo- +ken language modeling, textless NLP, speech LM +1. INTRODUCTION +Recently Self-Supervised Learning (SSL) methods for speech +have shown great success on plenty of downs stream tasks [1]. +From Automatic Speech Recognition [2, 3, 4] and speaker +diarization [5], to phone segmentation [6], these models have +shown remarkable results. +Specifically, these SSL models allow recent success in +Generative Spoken Language Modeling (GSLM) [7, 8, 9]. +In GSLM, we aim to learn a discrete representation of the +speech signal. This is often being done by applying the k- +means algorithm over the continuous representation obtained +from a SSL models. Then, we train a unit Language Model +(uLM) over such representation, and lastly we decode it back +to a time domain signal using neural vocoder [10]. During +inference time, we can sample from the uLM conditionally or +unconditionally. +Although these models are capable of generating mean- +ingful and coherent speech utterances, little is known about +the properties captured but these discrete representations. The +authors in [11] examined the purity between phonetics ele- +ments and the discrete units. The authors’ proposed method +is to analysis the discrete SSL representation considering +fine-grained linguistic properties, e.g., different articulatory +classes or closure and release portions. The authors in [12] +proposed a probing method to analyze the presence of phone +classes, gender and language information while comparing +monolingual and bilingual models. +In this work, we analyze quantitatively and visually dis- +crete representations obtained by HuBERT and CPC models +with respect to phoneme classes, gender and speaker identity. +Next, equipped with such an analysis we provide a metric to +identify redundancies in the k-means clustering, and propose +a method to improve upon it. +We find a high correlation between the units and the +phonemes, but with many redundancies in the units. +We +show that one reason may be the units’ context. In Addition, +we propose an unsupervised metric to measure these redun- +dancies and we use it to significant improvement the unit +clustering. +2. BACKGROUND +The general GSLM pipeline is comprised of three main mod- +ules: (i) Speech-to-unit, (ii) Unit language model, and (iii) +Unit-to-speech, where each of these modules is trained sepa- +rately. Speech resynthesis can be achieved while ignoring the +language model and directly feeding the quantized units into +the unit-to-speech module [10] +Speech To Unit (STU) module encodes the raw speech signal +into a discrete representation. The model first encodes the +speech into a continuous representation and then quantize the +representation to a sequence of discrete units [7, 13, 14]. +Formally, denote the domain of audio samples by x ⊂ R. +The representation for a raw signal is therefore a sequence of +samples x = (x1, . . . , xT ), where xt ∈ x for all 1 ≤ t ≤ T. +Consider an encoder network, f, that gets as input the speech +utterance and outputs a sequence of spectral representations +sampled at a low frequency as follows f(x) = (v1, . . . , vT ′). +Note that we do not assume anything about the structure of +the encoder network f. Since the representations learned by +such models are usually continuous, a k-means algorithm is +arXiv:2301.00591v1 [cs.CL] 2 Jan 2023 + +Fig. 1. Units visualization process. +applied over the models’ outputs to generate discrete units, +denoted as z = (z1, . . . , zT ′). Each element zi in z is a pos- +itive integer, zi ∈ {1, .., K} for 1 ≤ i ≤ T ′, where K is the +number of discrete units. +As the quantized representation, z, usually contain units +repetitions which degrade the performance of the language +modeling, a common approach is collapse repetitions and +generate a de-duplicated sequence while additionally storing +the units’ duration separately. +For instance, the sequence +12,12,25,31,31,31 will be converted into 12,25,31 +and the corresponding durations 2,1,3. +Unit Language Model (ULM) is trained on the extracted +and deduplicated discrete units, z. The language model can +be used, for example, to generate speech conditionally or un- +conditionally. +Unit To Speech module converts the discrete speech repre- +sentation, z, to a raw waveform. The authors in [7] used a +Tacotron2.0 [15] based model followed by WaveGlow [16] +vocoder. Later, [10] proposed a unit-based vocoder based on +the HiFi-GAN architecture to directly convert units to speech. +In this work, we focus on the latter setting. +3. METHOD +We analyze representations obtained by either HuEBRT [2] +or CPC [4] models considering various number of clusters. +All analysis code and the developed visualization tools will +be publicly available. +3.1. Analysis +Units Interpretation. We start by measuring the mutual in- +formation between the discrete representation and different +speech properties (i.e., phonemes, speaker id, and gender), +using the V-Measure score [17]. +For this purpose, we align each utterance with its corre- +sponding attribute. To get units-to-phonemes alignment we +use the TIMIT corpus [18]. The TIMIT dataset contains pairs +of audio - phonemes, which are time aligned. For speaker and +gender analysis we use the LibriSpeech corpus as it contains +large and diverse set of speaker. +Fig. 2. Circular Resynthesis evaluation metric. +Units Visualization. An additional point of view of the units +meaning is the spatial structure of units. For this purpose, +we create a 2D spatial view that contains information regard- +ing the relation between the continuous representation, the +discrete units, and their corresponding phonemes. Specifi- +cally, we apply the following two steps: (i) We project the +high-dimensional speech representation into 2d using the T- +SNE [19] algorithm. T-SNE is a nonlinear dimensionality re- +duction that intuitively preserves the non-linear distance re- +lations between neighbors in the high and low dimensions. +Then, we use the Voronoi diagram [20] that converts the scat- +ter plot into an area plot. Finally, we have left with a bounded +area in the 2D space for each unit; (ii) In the second part, +we create a single label to represents each cluster. We use +the units-phonemes alignment from the TIMIT (similarly to +the process in previous paragraph). Then, we assign for each +cluster the most represented phoneme in it. Finally, we re- +place the unit id with their corresponding phonemes and color +the area base on the phoneme and phoneme family. A visual +description of the proposed method can be seen in Figure 1. +Units Resynthesis. Next, we analyze the units’ information +from the opposite direction - that is, through the speech resyn- +thesis. We decode the units back to speech using a look-up- +table of the corresponding 20ms speech segments, then we +transcribe the generated audio and measure the transcription +error (e.g., the Character Error Rate). Intuitively, in case of +strong correlation between the units and the phonemes - we +can take a single “sound” to represent each unit - and apply +the UTS step using the concatenation of these sound pieces. +Notice, this approach is different than the one in [10] as there +is no neural vocoder. +Formally, let u, l be a sequences of deduplicated units and +their length obtained by applying STU on the input audio x. +and let xi be the part in x that is matched to deduped unit, ui. +Notice, xi can be of arbitrary length. +Lookup Vocoder defines as : +LV (u, l) = concat(F(u1, l1), . . . , F(un, ln)), +F(ui, li) = +� +T[Key(ui, li)], +if Key(ui, li) in T +xi, +else +, +(1) + +K-Means Centers +T-SNE +Voronoi Diagram +N × Multicdimensional Vectors +Acoustic-PhoneticCorpus +M +Units : +11313...17 +M +3-M +4 +Phonemes : A +wI +I.. SH +7 - SH +SHcR(i) +32 +UED +-> +-> +s<- hs +32 -> 28 +1, 7, 54.... +28Table 1. Units Interpretation results. For phonemes, higher is +better. While for the speaker and gender, lower score indicates +that the model manages to hide information about the speaker +and gender. +Model +Size +Speaker +Gender +Phoneme +CPC +50 +1.35 +0.66 +47.30 +100 +2.35 +0.54 +48.45 +200 +3.70 +1.62 +47.74 +2000 +10.39 +4.14 +44.06 +HuBERT +50 +0.73 +0.03 +42.49 +100 +1.41 +0.17 +45.48 +200 +1.95 +0.21 +46.64 +2000 +5.15 +0.65 +43.32 +MFCC +50 +9.11 +2.90 +8.57 +100 +11.54 +3.97 +8.73 +200 +13.81 +4.59 +8.96 +where T is a Look-up-table that stores for each key the corre- +sponding xi of the first appearance of this key, and Key maps +unit and length into key. +We explore four different types of Key : (i) Local-Single- +Key(ui) = (ui); (ii) Local-Full- Key(ui) = (ui, li); (iii) +Context-Single- Key(ui) = (ui−1, ui, ui+1); (iv) Context- +Full- Key(ui) = (ui−1, ui, ui+1, li). +3.2. Circular Resynthesis +We introduce the Circular Resynthesis (CR) method, an ut- +terly unsupervised evaluation metric that aims to measure the +redundancies in the discrete units. As described in Figure 2, +we first perform a full resynthesis procedure, in which we +encode and decode the speech signal. Then, we apply an ad- +ditional resynthesis stage and measure the Unit-Edit-Distance +(UED) between the first and the second units representing the +speech. This metric was recently proposed by [14] to evalu- +ate robustness of discrete speech representation against signal +variations. Intuitively, a high UED indicates redundancies in +the discrete units. To reach the final CR metric, for each pair +of units, we calculate the percentage of swaps between them +over all the dataset’s transcriptions. +3.3. Robust Clustering +Equipped with the CR metric, we explore three simple meth- +ods to improve the k-means clustering quality. In all three +methods, we start from the standard k-means with k = 2000 +and iterativly merge the clusters to reach the target number +of clusters. The first method, named Double K-means. In +which, we apply an additional k-means over the cluster cen- +torids from the first k-means step. The second method, de- +noted as K-means with Hierarchical Clustering, we apply +an an agglomerative clustering over the cluster centorids from +the first k-means step. The last method, named K-means with +Weighed Hierarchical Clustering, we use an agglomerative +Table 2. +Units Resynthesis results. +CER for UTS using +lookup and concatenate methods. The table contains results +for different lookup key types: Local-Single (L-S),Local-Full +(L-F) Context-Single (C-S) and Context-Full (C-F). +Model +Size +Hifi-GEN +Key Type +C-F +C-S +L-F +L-S +CPC +50 +5.95 +9.12 +25.36 +39.57 +60.98 +100 +5.67 +6.52 +15.21 +22.51 +53.59 +200 +5.37 +5.12 +10.16 +15.18 +40.65 +HuBERT +50 +7.31 +10.31 +14.96 +47.24 +58.42 +100 +4.39 +5.24 +6.26 +26.55 +57.49 +200 +4.10 +4.25 +4.69 +15.56 +19.88 +MFCC +50 +50.47 +33.85 +57.60 +71.43 +69.22 +100 +44.68 +15.79 +46.55 +67.54 +66.13 +200 +41.67 +6.22 +30.47 +61.46 +61.31 +clustering using a modified version of the euclidean distance, +weighted by the CR metric. Formally, the distance metric is +defined as follows: +D(i, j) = L2(ci, cj) · SWAP(ui, uj), +SWAP(ui, uj) = 1 +2 [CR(ui, uj) + CR(uj, ui)] , +(2) +while ci, cj are the ith and jth cluster continuous centroids, +and ui, uj are the ith and jth discrete unit. +4. RESULTS +4.1. Datasets +We use the the Librispeech[21] corpus to learn the k-means +clustering (train-clean-100), and the test-clean to +evaluate both the clustering methods and the look-up vocoder. +Additionally, we use the Librispeech corpus for calculating +the V-Measure for speaker and gender. For computing the +V-Measure over phonemes we use the TIMIT benchmark. +4.2. Units Interpretation +Table 1 presents the V-Measure results regarding three dif- +ferent attributes - speaker, gender, and phoneme. +The V- +Measure for the speaker and gender scores is lower than the +score of the phonemes- which indicates of high correlation +to the phonemes and a low correlation to the speaker or gen- +der. In addition, when we check the effect of the number +of the units- while for the speaker/gender, more units lead +to a higher score, in the phoneme score there is a max point +both for the HuBERT and CPC configurations. Therefore, we +claim that redundancies cause this trend in the units. Finally, +we can see that CPC has a higher score for the phonemes- but +also a higher score for speaker and gender. +4.3. Units Visualization +Figure 3 shows the spatial structure of the units. One can +see that there is a very consistent structure- first, units that + +Fig. 3. 2D view of the units’ centers. Each bounded area represents a single unit and is colored by the unit’s phoneme. We use +T-SNE and Voronoi diagram to get the units areas. The matching between the units and phonemes was made using the TIMIT +corpus, while each unit was labeled as a phoneme that represents her most commonly. +Table 3. Comparing the different clustering methods using ABX and speaker information.For all these metrics, lower is +better.The methods are : Regular k-means (K), Double K-means (K-K),K-means with Hierarchical Clustering (K-H) and K- +means with Weighed Hierarchical Clustering (K-WH) +Model +Size +ABX within +ABX across +Speaker probing +K +K-K +K-H +K-WH +K +K-K +K-H +K-WH +K +K-K +K-H +K-WH +CPC +50 +5.66 +5.38 +9.62 +8.80 +7.83 +6.77 +11.46 +10.56 +42.22 +32.96 +19.26 +18.15 +100 +5.42 +5.44 +6.66 +6.04 +7.07 +7.13 +8.26 +7.49 +52.96 +45.19 +20.37 +15.56 +200 +5.53 +5.27 +5.61 +5.68 +7.35 +7.10 +7.28 +7.13 +63.70 +49.63 +26.30 +22.59 +HuBERT +50 +7.23 +5.67 +5.94 +6.12 +8.93 +6.83 +7.43 +7.67 +30.37 +36.30 +36.67 +31.85 +100 +5.82 +5.01 +5.30 +5.29 +7.47 +6.50 +6.54 +6.32 +48.15 +48.89 +48.15 +46.67 +200 +5.79 +5.24 +5.18 +5.05 +7.49 +6.42 +6.46 +6.07 +65.19 +61.11 +54.81 +62.96 +represent the same phoneme are usually close to each other. +Moreover, phonemes from the same family (affricates, frica- +tives, Etc.’ ) tend also to be close to each other. In addition, +we can see that while for HuBERT and CPC, the space divide +between the different phonemes families is generally equal, in +the MFCC model, almost all the space uses for vowels. No- +tice, redundancies in the clusters can be also observed from +such figures. +4.4. Units Resynthesis +In Table 2, we shows the results for the units resynthesis. +We can see that for some configurations, there is slightly dif- +ference between the HiFi-GAN and the look-up scores- this +strengthens our understanding that units express fixed sounds +and are mainly correlative to phonemes. We can see that the +context of the units critically affects the results, while the +unit’s length has a lower effect. Finally, this understanding +may help in understand units’ redundancies, i.e., the same +phoneme in a different context will represent different units. +4.5. Robust Clustering +We evaluate the proposed approach along two different +axes: (i) phonetic measure in the form of ABX within and +across [22]; (ii) speaker information in the form of probing +similarly to [13]. Table 3 summarizes the results. We can see +that the proposed methods, although they are straightforward, +improve both the ABX and the speaker results for most of the +configurations. Furthermore, the best results for ABX-across +were obtained using CR- this strengthens our claim regarding +the unit’s redundancies. +5. CONCLUSION +In this work, we analyzed the GSLM discrete unit from three +different and complementary points of view: interpretation, +visualization, and resynthesis. The analysis showed a strong +correlation between the units and the phonemes. In addition, +we found redundancies in the units, which the units’ context +can explain. Finally, we proposed methods that improve the +unit’s clustering based on these understandings. + +fricatives +stops +affricates +nasals +semivowels +vowels +others +HuBERT +CPC +MFCC +laaan +awa +ao +aa +eh +aw +ae +s +回 +h# +sh +Inr +Ley +ux +ow +layl +h# +h# +可国 +w +In +ae +lae +sh] +dcl +PP +s +Th# +HiyNiy +Imm +pcll +iyng +Itcl +ep +h# +y +a +h# +aol +h# +Aht +h#6. 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IEEE, 2015, pp. 5206–5210. +[22] Jacob Kahn et al., “Libri-light: A benchmark for asr +with limited or no supervision,” in ICASSP 2020-2020 +IEEE International Conference on Acoustics, Speech +and Signal Processing (ICASSP). IEEE, 2020, pp. +7669–7673. + diff --git a/N9AyT4oBgHgl3EQftPlh/content/tmp_files/load_file.txt b/N9AyT4oBgHgl3EQftPlh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fca5972ca2bbe253db21dd23715c9ca6d2754163 --- /dev/null +++ b/N9AyT4oBgHgl3EQftPlh/content/tmp_files/load_file.txt @@ -0,0 +1,405 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf,len=404 +page_content='ANALYSING DISCRETE SELF SUPERVISED SPEECH REPRESENTATION FOR SPOKEN LANGUAGE MODELING Amitay Sicherman and Yossi Adi School of Engineering and Computer Science The Hebrew University of Jerusalem, Israel ABSTRACT This work profoundly analyzes discrete self-supervised speech representations through the eyes of Generative Spoken Lan- guage Modeling (GSLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Following the findings of such an analysis, we propose practical improvements to the discrete unit for the GSLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' First, we start comprehending these units by analyzing them in three axes: interpretation, visualization, and resynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Our analysis finds a high correlation between the speech units to phonemes and phoneme families, while their correlation with speaker or gender is weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Addition- ally, we found redundancies in the extracted units and claim that one reason may be the units’ context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Following this analysis, we propose a new, unsupervised metric to measure unit redundancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Finally, we use this metric to develop new methods that improve the robustness of units clustering and show significant improvement considering zero-resource speech metrics such as ABX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Code and analysis tools are available under the following link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Index Terms— self supervised learning, generative spo- ken language modeling, textless NLP, speech LM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' INTRODUCTION Recently Self-Supervised Learning (SSL) methods for speech have shown great success on plenty of downs stream tasks [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' From Automatic Speech Recognition [2, 3, 4] and speaker diarization [5], to phone segmentation [6], these models have shown remarkable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Specifically, these SSL models allow recent success in Generative Spoken Language Modeling (GSLM) [7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' In GSLM, we aim to learn a discrete representation of the speech signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' This is often being done by applying the k- means algorithm over the continuous representation obtained from a SSL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Then, we train a unit Language Model (uLM) over such representation, and lastly we decode it back to a time domain signal using neural vocoder [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' During inference time, we can sample from the uLM conditionally or unconditionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Although these models are capable of generating mean- ingful and coherent speech utterances, little is known about the properties captured but these discrete representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The authors in [11] examined the purity between phonetics ele- ments and the discrete units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The authors’ proposed method is to analysis the discrete SSL representation considering fine-grained linguistic properties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=', different articulatory classes or closure and release portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The authors in [12] proposed a probing method to analyze the presence of phone classes, gender and language information while comparing monolingual and bilingual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' In this work, we analyze quantitatively and visually dis- crete representations obtained by HuBERT and CPC models with respect to phoneme classes, gender and speaker identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Next, equipped with such an analysis we provide a metric to identify redundancies in the k-means clustering, and propose a method to improve upon it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' We find a high correlation between the units and the phonemes, but with many redundancies in the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' We show that one reason may be the units’ context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' In Addition, we propose an unsupervised metric to measure these redun- dancies and we use it to significant improvement the unit clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' BACKGROUND The general GSLM pipeline is comprised of three main mod- ules: (i) Speech-to-unit, (ii) Unit language model, and (iii) Unit-to-speech, where each of these modules is trained sepa- rately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Speech resynthesis can be achieved while ignoring the language model and directly feeding the quantized units into the unit-to-speech module [10] Speech To Unit (STU) module encodes the raw speech signal into a discrete representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The model first encodes the speech into a continuous representation and then quantize the representation to a sequence of discrete units [7, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Formally, denote the domain of audio samples by x ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The representation for a raw signal is therefore a sequence of samples x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' , xT ), where xt ∈ x for all 1 ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Consider an encoder network, f, that gets as input the speech utterance and outputs a sequence of spectral representations sampled at a low frequency as follows f(x) = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' , vT ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Note that we do not assume anything about the structure of the encoder network f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Since the representations learned by such models are usually continuous, a k-means algorithm is arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='00591v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='CL] 2 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Units visualization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' applied over the models’ outputs to generate discrete units, denoted as z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' , zT ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Each element zi in z is a pos- itive integer, zi ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='., K} for 1 ≤ i ≤ T ′, where K is the number of discrete units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' As the quantized representation, z, usually contain units repetitions which degrade the performance of the language modeling, a common approach is collapse repetitions and generate a de-duplicated sequence while additionally storing the units’ duration separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' For instance, the sequence 12,12,25,31,31,31 will be converted into 12,25,31 and the corresponding durations 2,1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Unit Language Model (ULM) is trained on the extracted and deduplicated discrete units, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The language model can be used, for example, to generate speech conditionally or un- conditionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Unit To Speech module converts the discrete speech repre- sentation, z, to a raw waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The authors in [7] used a Tacotron2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='0 [15] based model followed by WaveGlow [16] vocoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Later, [10] proposed a unit-based vocoder based on the HiFi-GAN architecture to directly convert units to speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' In this work, we focus on the latter setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' METHOD We analyze representations obtained by either HuEBRT [2] or CPC [4] models considering various number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' All analysis code and the developed visualization tools will be publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Analysis Units Interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' We start by measuring the mutual in- formation between the discrete representation and different speech properties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=', phonemes, speaker id, and gender), using the V-Measure score [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' For this purpose, we align each utterance with its corre- sponding attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' To get units-to-phonemes alignment we use the TIMIT corpus [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The TIMIT dataset contains pairs of audio - phonemes, which are time aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' For speaker and gender analysis we use the LibriSpeech corpus as it contains large and diverse set of speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Circular Resynthesis evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Units Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' An additional point of view of the units meaning is the spatial structure of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' For this purpose, we create a 2D spatial view that contains information regard- ing the relation between the continuous representation, the discrete units, and their corresponding phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Specifi- cally, we apply the following two steps: (i) We project the high-dimensional speech representation into 2d using the T- SNE [19] algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' T-SNE is a nonlinear dimensionality re- duction that intuitively preserves the non-linear distance re- lations between neighbors in the high and low dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Then, we use the Voronoi diagram [20] that converts the scat- ter plot into an area plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Finally, we have left with a bounded area in the 2D space for each unit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' (ii) In the second part, we create a single label to represents each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' We use the units-phonemes alignment from the TIMIT (similarly to the process in previous paragraph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Then, we assign for each cluster the most represented phoneme in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Finally, we re- place the unit id with their corresponding phonemes and color the area base on the phoneme and phoneme family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' A visual description of the proposed method can be seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Units Resynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Next, we analyze the units’ information from the opposite direction - that is, through the speech resyn- thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' We decode the units back to speech using a look-up- table of the corresponding 20ms speech segments, then we transcribe the generated audio and measure the transcription error (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=', the Character Error Rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Intuitively, in case of strong correlation between the units and the phonemes - we can take a single “sound” to represent each unit - and apply the UTS step using the concatenation of these sound pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Notice, this approach is different than the one in [10] as there is no neural vocoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Formally, let u, l be a sequences of deduplicated units and their length obtained by applying STU on the input audio x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' and let xi be the part in x that is matched to deduped unit, ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Notice, xi can be of arbitrary length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Lookup Vocoder defines as : LV (u, l) = concat(F(u1, l1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' , F(un, ln)), F(ui, li) = � T[Key(ui, li)], if Key(ui, li) in T xi, else , (1) K-Means Centers T-SNE Voronoi Diagram N × Multicdimensional Vectors Acoustic-PhoneticCorpus M Units : 11313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='17 M 3-M 4 Phonemes : A wI I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='. SH 7 - SH SHcR(i) 32 UED > > s<- hs 32 -> 28 1, 7, 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='. 28Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Units Interpretation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' For phonemes, higher is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' While for the speaker and gender, lower score indicates that the model manages to hide information about the speaker and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Model Size Speaker Gender Phoneme CPC 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='66 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='30 100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='54 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='45 200 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='62 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='74 2000 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='14 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='06 HuBERT 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='03 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='49 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='17 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='48 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='21 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='64 2000 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='65 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='32 MFCC 50 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='90 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='57 100 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='54 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='97 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='73 200 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='81 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='59 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='96 where T is a Look-up-table that stores for each key the corre- sponding xi of the first appearance of this key, and Key maps unit and length into key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' We explore four different types of Key : (i) Local-Single- Key(ui) = (ui);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' (ii) Local-Full- Key(ui) = (ui, li);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' (iii) Context-Single- Key(ui) = (ui−1, ui, ui+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' (iv) Context- Full- Key(ui) = (ui−1, ui, ui+1, li).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Circular Resynthesis We introduce the Circular Resynthesis (CR) method, an ut- terly unsupervised evaluation metric that aims to measure the redundancies in the discrete units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' As described in Figure 2, we first perform a full resynthesis procedure, in which we encode and decode the speech signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Then, we apply an ad- ditional resynthesis stage and measure the Unit-Edit-Distance (UED) between the first and the second units representing the speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' This metric was recently proposed by [14] to evalu- ate robustness of discrete speech representation against signal variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Intuitively, a high UED indicates redundancies in the discrete units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' To reach the final CR metric, for each pair of units, we calculate the percentage of swaps between them over all the dataset’s transcriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Robust Clustering Equipped with the CR metric, we explore three simple meth- ods to improve the k-means clustering quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' In all three methods, we start from the standard k-means with k = 2000 and iterativly merge the clusters to reach the target number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The first method, named Double K-means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' In which, we apply an additional k-means over the cluster cen- torids from the first k-means step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The second method, de- noted as K-means with Hierarchical Clustering, we apply an an agglomerative clustering over the cluster centorids from the first k-means step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The last method, named K-means with Weighed Hierarchical Clustering, we use an agglomerative Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Units Resynthesis results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' CER for UTS using lookup and concatenate methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The table contains results for different lookup key types: Local-Single (L-S),Local-Full (L-F) Context-Single (C-S) and Context-Full (C-F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Model Size Hifi-GEN Key Type C-F C-S L-F L-S CPC 50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='95 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='12 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='36 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='57 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='98 100 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='67 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='52 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='21 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='51 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='59 200 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='16 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='18 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='65 HuBERT 50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='31 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='31 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='96 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='24 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='42 100 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='39 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='26 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='55 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='49 200 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='69 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='56 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='88 MFCC 50 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='47 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='85 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='60 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='43 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='22 100 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='68 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='79 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='55 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='54 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='13 200 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='67 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='22 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='47 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='46 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='31 clustering using a modified version of the euclidean distance, weighted by the CR metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Formally, the distance metric is defined as follows: D(i, j) = L2(ci, cj) · SWAP(ui, uj), SWAP(ui, uj) = 1 2 [CR(ui, uj) + CR(uj, ui)] , (2) while ci, cj are the ith and jth cluster continuous centroids, and ui, uj are the ith and jth discrete unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Datasets We use the the Librispeech[21] corpus to learn the k-means clustering (train-clean-100), and the test-clean to evaluate both the clustering methods and the look-up vocoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Additionally, we use the Librispeech corpus for calculating the V-Measure for speaker and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' For computing the V-Measure over phonemes we use the TIMIT benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Units Interpretation Table 1 presents the V-Measure results regarding three dif- ferent attributes - speaker, gender, and phoneme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The V- Measure for the speaker and gender scores is lower than the score of the phonemes- which indicates of high correlation to the phonemes and a low correlation to the speaker or gen- der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' In addition, when we check the effect of the number of the units- while for the speaker/gender, more units lead to a higher score, in the phoneme score there is a max point both for the HuBERT and CPC configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Therefore, we claim that redundancies cause this trend in the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Finally, we can see that CPC has a higher score for the phonemes- but also a higher score for speaker and gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Units Visualization Figure 3 shows the spatial structure of the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' One can see that there is a very consistent structure- first, units that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 2D view of the units’ centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Each bounded area represents a single unit and is colored by the unit’s phoneme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' We use T-SNE and Voronoi diagram to get the units areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The matching between the units and phonemes was made using the TIMIT corpus, while each unit was labeled as a phoneme that represents her most commonly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Comparing the different clustering methods using ABX and speaker information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='For all these metrics, lower is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='The methods are : Regular k-means (K), Double K-means (K-K),K-means with Hierarchical Clustering (K-H) and K- means with Weighed Hierarchical Clustering (K-WH) Model Size ABX within ABX across Speaker probing K K-K K-H K-WH K K-K K-H K-WH K K-K K-H K-WH CPC 50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='66 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='38 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='62 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='80 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='83 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='77 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='46 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='81 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='96 represent the same phoneme are usually close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Moreover, phonemes from the same family (affricates, frica- tives, Etc.’ ) tend also to be close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' In addition, we can see that while for HuBERT and CPC, the space divide between the different phonemes families is generally equal, in the MFCC model, almost all the space uses for vowels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' No- tice, redundancies in the clusters can be also observed from such figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Units Resynthesis In Table 2, we shows the results for the units resynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' We can see that for some configurations, there is slightly dif- ference between the HiFi-GAN and the look-up scores- this strengthens our understanding that units express fixed sounds and are mainly correlative to phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' We can see that the context of the units critically affects the results, while the unit’s length has a lower effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Finally, this understanding may help in understand units’ redundancies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=', the same phoneme in a different context will represent different units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Robust Clustering We evaluate the proposed approach along two different axes: (i) phonetic measure in the form of ABX within and across [22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' (ii) speaker information in the form of probing similarly to [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Table 3 summarizes the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' We can see that the proposed methods, although they are straightforward, improve both the ABX and the speaker results for most of the configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Furthermore, the best results for ABX-across were obtained using CR- this strengthens our claim regarding the unit’s redundancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' CONCLUSION In this work, we analyzed the GSLM discrete unit from three different and complementary points of view: interpretation, visualization, and resynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' The analysis showed a strong correlation between the units and the phonemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' In addition, we found redundancies in the units, which the units’ context can explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' Finally, we proposed methods that improve the unit’s clustering based on these understandings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' fricatives stops affricates nasals semivowels vowels others HuBERT CPC MFCC laaan awa ao aa eh aw ae s 回 h# sh Inr Ley ux ow layl h# h# 可国 w In ae lae sh] dcl PP s Th# HiyNiy Imm pcll iyng Itcl ep h# y a h# aol h# Aht h#6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9AyT4oBgHgl3EQftPlh/content/2301.00591v1.pdf'} +page_content=' REFERENCES [1] Shu-wen Yang et al.' metadata={'source': 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Management∗ +Takahiro Mamiya, Shiu Mochiyama, and Takashi Hikihara +Department of Electrical Engineering, Kyoto University +Abstract +Power supply for small-scale battery-powered systems such as electric vehicles (EVs and mobile robots) is +being actively researched. +We are particularly interested in energy management, which considers the inter- +connection of such systems close to each other. This allows for overall redundancy to be maintained without +assuming excessive redundancy with individual power sources. Its implementation necessitates a high level of +integration between power management and information and communication technology. As one of these meth- +ods, this study investigates energy management based on power packetization. When the individual systems +to be connected have moving parts or are mobile, wireless power transmission is a promising method for power +sharing. However, power packetization has so far only been considered for wired transmission. In this paper, we +address the integration of power and information in wireless channels using power packetization. We propose a +power packet router circuit that can wirelessly transmit power over multiple channels selectively. Furthermore, +we demonstrate that the developed system can handle both wired intrasystem power management and wireless +intersystem power sharing in a unified manner. +1 +Introduction +Recent days have witnessed widespread use of electric power systems that are equipped with batteries and can thus +be driven without relying on an external and large power grid. Common examples include electric vehicles (EVs) +and mobile robots. While much effort has been dedicated to independent power management in such a system, +another research trend is the management of a network of such systems. We refer to a minimum element of a system +that can independently operate a local system throughout the paper. Constituting a networked system addresses +shared redundancy of power source capacity as a whole system, rather than as each individual system. That is, +when the power demand of one system temporarily increases, power can be supplied not only from the inside power +sources but also from the power sources of the other connected systems [1–3]. +Because local systems are spatially dispersed and can have a time-dependent supply/load profile, managing such +a network necessitates advanced sensing, computation, and communication technologies [4–6]. Several proposals for +power system management with ICTs support have been made [2,7,8]. Among them, a power packet dispatching +system is an encouraging proposal for the purpose. The system packetizes supplied power; that is, power is divided +into time segments, each of which is associated with an information tag via a voltage waveform [9, 10]. Power +packetization ensures that information exchange and power transmission occur concurrently in the physical layer, +allowing for power management in a network without causing an imbalance in information and physical quantity +processing. In the previous study, the authors’ group developed a circuit called a power packet router [9]. We +validated the concept of power packetization and routing with hardware configuration including the routers. +One advantage of power packetization is the ability to easily attach/detach local systems from a larger network. +The use of time-division multiplexing and physical tag attachment ensures that each packetized power transfer is +independent. In other words, power transfers between different pairs do not get mixed up even on the same power +line but can be differentiated physically. This leads to realizing what could be called a plug-and-play from the +perspective of power supply. +One difficulty here is that the power packet dispatching system has so far been developed using a wired connection +for power transfer. Wireless power transfer (WPT) is a revolutionary technology for supplying power to mobile +systems [11, 12]. It is beneficial for improved maneuverability of each local system to introduce the WPT to the +∗This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this +version may no longer be accessible. +1 +arXiv:2301.05368v1 [eess.SY] 13 Jan 2023 + +Packet +network +Packet +network +WPT +router +Load +Storage +Router +Packet +network +Wired connection +Wireless connection +Router +Router +Router +Router +Router +WPT +router +WPT +router +Power +source +Storage +Storage +Load +Load +Load +Load +Local system +Local +system +Local system +Connected system +Power packet +Figure 1: Surplus power supply via wireless power transfer between power packet networks. +power packet dispatching system for connections at the boundaries of the local systems. However, as discussed in +Section 3, simply connecting a WPT circuit to a power packet system does not work in conjunction with packet- +based power management in local systems. This research seeks to achieve on-demand power supply concentration +and dispersion in the connected network of local systems while ensuring easy attachment/detachment between +systems via a wireless connection. +Here, we investigate the following two points as fundamental studies to realize the wireless connection of mul- +tiple local systems powered by power packets. First, we suggest a dedicated router design in both software and +hardware configurations to ensure physical packetization with a wireless channel and collaboration with wired power +management. Then, in a connected system comprised of three local systems with the developed router installed, +we assert the selective transmission of power packets to only the local system designated by the tag. Second, we +demonstrate a power-sharing strategy for increasing power capacity redundancy via a wireless connection. With +wired and wireless connections, the parallel operation of intra- and intersystem power management is demonstrated. +Among a network of two local systems, each of which supplies a certain demand of its own load via wired connection, +surplus power at one system is transferred to another. +Many proposals for the duplex of multiple channels in WPT have been made, including multiplexing in time, +frequency, and spatial domain [13–15]. +Furthermore, several reports have addressed the simultaneous wireless +transmission of power and information [4, 16–18]. These proposals essentially assume that a power transmission +channel has already been established and that information is being transmitted concurrently, or vice versa. Our +proposal, on the other hand, attempts to go beyond the simple parallel transmission by integrating information that +manipulates the spatiotemporal distribution of power with power transmission itself at the physical layer. This, in +theory, eliminates the disparity between physical quantity and information, allowing us to achieve both wired and +wireless power transmission by cooperating for smart power management. +2 +Outline of power packet dispatching system +The basic configuration and operation of power packet dispatching systems are described in this section. +2.1 +Constitution of power packet +As depicted in Fig. 1, a power packet is a unit of power management in the system. A power packet comprises +pulse-shaped electric power called a payload and an information tag, a header, and a footer, which are attached +just before and after it. The information tag is a logic bitstream realized by a voltage waveform without current. +The tag can include any information, like the origin, destination, and length of the power packet. +The physical tag attachment enables power packet transmission to be time-division multiplexed. Power from +different sources and destinations is transmitted on the same channel while remaining distinct from one another. +2 + +owel Packc +voltage +01000011 +.1100101 +time +header +payload +: footerrouter input +router output +isolator +controller +gate driver +gate driver +storage1 +storage2 +demand signal +clock +input1 +input2 +output1 +output2 +power +signal +Figure 2: Circuit example of 2 input 2 output router [9]. +This feature sets the power packet dispatching system apart from conventional systems that treat power as a +continuous flow. +2.2 +Network configuration of power packet dispatching system +Each local system of Fig. 1 denotes a local system configuration example. Routers connect power sources, storage, +and loads to the network in this system. +A power packet router is installed as a node that connects multiple +transmission lines. The router forwards power packets by selecting a transmission line according to the packet’s tag +information [10]. +A power packet is routed from a source to a specific destination via several routers. The path to the load is not +required to be unique and can be changed dynamically depending on the situation. This feature facilitates flexible +power management in conjunction with a dynamic supply relationship. In the following section, we develop a router +that can perform this function even with a wireless connection. +2.3 +Routing method for power packets +Here we characterize the circuit configuration of a router and the principle of its routing operation [9, 19, 20]. +Figure 2 depicts the circuit configuration of a previously proposed, wire-connected router [21]. The circuit consists +of two sections: an input section that receives power packets from the transmission network and an output section +that forwards power packets to the transmission network. The operation of the input part is initialized when a +power packet reaches the router. The input section includes a signal reading circuit for reading the logic signals +of information tags. When the router recognizes that the incoming power packet is addressed to it, it turns on +the corresponding semiconductor switches to receive the payload power. For circuit protection, the signal reading +circuit electrically separates its signal output from the power supply lines using a device such as a photocoupler. +The incoming power packet is temporarily stored before being forwarded to the next hop. The output part generates +power packets from the temporal storage in response to the demand. In some cases, the circuit can be reduced to +just the input or output section. When installed just next to the source, for example, the output section with the +storage replaced by a power source is sufficient to produce a generated packet. Similarly, a circuit just before a load +can only be the input part, with the storage replaced by a load. +To read the logic signals of power packets, clocks corresponding to the one-bit width of a power packet must +be synchronized among adjacent routers. This can be accomplished by installing an additional wire for a common +clock input, or by adding another signal to the header for autonomous clock synchronization [22]. In this paper, we +employ a simple autonomous clock synchronization scheme, in which the clock period is fixed in advance, the first +three bits of the header are set to 010, and the phase is shifted if the 010 is not detected within a certain period. +The information tag consists of bits 1–3, which implies 010 for clock synchronization, and bits 4–7, which imply +the address of the output destination. Bits 8 – 100 correspond to the payload. For simplicity, the packet length is +fixed at 100 bits and this setting is shared by all routers. In this way, we exclude the footer. +3 + +3 +Router design for wireless transmission of power packets +In this section, we propose a router configuration for wireless transmission of power packets. We employ magnetic +resonant coupling for the wireless transmission. +This method is capable of transmitting large power over long +distances with high-efficiency [11]. This circuit is powered by AC, whereas the power packet dispatching system is +powered by DC. We must convert the current to incorporate WPT into power packet routing. Figure 3 depicts a +conceptual diagram of the voltage and magnetic flux density in the wireless power packet transmission. Using a +magnetic resonant coupling circuit that includes an inverter and a rectifier, DC is converted to AC and then back +to DC after wireless transmission. The following section describes the router design. +It should be noted that the inclusion of wireless transmission in the power packet dispatching system was first +proposed in the authors’ previous report [23]. +In the report, the wireless transmission was not packetized but +introduced as a one-to-one transmission channel without any tag attachment. In this paper, we propose a novel +router configuration that bring the functions of physical tag attachment and its reading to the wireless power +transfer. These functions not only realize the physical packetization of wireless power transmission but also extends +its use to packet-based power management as introduced in Section 5. +3.1 +Wireless transmitter of the power packet +Figure 4 depicts a router circuit for wireless power packet output. The configuration includes an inverter circuit +connected to the router’s output section, as described in Section 2.3. For DC/AC conversion, a class-E inverter [24] +is used. +The output circuit wirelessly transmits both the header signal and the payload power. +In this paper, the +inverter’s input is presented as a form of packetized power. The current flowing through the coil and the magnetic +flux density induced in the coil is thus modulated in an amplitude-shift-keying (ASK) manner according to the shape +of the power packet, as depicted in the middle of Fig. 3. It should be noted here that the header signal transmission +must minimize power consumption while the payload transmission must maximize the amount of power transferred. +The two requirements cannot be met solely through the transmitter’s operation, but rather through the design of +the receiver side. This point will be covered in greater detail in the following section. +3.2 +Wireless receiver of the power packet +To receive a wirelessly transmitted power packet, demodulation of the ASK-modulated header signal and highly +efficient AC/DC conversion of the payload are necessitated. +The proposed circuit shown in Fig. 5 meets both +requirements by dividing the demodulator into two circuits. The signal demodulation circuit reads the header, +and a class-E rectifier receives the payload. The detailed procedure is provided below. Initially, the switch Sd +connected to the signal demodulation circuit is turned on, while the SR connected to the rectifier circuit is turned +off. For signal demodulation, the envelope of the voltage across the secondary circuit’s resonant capacitor is passed +through an RC low-pass filter. The router’s controller then samples it at a predetermined clock cycle to convert +it into a logical sequence. The controller activates the switches that connect the coil to the rectifier circuit when +it determines from the tag that the power packet is addressed to itself. This causes a class-E rectifier to convert +the wirelessly transmitted payload into DC output. When the power packet is directed at another router, the +router’s controller disconnects both circuits and opens the coil. The detachment is used to avoid the influence of the +unintended connection and the resulting impedance change, which may degrade power transmission at the addressed +connection. At the end of the previous power packet, the controller turns on the switch to the demodulation circuit +to prepare for the next power packet. The end of a power packet is detected by simply counting the length of the +payload in 100-bit intervals. +Of course, simply connecting the signal demodulation circuit and the Class-E rectifier in parallel allows you to +read the header and receive the payload. However, when receiving the header, the current passes through the Class- +Encode +Power packet (DC) +ASK modulated AC +Packet's header +Packet's payload (DC pulse) +Decode +on Wire +on Wire +Wireless +Figure 3: A waveform concept during wireless transmission of power packets. +4 + +C2 +r1 +L1 +Lm +Lf1 +C1 +S1 +Controller +Gate driver +Source +Figure 4: Wireless transmitter of the power packet. +C3 +r2 +L2 +Lm +D1 +Lf2 +C4 +Cf +Rectifier +Rd +Dd +Cd2 +Demodulator +Isolator +Controller +Gate driver +Cd1 +Output +Sd +SR +Figure 5: Wireless receiver of the power packet. +E rectifier, and when receiving the payload, it passes through the demodulation circuit. Such a current contributes +nothing to the receiving operation but results in power loss. Because this type of loss is much greater than the loss +caused by the switching of the two demodulation circuits, the proposed scheme can greatly reduce the loss. +The frequency of the carrier wave used for magnetic resonant coupling is 1 MHz. The wireless router’s constants +are determined as shown in Table 1. The design is conducted in the following manner, regarding [24]. The coil has +a diameter of 100 mm, a wire diameter of 1 mm, several turns of 10, and a thickness of 12 mm. The transmission +circuit’s rise time was measured to be 25 µs. The rise time is defined as the time required for the output voltage to +attain 90 % of its steady-state value. The steady-state value was obtained under the test condition where the load +was 47 Ω resistor and the vertical distance between the coils was 50 mm. Based on this, we determined that the bit +width of the power packet should be 100 µs, which is sufficiently larger than the rise time. That is, the modulation +frequency is 10 kHz. The demodulation circuit is designed to demodulate signals with a cutoff frequency of about +100 kHz. +Table 1: Design values of circuit constants. +Primary side +Secondary side +Rectifier +Demodulator +f +1 MHz +L2 +19.2 µH +Cd1 +1.0 µF +Lf1 +100 µF +r2 +0.88 Ω +Cd2 +820 pF +C1 +3.3 nF +C3 +1.56 nF +Rd +12 kΩ +C2 +1.44 nF +C4 +1.68 nF +L1 +19.3 µH +Lf2 +100 µH +r1 +0.88 Ω +Cf +0.47 µF +Lm +1.75 µH +4 +Verification of selective reception of wirelessly transmitted power +packets +In this study, we consider one-to-many or many-to-many wireless power sharing among several local systems placed +close to each other. The packetization and time-division multiplexing methods enable simultaneous supplies between +different pairs of a transmitter and a receiver while completely distinguishing them. Here, we experiment with three +5 + +Local system 0 +Wireless +packet +encoder +V +Local system 1 +Local system 2 +R1 +Wireless +packet +decoder +Coil 0 +Coil 1 +Coil 2 +50mm +70mm +30mm +R2 +Wireless +packet +decoder +Figure 6: Network configuration with 3 local systems for verification of selectivity of wirelessly transmitted power +packet. +Figure 7: Verification of router operation mode for wirelessly transmitted power packets. +local systems, one transmitting and two receiving nodes. It is demonstrated that the two receivers can selectively +accept or reject power packets based on the attached information tag. The number of local systems and their +connection relationship can of course be easily expanded and modified due to packetization. +4.1 +Experimental setup for selective reception +The entire network configuration is depicted in Fig. 6. Local system 0 alternately sends power packets to local +systems 1 and 2, and systems 1 and 2 receive only those that match their addresses. Power packet header addresses +are set to 0001 and 0010 for systems 1 and 2, respectively. Local system 0 consists of a circuit from Fig. 4 and a +DC power supply of 12 V. Local systems 1 and 2 comprise a circuit of Fig. 4 with a load resistor of 47 Ω connected +to the output port. +Although the transmitting and receiving roles of the local systems are fixed for simplicity, it is possible to +transmit power packets bidirectionally by modifying the circuit configuration [25]. Therefore, this assumption will +not lose generality in power sharing. +To ensure that the router’s operation is not affected by the distance between the coils, the coil positions are set +as shown in Fig. 6. The coils of local systems 1 and 2 are placed at the same vertical distance 50 mm as the coil of +local system 0, but the horizontal distance is 30 mm and 70 mm, respectively. +4.2 +Receiving mode confirmation +First, we examine the switching behavior between the header signal demodulator and the payload rectifier circuit, +as designed in Section 3. Figure 7 depicts an internal signal of the router of local system 1 that represents the +receiver’s operation mode. The router was in the header mode every 10 ms, which corresponded to the transmission +cycle of the powder packets. Immediately after the header mode, the router switched to the payload mode every two +power packet deliveries. During the payload mode, power was supplied to the designated load. This suggests that +the controller received the header while connected to the demodulation circuit and then switched to the rectifier +6 + +payload mode +headermode +5.0 +2.5 +0.0 +-10 +0 +10 +20 +30 +time/ msFigure 8: Voltages at two loads in local systems 1 and 2. +Router m2 +Router l2 +Router rx +Router l1 +Router tx +Router m1 +part +� : Local system 1 +part � : Local system 2 +part � : Wireless power sharing +V1 +Rl1 +V2 +Rl2 +VCtx +Vm1 +VCl1 +VCrx +Vrx +VCl2 +Vm2 +VRl1 +VRl2 +Wireless +packet +encoder +Wireless +packet +reader +Wireless +packet +decoder +CNTL rx1 +Srx1 +CNTL m1 +Sm1 +CNTL m2 +Sm2 +CNTL tx2 +Stx2 +CNTL rx2 +Srx2 +CNTL l1 +Sl1 +CNTL l2 +Sl2 +CNTL tx1 +Stx1 +Ctx +Crx +Cl1 +Cl2 +Figure 9: Configuration of the network with 2 local systems connected wirelessly. +circuit in payload mode after recognizing the address. This result confirms that the proposed router can correctly +route power packets on the wireless channel. +4.3 +Confirmation of selective reception function +Second, we confirm that, according to the tag information, the local systems received time-division multiplexed +power packets. The load voltages of the routers of local systems 1 and 2 are depicted in Fig. 8. It can be seen that +local systems 1 and 2 received power alternately, indicating that they selectively accepted or denied receiving power +packets based on the attached destination address signal. Here, local system 1’s supply voltage was higher than +that of local system 2. This is because the output is proportional to the distance between the coils. This means +that, regardless of whether the output value is larger or smaller, the router’s selective reception is unaffected by the +difference in distance between the coils. +5 +Confirmation of power-sharing in the wirelessly connected systems +Next, in wirelessly connected local systems, we validate power management based on power packetization. We +consider two local systems where the local power supply is primarily managed via a wired connection. Every local +system consists of an internal power source, a capacitor, a wireless transmission circuit, and a resistive load. We +set a wirelessly connected networking system comprising two such local systems, as shown in Fig. 9. While each +local system supplies its source to its load, wireless power packet transmission compensates for excess or deficient +power. Each system’s goal is to keep the voltage supplied to the load above a certain level. +The proposed scheme deals with a connected system whose elements are subject to dynamic changes, such as +variable distance between local systems and time-dependent connection/disconnection of local systems. Dealing +with such dynamic changes altogether in a centralized controller is not ideal. Distributed control of power packet +transmission, however, is an effective method of accommodating such unpredictability. In this paper, we use a +distributed control scheme of packet-based power management [26], in which power packet transmission is managed +7 + +Load voltage/V +10 +local system 1 +local system 2 +0 +-20 +-10 +0 +10 +20 +time/msonly between adjacent routers. The following section describes the operation flow of the connected systems. +5.1 +Operation flow in connected systems +Capacitors are installed in the connected systems to generate and output power packets to the load. Power packets +are sent so that the voltages of these capacitors exceed a certain threshold. +The demand signal to the router for on-demand packet transmission can be given by information tags in power +packets or by using another channel such as radio signals [26]. In this paper, we use an external wire to transmit +demand signals for simplicity We designed an input high to the controller of the next router when the storage +voltage falls below the threshold. +We divide the configuration of Fig. 9 into the following three parts that are managed independently. +α Transmission from router m1 to router tx and router l1 +β Transmission from router tx to router rx +γ Transmission from router rx and router m2 to router l2 +The three parts’ basic operation principles are described below. +In part α, when the voltages across Ctx and Cl1 fall below the threshold, demand signals are transmitted to the +router m1 respectively. Router m1 generates and sends power packets to the destination from which the demand +signal is received. In the event of overlapping demand signals, priority is given to router l1 to keep the load voltage +stable. +In part β, router rx sends a demand signal to router tx when the voltage across Crx falls below the threshold. +Router tx generates and sends power packets to router rx based on the demand signal. +In part γ, when the voltage across Cl2 drops below the threshold, a demand signal is initially sent to router rx. +If a power packet is not delivered from router rx to router l2 within a certain amount of time, the demand signal +is sent to router m2, which generates and sends a power packet to router l2. +Besides the three principles, two constraints are imposed on the operation of routers tx and rx. First, they +do not output power packets if the voltages across its capacitor, Ctx or Crx, are lower than a certain value. To +transmit power packets, there must be an adequate potential difference between the source and the destination. +This constraint guarantees the possible difference between the source and destination capacitors and guarantees +the reliable transmission of power packets. Second, the routers are not enabled to input and output power packets +simultaneously. When both switches are switched on simultaneously, the circuits before and after the router are +linked parallel. In this case, the output impedance measured from the power supply (capacitor) located before the +router is lower than when only the input switch is turned on. This can result in an overcurrent at the source and +a rapid drop in capacitor voltage. The second constraint is levied to avoid this situation. This configuration may +prevent the router rx from emitting power on occasion. Even if this occurs, router m2 can supply power packets to +keep router l2’s voltage stable. +5.2 +Verification of connected systems operation +To test the operation of the connected systems, we set the supply voltages V1 =15 V and V2 =7 V. To create +a voltage gradient, the threshold voltages of capacitors Cl1, Ctx, Crx and Cl2 are set as 10 V, 9 V, 7 V and 5 V, +respectively. The parameters linked to wireless power transmission are set as depicted in Table 1 +It is worth noting that the routers’ wired channel switch units have been replaced with unidirectional ones. As +previously discussed, the symmetry of the circuit allows us to restrict the flow of power packets to one direction +without sacrificing generality. The circuit generates high by activating switch Sout−s, and low by activating switch +Sout−p. The diode prevents reverse current from flowing through the body diode of Sout−s. +5.2.1 +Confirmation of autonomous maintenance of capacitor voltage +We demonstrate the transmission of power packets and the modifications in voltages of each capacitor installed in +part α–γ. +Figure 10 depicts the voltages Vl1 and Vtx of the capacitors Cl1 and Ctx in part α and the gate signal of +the switches Sl1 and Stx1 that controlled the route of the power packets. It is observed that Vl1 and Vtx were +sustained above the threshold voltages. The voltages Vl1 and Vtx elevated when switches Sl1 and Stx1 were driven. +This demonstrates that capacitors Cl1 and Ctx effectively received power packets and were charged. Furthermore, +8 + +Figure 10: State of switches and voltage of capacitors in part α. +Figure 11: State of switches and voltage of capacitors in part β. +switching operation of Sl1 and Stx1 did not overlap at any time. +This result correlates to the setup that the +transmission of power packets to Cl1 is prioritized (see Section 5.1 for the details). +Figure 11 depicts the voltages Vtx and Vrx of the capacitors Ctx and Crx in part β and the gate signal of the +switch Srx that controlled the power packet reception of the router rx. Comparing the top and bottom graphs shows +that Vtx declined and Vrx elevated while Srx was on. This implies that power packets were wirelessly transmitted +successfully from router tx to router rx. It can also be validated that Srx turned off when Vtx attained the threshold +voltage. This implies that the system satisfied the constraints defined in Section 5.1, which hampers the output of +power packets under the threshold voltage. +Figure 12 depicts the voltages Vrx and Vl2 of the capacitors Crx and Cl2 in part γ and voltage waveforms of power +packets outputted from routers rx and m2. When Vl2 dropped below the threshold voltage, router rx transferred +power packets to router m2 so that Vl2 was kept above the threshold. Now let us concentrate on the operation +around t =25 ms when Vrx attains the threshold voltage. Router rx stopped outputting the power packets, and +simultaneously, router m2 started sending power packets. These findings suggest that the selective routing protocol +specified in Section 5.1 worked; the load sent the demand signal to the router rx at first, and if no packet was +transmitted, then sent to the router m2. +In Fig. 12, there exists a possible difference between the voltage of the power packet and Vl2. This was induced +by the forward voltage drop across the diode installed to prevent backflow current. This loss can be repressed by +using a switch instead of a diode. Thus, there is no impact on the verification of the principle. +Figure 13 depicts the gate signal of Stx1, output current from Ctx, input current to Crx, and the output voltage +waveforms of router rx. +When Ctx was outputting current, Crx was receiving current. +This implies that the +transmitted power packet was received without failure. Since router tx did not output power packets when Stx1 was +9 + +15 +10 +5 +Vcrx +V2 +0 +100 +50 +0 +50 +100 +6 +Gate signal / V +Sr1 +Stx1 +2 +0 +-100 +50 +0 +50 +100 +Time / msOutput voltage/V +15 +10 +5 +Vctx +Vcrx +100 +50 +0 +50 +100 +6 +signal /V +Srx1 +Gate +2 +0 +100 +50 +0 +50 +100 +Time/msFigure 12: Power packets and voltage of capacitors in part γ. +Figure 13: Input / output current and voltage of Ctx and Crx. +on, Stx1 and Stx2 were driven solely. Similarly, Srx1 and Srx2 were driven solely. From the above findings, it can +be deduced that the connected system achieves the load voltage maintenance with wireless power supply between +local systems 1 and 2 by following the control procedure defined in Section 5.1. +5.2.2 +Association between the percentage of power supply and the utilized power source +The percentage of power transferred on the wireless channel depends on the distance between the transceiver/receiver +coils. Hence, in the previous experiment’s setup, increasing the coil gap reduced the power supply capability from +the local system 1 to 2. The proposed control scheme of the routers can accommodate such a gap change by choosing +an appropriate supply channel. To test this operation, we compare the amount of wireless power transmission and +the power source selection in the local system 2 at various distances between the coils. We set three cases with +different distances: (i) 50 mm (the same as in the previous experiment), (ii) 100 mm, and (iii) > 250 mm. The setup +in case iii is supposed to be large enough to prevent wireless transmission. +Figures 14 and 15 depict the voltage Vrx and Vl2 and the power packets output by routers rx and m2 in cases ii +and iii. Please refer to Fig. 12 for the result in case i. The larger the distance between the coils, the less frequently +the router rx outputted power packets and the lower its average voltage got. On the other hand, Vl2 maintained +above the threshold in all cases. +Table 2 demonstrates the average of the input/output power of router rx and the output power of router m2 +10 + +Output voltage/V +15 +Vcrx +Vi2 +10 +5 +0 +-100 +50 +0 +50 +100 +Output voltage/V +15 +Vrx +Vm2 +10 +5 +0 +-100 +50 +0 +50 +100 +Time / ms5 +1.25 +Gate signal of Stx1 +Output current from Ctx +1.00 +signal +Current / A +3 +0.75 +2 +0.50 +Gate +1 +0.25 +0 +0.00 +100 +-50 +0 +50 +100 +12.5 +1.25 +Vcrx +10.0 +1.00 +A +7.5 +0.75 +Current / +5.0 +0.50 +2.5 +0.25 +0.0 +0.00 +-100 +-50 +0 +50 +100 +Time / msFigure 14: Power packets and voltage of capacitors in part γ of case (ii) : gap 100 mm. +Figure 15: Power packets and voltage of capacitors in part γ of case (ii) : gap 250 mm. +during the measured time 250 ms for different distances. The input/output power of router rx fell and the output +power of router m2 rose as the distance became larger. Meanwhile, the total output power of router rx and router +m2 had a slight change. This finding implies that the output power of router m2 compensates for the fall in the +output power of router rx. +From the above, it is asserted that the load voltage can be sustained autonomously by the proposed distributed +control scheme. Even when the amount of wireless transmission falls, the local system compensated for it with a +wired supply. +6 +Conclusion +In this paper, we developed a platform for wireless power packet transmission for power management among +numerous local systems. +First, we proposed a novel power packet router configuration capable of wireless transmission. The ASK modu- +lating circuit is installed on the router’s output side for both information and power transmission, with the power +packet serving as a power source. The input side includes a demodulation circuit for both information and power +receipt. The circuit shifts between a signal demodulation circuit and a power rectifier circuit to read the header +and receive the payload power, respectively. Not only does the switching configuration separate the incoming signal +and power, but it also reduces unnecessary power consumption during the receiving operation. +Using this router, we then verified the wireless power packet routing following the information tag. Physical tag +attachment and wireless power packet time-division multiplexing allowed receiving routers to distinguish the power +packet based on its destination address. The result shows that the proposed configuration allows for the selective +11 + +outputvoltage/V +15 +Vrx +Vi2 +10 +5 +0 +100 +50 +0 +50 +100 +output voltage/V +15 +Power packet from Crx +Power packet from V2 +10 +0 +-100 +50 +0 +50 +100 +time / msoutputvoltage/V +.5 +Vrx +Vi2 +10 +100 +50 +0 +50 +100 +outputvoltage/V +15 +Power packet from Crx +Powerpacketfrom V2 +5 +-100 +50 +0 +50 +100 +time / msTable 2: Input/output power of the routers in local system 2 at each gap. +Case +Gap +Router rx +Router rx +Router m2 +Total +input +output +output +output +i +50 mm +0.50 W +0.46 W +0.73 W +1.19 W +ii +100 mm +0.20 W +0.17 W +0.94 W +1.11 W +iii +> 250 mm +0.00 W +0.00 W +1.13 W +1.13 W +transmission of wireless power packets between multiple nearby local systems. This prevents interference with the +irrelevant power supply. +Next, we considered flexible coordination of inter- and intrasystem power management. The former was ac- +complished through the wireless transmission of power packets, while the latter was accomplished through a wired +supply. For this purpose, we created a distributed control scheme for the routers. A local system transmitted power +packets wirelessly to another when it had enough power while keeping the voltage of its load as a top priority. The +experiments revealed that the two types of operation were coordinated successfully. Furthermore, the proposed +distributed control scheme chose an appropriate supply channel based on the power interaction availability between +the local systems. We validated this operation by altering the gap between the coils of the two local systems, +demonstrating that the inter- or intrasystem power management was successfully chosen to satisfy the local loads’ +demand. +From the above verifications, we deduce that wireless power packet transmission can improve power management +capability in a connected power packet dispatching system by selectively cooperating wired and wireless power packet +transmission. +Acknowledgments +This work was partially supported by JSPS KAKENHI 20H02151, JST-OPERA Program no. JPMJOP1841, and +SIP Cross Ministerial Strategic Innovation Promotion Program no.18088028. +References +[1] E. Dialynas and N. D. Hatziargyriou, “Impact of microgrids on service quality,” 2007 IEEE Power Engineering +Society General Meeting, PES, pp. 1–5, 2007. +[2] M. M. He, E. M. Reutzel, X. Jiang, R. H. Katz, S. R. Sanders, D. E. Culler, and K. Lutz, “An architecture +for local energy generation, distribution, and sharing,” in Proc. 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(INTELEC). +IEEE, Oct. 2018, pp. 1–5. +13 + diff --git a/Q9E4T4oBgHgl3EQf_A6p/content/tmp_files/load_file.txt b/Q9E4T4oBgHgl3EQf_A6p/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..716538319e3e061f552c9c37744ad54deb955b59 --- /dev/null +++ b/Q9E4T4oBgHgl3EQf_A6p/content/tmp_files/load_file.txt @@ -0,0 +1,650 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf,len=649 +page_content='Router for Wireless Power Packet Transmission: Design and Application to Intersystem Power Management∗ Takahiro Mamiya, Shiu Mochiyama, and Takashi Hikihara Department of Electrical Engineering, Kyoto University Abstract Power supply for small-scale battery-powered systems such as electric vehicles (EVs and mobile robots) is being actively researched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' We are particularly interested in energy management, which considers the inter- connection of such systems close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This allows for overall redundancy to be maintained without assuming excessive redundancy with individual power sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Its implementation necessitates a high level of integration between power management and information and communication technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' As one of these meth- ods, this study investigates energy management based on power packetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' When the individual systems to be connected have moving parts or are mobile, wireless power transmission is a promising method for power sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' However, power packetization has so far only been considered for wired transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In this paper, we address the integration of power and information in wireless channels using power packetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' We propose a power packet router circuit that can wirelessly transmit power over multiple channels selectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Furthermore, we demonstrate that the developed system can handle both wired intrasystem power management and wireless intersystem power sharing in a unified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 1 Introduction Recent days have witnessed widespread use of electric power systems that are equipped with batteries and can thus be driven without relying on an external and large power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Common examples include electric vehicles (EVs) and mobile robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' While much effort has been dedicated to independent power management in such a system, another research trend is the management of a network of such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' We refer to a minimum element of a system that can independently operate a local system throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Constituting a networked system addresses shared redundancy of power source capacity as a whole system, rather than as each individual system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' That is, when the power demand of one system temporarily increases, power can be supplied not only from the inside power sources but also from the power sources of the other connected systems [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Because local systems are spatially dispersed and can have a time-dependent supply/load profile, managing such a network necessitates advanced sensing, computation, and communication technologies [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Several proposals for power system management with ICTs support have been made [2,7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Among them, a power packet dispatching system is an encouraging proposal for the purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The system packetizes supplied power;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' that is, power is divided into time segments, each of which is associated with an information tag via a voltage waveform [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Power packetization ensures that information exchange and power transmission occur concurrently in the physical layer, allowing for power management in a network without causing an imbalance in information and physical quantity processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In the previous study, the authors’ group developed a circuit called a power packet router [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' We validated the concept of power packetization and routing with hardware configuration including the routers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' One advantage of power packetization is the ability to easily attach/detach local systems from a larger network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The use of time-division multiplexing and physical tag attachment ensures that each packetized power transfer is independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In other words, power transfers between different pairs do not get mixed up even on the same power line but can be differentiated physically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This leads to realizing what could be called a plug-and-play from the perspective of power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' One difficulty here is that the power packet dispatching system has so far been developed using a wired connection for power transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Wireless power transfer (WPT) is a revolutionary technology for supplying power to mobile systems [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' It is beneficial for improved maneuverability of each local system to introduce the WPT to the ∗This work has been submitted to the IEEE for possible publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Copyright may be transferred without notice, after which this version may no longer be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='05368v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='SY] 13 Jan 2023 Packet network Packet network WPT router Load Storage Router Packet network Wired connection Wireless connection Router Router Router Router Router WPT router WPT router Power source Storage Storage Load Load Load Load Local system Local system Local system Connected system Power packet Figure 1: Surplus power supply via wireless power transfer between power packet networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' power packet dispatching system for connections at the boundaries of the local systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' However, as discussed in Section 3, simply connecting a WPT circuit to a power packet system does not work in conjunction with packet- based power management in local systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This research seeks to achieve on-demand power supply concentration and dispersion in the connected network of local systems while ensuring easy attachment/detachment between systems via a wireless connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Here, we investigate the following two points as fundamental studies to realize the wireless connection of mul- tiple local systems powered by power packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' First, we suggest a dedicated router design in both software and hardware configurations to ensure physical packetization with a wireless channel and collaboration with wired power management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Then, in a connected system comprised of three local systems with the developed router installed, we assert the selective transmission of power packets to only the local system designated by the tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Second, we demonstrate a power-sharing strategy for increasing power capacity redundancy via a wireless connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' With wired and wireless connections, the parallel operation of intra- and intersystem power management is demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Among a network of two local systems, each of which supplies a certain demand of its own load via wired connection, surplus power at one system is transferred to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Many proposals for the duplex of multiple channels in WPT have been made, including multiplexing in time, frequency, and spatial domain [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Furthermore, several reports have addressed the simultaneous wireless transmission of power and information [4, 16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' These proposals essentially assume that a power transmission channel has already been established and that information is being transmitted concurrently, or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Our proposal, on the other hand, attempts to go beyond the simple parallel transmission by integrating information that manipulates the spatiotemporal distribution of power with power transmission itself at the physical layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This, in theory, eliminates the disparity between physical quantity and information, allowing us to achieve both wired and wireless power transmission by cooperating for smart power management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 2 Outline of power packet dispatching system The basic configuration and operation of power packet dispatching systems are described in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='1 Constitution of power packet As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 1, a power packet is a unit of power management in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' A power packet comprises pulse-shaped electric power called a payload and an information tag, a header, and a footer, which are attached just before and after it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The information tag is a logic bitstream realized by a voltage waveform without current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The tag can include any information, like the origin, destination, and length of the power packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The physical tag attachment enables power packet transmission to be time-division multiplexed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Power from different sources and destinations is transmitted on the same channel while remaining distinct from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 2 owel Packc voltage 01000011 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='1100101 time header payload : footerrouter input router output isolator controller gate driver gate driver storage1 storage2 demand signal clock input1 input2 output1 output2 power signal Figure 2: Circuit example of 2 input 2 output router [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This feature sets the power packet dispatching system apart from conventional systems that treat power as a continuous flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='2 Network configuration of power packet dispatching system Each local system of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 1 denotes a local system configuration example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Routers connect power sources, storage, and loads to the network in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' A power packet router is installed as a node that connects multiple transmission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The router forwards power packets by selecting a transmission line according to the packet’s tag information [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' A power packet is routed from a source to a specific destination via several routers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The path to the load is not required to be unique and can be changed dynamically depending on the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This feature facilitates flexible power management in conjunction with a dynamic supply relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In the following section, we develop a router that can perform this function even with a wireless connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='3 Routing method for power packets Here we characterize the circuit configuration of a router and the principle of its routing operation [9, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Figure 2 depicts the circuit configuration of a previously proposed, wire-connected router [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The circuit consists of two sections: an input section that receives power packets from the transmission network and an output section that forwards power packets to the transmission network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The operation of the input part is initialized when a power packet reaches the router.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The input section includes a signal reading circuit for reading the logic signals of information tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' When the router recognizes that the incoming power packet is addressed to it, it turns on the corresponding semiconductor switches to receive the payload power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' For circuit protection, the signal reading circuit electrically separates its signal output from the power supply lines using a device such as a photocoupler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The incoming power packet is temporarily stored before being forwarded to the next hop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The output part generates power packets from the temporal storage in response to the demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In some cases, the circuit can be reduced to just the input or output section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' When installed just next to the source, for example, the output section with the storage replaced by a power source is sufficient to produce a generated packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Similarly, a circuit just before a load can only be the input part, with the storage replaced by a load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' To read the logic signals of power packets, clocks corresponding to the one-bit width of a power packet must be synchronized among adjacent routers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This can be accomplished by installing an additional wire for a common clock input, or by adding another signal to the header for autonomous clock synchronization [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In this paper, we employ a simple autonomous clock synchronization scheme, in which the clock period is fixed in advance, the first three bits of the header are set to 010, and the phase is shifted if the 010 is not detected within a certain period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The information tag consists of bits 1–3, which implies 010 for clock synchronization, and bits 4–7, which imply the address of the output destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Bits 8 – 100 correspond to the payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' For simplicity, the packet length is fixed at 100 bits and this setting is shared by all routers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In this way, we exclude the footer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 3 3 Router design for wireless transmission of power packets In this section, we propose a router configuration for wireless transmission of power packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' We employ magnetic resonant coupling for the wireless transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This method is capable of transmitting large power over long distances with high-efficiency [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This circuit is powered by AC, whereas the power packet dispatching system is powered by DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' We must convert the current to incorporate WPT into power packet routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Figure 3 depicts a conceptual diagram of the voltage and magnetic flux density in the wireless power packet transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Using a magnetic resonant coupling circuit that includes an inverter and a rectifier, DC is converted to AC and then back to DC after wireless transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The following section describes the router design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' It should be noted that the inclusion of wireless transmission in the power packet dispatching system was first proposed in the authors’ previous report [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In the report, the wireless transmission was not packetized but introduced as a one-to-one transmission channel without any tag attachment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In this paper, we propose a novel router configuration that bring the functions of physical tag attachment and its reading to the wireless power transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' These functions not only realize the physical packetization of wireless power transmission but also extends its use to packet-based power management as introduced in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='1 Wireless transmitter of the power packet Figure 4 depicts a router circuit for wireless power packet output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The configuration includes an inverter circuit connected to the router’s output section, as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' For DC/AC conversion, a class-E inverter [24] is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The output circuit wirelessly transmits both the header signal and the payload power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In this paper, the inverter’s input is presented as a form of packetized power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The current flowing through the coil and the magnetic flux density induced in the coil is thus modulated in an amplitude-shift-keying (ASK) manner according to the shape of the power packet, as depicted in the middle of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' It should be noted here that the header signal transmission must minimize power consumption while the payload transmission must maximize the amount of power transferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The two requirements cannot be met solely through the transmitter’s operation, but rather through the design of the receiver side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This point will be covered in greater detail in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='2 Wireless receiver of the power packet To receive a wirelessly transmitted power packet, demodulation of the ASK-modulated header signal and highly efficient AC/DC conversion of the payload are necessitated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The proposed circuit shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 5 meets both requirements by dividing the demodulator into two circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The signal demodulation circuit reads the header, and a class-E rectifier receives the payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The detailed procedure is provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Initially, the switch Sd connected to the signal demodulation circuit is turned on, while the SR connected to the rectifier circuit is turned off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' For signal demodulation, the envelope of the voltage across the secondary circuit’s resonant capacitor is passed through an RC low-pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The router’s controller then samples it at a predetermined clock cycle to convert it into a logical sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The controller activates the switches that connect the coil to the rectifier circuit when it determines from the tag that the power packet is addressed to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This causes a class-E rectifier to convert the wirelessly transmitted payload into DC output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' When the power packet is directed at another router, the router’s controller disconnects both circuits and opens the coil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The detachment is used to avoid the influence of the unintended connection and the resulting impedance change, which may degrade power transmission at the addressed connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' At the end of the previous power packet, the controller turns on the switch to the demodulation circuit to prepare for the next power packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The end of a power packet is detected by simply counting the length of the payload in 100-bit intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Of course, simply connecting the signal demodulation circuit and the Class-E rectifier in parallel allows you to read the header and receive the payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=" However, when receiving the header, the current passes through the Class- Encode Power packet (DC) ASK modulated AC Packet's header Packet's payload (DC pulse) Decode on Wire on Wire Wireless Figure 3: A waveform concept during wireless transmission of power packets." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 4 C2 r1 L1 Lm Lf1 C1 S1 Controller Gate driver Source Figure 4: Wireless transmitter of the power packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' C3 r2 L2 Lm D1 Lf2 C4 Cf Rectifier Rd Dd Cd2 Demodulator Isolator Controller Gate driver Cd1 Output Sd SR Figure 5: Wireless receiver of the power packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' E rectifier, and when receiving the payload, it passes through the demodulation circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Such a current contributes nothing to the receiving operation but results in power loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Because this type of loss is much greater than the loss caused by the switching of the two demodulation circuits, the proposed scheme can greatly reduce the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The frequency of the carrier wave used for magnetic resonant coupling is 1 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The wireless router’s constants are determined as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The design is conducted in the following manner, regarding [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The coil has a diameter of 100 mm, a wire diameter of 1 mm, several turns of 10, and a thickness of 12 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The transmission circuit’s rise time was measured to be 25 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The rise time is defined as the time required for the output voltage to attain 90 % of its steady-state value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The steady-state value was obtained under the test condition where the load was 47 Ω resistor and the vertical distance between the coils was 50 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Based on this, we determined that the bit width of the power packet should be 100 µs, which is sufficiently larger than the rise time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' That is, the modulation frequency is 10 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The demodulation circuit is designed to demodulate signals with a cutoff frequency of about 100 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Table 1: Design values of circuit constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Primary side Secondary side Rectifier Demodulator f 1 MHz L2 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='2 µH Cd1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='0 µF Lf1 100 µF r2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='88 Ω Cd2 820 pF C1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='3 nF C3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='56 nF Rd 12 kΩ C2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='44 nF C4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='68 nF L1 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='3 µH Lf2 100 µH r1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='88 Ω Cf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='47 µF Lm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='75 µH 4 Verification of selective reception of wirelessly transmitted power packets In this study, we consider one-to-many or many-to-many wireless power sharing among several local systems placed close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The packetization and time-division multiplexing methods enable simultaneous supplies between different pairs of a transmitter and a receiver while completely distinguishing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Here, we experiment with three 5 Local system 0 Wireless packet encoder V Local system 1 Local system 2 R1 Wireless packet decoder Coil 0 Coil 1 Coil 2 50mm 70mm 30mm R2 Wireless packet decoder Figure 6: Network configuration with 3 local systems for verification of selectivity of wirelessly transmitted power packet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Figure 7: Verification of router operation mode for wirelessly transmitted power packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' local systems, one transmitting and two receiving nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' It is demonstrated that the two receivers can selectively accept or reject power packets based on the attached information tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The number of local systems and their connection relationship can of course be easily expanded and modified due to packetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='1 Experimental setup for selective reception The entire network configuration is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Local system 0 alternately sends power packets to local systems 1 and 2, and systems 1 and 2 receive only those that match their addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Power packet header addresses are set to 0001 and 0010 for systems 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Local system 0 consists of a circuit from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 4 and a DC power supply of 12 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Local systems 1 and 2 comprise a circuit of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 4 with a load resistor of 47 Ω connected to the output port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Although the transmitting and receiving roles of the local systems are fixed for simplicity, it is possible to transmit power packets bidirectionally by modifying the circuit configuration [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Therefore, this assumption will not lose generality in power sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' To ensure that the router’s operation is not affected by the distance between the coils, the coil positions are set as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The coils of local systems 1 and 2 are placed at the same vertical distance 50 mm as the coil of local system 0, but the horizontal distance is 30 mm and 70 mm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='2 Receiving mode confirmation First, we examine the switching behavior between the header signal demodulator and the payload rectifier circuit, as designed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Figure 7 depicts an internal signal of the router of local system 1 that represents the receiver’s operation mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The router was in the header mode every 10 ms, which corresponded to the transmission cycle of the powder packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Immediately after the header mode, the router switched to the payload mode every two power packet deliveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' During the payload mode, power was supplied to the designated load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This suggests that the controller received the header while connected to the demodulation circuit and then switched to the rectifier 6 payload mode headermode 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='0 10 0 10 20 30 time/ msFigure 8: Voltages at two loads in local systems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Router m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Router l2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Router rx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Router l1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Router tx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Router m1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='part ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='� : Local system 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='part � : Local system 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='part � : Wireless power sharing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='V1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Rl1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='V2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Rl2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='VCtx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Vm1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='VCl1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='VCrx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Vrx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='VCl2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Vm2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='VRl1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='VRl2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Wireless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='packet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Wireless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='packet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='reader ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Wireless ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='packet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='decoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='CNTL rx1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Srx1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='CNTL m1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Sm1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='CNTL m2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Sm2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='CNTL tx2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Stx2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='CNTL rx2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Srx2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='CNTL l1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Sl1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='CNTL l2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Sl2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='CNTL tx1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Stx1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Ctx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Crx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Cl1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Cl2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='Figure 9: Configuration of the network with 2 local systems connected wirelessly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' circuit in payload mode after recognizing the address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This result confirms that the proposed router can correctly route power packets on the wireless channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='3 Confirmation of selective reception function Second, we confirm that, according to the tag information, the local systems received time-division multiplexed power packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The load voltages of the routers of local systems 1 and 2 are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' It can be seen that local systems 1 and 2 received power alternately, indicating that they selectively accepted or denied receiving power packets based on the attached destination address signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Here, local system 1’s supply voltage was higher than that of local system 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This is because the output is proportional to the distance between the coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This means that, regardless of whether the output value is larger or smaller, the router’s selective reception is unaffected by the difference in distance between the coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 5 Confirmation of power-sharing in the wirelessly connected systems Next, in wirelessly connected local systems, we validate power management based on power packetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' We consider two local systems where the local power supply is primarily managed via a wired connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Every local system consists of an internal power source, a capacitor, a wireless transmission circuit, and a resistive load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' We set a wirelessly connected networking system comprising two such local systems, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' While each local system supplies its source to its load, wireless power packet transmission compensates for excess or deficient power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Each system’s goal is to keep the voltage supplied to the load above a certain level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The proposed scheme deals with a connected system whose elements are subject to dynamic changes, such as variable distance between local systems and time-dependent connection/disconnection of local systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Dealing with such dynamic changes altogether in a centralized controller is not ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Distributed control of power packet transmission, however, is an effective method of accommodating such unpredictability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In this paper, we use a distributed control scheme of packet-based power management [26], in which power packet transmission is managed 7 Load voltage/V 10 local system 1 local system 2 0 20 10 0 10 20 time/msonly between adjacent routers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The following section describes the operation flow of the connected systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='1 Operation flow in connected systems Capacitors are installed in the connected systems to generate and output power packets to the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Power packets are sent so that the voltages of these capacitors exceed a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The demand signal to the router for on-demand packet transmission can be given by information tags in power packets or by using another channel such as radio signals [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In this paper, we use an external wire to transmit demand signals for simplicity We designed an input high to the controller of the next router when the storage voltage falls below the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' We divide the configuration of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 9 into the following three parts that are managed independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' α Transmission from router m1 to router tx and router l1 β Transmission from router tx to router rx γ Transmission from router rx and router m2 to router l2 The three parts’ basic operation principles are described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In part α, when the voltages across Ctx and Cl1 fall below the threshold, demand signals are transmitted to the router m1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Router m1 generates and sends power packets to the destination from which the demand signal is received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In the event of overlapping demand signals, priority is given to router l1 to keep the load voltage stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In part β, router rx sends a demand signal to router tx when the voltage across Crx falls below the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Router tx generates and sends power packets to router rx based on the demand signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In part γ, when the voltage across Cl2 drops below the threshold, a demand signal is initially sent to router rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' If a power packet is not delivered from router rx to router l2 within a certain amount of time, the demand signal is sent to router m2, which generates and sends a power packet to router l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Besides the three principles, two constraints are imposed on the operation of routers tx and rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' First, they do not output power packets if the voltages across its capacitor, Ctx or Crx, are lower than a certain value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' To transmit power packets, there must be an adequate potential difference between the source and the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This constraint guarantees the possible difference between the source and destination capacitors and guarantees the reliable transmission of power packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Second, the routers are not enabled to input and output power packets simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' When both switches are switched on simultaneously, the circuits before and after the router are linked parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In this case, the output impedance measured from the power supply (capacitor) located before the router is lower than when only the input switch is turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This can result in an overcurrent at the source and a rapid drop in capacitor voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The second constraint is levied to avoid this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This configuration may prevent the router rx from emitting power on occasion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Even if this occurs, router m2 can supply power packets to keep router l2’s voltage stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='2 Verification of connected systems operation To test the operation of the connected systems, we set the supply voltages V1 =15 V and V2 =7 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' To create a voltage gradient, the threshold voltages of capacitors Cl1, Ctx, Crx and Cl2 are set as 10 V, 9 V, 7 V and 5 V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The parameters linked to wireless power transmission are set as depicted in Table 1 It is worth noting that the routers’ wired channel switch units have been replaced with unidirectional ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' As previously discussed, the symmetry of the circuit allows us to restrict the flow of power packets to one direction without sacrificing generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The circuit generates high by activating switch Sout−s, and low by activating switch Sout−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The diode prevents reverse current from flowing through the body diode of Sout−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='1 Confirmation of autonomous maintenance of capacitor voltage We demonstrate the transmission of power packets and the modifications in voltages of each capacitor installed in part α–γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Figure 10 depicts the voltages Vl1 and Vtx of the capacitors Cl1 and Ctx in part α and the gate signal of the switches Sl1 and Stx1 that controlled the route of the power packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' It is observed that Vl1 and Vtx were sustained above the threshold voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The voltages Vl1 and Vtx elevated when switches Sl1 and Stx1 were driven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This demonstrates that capacitors Cl1 and Ctx effectively received power packets and were charged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Furthermore, 8 Figure 10: State of switches and voltage of capacitors in part α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Figure 11: State of switches and voltage of capacitors in part β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' switching operation of Sl1 and Stx1 did not overlap at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This result correlates to the setup that the transmission of power packets to Cl1 is prioritized (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='1 for the details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Figure 11 depicts the voltages Vtx and Vrx of the capacitors Ctx and Crx in part β and the gate signal of the switch Srx that controlled the power packet reception of the router rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Comparing the top and bottom graphs shows that Vtx declined and Vrx elevated while Srx was on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This implies that power packets were wirelessly transmitted successfully from router tx to router rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' It can also be validated that Srx turned off when Vtx attained the threshold voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This implies that the system satisfied the constraints defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='1, which hampers the output of power packets under the threshold voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Figure 12 depicts the voltages Vrx and Vl2 of the capacitors Crx and Cl2 in part γ and voltage waveforms of power packets outputted from routers rx and m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' When Vl2 dropped below the threshold voltage, router rx transferred power packets to router m2 so that Vl2 was kept above the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Now let us concentrate on the operation around t =25 ms when Vrx attains the threshold voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Router rx stopped outputting the power packets, and simultaneously, router m2 started sending power packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' These findings suggest that the selective routing protocol specified in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='1 worked;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' the load sent the demand signal to the router rx at first, and if no packet was transmitted, then sent to the router m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 12, there exists a possible difference between the voltage of the power packet and Vl2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This was induced by the forward voltage drop across the diode installed to prevent backflow current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This loss can be repressed by using a switch instead of a diode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Thus, there is no impact on the verification of the principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Figure 13 depicts the gate signal of Stx1, output current from Ctx, input current to Crx, and the output voltage waveforms of router rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' When Ctx was outputting current, Crx was receiving current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This implies that the transmitted power packet was received without failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Since router tx did not output power packets when Stx1 was 9 15 10 5 Vcrx V2 0 100 50 0 50 100 6 Gate signal / V Sr1 Stx1 2 0 100 50 0 50 100 Time / msOutput voltage/V 15 10 5 Vctx Vcrx 100 50 0 50 100 6 signal /V Srx1 Gate 2 0 100 50 0 50 100 Time/msFigure 12: Power packets and voltage of capacitors in part γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Figure 13: Input / output current and voltage of Ctx and Crx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' on, Stx1 and Stx2 were driven solely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Similarly, Srx1 and Srx2 were driven solely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' From the above findings, it can be deduced that the connected system achieves the load voltage maintenance with wireless power supply between local systems 1 and 2 by following the control procedure defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='2 Association between the percentage of power supply and the utilized power source The percentage of power transferred on the wireless channel depends on the distance between the transceiver/receiver coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Hence, in the previous experiment’s setup, increasing the coil gap reduced the power supply capability from the local system 1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The proposed control scheme of the routers can accommodate such a gap change by choosing an appropriate supply channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' To test this operation, we compare the amount of wireless power transmission and the power source selection in the local system 2 at various distances between the coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' We set three cases with different distances: (i) 50 mm (the same as in the previous experiment), (ii) 100 mm, and (iii) > 250 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The setup in case iii is supposed to be large enough to prevent wireless transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Figures 14 and 15 depict the voltage Vrx and Vl2 and the power packets output by routers rx and m2 in cases ii and iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Please refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 12 for the result in case i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The larger the distance between the coils, the less frequently the router rx outputted power packets and the lower its average voltage got.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' On the other hand, Vl2 maintained above the threshold in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Table 2 demonstrates the average of the input/output power of router rx and the output power of router m2 10 Output voltage/V 15 Vcrx Vi2 10 5 0 100 50 0 50 100 Output voltage/V 15 Vrx Vm2 10 5 0 100 50 0 50 100 Time / ms5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='25 Gate signal of Stx1 Output current from Ctx 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='00 signal Current / A 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='75 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='50 Gate 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='25 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='00 100 50 0 50 100 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='25 Vcrx 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='00 A 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='75 Current / 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='00 100 50 0 50 100 Time / msFigure 14: Power packets and voltage of capacitors in part γ of case (ii) : gap 100 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Figure 15: Power packets and voltage of capacitors in part γ of case (ii) : gap 250 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' during the measured time 250 ms for different distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The input/output power of router rx fell and the output power of router m2 rose as the distance became larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Meanwhile, the total output power of router rx and router m2 had a slight change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This finding implies that the output power of router m2 compensates for the fall in the output power of router rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' From the above, it is asserted that the load voltage can be sustained autonomously by the proposed distributed control scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Even when the amount of wireless transmission falls, the local system compensated for it with a wired supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' 6 Conclusion In this paper, we developed a platform for wireless power packet transmission for power management among numerous local systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' First, we proposed a novel power packet router configuration capable of wireless transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The ASK modu- lating circuit is installed on the router’s output side for both information and power transmission, with the power packet serving as a power source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The input side includes a demodulation circuit for both information and power receipt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The circuit shifts between a signal demodulation circuit and a power rectifier circuit to read the header and receive the payload power, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Not only does the switching configuration separate the incoming signal and power, but it also reduces unnecessary power consumption during the receiving operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Using this router, we then verified the wireless power packet routing following the information tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Physical tag attachment and wireless power packet time-division multiplexing allowed receiving routers to distinguish the power packet based on its destination address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The result shows that the proposed configuration allows for the selective 11 outputvoltage/V 15 Vrx Vi2 10 5 0 100 50 0 50 100 output voltage/V 15 Power packet from Crx Power packet from V2 10 0 100 50 0 50 100 time / msoutputvoltage/V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='5 Vrx Vi2 10 100 50 0 50 100 outputvoltage/V 15 Power packet from Crx Powerpacketfrom V2 5 100 50 0 50 100 time / msTable 2: Input/output power of the routers in local system 2 at each gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Case Gap Router rx Router rx Router m2 Total input output output output i 50 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='50 W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='46 W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='73 W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='19 W ii 100 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='20 W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='17 W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='94 W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='11 W iii > 250 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='00 W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='00 W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='13 W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='13 W transmission of wireless power packets between multiple nearby local systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' This prevents interference with the irrelevant power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Next, we considered flexible coordination of inter- and intrasystem power management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The former was ac- complished through the wireless transmission of power packets, while the latter was accomplished through a wired supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' For this purpose, we created a distributed control scheme for the routers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' A local system transmitted power packets wirelessly to another when it had enough power while keeping the voltage of its load as a top priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' The experiments revealed that the two types of operation were coordinated successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Furthermore, the proposed distributed control scheme chose an appropriate supply channel based on the power interaction availability between the local systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' We validated this operation by altering the gap between the coils of the two local systems, demonstrating that the inter- or intrasystem power management was successfully chosen to satisfy the local loads’ demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' From the above verifications, we deduce that wireless power packet transmission can improve power management capability in a connected power packet dispatching system by selectively cooperating wired and wireless power packet transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' Acknowledgments This work was partially supported by JSPS KAKENHI 20H02151, JST-OPERA Program no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content=' JPMJOP1841, and SIP Cross Ministerial Strategic Innovation Promotion Program no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} +page_content='18088028.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQf_A6p/content/2301.05368v1.pdf'} diff --git a/QNE0T4oBgHgl3EQf1QJC/content/tmp_files/2301.02696v1.pdf.txt b/QNE0T4oBgHgl3EQf1QJC/content/tmp_files/2301.02696v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c5fe80831a9d8e38784f8c9da7d2ad4a6ed951a --- /dev/null +++ b/QNE0T4oBgHgl3EQf1QJC/content/tmp_files/2301.02696v1.pdf.txt @@ -0,0 +1,1254 @@ +Band unfolding with a general transformation matrix: from code +implementation to interpretation of photoemission spectra +Oleg Rubel,1, ∗ Jean-Baptiste Moussy,2 Paul Foulquier,2, 3 and V´eronique Brouet3, † +1Department of Materials Science and Engineering, +McMaster University, 1280 Main Street West, +Hamilton, Ontario L8S 4L8, Canada +2Universit´e Paris-Saclay, CEA, CNRS, +SPEC, 91191, Gif-sur-Yvette, France +3Universit´e Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France. +(Dated: January 10, 2023) +1 +arXiv:2301.02696v1 [physics.comp-ph] 6 Jan 2023 + +Abstract +Unfolding of a supercell band structure into a primitive Brillouin zone is important for un- +derstanding implications of structural distortions, disorder, defects, solid solutions on materials +electronic structure. Necessity of the band unfolding is also recognised in interpretation of angle- +resolved photoemission spectroscopy (ARPES) measurements. We describe an extension of the +fold2Bloch package by implementing an arbitrary transformation matrix used to establish a rela- +tion between primitive cell and supercell. This development allows us to overcome limitations of +supercells constructed exclusively by scaling of primitive cell lattice vectors. It becomes possible to +transform between primitive and conventional cells as well as include rotations. The fold2Bloch +is publicaly available from a GitHub repository as a FORTRAN code. It interfaces with the all- +electron full-potential WIEN2k and the pseudopotential VASP density functional theory packages. +The fold2Bloch is supplemented by additional pre- and post-processing utilities that aid in gen- +erating k points in the supercell (such that they later fall onto a desired path in the primitive +Brillouin zone after unfolding) and plotting the unfolded band structure. We selected Sr2IrO4 as +an illustrative example and, for the first time, present its properly unfolded band structure in direct +comparison with ARPES measurements. In addition, critical importance of the band unfolding for +interpretation of SrIrO3 ARPES data is illustrated and discussed as a perspective. +I. +INTRODUCTION +The band dispersion E(k) obtained from electronic structure calculations provides infor- +mation about Fermi surface of metals, direct/indirect transition in semiconductors, effective +masses, etc. When periodicity is perturbed (e.g., solid solutions, defects, magnetic order) +the Brillouin zone (BZ) shrinks and bands get folded. As a result, E(k) becomes obscured +and difficult to interpret. +Even when the periodicity is formally perturbed, it is still possible to recover an effective +band structure in a BZ of the primitive cell using a k-spectral decomposition also known as +‘band unfolding’. There are numerous examples of band unfolding. The notable milestones +include an electronic structure of aperiodic solids [1–3], solid solutions described within a +∗ O.R. email: rubelo@mcmaster.ca, ORCID: 0000-0001-5104-5602 +† V.B. email: veronique.brouet@u-psud.fr +2 + +tight-binding approximation [4–6] or pseudopotentials with a plane-wave basis set [7–9], +and Wannier functions derived from first-principle calculations to interpret angle-resolved +photoemission spectroscopy (ARPES) experiments [10]. There are a number of public im- +plementations of band unfolding in the density functional theory (DFT) community, for +instance, BandsUP [11, 12], fold2Bloch [13], VASPKIT [14], vaspwfc [15]. +The simplest scenario to unfold the band structure involves the supercell constructed +from primitive real-space lattice vectors ai (i = 1, 2, 3) by scaling them with an integer niai +(ni ∈ Z+). However, it does not cover all possibilities. There are lattices constructed by +a combination of rotation and scaling of the primitive structural unit. Examples include a +conventional vs reduced unit cell [16, 17] or an octahedral rotation and tilting in perovskite +structures [18]. Popescu and Zunger [9] suggested a more general approach to unfolding +that involves a transformation matrix. It was already implemented in BandsUP, VASPKIT, +and vaspwfc but not in fold2Bloch. +II. +METHOD +A. +Lattice transformations +Following the notations in Ref. 9 we denote by small (capital) symbols quantities referring +to the primitive cell (supercell). Transformation of real-space primitive ai to supercell Ai +lattice vectors can be expressed as [19] +Ai = +3 +� +j=1 +Pji aj +(i = 1, 2, 3) +(1) +or in the matrix form as +A = P ⊺ a, +(2) +where the lattice vectors matrices A and a are constructed of rows being individual vectors +and columns being their Cartesian components (x, y, z), e.g., +A = +� +� +� +� +� +A1 +A2 +A3 +� +� +� +� +� ≡ +� +���� +A11 A12 A13 +A21 A22 A23 +A31 A32 A33 +� +���� . +(3) +Here P is a transformation matrix of the size 3 × 3 compatible with conventions recom- +mended by the International tables for crystallography [19] (same as in VESTA structure +3 + +visualization software [20] or Bilbao crystallographic server [21] but different from Ref. 9). +The transformation matrix is obtained by solving the linear Eq. (2), which yields +P = (A a−1)⊺. +(4) +The reverse transformation of a supercell to the primitive cell is obtained using the inverse +matrix P −1 +a = (P −1)⊺ A. +(5) +Scaling of the cell volume as a result of the primitive cell to supercell transformation is given +by [19] +nv = det(P), +(6) +which imposes two constrains: P should be positive defined and Pij ∈ Z, from which it +follows that det(P) ∈ Z+. +Reciprocal lattice vectors are also transformed using P. +Owing to the relation B = +(A−1)⊺, reciprocal lattice vectors of a supercell Bi can be transformed to the reciprocal +primitive vectors bi as [19] +b = P B +(7) +and back as +B = P −1 b. +(8) +Note that the transposition of P is not required for converting reciprocal lattice vectors in +the matrix form contrary to the conversions proposed in Ref. 9 (see Eqs. (1) and (2) therein). +B. +Band unfolding +We already outlined in Ref. 13 an unfolding procedure used in the fold2Bloch imple- +mentation when a supercell is constructed by simple scaling of the primitive cell . Here we +present an extended (more general) version. +DFT codes for solids internally operate with wave vectors in fractional coordinates. Here +we will use tilde to denote primitive (supercell) fractional coordinates ˜k ( ˜K) where com- +ponents of each vector span a range between 0 and 1. Cartesian components of the wave +vector are obtained by a multiplication with the reciprocal lattice matrix +k = ˜k b +(9) +4 + +and similarly for the supercell K. +Transformation of reciprocal-space supercell ˜K to primitive ˜k wave vectors (fractional +coordinates) is given by [19] +˜k = [ ˜K + (m1, m2, m3)] P −1 +mod 1. +(10) +With all possible combinations of mi ∈ Z, the number of unique k points generated within +the first BZ of the primitive cell (Fig. 1) is given by the volume scale (Eq. (6)). The new +(unfolded) k points in Fig. 1b form two subsets k1 (open markers) and k2 (red) of the original +grid K + G in Fig. 1a. The two subsets were created since the volume change is nv = 2 in +the example shown. More generally, the subset property is expressed as +kl + g ⊂ K + G +(l = 1, . . . , nv), +(11) +with G(m1, m2, m3) = m1B1 + m2B2 + m3B3 and g(m1, m2, m3) = m1b1 + m2b2 + m3b3 +being commensurate vectors of the plane wave expansion. +The wave function in WIEN2k is split into two regions: the atomic spheres and the +interstitial region [22]. A plane wave basis set is used in the interstitial region +Ψ(int) +σ,n,K(r) = +� +G +Cσ,n,K(G) ei(K+G)·r, +(12) +where C are plane wave coefficients for a specific electronic state with the wave vector K, +spin channel σ, band index n. +Spectral weight of the new unfolded k point is evaluated from the subset of plain wave +coefficients [9] +wσ,n(kl) = +� +g +|Cσ,n,K(kl + g)|2 +(l = 1, . . . , nv). +(13) +Weights are normalized such that +nv +� +l=1 +wσ,n(kl) = 1 +(14) +In the case of a spinor wave function, weights of spin up and down components are mixed +wn(k) = α2w↑,n(k) + β2w↓,n(k), +(15) +where α and β are components of the spinor wave function (α2 + β2 = 1). This result is +similar to decomposition of partial spectral weights proposed by Medeiros et al. [12], yet +it is additionally augmented by the relative contribution of each spin channel to the wave +function. +5 + +C. +Electronic structure calculations +All electronic structure calculations were performed with WIEN2k DFT package [22, 23]. +Relevant parameters are listed in Table I. We used the Perdew, Burke, and Ernzerhof [24] +(PBE) exchange-correlation functional in combination with the onsite Hubbard correction +U [25] for Ir-d electrons in the case of Sr2IrO4. The spin-orbit coupling was included in all +calculations. The spin polarization was enabled only in the case of Sr2IrO4, where we used +a collinear antiferromagnetic ordering as in Ref. 26 (Fig. 2d) initialized with the magnetic +moment of ±1 Bohr magneton (µB) per Ir site. +D. +Implementation and execution workflow +WIEN2k: First we initialize the calculation and complete the self-consistent field (SCF) +cycle using the case.struct structure file as an input to obtain a self-consistent potential +and a charge density. Once the calculation converges all necessary files are saves in the SOC +folder. A detailed workflow can be found in the supporting information (SI) section. +Utils/fold.m: Next we generate a list of folded ˜K points within the supercell BZ using +the Octave/MATLAB script that takes a desired ˜k point path in the primitive BZ, the +number of intermediate points for each section of the path, as well as the transformation +matrix P as inputs. +$ octave fold.m +The folding is achieved by the following matrix product [19] +˜K = ˜k P +mod 1. +(16) +The generated unique ˜K points are stored in the case.klist band file in a WIEN2k na- +tive format as three integer numbers per k point with a common integer divisor. +It is +important to note conventions used within WIEN2k to interpret k point coordinates in the +case.klist band file. The BZ of a conventional lattice is implied for F, B, CXY, CXZ, and +CXZ orthorhombic lattices. The BZ of a primitive lattice is used for other lattice types (P, +H, R, CXZ monoclinic). Those peculiarities affect the selection of supercell lattice vectors +A in Eq. (3) and the construction of P matrix using Eq. (4). In practice, we expect that +majority of users will have P-type supercells due the symmetry broken by disorder/defects. +6 + +WIEN2k: Now we generate eigenvalues and eigenvectors (wave functions or vector files) +for the list of k points in the case.klist band file. We use files saved in the SOC folder after +the previous SCF step. +$ x lapw1 -band -up [-p] +$ x lapw1 -band -dn [-p] +$ x lapwso -up [-orb] [-p] +The -orb switch is activated for the DFT+U calculation. Here we produce files that are +essential for unfolding: case.vectorso[up/dn] (wave functions and energy eigenvalues) +and case.normso[up/dn] (spinor components α2 and β2). +fold2Bloch: fold2Bloch is a FORTRAN code that can be compiled with either Intel or +GNU FORTRAN compilers. It takes wave functions, spinor components, and the transfor- +mation matrix P as input arguments +$ fold2Bloch -so case.vectorsoup[ 1] case.vectorsodn[ 1] +... +case.normsoup[ 1] case.normsodn[ 1] +... +"’P11 P12 P13:P21 P22 P23:P31 P32 P33’" +and generates a case.f2b file. If WIEN2k calculations run in a k-parallel mode ([-p] op- +tion), output vector and norm files will be marked with XX for each parallel stream. These +files can be processed individually, and the output can be concatenated into one case.f2b +file. The sample listing of the output file is given below +k_1 +k_2 +k_3 +E (Ry) +w +... +0.000000 +0.000000 +0.000000 +0.393800 +0.001619 +0.000000 +0.000000 +0.250000 +0.393800 +0.000001 +0.000000 +0.000000 +0.500000 +0.393800 +0.487296 +0.000000 +0.000000 +0.750000 +0.393800 +0.000001 +0.500000 +0.500000 +0.000000 +0.393800 +0.000000 +0.500000 +0.500000 +0.250000 +0.393800 +0.255542 +0.500000 +0.500000 +0.500000 +0.393800 +0.000000 +0.500000 +0.500000 +0.750000 +0.393800 +0.255542 +0.000000 +0.000000 +0.000000 +0.394518 +0.374261 +0.000000 +0.000000 +0.250000 +0.394518 +0.000001 +0.000000 +0.000000 +0.500000 +0.394518 +0.001956 +7 + +0.000000 +0.000000 +0.750000 +0.394518 +0.000001 +0.500000 +0.500000 +0.000000 +0.394518 +0.000000 +0.500000 +0.500000 +0.250000 +0.394518 +0.311889 +0.500000 +0.500000 +0.500000 +0.394518 +0.000000 +0.500000 +0.500000 +0.750000 +0.394518 +0.311893 +... +Here one can see results of band unfolding with the transformation matrix P = [1, ¯1, ¯2; 1, 1, ¯2; 0, 0, 4] +and the volume scale of nv = 8. Two groups of eigenvalues are shown each unfolded into +eight new k points (fractional coordinates) in the primitive BZ. Even though both eigen- +values belong to Γ point in the supercell BZ, after unfolding the first does not have any +notable Γ character (only ca. 0.16%), while the second eigenvalue retains about 37% of its +Γ character. +Utils/ubs bmp.m: This Octave script is used to prepare a binary file case.f2b.bin +for band structure plotting. The inputs are the case.f2b file, the desired ˜k path in the +primitive BZ (it has to match that used as input in fold.m), reciprocal lattice vectors of the +supercell (can be read from the case.outputkgen file in a column-wise manner and later +transposed within the script), the Fermi energy from case.scf file, smearing for a Gaussian +function in energy and k space. +$ octave ubs bmp.m +Sensible values for the energy and k space smearing are about 1/50 of the energy range and +the k path length selected for the band structure plot. At the end of execution, XTICKS and +KLABEL vectors are printed. They need to be noted and used in the next stage. +Utils/f2b-band-structure.plt: Finally, the unfolded band structure is plotted using +Gnuplot with the input binary file case.f2b.bin. +$ gnuplot f2b-band-structure.plt +The Gnuplot script incorporates data from XTICKS and KLABEL vectors used for labeling the +high-symmetry points along the path and generates a publication-quality plot (case.eps). +E. +Experimental details +The experimental structure of Sr2IrO4 was measured with ARPES on the Cassiop´ee +beamline of the SOLEIL synchrotron, with a SCIENTA R-4000 analyzer and an overall +8 + +resolution better than 15 meV. The temperature was 20 K, the photon energy was set to +100 eV and linear polarization in the plane containing ΓM was used. The samples were +prepared using a self-flux method, as reported before [27, 28]. +We have grown SrIrO3 thin films on SrTiO3 (001) substrate using pulsed laser deposition. +A frequency-tripled Nd:YAG laser (λ = 355 nm, f = 2.5 Hz, pulse duration 15 ns) was fo- +cused on a polycrystalline SrIrO3 target made by solid state synthesis. The substrate surface +was prepared following the process described in Ref. 29 to obtain a uniform TiO2 surface +termination. The deposition was performed with the substrate heated at T = 600 ◦C and +with oxygen partial pressure P = 2.5 × 10−1 mbar and monitored by in-situ reflection high- +energy electron diffraction. Then, the thin films were cooled down to room temperature with +the same oxygen partial pressure to compensate for any oxygen vacancy. Finally, we fully +structurally and physically characterized our thin films by X-ray diffraction and reflectiv- +ity using a Br¨uker D8 advance diffractometer and by surface diffraction on SIXS beamline +at SOLEIL synchrotron, atomic force microscopy and electronic transport, concluding to +similar characteristics and properties to previous reports in the literature [30–32]. +III. +RESULTS AND DISCUSSION +A. +Sr2IrO4: Theoretical calculations +The structure of Sr2IrO4 was imported from Springer Materials [33] (dataset ID sd 1945591). +The original structure is BCT and includes a tilting of IrO6 octahedra (Fig. 2a-d). The +magnetic ordering further reduces the symmetry to a tetragonal cell with 8-Ir atoms. Its +lattice vectors matrix (˚A) is +A = +� +���� +5.485 +0 +0 +0 +5.485 +0 +0 +0 +25.775 +� +���� . +(17) +For verification purposes we also created an idealized structure by eliminating the octahedral +tilting and magnetic ordering. Its elementary structural unit (a 1-Ir primitive BCT unit cell) +9 + +is shown in Fig. 2e. The corresponding matrix of primitive lattice vectors is +a = +� +���� +A1/2 −A2/2 +0 +A1/2 +A2/2 +0 +A1/2 +0 +A3/4 +� +���� . +(18) +Following Eq. (4) we obtain the transformation matrix +P = +� +���� +1 −1 −2 +1 +1 +−2 +0 +0 +4 +� +���� , +(19) +which establishes the relation between 1-Ir primitive cell and 8-Ir supercell (without distor- +tions). The structure files (in a WIEN2k native format) can be found in the SI section and +visualized with XcrySDen [34] or VESTA [20]. +The BZ of 8-Ir and 1-Ir cell is shown in Fig. 3a,b, respectively. For plotting band struc- +tures we selected the Γ−M−X−Γ path. Since we made a non-standard choice for BCT +primitive lattice vectors (see Table 9.1.7.2, tI Bravais lattice in the International tables for +crystallography [35]), special care should be taken to map k point coordinates from the +conventional to the primitive cell (Fig. 3b) to ensure that the coordinates are compatible +with the chosen definition for lattice vectors (Table II). +For verification purposes we first need to calculate the band structure without tilting of +IrO6 octehedra. This way we can establish a direct comparison between the 8-Ir cell and the +primitive 1-Ir BCT cell. It is convenient to do this comparison at the PBE level of theory, +since it leads to a non-magnetic solution with a more simple band structure (Fig. 4a). Bands +near the Fermi energy are due to Ir-d electrons: the dispersive bands starting off Γ near EF +correspond to the dx2−y2 orbital; the remaining bands are due to t2g states (dxy, dyz, and dxz +orbitals); the dz2 orbital belongs to eg states located at higher energies outside the energy +range of interest. +Figure 4b shows the band structure of the 8-Ir supercell without octahedral tilting. This +band structure is obscured by the zone folding. +Figure 3c can help to rationalize zone +folding at high-symmetry k points (prime indices are for the supercell). Eigenvalues at both +Γ and X points are folded into Γ′ of the supercell. Eigenvalues at M point fall into X′. The +Dirac-like band crossings along the Γ′−X′ segment are folding artefacts not observed in the +10 + +primitive cell. The unfolded and primitive band structures are identical (Fig. 4a,c) that +gives confidence in our approach and its implementation. +The realistic structure of Sr2IrO4 includes tilting of IrO6 octahedra. Those distortions +cause perturbations in the band structure, which can be assessed thanks to the new func- +tionality of fold2Bloch. The unfolded band structure of 8-Ir cell with octahedral tiltings +(Fig. 5b) can now be compared to Fig. 4c (both calculated at the PBE level of theory). The +most notable change is that the rotation allows hybridization between dx2−y2 and dxy, as is +also the case in Sr2RhO4 [36]. Therefore, new dx2−y2 states move to higher energies (not +visible in Fig. 5) and dxy states are now pushed below the Fermi energy resulting in a new +bright ‘spot’ at X below the Fermi energy (Fig. 5b), which is unfolded from Γ′′ leaving a +weak replica at Γ. Interestingly, states at M point are immune to the distortions. +To account for correlation effects we added the Hubbard Ueff = 3 eV correction for Ir- +d states. The magnitude of Ueff was chosen to reproduce the experimental band gap of +0.5 eV [37, 38]. +Unlike PBE, PBE+U favours a magnetic solution with the moment of +µ(Ir) ≈ 0.24µB per Ir site (the ordering is shown in Fig. 2d with the [001] spin quantization +direction for simplicity). Experimental µ(Ir) moments are 0.21 [26], 0.29 [39], and 0.37 [40]. +A gap opens up between the on-site spin up and down states, which is particularly clear at +M and at the middle of ΓX (Fig. 5c-d). +B. +Sr2IrO4: Comparison with experiment +Figure 6(a) shows the ARPES spectral intensity along the k-path Γ-M-X-Γ, which was +extracted from our measurement using a photon energy of 100 eV and a linear polarization +along ΓM [27]. The different bands observed in this plot are sketched in Fig. 6(b) by red +and blue guides to the eyes, the colour corresponds to their dominant orbital character, +characterized by the effective value of the total electronic angular momentum J, either +J = 1/2 (mJ = ±1/2) or J = 3/2 (mJ = ±1/2, ±3/2). These data are similar to several +previous reports [28, 41, 42], where more details can also be found. +The most intense band is the J = 3/2 band at X (blue star). Although it should also be +present at Γ, which is equivalent in the supercell (see Fig. 3(c)), it can hardly be distinguished +there. This is perfectly captured by the unfolded calculation of Fig. 6(d), which features +much lower intensity for this band at Γ compared to X. A comparison of the measured +11 + +and calculated intensity along Γ and X is also shown in Fig. 6(c) and (e), respectively. +ARPES spectra in Fig. 6(e) were calculated based on the discrete spectral weights wn(k) +defined by Eq. (15) and energies En(k) of the unfolded band structure at specific k points. +A Gaussian broadening of the width σ = 0.2 eV was applied. However, it is still a very +crude approximation as other relevant details were omitted (matrix elements for initial- +and final-state crystal wave functions, finite-lifetime effects, surface discontinuity, multiple +scattering [43]). These matrix elements will further modulate the measured intensity, but +the qualitative difference is well captured by the calculation. However, the relative position +of the J = 3/2 band at X and J = 1/2 band at M is quite different in the two cases. In +experiment, those bands are ca. 0.22 eV apart, while in calculations the energy difference +between the valence band maxima at M and X is ca. 0.11 eV (compare red and blue stars +on Fig. 6 panels (c) and (e)). This discrepancy is due to an underestimation of the effective +spin-orbit coupling already noticed and discussed in the literature [44, 45]. +The J = 1/2 band is the one where the magnetic gap opens [42] (Fig. 6(d), red arrows). +This band is clearly visible along ΓM, but much weaker along MX, two paths expected to be +equivalent in the supercell. Again, this fits with the theoretical calculation. Similarly, the +J = 1/2 band drastically loses weight on the second half of ΓX, both in the experiment and +in calculation. Note that, as the relative positions of J = 1/2 and J = 3/2 are not correctly +captured, the break in the dispersion due to hybridization between J = 1/2 and J = 3/2 +where they cross is also shifted. +The other bands are more difficult to isolate in ARPES, either because they become too +broad at high binding energies or because they have low intensity in these experimental +conditions. However, it is clear that another band is present at −0.8 eV at M and a trace +of a second band at X can be seen. They are marked by cyan stars and correspond well to +the other J = 3/2 band (mJ = ±1/2), which is dominated by dxy weight, and therefore has +a lower cross section in ARPES [28]. +C. +SrIrO3: Perspective +We illustrated the unfolding process with the case of Sr2IrO4, where the connection to a +primitive unit cell, without rotation of the oxygen octahedra, is relatively easy to anticipate. +However, in more complicated cases, it can become totally impossible to understand the band +12 + +structure and compare it to ARPES data, without the help of the unfolding calculation. +A very interesting case is the related 3D iridate SrIrO3. Its structure is shown in Fig. 7(a). +In addition to in-plane rotation of the oxygen octahedra, similar to Sr2IrO4, it also exhibits +a tilt of the oxygen octahedra from the c axis, inducing another type of folding along kz. The +resulting BZ is shown in Fig. 7(b), it is rotated 45◦ in the (kx, ky, 0) plane, as for Sr2IrO4, but +also halved along kz. The calculated band structure is semimetallic and the four J = 1/2 +folded bands are expected to form a Dirac nodal line around the U point (1/2, 0, 1/2) [46]. As +topological features are rare in correlated oxides, this occurrence generated great interest. +However, the orthorhombic structure of SrIrO3 is only stable in thin films, which adds +questions on the role of the epitaxial constraint and the survival of topological features in +real systems [47]. It would therefore be very interesting to look for these features directly +with ARPES, but the situation is not yet concluding. The only ARPES studies available +to date were performed at a fixed photon energy [48, 49], which may not allow to precisely +locate the U point, or at high photon energies, where the energy resolution is lower [50]. +We only sketch here the help of unfolded calculations for deciphering the electronic struc- +ture of SrIrO3, more details will be published later. Figure 7(c) shows the images of the +dispersion along ΓX direction with a photon energy of 100 eV obtained on a SrIrO3(001) +thin film (t = 10.4 nm) grown on SrTiO3(001) substrates (for clarity, we keep the same +labelling as previously for the (kx, ky, 0) plane of Sr2IrO4 in Fig. +3(c), i.e., X labels the +corner of the primitive 2D BZ similar to Fig. 3). The ARPES image looks quite simple, with +a prominent band at X (almost invisible at Γ), resembling the J = 3/2 in Sr2IrO4 shifted up +by 0.2 eV and a band reaching the Fermi level at the middle of ΓX, looking like J = 1/2 in +Sr2IrO4 shifted up by 0.15 eV. Quantitatively, the comparison with raw calculations in the +supercell is usually very confusing, as a much more complicated band structure is predicted +[Fig. 7(f)]. We will show how unfolding of the band structure can simplify the picture. +We calculated with WIEN2k the electronic structure for the structure given in Table III +of Ref. 51. The space group is Pbnm (#62), and the parameters are given in Table III. As +for Sr2IrO4, we can define the lattice vector matrix of the supercell (˚A) and the fictitious +13 + +primitive cell, as well as the transformation matrix: +A = +� +���� +5.597 +0 +0 +0 +5.568 +0 +0 +0 +7.892 +� +���� , +a = +� +���� +A1/2 −A3/2 +0 +A1/2 +A2/2 +0 +0 +0 +A3/2 +� +���� , +P = +� +���� +1 −1 0 +1 +1 +0 +0 +0 +2 +� +���� +(20) +No Coulomb repulsion U was used and a semi-metallic non-magnetic state is obtained, as it +is the case experimentally [52]. +Assuming our ARPES data at this photon energy correspond to kz = 0, we show the +calculated electronic structure in Fig. 7(d,f) with and without SOC. Clearly, the calculation +looks much more complicated than ARPES data, and it is extremely difficult to understand +which bands should be compared to the experiment. +After unfolding [Fig. 7(e,g)], a set of 3 bands is clearly emphasized, although their dis- +persions are affected by their mutual interactions. Using a color code for orbital characters +(the case.inq file was modified to rotate a and b by 45◦ and align them with Ir−O−Ir +bonds), one can clearly recognize the original dxz/dyz/dxy bands [panel (e), without SOC] +and J = 1/2, J = 3/2 after their interaction with SOC [panel (g)]. The places where a +SOC induced hybridization gap openings are noted as empty circles. Obviously the band +marked by a red star is at significantly different position in the measurement, compared +to the calculation. On the other hand, the band at X (blue star) exhibits a well-defined +parabolic shape in the measurement over 0.4 eV, while there is a large SOC induced gap +near the top of the dispersion in the calculation. Similar to Sr2IrO4, we can anticipate that +the effective spin-orbit coupling may be underestimated in the calculation, which will change +the splitting of the two bands and also the position of their crossing, hence the break in +the dispersion. Moreover, as the shape of dxy depends sensitively on the rotations, it may +be necessary to tune the structure to get the right interaction pattern. This has extremely +important consequences for the formation of the Dirac nodal line at kz = 1/2, which we will +not discuss here. This examples shows how adjusting the structure to get a better descrip- +tion of the experimental data will be possible only when a basic understanding of the origin +of the different bands is reached, thanks to the unfolding scheme. +Let us stress that the intensity of the folded bands is also a fine marker of the strength +of the interactions at the origin of the supercell, which is here the rotations of the oxygen +octahedra. In a similar spirit, the information contained in the intensity of the folded bands +14 + +on these interactions was recently used to refine the structure of SrRuO3 thin films [53]. +The near absence of the folded bands in our measurement suggests a small coupling. From +the topological point of view, the meaning of the crossing of two bands with very different +spectral weight is also a question that may deserve further work. +IV. +CONCLUSION +Unfolding of a supercell band structure into a primitive Brillouin zone is important for +understanding implications of structural distortions, disorder, defects, solid solutions on +materials electronic structure. Necessity of the band unfolding is also recognised in interpre- +tation of angle-resolved photoemission spectroscopy (ARPES) measurements. We described +an extension of the fold2Bloch package by implementing an arbitrary transformation ma- +trix used to establish a relation between primitive cell and supercell. The convention selected +for the transformation matrix is compatible with that recommended by the International +tables for crystallography. This development allows us to overcome limitations of supercells +constructed exclusively by scaling of primitive cell lattice vectors. For instance, it becomes +possible to transform between primitive and conventional cells as well as include rotations. +The updated fold2Bloch package is available from a GitHub repository as a FORTRAN +code. +It interfaces with the all-electron full-potential WIEN2k and the pseudopotential +VASP density functional theory packages. The fold2Bloch is supplemented by additional +pre- and post-processing utilities that aid in generating k points in the supercell (such that +they later fall onto a desired path in the primitive Brillouin zone after unfolding) and plot- +ting the unfolded band structure. We selected Sr2IrO4 as an illustrative example and, for the +first time, present its properly unfolded band structure in direct comparison with ARPES +measurements. In addition, critical importance of the band unfolding for interpretation of +SrIrO3 ARPES data is illustrated and discussed as a perspective. 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Horak, D. Puggioni, C. R. Serrao, R. Chen, D. Yi, C. Frontera, V. Holy, +A. Vishwanath, J. M. Rondinelli, X. Marti, and R. Ramesh, Strain-induced nonsymmorphic +symmetry breaking and removal of Dirac semimetallic nodal line in an orthoperovskite iridate, +Phys. Rev. B 93, 085118 (2016). +[48] Y. F. Nie, P. D. C. King, C. H. Kim, M. Uchida, H. I. Wei, B. D. Faeth, J. P. Ruf, J. P. C. +Ruff, L. Xie, X. Pan, C. J. Fennie, D. G. Schlom, and K. M. Shen, Interplay of spin-orbit +interactions, dimensionality, and octahedral rotations in semimetallic SrIrO3, Phys. Rev. Lett. +114, 016401 (2015). +[49] Z. T. Liu, M. Y. Li, Q. F. Li, J. S. Liu, W. Li, H. F. Yang, Q. Yao, C. C. Fan, X. G. +Wan, Z. Wang, and D. W. Shen, Direct observation of the Dirac nodes lifting in semimetallic +perovskite SrIrO3 thin films, Sci. Rep. 6, 30309 (2016). +[50] P. Sch¨utz, D. D. Sante, L. Dudy, J. Gabel, M. St¨ubinger, M. Kamp, Y. Huang, M. Capone, +M.-A. Husanu, V. N. Strocov, G. Sangiovanni, M. Sing, and R. Claessen, Dimensionality- +driven metal-insulator transition in spin-orbit-coupled SrIrO3, Phys. Rev. Lett. 119, 256404 +(2017). +[51] D. Puggioni and J. M. Rondinelli, Comment on “High-pressure synthesis of orthorhombic +SrIrO3 perovskite and its positive magnetoresistance” [J. Appl. Phys. 103, 103706 (2008)], J. +Appl. Phys. 119, 086102 (2016). +[52] J. G. Zhao, L. X. Yang, Y. Yu, F. Y. Li, R. C. Yu, Z. Fang, L. C. Chen, and C. Q. Jin, High- +pressure synthesis of orthorhombic SrIrO3 perovskite and its positive magnetoresistance, J. +Appl. Phys. 103, 103706 (2008). +[53] B. Sohn and C. Kim, Evolution of electronic band reconstruction in thickness-controlled per- +ovskite SrRuO3 thin films, J. Korean Phys. Soc. 10.1007/s40042-022-00633-5 (2022). +20 + +SUPPORTING INFORMATION +We include a ZIP archive with WIEN2k structures, relevant scripts, initialization and +unfolding workflows. See README file within the archive for additional description of the +content. +21 + +(a) +(b) +Γ +K +K + G(1, 0, 0) +B1 +B2 +k1 +k2 +b1 +b2 +Γ +k2 + g(1, 0, 0) +k1 + g(0, −1, 0) +FIG. 1. Unfolding in reciprocal space. (a) BZ of a supercell with an arbitrary K point. (b) BZ +of a primitive cell obtained by the two-dimensional transformation matrix of P = [1, 1; ¯1, 1]. The +cell volume changes twice (nv = 2), thus the K point transforms into two new point k1 and k2 +that form two groups of plane wave coefficients (open and red markers). The dashed line shows +the supercell BZ inside of the primitive BZ. +22 + +(a) +(d) +(e) +O +Sr +Ir +(c) +(b) +1 Ir primitive cell +1 Ir primitive cell +FIG. 2. Structure of Sr2IrO4 (space group 88, I41/a (non-magnetic) or 13, P2/c (magnetic)): (a) +top view, (b) top view with the first Ir layer only, (c) top view with the second Ir layer only, (d) side +view with arrows showing the magnetic ordering, and (e) primitive 1-Ir unit cell (space group 139, +I4/mmm) obtained from the supercell without octehedral tilting using the transformation matrix +of P −1 = (1/2, 1/2, 1/2; 1/2, 1/2, 0; 0, 0, 1/4). The octahedral tilting is present on panels (a)–(d). +23 + +ab +ab +ab +a(a) +(b) +(c) +Γ +M +primitive +conventional +X +b2 +b3 +b1 +primitive +supercell +Γ’ +M’ +X’ +B2 +B3 +B1 +Γ, Γ’ +X, Γ’’ +M, X’ +M’ +FIG. 3. (a) BZ of Sr2IrO4 tetragonal cell (space group 88). (b) Primitive BZ of a body-centred +tetragonal lattice (black) and the (kx, ky, 0) plane of a conventional BZ (red). (c) Overlay of the +primitive and supercell BZ (top view). The k path of interest within the (kx, ky, 0) plane is shown +by green arrows. +24 + +(a) +(b) +(c) +Energy (eV) +Wave vector +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +0.5 +1.0 +Γ +M +X +Γ +Γ +M +X +Γ +EF +Energy (eV) + 0.0 + 1.0 + -1.0 + -2.0 +* +* +* +* +X’ +Γ’’ +M’ +EF +Energy (eV) + 0.0 + 1.0 + -1.0 + -2.0 +Γ’ +Γ’ +Wave vector +Wave vector +FIG. 4. Band structure of Sr2IrO4 at PBE+SOC level of theory without octehedral tilting: (a) +body-centred tetragonal primitive cell (space group 139, 1-Ir atom), (b) tetragonal supercell (space +group 88, 8-Ir atoms), (c) supercell unfolded into the primitive cell using P = [1, ¯1, ¯2; 1, 1, ¯2; 0, 0, 4]. +The asterisk (*) marks dx2−y2 states. Energies are plotted relative to the Fermi energy. Red (black) +labels for high-symmetry points in the reciprocal space refer to the primitive cell (supercell) BZ. +25 + +(a) +(b) +Energy (eV) +Wave vector +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +0.5 +1.0 +Γ +M +X +Γ +(d) +(c) +X’ +Γ’’ +M’ +E F +Energy (eV) + 0.0 + 1.0 + -1.0 + -2.0 +Γ’ +Γ’ +X’ +Γ’’ +M’ +E F +Energy (eV) + 0.0 + 1.0 + -1.0 + -2.0 +Γ’ +Γ’ +Energy (eV) +Wave vector +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +0.5 +1.0 +Γ +M +X +Γ +Wave vector +Wave vector +* +* +* +* +* +* +* +* +FIG. 5. Unfolded band structure of Sr2IrO4 with octehedral tilting: (a,b) PBE+SOC folded and +unfolded, (c,d) PBE+SOC with onsite Ueff = 3 eV for Ir-d. Energies are plotted relative to the +Fermi energy. The asterisk (*) marks dxy states. +26 + +(e) +Energy (eV) +Wave vector +−2.0 +−1.5 +−1.0 +−0.5 +0.0 +0.5 +Γ +M +X +Γ +* +* +* +* +−1.0 +−0.5 +0.0 +E − EF (eV) +Γ +Γ +M +X +−1.0 +−0.5 +0.0 +Γ +X +Μ +Γ +* +* +* +* +* +* +* +* + J=1/2 + J=3/2 (mJ=±3/2) + J=3/2 (mJ=±1/2) +(a) +(b) +(d) +(c) +E − EF (eV) +Γ +M +X +−1 +0.5 +0 +E − EF (eV) +0 +2 +4 +6 +8 +ARPES intensity (arb. units) +* +* +* +* +E − EF (eV) +Intensity (arb. units) +1.4 +1.2 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +−1.0 +−0.5 +0.0 +Γ +M +X +FIG. 6. (a) Energy-momentum plot of the ARPES intensity as a function of the path Γ-M-X-Γ. +(b) Sketch of the dispersion of the main bands visible in ARPES. The lines are guides to the eyes +extracted from the data, the colors are given by comparison to the calculation. (c) Energy distri- +bution curves at X (blue), Γ (black) and M (red). (d) Comparison with the unfolded calculation +along the same path. The red arrow mark the opening of the magnetic gap. (e) Calculated ARPES +spectra at X, Γ, and M (see text for details). Position of the Fermi energy in calculations was ad- +justed to match the experimental energy distance between the Fermi energy and the J = 1/2 band +maximum at M (red star) on the panel (c). +27 + +-1.5 +-1.0 +-0.5 +0.0 +0.5 +Energy (eV) +Γ +X +-1.5 +-1.0 +-0.5 +0.0 +0.5 +Energy (eV) +Γ +X +-1.5 +-1.0 +-0.5 +0.0 +0.5 +Γ +X +y axis +x axis +xz +xy +yz +b +a +c +(a) +(c) +(b) +(d) +no SOC +no SOC +with SOC +with SOC +(e) +(f) +(g) +-0.6 +-0.4 +-0.2 +0.0 +Energy (eV) +Γ +X +X +* +* +-1.5 +-1.0 +-0.5 +0.0 +0.5 +Γ +X +* +* +kz +kx +ky +U’ +T’ +Z’ +X’ +X +Γ +Y’ +S’ +R’ +FIG. 7. (a) Sketch of the SrIrO3 structure. There are 4 inequivalent Ir, at the center of oxygen +octahedra of a different color. (b) Sketch of the corresponding BZ. The black cube corresponds to +the primitive BZ (pseudocubic unit cell) and the shaded region to the supercell BZ. (c) Energy- +momentum image of the dispersion along ΓX for a thin film of SrIrO3/SrTiO3, measured with +100 eV photon energy. The ΓX path is the diagonal of the primitive unit cell similar to Fig. 3(c). +(d) Calculated band structure along ΓX for kz = 0 without SOC. (e) Same as (d) with unfolding +weight as marker size and color scale indicating dxz (red), dyz (green) and dxy (blue) character. +(f,g) Same as (d,e) with SOC. The circles highlight the regions where a large SOC-induced gap +opens. The stars highlight bands discussed in the paper. +28 + +TABLE I. Structural and calculation parameters. +Parameters +Sr2IrO4 +SrIrO3 +Space group (non-magnetic) +88 (I41/a) +62 (Pbnm) +Space group (magnetic) +13 (P2/c) +n/a +Lattice param. (˚A) +5.485, 25.775 +5.568, 5.597, 7.892 +RMT (bohr) +2.24 (Sr) +2.26 (Sr) +1.98 (Ir) +2.09 (Ir) +1.62 (O) +1.71 (O) +nval +10 (Sr) +10 (Sr) +31 (Ir) +31 (Ir) +2 (O) +2 (O) +RMTminKmax +7.0 +7.0 +Gmax +12 +12 +lmax +10 +10 +lvnsmax +6 +4 +k mesh +12 × 12 × 2 +11 × 11 × 7 +(shifted) +Energy (Ry) and +10−4 +10−4 +charge converg. +10−3 +10−3 +29 + +TABLE II. BCT reciprocal lattice points. +Label +Conventional [21] +Primitive +Primitive +(standard [21]) +(Fig. 3c) +Γ +(0, 0, 0) +(0, 0, 0) +(0, 0, 0) +X +(1/2, 1/2, 0) +(0, 0, 1/2) +(1/2, 1/2, 1/2) +M +(1/2, 0, 0) +(−1/4, 1/4, 1/4) +(1/2, 0, 1/4) +30 + diff --git a/QNE0T4oBgHgl3EQf1QJC/content/tmp_files/load_file.txt b/QNE0T4oBgHgl3EQf1QJC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..212e1d3d901b926c708b1aa799d0f8be47b832dc --- /dev/null +++ b/QNE0T4oBgHgl3EQf1QJC/content/tmp_files/load_file.txt @@ -0,0 +1,1210 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf,len=1209 +page_content='Band unfolding with a general transformation matrix: from code implementation to interpretation of photoemission spectra Oleg Rubel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' ∗ Jean-Baptiste Moussy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='2 Paul Foulquier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 3 and V´eronique Brouet3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' † 1Department of Materials Science and Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' McMaster University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 1280 Main Street West,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Hamilton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Ontario L8S 4L8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Canada 2Universit´e Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' CEA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' SPEC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 91191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Gif-sur-Yvette,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' France 3Universit´e Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Laboratoire de Physique des Solides,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 91405,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Orsay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (Dated: January 10, 2023) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='02696v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='comp-ph] 6 Jan 2023 Abstract Unfolding of a supercell band structure into a primitive Brillouin zone is important for un- derstanding implications of structural distortions, disorder, defects, solid solutions on materials electronic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Necessity of the band unfolding is also recognised in interpretation of angle- resolved photoemission spectroscopy (ARPES) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' We describe an extension of the fold2Bloch package by implementing an arbitrary transformation matrix used to establish a rela- tion between primitive cell and supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' This development allows us to overcome limitations of supercells constructed exclusively by scaling of primitive cell lattice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' It becomes possible to transform between primitive and conventional cells as well as include rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The fold2Bloch is publicaly available from a GitHub repository as a FORTRAN code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' It interfaces with the all- electron full-potential WIEN2k and the pseudopotential VASP density functional theory packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The fold2Bloch is supplemented by additional pre- and post-processing utilities that aid in gen- erating k points in the supercell (such that they later fall onto a desired path in the primitive Brillouin zone after unfolding) and plotting the unfolded band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' We selected Sr2IrO4 as an illustrative example and, for the first time, present its properly unfolded band structure in direct comparison with ARPES measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' In addition, critical importance of the band unfolding for interpretation of SrIrO3 ARPES data is illustrated and discussed as a perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' INTRODUCTION The band dispersion E(k) obtained from electronic structure calculations provides infor- mation about Fermi surface of metals, direct/indirect transition in semiconductors, effective masses, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' When periodicity is perturbed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=', solid solutions, defects, magnetic order) the Brillouin zone (BZ) shrinks and bands get folded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' As a result, E(k) becomes obscured and difficult to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Even when the periodicity is formally perturbed, it is still possible to recover an effective band structure in a BZ of the primitive cell using a k-spectral decomposition also known as ‘band unfolding’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' There are numerous examples of band unfolding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The notable milestones include an electronic structure of aperiodic solids [1–3], solid solutions described within a ∗ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' email: rubelo@mcmaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='ca, ORCID: 0000-0001-5104-5602 † V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' email: veronique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='brouet@u-psud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='fr 2 tight-binding approximation [4–6] or pseudopotentials with a plane-wave basis set [7–9], and Wannier functions derived from first-principle calculations to interpret angle-resolved photoemission spectroscopy (ARPES) experiments [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' There are a number of public im- plementations of band unfolding in the density functional theory (DFT) community, for instance, BandsUP [11, 12], fold2Bloch [13], VASPKIT [14], vaspwfc [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The simplest scenario to unfold the band structure involves the supercell constructed from primitive real-space lattice vectors ai (i = 1, 2, 3) by scaling them with an integer niai (ni ∈ Z+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' However, it does not cover all possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' There are lattices constructed by a combination of rotation and scaling of the primitive structural unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Examples include a conventional vs reduced unit cell [16, 17] or an octahedral rotation and tilting in perovskite structures [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Popescu and Zunger [9] suggested a more general approach to unfolding that involves a transformation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' It was already implemented in BandsUP, VASPKIT, and vaspwfc but not in fold2Bloch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Lattice transformations Following the notations in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 9 we denote by small (capital) symbols quantities referring to the primitive cell (supercell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Transformation of real-space primitive ai to supercell Ai lattice vectors can be expressed as [19] Ai = 3 � j=1 Pji aj (i = 1, 2, 3) (1) or in the matrix form as A = P ⊺ a, (2) where the lattice vectors matrices A and a are constructed of rows being individual vectors and columns being their Cartesian components (x, y, z), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=', A = � � � � � A1 A2 A3 � � � � � ≡ � ���� A11 A12 A13 A21 A22 A23 A31 A32 A33 � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (3) Here P is a transformation matrix of the size 3 × 3 compatible with conventions recom- mended by the International tables for crystallography [19] (same as in VESTA structure 3 visualization software [20] or Bilbao crystallographic server [21] but different from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The transformation matrix is obtained by solving the linear Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (2), which yields P = (A a−1)⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (4) The reverse transformation of a supercell to the primitive cell is obtained using the inverse matrix P −1 a = (P −1)⊺ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (5) Scaling of the cell volume as a result of the primitive cell to supercell transformation is given by [19] nv = det(P), (6) which imposes two constrains: P should be positive defined and Pij ∈ Z, from which it follows that det(P) ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Reciprocal lattice vectors are also transformed using P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Owing to the relation B = (A−1)⊺, reciprocal lattice vectors of a supercell Bi can be transformed to the reciprocal primitive vectors bi as [19] b = P B (7) and back as B = P −1 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (8) Note that the transposition of P is not required for converting reciprocal lattice vectors in the matrix form contrary to the conversions proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 9 (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (1) and (2) therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Band unfolding We already outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 13 an unfolding procedure used in the fold2Bloch imple- mentation when a supercell is constructed by simple scaling of the primitive cell .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Here we present an extended (more general) version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' DFT codes for solids internally operate with wave vectors in fractional coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Here we will use tilde to denote primitive (supercell) fractional coordinates ˜k ( ˜K) where com- ponents of each vector span a range between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Cartesian components of the wave vector are obtained by a multiplication with the reciprocal lattice matrix k = ˜k b (9) 4 and similarly for the supercell K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Transformation of reciprocal-space supercell ˜K to primitive ˜k wave vectors (fractional coordinates) is given by [19] ˜k = [ ˜K + (m1, m2, m3)] P −1 mod 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (10) With all possible combinations of mi ∈ Z, the number of unique k points generated within the first BZ of the primitive cell (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 1) is given by the volume scale (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The new (unfolded) k points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 1b form two subsets k1 (open markers) and k2 (red) of the original grid K + G in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The two subsets were created since the volume change is nv = 2 in the example shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' More generally, the subset property is expressed as kl + g ⊂ K + G (l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' , nv), (11) with G(m1, m2, m3) = m1B1 + m2B2 + m3B3 and g(m1, m2, m3) = m1b1 + m2b2 + m3b3 being commensurate vectors of the plane wave expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The wave function in WIEN2k is split into two regions: the atomic spheres and the interstitial region [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' A plane wave basis set is used in the interstitial region Ψ(int) σ,n,K(r) = � G Cσ,n,K(G) ei(K+G)·r, (12) where C are plane wave coefficients for a specific electronic state with the wave vector K, spin channel σ, band index n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Spectral weight of the new unfolded k point is evaluated from the subset of plain wave coefficients [9] wσ,n(kl) = � g |Cσ,n,K(kl + g)|2 (l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' , nv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (13) Weights are normalized such that nv � l=1 wσ,n(kl) = 1 (14) In the case of a spinor wave function, weights of spin up and down components are mixed wn(k) = α2w↑,n(k) + β2w↓,n(k), (15) where α and β are components of the spinor wave function (α2 + β2 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' This result is similar to decomposition of partial spectral weights proposed by Medeiros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' [12], yet it is additionally augmented by the relative contribution of each spin channel to the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Electronic structure calculations All electronic structure calculations were performed with WIEN2k DFT package [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Relevant parameters are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' We used the Perdew, Burke, and Ernzerhof [24] (PBE) exchange-correlation functional in combination with the onsite Hubbard correction U [25] for Ir-d electrons in the case of Sr2IrO4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The spin-orbit coupling was included in all calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The spin polarization was enabled only in the case of Sr2IrO4, where we used a collinear antiferromagnetic ordering as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 26 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 2d) initialized with the magnetic moment of ±1 Bohr magneton (µB) per Ir site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Implementation and execution workflow WIEN2k: First we initialize the calculation and complete the self-consistent field (SCF) cycle using the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='struct structure file as an input to obtain a self-consistent potential and a charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Once the calculation converges all necessary files are saves in the SOC folder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' A detailed workflow can be found in the supporting information (SI) section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Utils/fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='m: Next we generate a list of folded ˜K points within the supercell BZ using the Octave/MATLAB script that takes a desired ˜k point path in the primitive BZ, the number of intermediate points for each section of the path, as well as the transformation matrix P as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' $ octave fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='m The folding is achieved by the following matrix product [19] ˜K = ˜k P mod 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (16) The generated unique ˜K points are stored in the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='klist band file in a WIEN2k na- tive format as three integer numbers per k point with a common integer divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' It is important to note conventions used within WIEN2k to interpret k point coordinates in the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='klist band file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The BZ of a conventional lattice is implied for F, B, CXY, CXZ, and CXZ orthorhombic lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The BZ of a primitive lattice is used for other lattice types (P, H, R, CXZ monoclinic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Those peculiarities affect the selection of supercell lattice vectors A in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (3) and the construction of P matrix using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' In practice, we expect that majority of users will have P-type supercells due the symmetry broken by disorder/defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 6 WIEN2k: Now we generate eigenvalues and eigenvectors (wave functions or vector files) for the list of k points in the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='klist band file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' We use files saved in the SOC folder after the previous SCF step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' $ x lapw1 -band -up [-p] $ x lapw1 -band -dn [-p] $ x lapwso -up [-orb] [-p] The -orb switch is activated for the DFT+U calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Here we produce files that are essential for unfolding: case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='vectorso[up/dn] (wave functions and energy eigenvalues) and case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='normso[up/dn] (spinor components α2 and β2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' fold2Bloch: fold2Bloch is a FORTRAN code that can be compiled with either Intel or GNU FORTRAN compilers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' It takes wave functions, spinor components, and the transfor- mation matrix P as input arguments $ fold2Bloch -so case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='vectorsoup[ 1] case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='vectorsodn[ 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='normsoup[ 1] case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='normsodn[ 1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' "’P11 P12 P13:P21 P22 P23:P31 P32 P33’" and generates a case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='f2b file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' If WIEN2k calculations run in a k-parallel mode ([-p] op- tion), output vector and norm files will be marked with XX for each parallel stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' These files can be processed individually, and the output can be concatenated into one case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='f2b file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The sample listing of the output file is given below k_1 k_2 k_3 E (Ry) w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='000000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='000000 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='500000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='394518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='000000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='500000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='500000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='750000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='394518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='311893 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Here one can see results of band unfolding with the transformation matrix P = [1, ¯1, ¯2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 1, 1, ¯2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 0, 0, 4] and the volume scale of nv = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Two groups of eigenvalues are shown each unfolded into eight new k points (fractional coordinates) in the primitive BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Even though both eigen- values belong to Γ point in the supercell BZ, after unfolding the first does not have any notable Γ character (only ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='16%), while the second eigenvalue retains about 37% of its Γ character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Utils/ubs bmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='m: This Octave script is used to prepare a binary file case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='f2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='bin for band structure plotting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The inputs are the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='f2b file, the desired ˜k path in the primitive BZ (it has to match that used as input in fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='m), reciprocal lattice vectors of the supercell (can be read from the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='outputkgen file in a column-wise manner and later transposed within the script), the Fermi energy from case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='scf file, smearing for a Gaussian function in energy and k space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' $ octave ubs bmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='m Sensible values for the energy and k space smearing are about 1/50 of the energy range and the k path length selected for the band structure plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' At the end of execution, XTICKS and KLABEL vectors are printed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' They need to be noted and used in the next stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Utils/f2b-band-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='plt: Finally, the unfolded band structure is plotted using Gnuplot with the input binary file case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='f2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' $ gnuplot f2b-band-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='plt The Gnuplot script incorporates data from XTICKS and KLABEL vectors used for labeling the high-symmetry points along the path and generates a publication-quality plot (case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='eps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Experimental details The experimental structure of Sr2IrO4 was measured with ARPES on the Cassiop´ee beamline of the SOLEIL synchrotron, with a SCIENTA R-4000 analyzer and an overall 8 resolution better than 15 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The temperature was 20 K, the photon energy was set to 100 eV and linear polarization in the plane containing ΓM was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The samples were prepared using a self-flux method, as reported before [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' We have grown SrIrO3 thin films on SrTiO3 (001) substrate using pulsed laser deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' A frequency-tripled Nd:YAG laser (λ = 355 nm, f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 Hz, pulse duration 15 ns) was fo- cused on a polycrystalline SrIrO3 target made by solid state synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The substrate surface was prepared following the process described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 29 to obtain a uniform TiO2 surface termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The deposition was performed with the substrate heated at T = 600 ◦C and with oxygen partial pressure P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 × 10−1 mbar and monitored by in-situ reflection high- energy electron diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Then, the thin films were cooled down to room temperature with the same oxygen partial pressure to compensate for any oxygen vacancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Finally, we fully structurally and physically characterized our thin films by X-ray diffraction and reflectiv- ity using a Br¨uker D8 advance diffractometer and by surface diffraction on SIXS beamline at SOLEIL synchrotron, atomic force microscopy and electronic transport, concluding to similar characteristics and properties to previous reports in the literature [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Sr2IrO4: Theoretical calculations The structure of Sr2IrO4 was imported from Springer Materials [33] (dataset ID sd 1945591).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The original structure is BCT and includes a tilting of IrO6 octahedra (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 2a-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The magnetic ordering further reduces the symmetry to a tetragonal cell with 8-Ir atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Its lattice vectors matrix (˚A) is A = � ���� 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='485 0 0 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='485 0 0 0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='775 � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (17) For verification purposes we also created an idealized structure by eliminating the octahedral tilting and magnetic ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Its elementary structural unit (a 1-Ir primitive BCT unit cell) 9 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The corresponding matrix of primitive lattice vectors is a = � ���� A1/2 −A2/2 0 A1/2 A2/2 0 A1/2 0 A3/4 � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (18) Following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (4) we obtain the transformation matrix P = � ���� 1 −1 −2 1 1 −2 0 0 4 � ���� , (19) which establishes the relation between 1-Ir primitive cell and 8-Ir supercell (without distor- tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The structure files (in a WIEN2k native format) can be found in the SI section and visualized with XcrySDen [34] or VESTA [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The BZ of 8-Ir and 1-Ir cell is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 3a,b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' For plotting band struc- tures we selected the Γ−M−X−Γ path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Since we made a non-standard choice for BCT primitive lattice vectors (see Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='2, tI Bravais lattice in the International tables for crystallography [35]), special care should be taken to map k point coordinates from the conventional to the primitive cell (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 3b) to ensure that the coordinates are compatible with the chosen definition for lattice vectors (Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' For verification purposes we first need to calculate the band structure without tilting of IrO6 octehedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' This way we can establish a direct comparison between the 8-Ir cell and the primitive 1-Ir BCT cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' It is convenient to do this comparison at the PBE level of theory, since it leads to a non-magnetic solution with a more simple band structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Bands near the Fermi energy are due to Ir-d electrons: the dispersive bands starting off Γ near EF correspond to the dx2−y2 orbital;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' the remaining bands are due to t2g states (dxy, dyz, and dxz orbitals);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' the dz2 orbital belongs to eg states located at higher energies outside the energy range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Figure 4b shows the band structure of the 8-Ir supercell without octahedral tilting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' This band structure is obscured by the zone folding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Figure 3c can help to rationalize zone folding at high-symmetry k points (prime indices are for the supercell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Eigenvalues at both Γ and X points are folded into Γ′ of the supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Eigenvalues at M point fall into X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The Dirac-like band crossings along the Γ′−X′ segment are folding artefacts not observed in the 10 primitive cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The unfolded and primitive band structures are identical (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 4a,c) that gives confidence in our approach and its implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The realistic structure of Sr2IrO4 includes tilting of IrO6 octahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Those distortions cause perturbations in the band structure, which can be assessed thanks to the new func- tionality of fold2Bloch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The unfolded band structure of 8-Ir cell with octahedral tiltings (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 5b) can now be compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 4c (both calculated at the PBE level of theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The most notable change is that the rotation allows hybridization between dx2−y2 and dxy, as is also the case in Sr2RhO4 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Therefore, new dx2−y2 states move to higher energies (not visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 5) and dxy states are now pushed below the Fermi energy resulting in a new bright ‘spot’ at X below the Fermi energy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 5b), which is unfolded from Γ′′ leaving a weak replica at Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Interestingly, states at M point are immune to the distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' To account for correlation effects we added the Hubbard Ueff = 3 eV correction for Ir- d states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The magnitude of Ueff was chosen to reproduce the experimental band gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 eV [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Unlike PBE, PBE+U favours a magnetic solution with the moment of µ(Ir) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='24µB per Ir site (the ordering is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 2d with the [001] spin quantization direction for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Experimental µ(Ir) moments are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='21 [26], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='29 [39], and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='37 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' A gap opens up between the on-site spin up and down states, which is particularly clear at M and at the middle of ΓX (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 5c-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Sr2IrO4: Comparison with experiment Figure 6(a) shows the ARPES spectral intensity along the k-path Γ-M-X-Γ, which was extracted from our measurement using a photon energy of 100 eV and a linear polarization along ΓM [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The different bands observed in this plot are sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 6(b) by red and blue guides to the eyes, the colour corresponds to their dominant orbital character, characterized by the effective value of the total electronic angular momentum J, either J = 1/2 (mJ = ±1/2) or J = 3/2 (mJ = ±1/2, ±3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' These data are similar to several previous reports [28, 41, 42], where more details can also be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The most intense band is the J = 3/2 band at X (blue star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Although it should also be present at Γ, which is equivalent in the supercell (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 3(c)), it can hardly be distinguished there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' This is perfectly captured by the unfolded calculation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 6(d), which features much lower intensity for this band at Γ compared to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' A comparison of the measured 11 and calculated intensity along Γ and X is also shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 6(c) and (e), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' ARPES spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 6(e) were calculated based on the discrete spectral weights wn(k) defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (15) and energies En(k) of the unfolded band structure at specific k points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' A Gaussian broadening of the width σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='2 eV was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' However, it is still a very crude approximation as other relevant details were omitted (matrix elements for initial- and final-state crystal wave functions, finite-lifetime effects, surface discontinuity, multiple scattering [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' These matrix elements will further modulate the measured intensity, but the qualitative difference is well captured by the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' However, the relative position of the J = 3/2 band at X and J = 1/2 band at M is quite different in the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' In experiment, those bands are ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='22 eV apart, while in calculations the energy difference between the valence band maxima at M and X is ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='11 eV (compare red and blue stars on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 6 panels (c) and (e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' This discrepancy is due to an underestimation of the effective spin-orbit coupling already noticed and discussed in the literature [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The J = 1/2 band is the one where the magnetic gap opens [42] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 6(d), red arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' This band is clearly visible along ΓM, but much weaker along MX, two paths expected to be equivalent in the supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Again, this fits with the theoretical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Similarly, the J = 1/2 band drastically loses weight on the second half of ΓX, both in the experiment and in calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Note that, as the relative positions of J = 1/2 and J = 3/2 are not correctly captured, the break in the dispersion due to hybridization between J = 1/2 and J = 3/2 where they cross is also shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The other bands are more difficult to isolate in ARPES, either because they become too broad at high binding energies or because they have low intensity in these experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' However, it is clear that another band is present at −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='8 eV at M and a trace of a second band at X can be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' They are marked by cyan stars and correspond well to the other J = 3/2 band (mJ = ±1/2), which is dominated by dxy weight, and therefore has a lower cross section in ARPES [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' SrIrO3: Perspective We illustrated the unfolding process with the case of Sr2IrO4, where the connection to a primitive unit cell, without rotation of the oxygen octahedra, is relatively easy to anticipate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' However, in more complicated cases, it can become totally impossible to understand the band 12 structure and compare it to ARPES data, without the help of the unfolding calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' A very interesting case is the related 3D iridate SrIrO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Its structure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 7(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' In addition to in-plane rotation of the oxygen octahedra, similar to Sr2IrO4, it also exhibits a tilt of the oxygen octahedra from the c axis, inducing another type of folding along kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The resulting BZ is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 7(b), it is rotated 45◦ in the (kx, ky, 0) plane, as for Sr2IrO4, but also halved along kz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The calculated band structure is semimetallic and the four J = 1/2 folded bands are expected to form a Dirac nodal line around the U point (1/2, 0, 1/2) [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' As topological features are rare in correlated oxides, this occurrence generated great interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' However, the orthorhombic structure of SrIrO3 is only stable in thin films, which adds questions on the role of the epitaxial constraint and the survival of topological features in real systems [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' It would therefore be very interesting to look for these features directly with ARPES, but the situation is not yet concluding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The only ARPES studies available to date were performed at a fixed photon energy [48, 49], which may not allow to precisely locate the U point, or at high photon energies, where the energy resolution is lower [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' We only sketch here the help of unfolded calculations for deciphering the electronic struc- ture of SrIrO3, more details will be published later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Figure 7(c) shows the images of the dispersion along ΓX direction with a photon energy of 100 eV obtained on a SrIrO3(001) thin film (t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='4 nm) grown on SrTiO3(001) substrates (for clarity, we keep the same labelling as previously for the (kx, ky, 0) plane of Sr2IrO4 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 3(c), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=', X labels the corner of the primitive 2D BZ similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The ARPES image looks quite simple, with a prominent band at X (almost invisible at Γ), resembling the J = 3/2 in Sr2IrO4 shifted up by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='2 eV and a band reaching the Fermi level at the middle of ΓX, looking like J = 1/2 in Sr2IrO4 shifted up by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='15 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Quantitatively, the comparison with raw calculations in the supercell is usually very confusing, as a much more complicated band structure is predicted [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 7(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' We will show how unfolding of the band structure can simplify the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' We calculated with WIEN2k the electronic structure for the structure given in Table III of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The space group is Pbnm (#62), and the parameters are given in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' As for Sr2IrO4, we can define the lattice vector matrix of the supercell (˚A) and the fictitious 13 primitive cell, as well as the transformation matrix: A = � ���� 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='597 0 0 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='568 0 0 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='892 � ���� , a = � ���� A1/2 −A3/2 0 A1/2 A2/2 0 0 0 A3/2 � ���� , P = � ���� 1 −1 0 1 1 0 0 0 2 � ���� (20) No Coulomb repulsion U was used and a semi-metallic non-magnetic state is obtained, as it is the case experimentally [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Assuming our ARPES data at this photon energy correspond to kz = 0, we show the calculated electronic structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 7(d,f) with and without SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Clearly, the calculation looks much more complicated than ARPES data, and it is extremely difficult to understand which bands should be compared to the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' After unfolding [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 7(e,g)], a set of 3 bands is clearly emphasized, although their dis- persions are affected by their mutual interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Using a color code for orbital characters (the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='inq file was modified to rotate a and b by 45◦ and align them with Ir−O−Ir bonds), one can clearly recognize the original dxz/dyz/dxy bands [panel (e), without SOC] and J = 1/2, J = 3/2 after their interaction with SOC [panel (g)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The places where a SOC induced hybridization gap openings are noted as empty circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Obviously the band marked by a red star is at significantly different position in the measurement, compared to the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' On the other hand, the band at X (blue star) exhibits a well-defined parabolic shape in the measurement over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='4 eV, while there is a large SOC induced gap near the top of the dispersion in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Similar to Sr2IrO4, we can anticipate that the effective spin-orbit coupling may be underestimated in the calculation, which will change the splitting of the two bands and also the position of their crossing, hence the break in the dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Moreover, as the shape of dxy depends sensitively on the rotations, it may be necessary to tune the structure to get the right interaction pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' This has extremely important consequences for the formation of the Dirac nodal line at kz = 1/2, which we will not discuss here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' This examples shows how adjusting the structure to get a better descrip- tion of the experimental data will be possible only when a basic understanding of the origin of the different bands is reached, thanks to the unfolding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Let us stress that the intensity of the folded bands is also a fine marker of the strength of the interactions at the origin of the supercell, which is here the rotations of the oxygen octahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' In a similar spirit, the information contained in the intensity of the folded bands 14 on these interactions was recently used to refine the structure of SrRuO3 thin films [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The near absence of the folded bands in our measurement suggests a small coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' From the topological point of view, the meaning of the crossing of two bands with very different spectral weight is also a question that may deserve further work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' CONCLUSION Unfolding of a supercell band structure into a primitive Brillouin zone is important for understanding implications of structural distortions, disorder, defects, solid solutions on materials electronic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Necessity of the band unfolding is also recognised in interpre- tation of angle-resolved photoemission spectroscopy (ARPES) measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' We described an extension of the fold2Bloch package by implementing an arbitrary transformation ma- trix used to establish a relation between primitive cell and supercell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The convention selected for the transformation matrix is compatible with that recommended by the International tables for crystallography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' This development allows us to overcome limitations of supercells constructed exclusively by scaling of primitive cell lattice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' For instance, it becomes possible to transform between primitive and conventional cells as well as include rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The updated fold2Bloch package is available from a GitHub repository as a FORTRAN code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' It interfaces with the all-electron full-potential WIEN2k and the pseudopotential VASP density functional theory packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The fold2Bloch is supplemented by additional pre- and post-processing utilities that aid in generating k points in the supercell (such that they later fall onto a desired path in the primitive Brillouin zone after unfolding) and plot- ting the unfolded band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' We selected Sr2IrO4 as an illustrative example and, for the first time, present its properly unfolded band structure in direct comparison with ARPES measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' In addition, critical importance of the band unfolding for interpretation of SrIrO3 ARPES data is illustrated and discussed as a perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Of particular interest for iridates is the ability of ARPES to sense imprints that tilting of IrO6 octahedra leaves on the materials’ electronic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' ARPES measurements teach us an important lesson: small structural perturbations neither lead to a sudden change in the electronic structure nor redefine the associated primitive Brillouin zone, contrary to what is expected from the formal symmetry of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 15 ACKNOWLEDGMENTS Calculations were performed using the Compute Canada infrastructure supported by the Canada Foundation for Innovation under John R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Evans Leaders Fund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' ARPES work was supported by the Agence Nationale de la Recherche grant “SOCRATE” (Grant No.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Jin, High- pressure synthesis of orthorhombic SrIrO3 perovskite and its positive magnetoresistance, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 103, 103706 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' [53] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Sohn and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Kim, Evolution of electronic band reconstruction in thickness-controlled per- ovskite SrRuO3 thin films, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Korean Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='1007/s40042-022-00633-5 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 20 SUPPORTING INFORMATION We include a ZIP archive with WIEN2k structures, relevant scripts, initialization and unfolding workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' See README file within the archive for additional description of the content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 21 (a) (b) Γ K K + G(1, 0, 0) B1 B2 k1 k2 b1 b2 Γ k2 + g(1, 0, 0) k1 + g(0, −1, 0) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Unfolding in reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (a) BZ of a supercell with an arbitrary K point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (b) BZ of a primitive cell obtained by the two-dimensional transformation matrix of P = [1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' ¯1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The cell volume changes twice (nv = 2), thus the K point transforms into two new point k1 and k2 that form two groups of plane wave coefficients (open and red markers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The dashed line shows the supercell BZ inside of the primitive BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 22 (a) (d) (e) O Sr Ir (c) (b) 1 Ir primitive cell 1 Ir primitive cell FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Structure of Sr2IrO4 (space group 88, I41/a (non-magnetic) or 13, P2/c (magnetic)): (a) top view, (b) top view with the first Ir layer only, (c) top view with the second Ir layer only, (d) side view with arrows showing the magnetic ordering, and (e) primitive 1-Ir unit cell (space group 139, I4/mmm) obtained from the supercell without octehedral tilting using the transformation matrix of P −1 = (1/2, 1/2, 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 1/2, 1/2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 0, 0, 1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The octahedral tilting is present on panels (a)–(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 23 ab ab ab a(a) (b) (c) Γ M primitive conventional X b2 b3 b1 primitive supercell Γ’ M’ X’ B2 B3 B1 Γ, Γ’ X, Γ’’ M, X’ M’ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (a) BZ of Sr2IrO4 tetragonal cell (space group 88).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (b) Primitive BZ of a body-centred tetragonal lattice (black) and the (kx, ky, 0) plane of a conventional BZ (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (c) Overlay of the primitive and supercell BZ (top view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The k path of interest within the (kx, ky, 0) plane is shown by green arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 24 (a) (b) (c) Energy (eV) Wave vector −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 Γ M X Γ Γ M X Γ EF Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 X’ Γ’’ M’ EF Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 Γ’ Γ’ Wave vector Wave vector FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Band structure of Sr2IrO4 at PBE+SOC level of theory without octehedral tilting: (a) body-centred tetragonal primitive cell (space group 139, 1-Ir atom), (b) tetragonal supercell (space group 88, 8-Ir atoms), (c) supercell unfolded into the primitive cell using P = [1, ¯1, ¯2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 1, 1, ¯2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 0, 0, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The asterisk (*) marks dx2−y2 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Energies are plotted relative to the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Red (black) labels for high-symmetry points in the reciprocal space refer to the primitive cell (supercell) BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 25 (a) (b) Energy (eV) Wave vector −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 Γ M X Γ (d) (c) X’ Γ’’ M’ E F Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 Γ’ Γ’ X’ Γ’’ M’ E F Energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 Γ’ Γ’ Energy (eV) Wave vector −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 Γ M X Γ Wave vector Wave vector FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Unfolded band structure of Sr2IrO4 with octehedral tilting: (a,b) PBE+SOC folded and unfolded, (c,d) PBE+SOC with onsite Ueff = 3 eV for Ir-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Energies are plotted relative to the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The asterisk (*) marks dxy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 26 (e) Energy (eV) Wave vector −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 Γ M X Γ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 E − EF (eV) Γ Γ M X −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 Γ X Μ Γ J=1/2 J=3/2 (mJ=±3/2) J=3/2 (mJ=±1/2) (a) (b) (d) (c) E − EF (eV) Γ M X −1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 0 E − EF (eV) 0 2 4 6 8 ARPES intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' units) E − EF (eV) Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' units) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 Γ M X FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (a) Energy-momentum plot of the ARPES intensity as a function of the path Γ-M-X-Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (b) Sketch of the dispersion of the main bands visible in ARPES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The lines are guides to the eyes extracted from the data, the colors are given by comparison to the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (c) Energy distri- bution curves at X (blue), Γ (black) and M (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (d) Comparison with the unfolded calculation along the same path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The red arrow mark the opening of the magnetic gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (e) Calculated ARPES spectra at X, Γ, and M (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Position of the Fermi energy in calculations was ad- justed to match the experimental energy distance between the Fermi energy and the J = 1/2 band maximum at M (red star) on the panel (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 Energy (eV) Γ X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 Energy (eV) Γ X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 Γ X y axis x axis xz xy yz b a c (a) (c) (b) (d) no SOC no SOC with SOC with SOC (e) (f) (g) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 Energy (eV) Γ X X 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='5 Γ X kz kx ky U’ T’ Z’ X’ X Γ Y’ S’ R’ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (a) Sketch of the SrIrO3 structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' There are 4 inequivalent Ir, at the center of oxygen octahedra of a different color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (b) Sketch of the corresponding BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The black cube corresponds to the primitive BZ (pseudocubic unit cell) and the shaded region to the supercell BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (c) Energy- momentum image of the dispersion along ΓX for a thin film of SrIrO3/SrTiO3, measured with 100 eV photon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The ΓX path is the diagonal of the primitive unit cell similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (d) Calculated band structure along ΓX for kz = 0 without SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (e) Same as (d) with unfolding weight as marker size and color scale indicating dxz (red), dyz (green) and dxy (blue) character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (f,g) Same as (d,e) with SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The circles highlight the regions where a large SOC-induced gap opens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' The stars highlight bands discussed in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 28 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Structural and calculation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Parameters Sr2IrO4 SrIrO3 Space group (non-magnetic) 88 (I41/a) 62 (Pbnm) Space group (magnetic) 13 (P2/c) n/a Lattice param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' (˚A) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='485, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='775 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='568, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='597, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='892 RMT (bohr) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='24 (Sr) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='26 (Sr) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='98 (Ir) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='09 (Ir) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='62 (O) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='71 (O) nval 10 (Sr) 10 (Sr) 31 (Ir) 31 (Ir) 2 (O) 2 (O) RMTminKmax 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content='0 Gmax 12 12 lmax 10 10 lvnsmax 6 4 k mesh 12 × 12 × 2 11 × 11 × 7 (shifted) Energy (Ry) and 10−4 10−4 charge converg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 10−3 10−3 29 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' BCT reciprocal lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' Label Conventional [21] Primitive Primitive (standard [21]) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} +page_content=' 3c) Γ (0, 0, 0) (0, 0, 0) (0, 0, 0) X (1/2, 1/2, 0) (0, 0, 1/2) (1/2, 1/2, 1/2) M (1/2, 0, 0) (−1/4, 1/4, 1/4) (1/2, 0, 1/4) 30' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNE0T4oBgHgl3EQf1QJC/content/2301.02696v1.pdf'} diff --git a/QdFJT4oBgHgl3EQf2y0y/content/tmp_files/2301.11657v1.pdf.txt b/QdFJT4oBgHgl3EQf2y0y/content/tmp_files/2301.11657v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e1eed22f4aaf6aa0fba9fc4393ff11a47b1d2983 --- /dev/null +++ b/QdFJT4oBgHgl3EQf2y0y/content/tmp_files/2301.11657v1.pdf.txt @@ -0,0 +1,1051 @@ +arXiv:2301.11657v1 [math.ST] 27 Jan 2023 +Multilayer hypergraph clustering +using the aggregate similarity matrix +Kalle Alaluusua1, Konstantin Avrachenkov2, B. R. Vinay Kumar2, and +Lasse Leskelä1 +1 Aalto University, Espoo, Finland +{kalle.alaluusua, lasse.leskela}@aalto.fi +2 INRIA, Sophia Antipolis, Valbonne, France +{vinay-kumar.bindiganavile-ramadas, k.avrachenkov}@inria.fr +January 27, 2023 +Abstract +We consider the community recovery problem on a multilayer variant of +the hypergraph stochastic block model (HSBM). Each layer is associated with +an independent realization of a d-uniform HSBM on N vertices. Given the +aggregated number of hyperedges incident to each pair of vertices, represented +using a similarity matrix, the goal is to obtain a partition of the N vertices +into disjoint communities. +In this work, we investigate a semidefinite pro- +gramming (SDP) approach and obtain information–theoretic conditions on +the model parameters that guarantee exact recovery both in the assortative +and the disassortative cases. +Keywords: hypergraph SBM, community detection, semidefinite program- +ming, multilayer, clustering, planted partition +MSC2020: 05C65, 05C80, 62H30, 90B15, 90C22, 90C35, 94A16 +1 +Introduction +Traditional network data are observed as interactions between node pairs, repre- +sented as a graph, or equivalently as an adjacency matrix. More refined forms of +network data may involve multiple types of higher-order interactions simultaneously +involving multiple nodes. Such a data set can be viewed as a binary array A(m) +e +in- +dexed by node sets e and positive integers m so that A(m) +e += 1 indicates that an +interaction of type m occurs among node set e. Such an array may also be viewed +as a multilayer hypergraph where the entries of the array indicate the presence +of hyperedge e in the m-th layer. Examples of such data could arise in a variety +of scenarios. Researchers attending conferences, table reservations at restaurants, +processor sharing etc. Stochastic block models (SBMs) are a popular choice for gen- +erative models with a community structure for such applications. Hypergraphical +stochastic block models (HSBMs) introduce hyperedges into SBMs, thus extending +their modelling capabilities. +1 + +We will next describe a generative model of a hypergraph with N ≥ 1 nodes and +M ≥ 1 layers, where each hyperedge has size d ≥ 2. The set of nodes is divided into +two communities of equal sizes (we assume N is even), and the resulting community +structure, denoted by σ(N), is uniformly distributed on the set {(σ1, σ2, · · · , σN) ∈ +{−1, +1}N : |{i : σi = +1}| = |{i : σi = −1}|}. The community profile of a node set +e is defined as a vector τ(e) = (τ−1(e), τ+1(e)), where τk(e) is equal to the number +of nodes in e with community membership k. We will then sample a multilayer +hypergraph on node set [N] = {1, . . . , N} so that each node set e ⊂ [N] having +size d and community profile τ(e) = t is linked by a hyperedge in layer m with +probability +p(m) +t += α(m) +t +log N +�N−1 +d−1 +� +, +(1) +independently of other node sets and layers. This scaling of the hyperedge probabil- +ities ensures that the expected average degree of each node is Θ(log N). References +[2, 10, 26] show that the phase-transition for exact recovery occurs in this regime, +and this regime is also critical for connectivity in general hypergraph models [7]. +The d-uniform multilayer hypergraph can be represented as a binary array A = +(A(m) +e +) in which the entries are mutually independent Bernoulli random variables. +The event {A(m) +e += 1} has probability p(m) +t +when τ(e) = t. To indicate that A is +sampled from the model, we abbreviate A ∼ HSBM(N, M, d, (α(m) +t +)). We will focus +on a symmetric model in which +α(m) +(r,d−r) = α(m) +(d−r,r) +for all 0 ≤ r ≤ d and all m. This means that the presence of the hyperedge depends +only on the number of nodes of each community rather than the community label. +The problem of community detection is to output an estimate ˆσ(N) of the un- +derlying node communities. The estimate is said to achieve exact recovery if, +lim +N→∞ P +� +ˆσ(N) ∈ {±σ(N)} +� += 1. +In this work, the main focus is to study the community recovery problem based on +a layer-aggregated similarity matrix Wij = � +m W (m) +ij +where (W (m) +ij +) =: W (m) is a +zero-diagonal matrix with off-diagonal entries +W (m) +ij += +� +e:e∋i,j +A(m) +e +counting the number of hyperedges in layer m incident to nodes i and j. Commu- +nity recovery based on the similarity matrix W instead of the full data set A is +motivated by two aspects: privacy and computational tractability. For example, in +an application where N individuals visit M restaurants, providing the full hyper- +graph could reveal the frequency a particular individual visits a restaurant. This +could violate the privacy of the individual. On the other hand, providing the sim- +ilarity matrix obfuscates such individual information, since information regarding +the restaurants that are visited is not revealed. Additionally, the similarity matrix +provides a compact matrix representation of the hypergraph that is easier to ma- +nipulate using matrix algebra. Nevertheless, it is clear that the similarity matrix +contains less information than the complete hypergraph. +2 + +In this work, we investigate a semidefinite programming (SDP) approach for +solving the community detection problem using W . Our main result in Section 3 +provides an information quantity that characterizes the performance of the SDP +approach. To be more specific, it gives a sufficient condition relating the different +parameters of the model for exact recovery using the SDP technique. +The paper is organized as follows. In Section 2, we provide recent literature in +the area and highlight the key differences in our work. In Section 3, we describe +the semidefinite programming algorithm and state our main result concerning the +information theoretic conditions on the parameters of the model that guarantee exact +recovery. Section 4 contains some simulation results of our algorithm on synthetic +data generated using the HSBM model. The proofs of our results are provided in +Section 5. Section 6 concludes the paper and provides some future directions. +2 +Related work +The usual paradigm for theoretical research in community detection is to first pro- +pose a generative model that captures the application (data) as a graph or a net- +work, followed by the analysis of a clustering algorithm for the proposed model. +Community recovery algorithms on stochastic block models (SBMs) have attracted +considerable attention in the past. A comprehensive survey of the field is provided +in [1], see also a review of recent results on graph clustering in [5]. Our interest in +this work is on multilayer hypergraphs. We first provide a brief survey of recent +work in clustering on multilayer networks followed by hypergraph SBMs. +A seminal work for multilayer networks is the review article [27]. Subsequently, +in [34], the authors consider estimating the membership for each individual layer +in a multilayer SBM. In the special case that memberships do not change, their +method works on the normalized aggregate adjacency matrix. The authors in [31] +establish that increasing the number of network layers guarantees consistent com- +munity recovery (by a least squares estimator) even when the graphs get sparser. +SBMs with general interactions allow an alternate model for multilayer networks. +These are studied in [6] where the authors address the community recovery problem +using aggregate adjacency matrix as well as the full graph. The authors in [3] study +Bayesian community recovery in a regime where both the number of nodes and the +number of layers increase. +With regards to literature on hypergraphs, the hypergraph stochastic block +model (HSBM) was first introduced by Ghoshdastidar and Dukkipati in [13] to +capture higher order interactions. They also show strong consistency of spectral +methods in dense uniform hypergraphs. In subsequent works [14, 15, 16], they ex- +tend their results to partial recovery in sparse non-uniform hypergraphs. Some other +works on the spectral algorithm for hypergraphs include [2, 33]. +The recent work by Zhang and Tan [36] considers the general d-uniform HSBM +with multiple communities. They establish a sharp phase transition for exact recov- +ery when the knowledge of the whole hypergraph is given. They recover results from +several previous works including [9, 10, 26, 33]. In the process of solving the exact re- +covery problem, they do show almost exact recovery using only the similarity matrix +through a spectral approach. Another general hypergraph model with theoretical +guarantees by [37] employs a latent space representation of nodes to cover HSBM, +non-uniform hypergraphs, and hypergraphs with heterogeneity among nodes. +3 + +Some of the other approaches used in the literature for the community recovery +problem on hypergraphs are based on modularity [21, 22, 23, 28, 29], tensor decom- +position [24, 37], random walk based methods [33, 35, 38], variational inference [8], +and approximate message passing [4, 32]. +In this work, we investigate the problem of exact recovery on the HSBM through +the lens of semidefinite programming (SDP). Our work is closest in spirit to [12] and +[26] that discuss the SDP approach. The SDP formulation (described in Section 3) +arises as a relaxation of the computationally hard procedure of finding a nodes’ par- +tition with minimum number of edges crossing it. In [26], the authors show that for +a d-uniform homogeneous HSBM with two equal-sized and symmetric communities, +exact recovery using the full hypergraph shows a sharp phase transition behavior. +They go on to propose a ‘truncate-and-relax’ algorithm that utilizes the structure of +the similarity matrix. An SDP approach then guarantees exact recovery with high +probability, albeit in a parameter regime which is slightly sub-optimal. This gap is +bridged in [12] who consider the community recovery problem with the knowledge of +only the similarity matrix. Below, we highlight the differences from these previous +works: +1. The authors in both [12] and [26] consider the homogeneous model in which +the hyperedge parameters take just two values corresponding to all nodes of +a hyperedge being in the same community, and at least one of them being +in a different community. Related works with the same assumption include +[2, 11, 30]. In this work, we allow for hyperedge parameters to depend on +the number of nodes of each community in the hyperedge resulting in an +inhomogeneous HSBM as in [36], albeit with the symmetric assumption. A +similar assumption is present in other works such as [13, 15] as well. +2. Much of the earlier work assumes that the data is assortative or homophilic, i.e. +nodes in the same community are more likely to be adjacent to each other than +to nodes in different communities. Our results incorporate the disassortative +or heterophilic case where the opposite is true. This could be of interest for +some applications: reputation of a research institute is partly assessed based +on the amount of collaboration with experts from external institutions (see +e.g. [18]); so, one might expect certain research networks to be disassortative. +3. Our model targets multilayer HSBMs that can be seen as a generalization of +previous models. Moreover, these layers could individually be assortative or +disassortative which could then capture a plethora of applications. +3 +Algorithm and main results +A first approach to obtain an estimate of the node communities given the similarity +matrix W is to solve the min-bisection problem: +max +� +i,j +Wijxixj +subject to x ∈ {±1}N, 1Tx = 0. +(2) +This formulation assumes that the data is assortative. In the disassortative case +the opposite is true, and we replace the maximization in (2) with minimization, or +equivalently change the sign of the objective function. +4 + +However, the problem in (2) is known to be NP-hard in general (see [25]) and +therefore, we consider a semidefinite programming (SDP) relaxation of it whose +generalisation is described in Algorithm 1. +Algorithm 1 [SDP] +Input: N × N similarity matrix W and s ∈ {±1}. +Output: Community estimate ˆσ +1: Solve the following optimization problem: +maximize +� +ij∈([N] +2 ) +sWijXij +subject to +� +ij∈([N] +2 ) +Xij = 0, +Xii = 1 for all i ∈ [N] +X ⪰ 0. +(3) +2: Let X∗ ∈ RN×N be the optimal solution of (3) and let it have an eigendecom- +position X∗ = �N +i=1 λivivT +i with λ1 ≥ λ2 ≥ · · · ≥ λN. +3: Output ˆσ = sgn(v1) +Remark 1. An alternate approach in the disassortative case is to consider the com- +plement of the hypergraph, which is assortative. A similarity matrix of the comple- +ment is equivalent to +�N−2 +d−2 +� +11T − W . However, owing to our scaling assumption +in (1), the resulting similarity matrix of the complement is no longer in the same +regime, and requires a different approach to analyze. +To capture the level of assortativity of our model, we define the following quan- +tity, accordingly referred to as the assortativity +ξ := +M +� +m=1 +d−1 +� +r=0 +�d − 1 +r +� +(d − 1 − 2r)α(m) +(r,d−r). +(4) +The summation over the different layers implies that the full hypergraph can be +assortative even if individual layers W (m) are not. Table 1 displays this formula +for selected values of d. In Section 5.5, it is shown that ξ is a normalized expected +difference between the number of hyperedges shared between two nodes when they +are of the same community and when they are of different communities. Therefore, +a model is said to be assortative if ξ > 0, and disassortative if ξ < 0. +In order to state our main result concerning the performance of Algorithm 1, we +will need the following information quantity +I = +sup +λ∈R +M +� +m=1 +d−1 +� +r=0 +2−(d−1) +�d − 1 +r +� +α(m) +(r,d−r) +� +1 − e−λ(d−1−2r)� +(5) +To be concise, we omit the dependence on model parameters in the definition of +both ξ and I. +We now state the main theorem that characterizes Algorithm 1 and provides +a sufficient condition for exact recovery on the aggregate similarity matrix of a +symmetric multilayer HSBM using the SDP approach. +5 + +Table 1: +Assortativity ξ in a d-uniform HSBM with M layers, where we denote +αt = �M +m=1 α(m) +t +. +d +ξ +2 +α(0,2) − α(1,1) +3 +2α(0,3) − 2α(1,2) +4 +3α(0,4) − 3α(2,2) +5 +4α(0,5) + 4α(1,4) − 8α(2,3) +6 +5α(0,6) + 10α(1,5) − 5α(2,4) − 10α(3,3) +Theorem 1. Suppose A ∼ HSBM(N, M, d, (α(m) +t +)), and let W be the aggregate +similarity matrix of A. When I > 1, Algorithm 1 with s = sgn(ξ) achieves exact +recovery. +The proof of Theorem 1 is provided in Section 5. Taking M = 1 with parameters +α(r,d−r) = +� +α +for r = 0 and r = d +β +for 1 ≤ r ≤ d − 1 +reduces to a homogeneous model that has been studied earlier in the assortative +case with ξ = (d − 1)(α − β) > 0. In this setting Kim, Bandeira, and Goemans [26] +showed that the SDP algorithm does not achieve exact recovery when I < 1 and +Gaudio and Joshi [12] proved that the SDP algorithm achieves exact recovery when +I > 1, as conjectured in [26]. +4 +Numerical illustrations +We perform numerical simulations to demonstrate the effect of the number of ob- +served hypergraph layers on the classification accuracy of Algorithm 1 1. The syn- +thetic data is sampled from a 4-uniform HSBM(50, M, 4, (α(m) +t +)), with 1 ≤ M ≤ 3. +We let the hypergraph layers be identically distributed giving α(m) +t +=: αt for all +m ∈ [M]. Table 2 provides the parameter values used in the simulations. These +values are chosen such that the expected degree, i.e. the number of hyperedges a +node is incident to, is the same in both the homogeneous and the inhomogeneous +cases. The associated hyperedge probabilities are computed from (1). +Table 2: +Four sets of parameter values (αt) used in simulation experiments. +Homogeneous +Inhomogeneous +t +Assortat. +Disassort. +Assortat. +Disassort. +(4, 0) +18.8 +7.3 +18.8 +4.7 +(3, 1) +7.3 +18.8 +9.4 +9.4 +(2, 2) +7.3 +18.8 +4.7 +18.8 +To evaluate the accuracy of our estimate given σ, we use the Hubert-Arabie +adjusted Rand index (AR) [17, 20], which is a measure of similarity between two +1Source code: https://github.com/kalaluusua/ +6 + +community assignments. The index is equal to 1 when the assignments are iden- +tical, and 0 when they are independent. For each simulated hypergraph, we also +compute the classification error (CE), which we define as the fraction of misclassi- +fied nodes N−1 min{Ham(ˆσ, σ), Ham(ˆσ, −σ)}, where Ham denotes the Hamming +distance. The results (averaged over five different random seed initializations) are +depicted in Table 3. +Table 3: Classification error (CE) and Adjusted Rand index (AR) of the community +assignment estimate. +Homogeneous +Inhomogeneous +Assortat. +Disassort. +Assortat. +Disassort. +M +|ξ| +I +CE +AR +CE +AR +|ξ| +I +CE +AR +CE +AR +1 +34.5 0.58 0.160 0.464 0.184 0.475 +42.4 0.41 0.052 0.799 0.012 0.952 +2 +68.9 1.08 0.024 0.906 0.052 0.800 +84.8 0.83 0.008 0.969 0.000 1.000 +3 +103.4 1.62 0.004 0.984 0.012 0.953 +127.2 1.24 0.000 1.000 0.000 1.000 +Based on the I-values, we expect that the community detection performance im- +proves as M increases and is the same for the assortative and disassortative cases. +This is precisely what Table 3 shows. Moreover, the larger I-values of the homo- +geneous case lead us to expect an overall better performance in comparison to the +homogeneous case. Surprisingly, this is not the case. An inspection of the ξ-values +reveals a larger level of (dis)assortativity in the inhomogeneous case. In small to +moderate hypergraph sizes, we suspect that the level of assortativity may predict the +detection performance of Algorithm 1 better than the information-theoretic quantity +I, whose effect is more profound in the asymptotic regime. +5 +Analysis of the algorithm +In this section, we provide a detailed proof of Theorem 1. We follow the procedure +of [12, 19, 26] to analyze the SDP framework, and extend it to a more general +model HSBM(N, M, d, (α(m) +t +)) that addresses multiple layers, disassortativity, and +(symmetric) inhomogeneity. +An outline of this section is as follows. Section 5.1 constructs a dual certificate +strategy to solve the SDP in (3) and specializes it to the assortative and disas- +sortative cases. +Bounds on certain quantities that arise as part of this strategy +are provided in Sections 5.2 and 5.3. Section 5.4 puts the parts together to com- +plete the proof of Theorem 1. Finally, in Section 5.5, we comment on the assorta- +tive/disassortative nature of the model and its manifestation in our analysis. +5.1 +SDP analysis +To begin, we state a sufficient condition for optimality of Algorithm 1. This is a +corollary of [12, Lemma 2.2] that asserts strong duality for (3) with s = 1. +Lemma 1. Fix s ∈ {±1}. Suppose there is a diagonal matrix D ∈ RN×N and +ν ∈ R such that the matrix S := D + ν11T −sW is positive semidefinite, its second +7 + +smallest eigenvalue λN−1(S) is strictly positive, and Sσ = 0, then X∗ = σσT is +the unique optimal solution to (3) (with the same s). +For the HSBM(N, M, d, (α(m) +t +)) model with node communities σ and the aggre- +gate similarity matrix W , define +Dii := s +� +j +Wijσiσj, +(6) +where s = sgn(ξ). With D = diag(Dii), it is easy to verify that Sσ = 0 and, +therefore, it suffices to show that +P +� +inf +x⊥σ:∥x∥2=1 xTSx > 0 +� += 1 − o(1) +(7) +for Lemma 1 to hold. Using a similar methodology as in [19, Theorem 2], we obtain +the following complementary lemmas for the assortative and disassortative cases, +respectively. +Lemma 2. Let ξ > 0. With D defined via (6) with s = sgn(ξ) = +1 and S := +D + 11T − W , for all x⊥σ such that ∥x∥2 = 1, we have +xTSx ≥ min +i +Dii − ∥W − EW ∥2, +where EW is the expected aggregate similarity matrix conditioned on σ. +Proof. The expected similarity matrix for a symmetric HSBM admits the following +rank-2 decomposition: +EW = +�win + wout +2 +� +11T + +�win − wout +2 +� +σσT − winI, +where win = E[Wij|σi = σj], wout = E[Wij|σi ̸= σj] and I is the N × N identity +matrix. We can then write +xTSx = xTDx + +� +1Tx +�2 − xT(W − EW )x − xTEW x += xTDx + +� +1Tx +�2 − xT(W − EW )x +− +�win + wout +2 +� � +1Tx +�2 − +�win − wout +2 +� � +σT x +�2 + win||x||2 +2. +Because of the definition of the spectral norm and the facts that x⊥σ, and win, wout = +Θ( log N +N ) as shown in the proof of Proposition 1, we obtain +xTSx ≥ Dii∥x∥2 +2 + +� +1Tx +�2 +� +1 − win + wout +2 +� +− xT(W − EW )x +≥ min +i +Dii − ∥W − EW ∥2, +which proves the lemma. +Lemma 3. Let ξ < 0. With D defined via (6) with s = sgn(ξ) = −1 and S := +D + W , for all x⊥σ such that ∥x∥2 = 1, we have +xTSx ≥ min +i +Dii − ∥W − EW ∥2 − win||x||2 +2. +8 + +Proof. The claim follows from applying the techniques from the proof of Lemma 2 +on +xTSx = xTDx − xT(EW − W )x + xTEW x. +5.2 +Upper bound on ∥W − EW ∥2 +Let E be the set of all node sets (hyperedges) e ⊂ [N] having size d. We denote +by H(d, N, (fe)) = ([N], (we, e ∈ E)) a weighted d-uniform hypergraph, where each +hyperedge e ∈ E is independently assigned an fe distributed weight we. +Lemma 4 (Theorem 4, [30]). Let G ∼ H(d, N, (fe)) such that supp(fe) = [0, 1], and +denote by W the similarity matrix of G. Assume that maxe∈E Efe ≤ c0 log N +(N−1 +d−1) . Then +there exists a constant C = C(d, c0) > 0 such that +P +� +∥W − EW ∥2 ≤ C +� +log N +� +≥ 1 − O(N−11). +To apply Lemma 4, we note that the frequency of occurrences of hyperedge +e in G ∼ HSBM(N, M, d, (α(m) +t +)) over the M layers follows the distribution fe = +M−1 �M +m=1 Ber(p(m) +t +) with supp(fe) = [0, 1] and maxe∈E Efe ≤ α∗ log N +(N−1 +d−1) , where α∗ = +arg maxt,m α(m) +t +. The aggregate similarity matrix of G whose elements are normal- +ized between zero and one, W/M, is equal in distribution to the similarity matrix +of H ∼ H(d, N, (fe)). Finally, by Lemma 4, and by the definition of the spectral +norm, +P +� +∥W − EW ∥2 ≤ CM +� +log N +� +≥ 1 − O(N−11). +5.3 +Lower bound on Dii +Lemma 5. Let I > 1. Then there exists a constant ǫ > 0 dependent on model +parameters such that for all i ∈ [N], +P(Dii ≤ ǫ log N) = o(N−1). +Proof. We can write Dii in (6) as +Dii = s +� +m +� +j:j̸=i +� +e∈E:e∋i,j +A(m) +e +σiσj = s +� +m +� +e∈E:e∋i +A(m) +e +� +j∈e\{i} +σiσj. +We will split the sum on the right based on the community profile of the node set +e \ {i}. Denote by Td−1 the set of vectors t = (t−1, t+1) with nonnegative integer- +valued coordinates summing up to t−1 + t+1 = d − 1. For each t ∈ Td−1, denote by +Ei,t the collection of node sets e of size d such that e ∋ i and such that the number +of nodes j ∈ e\{i} with community membership σj = k equals tk for k = {−1, +1}. +Then for any node i with block membership σi = k and any e ∈ Ei,t, +� +j∈e\{i} +σiσj = tk − t−k. +9 + +Therefore, for any i with block membership σi = k, we find that +Dii = s +� +m +� +t∈Td−1 +� +e∈Ei,t +A(m) +e +(tk − t−k) = s +� +m +� +t∈Td−1 +(tk − t−k)Y (m) +i,t , +(8) +where Y (m) +i,t += � +e∈Ei,t A(m) +e +equals the number of hyperedges e in layer m that contain +i and for which the i-excluded community profile equals t ∈ Td−1. For any such e, +the full community profile equals t+ek, where ek is a basis vector for the coordinate +k ∈ {−1, 1}. Furthermore, the size of the set Ei,t equals Rk,t := |Ei,t| = +� N +2 −1 +tk +�� N +2 +t−k +� +. +It follows that the random variables Y (m) +i,t +are mutually independent and binomially +distributed according to Y (m) +i,t +∼ Bin(Rk,t, p(m) +t+ek). Fix λ ≥ 0. By independence and +the inequality 1 − x ≤ ex, we find that the moment-generating function of Dii is +bounded by +E +� +e−λDii� += +� +m +� +t∈Td−1 +E +� +e−sλ(tk−t−k)Y (m) +i,t +� += +� +m +� +t∈Td−1 +� +1 − p(m) +t+ek +� +1 − e−sλ(tk−t−k)��Rk,t +≤ +� +t∈Td−1 +� +m +exp +� +−Rk,tp(m) +t+ek +� +1 − e−sλ(tk−t−k)�� +. +Using the bounds (1 − j−1 +n ) nj +j! ≤ +�n +j +� +≤ nj +j! and the scaling assumption (1), we find +that +Rk,tp(m) +t+ek = (1 + o(1))2−(d−1) +�d − 1 +t−k +� +α(m) +t+ek log N. +(9) +We conclude that +E +� +e−λDii� +≤ e−(1+o(1))ψk(sλ) log N, +where, for x ∈ R, +ψk(x) := 2−(d−1) � +m +� +t∈Td−1 +�d − 1 +t−k +� +α(m) +t+ek +� +1 − e−x(tk−t−k)� +. +For the inner summation, taking t−k = r, we have that tk = d − 1 − r and αt+ek = +α(r,d−r) = α(d−r,r), thus giving +ψk(x) = 2−(d−1) � +m +d−1 +� +r=0 +�d − 1 +r +� +α(m) +(r,d−r) +� +1 − e−x(d−1−2r)� +=: ψ(x). +Note that the above expression is independent of the community of node i owing +to the symmetry inherent in our model. Markov’s inequality applied to the random +variable e−λDii then implies that for any ǫ > 0, +P(Dii ≤ ǫ log N) ≤ eλǫ log NEe−λDii ≤ Nλǫ−(1+o(1))ψ(sλ). +(10) +We note that ψ(x) is a concave function with ψ(0) = 0 and +ψ′(0) = 2−(d−1) � +m +d−1 +� +r=0 +�d − 1 +r +� +(d − 1 − 2r)α(m) +(r,d−r) = 2−(d−1)ξ, +10 + +where ξ is the assortativity defined by (4). Letting s = sgn(ξ), it follows that +I := sup +x∈R +ψ(x) = sup +λ≥0 +ψ(sλ), +where I is the information quantity defined by (5). Given d ≥ 2, we note that +−(d − 1 − 2r) is positive for at least one 0 ≤ r ≤ d − 1, and the corresponding term +in ψ(x) decreases to −∞ as x increases to ∞. On the other hand, −(d − 1 − 2r) +is negative for at least one 0 ≤ r ≤ d − 1, and the corresponding term decreases to +−∞ as x decreases to −∞. Moreover, all of the terms are bounded from above. It +follows that ψ(x) attains its supremum on R. If we assume that I > 1 and choose +a small enough ǫ > 0, then (10) implies that +P(Dii ≤ ǫ log N) ≤ Nλ∗ǫ−(1+o(1))I = o(N−1), +where λ∗ = arg maxλ≥0 ψ(sλ). +5.4 +Proof of Theorem 1 +Lemma 5 shows that Dii ≤ ǫ log N with probability o(N−1). Taking union bound +over i, we obtain mini∈[N] Dii ≤ ǫ log N with probability o(1). By Lemma 4, ∥W − +EW ∥2 ≤ CM√log N with probability 1−O(N−11). Moreover, win = Θ(N−1 log N) +as shown in the proof of Proposition 1. By Lemmas 2 and 3, we then have xTSx ≥ +ǫ log N − CM√log N − N−1 log N > 0 with probability o(1) for all x⊥σ such that +∥x∥2 = 1. Application of Lemma 1 then concludes the proof. +5.5 +Assortativity +In this section, we provide an alternate interpretation of assortativity in terms of the +aggregate similarity matrix W in the asymptotic regime. The following proposition +shows that ξ is a normalized expected difference between the number of hyperedges +shared between two nodes when they are of the same community and when they are +of different communities. +Proposition 1. Let win = E[Wij|σi = σj] and wout = E[Wij|σi ̸= σj]. Then, for +i ̸= j +win − wout = log N +2d−2N ξ + o +�log N +N +� +. +Proof. First, we note that for k ∈ {−1, 1} and i ̸= j +win = E[Wij | σi = k, σj = k] = +� +m=1 +� +t∈Td−2 +�N/2 − 2 +tk +��N/2 +t−k +� +p(m) +t+ek+ek, +and +wout = E[Wij | σi = k, σj = −k] = +� +m=1 +� +t∈Td−2 +�N/2 − 1 +tk +��N/2 − 1 +t−k +� +p(m) +t+ek+e−k. +Applying the bounds (1 − j−1 +n ) nj +j! ≤ +�n +j +� +≤ nj +j! and the scaling assumption (1), +win = log N +N +· (1 + o(1))(d − 1) +2d−1 +� +m=1 +� +t∈Td−2 +�d − 2 +t−k +� +α(m) +t+ek+ek = Θ +�log N +N +� +. +11 + +Similarly, wout = Θ(log N/N). By (6), for two communities of equal size we have +EDii = +� +j̸=i:σi=σj +win − +� +j:σi̸=σj +wout = N +2 (win − wout) − o(1). +(11) +Using (8) and (9), the expected value of Dii can also be written as +EDii = (1 + o(1))2−(d−1)ξ log N > 0 +which combined with (11) implies the statement of the proposition. +6 +Conclusions +In this work, we motivated and described the d-uniform multilayer inhomogeneous +HSBM. We studied the problem of exact community recovery for the model using +an SDP approach and the aggregate similarity matrix. For the symmetric case, our +analysis provided a sufficient condition in terms of the information quantity I for +community recovery. The generality of our model allows us to recover the sufficient +conditions for some earlier models proposed in the literature. +Our treatment of the problem brings to the fore numerous related questions +which are listed below: +- The assumption of symmetry on the parameters could be relaxed to make +the hyperedge probabilities depend on the community labels. Additionally, it +could be worthwhile to investigate asymmetry brought about by an imbalance +in the community sizes. +- This work provides sufficient conditions for exact-recovery based on the SDP +approach. +Necessary conditions for the multilayer HSBM model with the +knowledge of the similarity matrix can be obtained using a methodology sim- +ilar to [26] which will be addressed in a future publication. +- In this paper, the number of layers, M, is taken to be a constant. 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Advances in Neural Information Processing Sys- +tems (NeurIPS) (2006) +15 + diff --git a/QdFJT4oBgHgl3EQf2y0y/content/tmp_files/load_file.txt b/QdFJT4oBgHgl3EQf2y0y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6372769db7275633438ebdb85c3ab2cc970b8b02 --- /dev/null +++ b/QdFJT4oBgHgl3EQf2y0y/content/tmp_files/load_file.txt @@ -0,0 +1,495 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf,len=494 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='11657v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='ST] 27 Jan 2023 Multilayer hypergraph clustering using the aggregate similarity matrix Kalle Alaluusua1, Konstantin Avrachenkov2, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Vinay Kumar2, and Lasse Leskelä1 1 Aalto University, Espoo, Finland {kalle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='alaluusua, lasse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='leskela}@aalto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='fi 2 INRIA, Sophia Antipolis, Valbonne, France {vinay-kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='bindiganavile-ramadas, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='avrachenkov}@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='fr January 27, 2023 Abstract We consider the community recovery problem on a multilayer variant of the hypergraph stochastic block model (HSBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Each layer is associated with an independent realization of a d-uniform HSBM on N vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Given the aggregated number of hyperedges incident to each pair of vertices, represented using a similarity matrix, the goal is to obtain a partition of the N vertices into disjoint communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In this work, we investigate a semidefinite pro- gramming (SDP) approach and obtain information–theoretic conditions on the model parameters that guarantee exact recovery both in the assortative and the disassortative cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Keywords: hypergraph SBM, community detection, semidefinite program- ming, multilayer, clustering, planted partition MSC2020: 05C65, 05C80, 62H30, 90B15, 90C22, 90C35, 94A16 1 Introduction Traditional network data are observed as interactions between node pairs, repre- sented as a graph, or equivalently as an adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' More refined forms of network data may involve multiple types of higher-order interactions simultaneously involving multiple nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Such a data set can be viewed as a binary array A(m) e in- dexed by node sets e and positive integers m so that A(m) e = 1 indicates that an interaction of type m occurs among node set e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Such an array may also be viewed as a multilayer hypergraph where the entries of the array indicate the presence of hyperedge e in the m-th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Examples of such data could arise in a variety of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Researchers attending conferences, table reservations at restaurants, processor sharing etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Stochastic block models (SBMs) are a popular choice for gen- erative models with a community structure for such applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Hypergraphical stochastic block models (HSBMs) introduce hyperedges into SBMs, thus extending their modelling capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 1 We will next describe a generative model of a hypergraph with N ≥ 1 nodes and M ≥ 1 layers, where each hyperedge has size d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The set of nodes is divided into two communities of equal sizes (we assume N is even), and the resulting community structure, denoted by σ(N), is uniformly distributed on the set {(σ1, σ2, · · · , σN) ∈ {−1, +1}N : |{i : σi = +1}| = |{i : σi = −1}|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The community profile of a node set e is defined as a vector τ(e) = (τ−1(e), τ+1(e)), where τk(e) is equal to the number of nodes in e with community membership k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' We will then sample a multilayer hypergraph on node set [N] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' , N} so that each node set e ⊂ [N] having size d and community profile τ(e) = t is linked by a hyperedge in layer m with probability p(m) t = α(m) t log N �N−1 d−1 � , (1) independently of other node sets and layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' This scaling of the hyperedge probabil- ities ensures that the expected average degree of each node is Θ(log N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' References [2, 10, 26] show that the phase-transition for exact recovery occurs in this regime, and this regime is also critical for connectivity in general hypergraph models [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The d-uniform multilayer hypergraph can be represented as a binary array A = (A(m) e ) in which the entries are mutually independent Bernoulli random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The event {A(m) e = 1} has probability p(m) t when τ(e) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' To indicate that A is sampled from the model, we abbreviate A ∼ HSBM(N, M, d, (α(m) t )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' We will focus on a symmetric model in which α(m) (r,d−r) = α(m) (d−r,r) for all 0 ≤ r ≤ d and all m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' This means that the presence of the hyperedge depends only on the number of nodes of each community rather than the community label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The problem of community detection is to output an estimate ˆσ(N) of the un- derlying node communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The estimate is said to achieve exact recovery if, lim N→∞ P � ˆσ(N) ∈ {±σ(N)} � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In this work, the main focus is to study the community recovery problem based on a layer-aggregated similarity matrix Wij = � m W (m) ij where (W (m) ij ) =: W (m) is a zero-diagonal matrix with off-diagonal entries W (m) ij = � e:e∋i,j A(m) e counting the number of hyperedges in layer m incident to nodes i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Commu- nity recovery based on the similarity matrix W instead of the full data set A is motivated by two aspects: privacy and computational tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' For example, in an application where N individuals visit M restaurants, providing the full hyper- graph could reveal the frequency a particular individual visits a restaurant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' This could violate the privacy of the individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' On the other hand, providing the sim- ilarity matrix obfuscates such individual information, since information regarding the restaurants that are visited is not revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Additionally, the similarity matrix provides a compact matrix representation of the hypergraph that is easier to ma- nipulate using matrix algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Nevertheless, it is clear that the similarity matrix contains less information than the complete hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 2 In this work, we investigate a semidefinite programming (SDP) approach for solving the community detection problem using W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Our main result in Section 3 provides an information quantity that characterizes the performance of the SDP approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' To be more specific, it gives a sufficient condition relating the different parameters of the model for exact recovery using the SDP technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In Section 2, we provide recent literature in the area and highlight the key differences in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In Section 3, we describe the semidefinite programming algorithm and state our main result concerning the information theoretic conditions on the parameters of the model that guarantee exact recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Section 4 contains some simulation results of our algorithm on synthetic data generated using the HSBM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The proofs of our results are provided in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Section 6 concludes the paper and provides some future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 2 Related work The usual paradigm for theoretical research in community detection is to first pro- pose a generative model that captures the application (data) as a graph or a net- work, followed by the analysis of a clustering algorithm for the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Community recovery algorithms on stochastic block models (SBMs) have attracted considerable attention in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' A comprehensive survey of the field is provided in [1], see also a review of recent results on graph clustering in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Our interest in this work is on multilayer hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' We first provide a brief survey of recent work in clustering on multilayer networks followed by hypergraph SBMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' A seminal work for multilayer networks is the review article [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Subsequently, in [34], the authors consider estimating the membership for each individual layer in a multilayer SBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In the special case that memberships do not change, their method works on the normalized aggregate adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The authors in [31] establish that increasing the number of network layers guarantees consistent com- munity recovery (by a least squares estimator) even when the graphs get sparser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' SBMs with general interactions allow an alternate model for multilayer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' These are studied in [6] where the authors address the community recovery problem using aggregate adjacency matrix as well as the full graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The authors in [3] study Bayesian community recovery in a regime where both the number of nodes and the number of layers increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' With regards to literature on hypergraphs, the hypergraph stochastic block model (HSBM) was first introduced by Ghoshdastidar and Dukkipati in [13] to capture higher order interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' They also show strong consistency of spectral methods in dense uniform hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In subsequent works [14, 15, 16], they ex- tend their results to partial recovery in sparse non-uniform hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Some other works on the spectral algorithm for hypergraphs include [2, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The recent work by Zhang and Tan [36] considers the general d-uniform HSBM with multiple communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' They establish a sharp phase transition for exact recov- ery when the knowledge of the whole hypergraph is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' They recover results from several previous works including [9, 10, 26, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In the process of solving the exact re- covery problem, they do show almost exact recovery using only the similarity matrix through a spectral approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Another general hypergraph model with theoretical guarantees by [37] employs a latent space representation of nodes to cover HSBM, non-uniform hypergraphs, and hypergraphs with heterogeneity among nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 3 Some of the other approaches used in the literature for the community recovery problem on hypergraphs are based on modularity [21, 22, 23, 28, 29], tensor decom- position [24, 37], random walk based methods [33, 35, 38], variational inference [8], and approximate message passing [4, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In this work, we investigate the problem of exact recovery on the HSBM through the lens of semidefinite programming (SDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Our work is closest in spirit to [12] and [26] that discuss the SDP approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The SDP formulation (described in Section 3) arises as a relaxation of the computationally hard procedure of finding a nodes’ par- tition with minimum number of edges crossing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In [26], the authors show that for a d-uniform homogeneous HSBM with two equal-sized and symmetric communities, exact recovery using the full hypergraph shows a sharp phase transition behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' They go on to propose a ‘truncate-and-relax’ algorithm that utilizes the structure of the similarity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' An SDP approach then guarantees exact recovery with high probability, albeit in a parameter regime which is slightly sub-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' This gap is bridged in [12] who consider the community recovery problem with the knowledge of only the similarity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Below, we highlight the differences from these previous works: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The authors in both [12] and [26] consider the homogeneous model in which the hyperedge parameters take just two values corresponding to all nodes of a hyperedge being in the same community, and at least one of them being in a different community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Related works with the same assumption include [2, 11, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In this work, we allow for hyperedge parameters to depend on the number of nodes of each community in the hyperedge resulting in an inhomogeneous HSBM as in [36], albeit with the symmetric assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' A similar assumption is present in other works such as [13, 15] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Much of the earlier work assumes that the data is assortative or homophilic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' nodes in the same community are more likely to be adjacent to each other than to nodes in different communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Our results incorporate the disassortative or heterophilic case where the opposite is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' This could be of interest for some applications: reputation of a research institute is partly assessed based on the amount of collaboration with experts from external institutions (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' [18]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' so, one might expect certain research networks to be disassortative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Our model targets multilayer HSBMs that can be seen as a generalization of previous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Moreover, these layers could individually be assortative or disassortative which could then capture a plethora of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 3 Algorithm and main results A first approach to obtain an estimate of the node communities given the similarity matrix W is to solve the min-bisection problem: max � i,j Wijxixj subject to x ∈ {±1}N, 1Tx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' (2) This formulation assumes that the data is assortative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In the disassortative case the opposite is true, and we replace the maximization in (2) with minimization, or equivalently change the sign of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 4 However, the problem in (2) is known to be NP-hard in general (see [25]) and therefore, we consider a semidefinite programming (SDP) relaxation of it whose generalisation is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Algorithm 1 [SDP] Input: N × N similarity matrix W and s ∈ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Output: Community estimate ˆσ 1: Solve the following optimization problem: maximize � ij∈([N] 2 ) sWijXij subject to � ij∈([N] 2 ) Xij = 0, Xii = 1 for all i ∈ [N] X ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' (3) 2: Let X∗ ∈ RN×N be the optimal solution of (3) and let it have an eigendecom- position X∗ = �N i=1 λivivT i with λ1 ≥ λ2 ≥ · · · ≥ λN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 3: Output ˆσ = sgn(v1) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' An alternate approach in the disassortative case is to consider the com- plement of the hypergraph, which is assortative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' A similarity matrix of the comple- ment is equivalent to �N−2 d−2 � 11T − W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' However, owing to our scaling assumption in (1), the resulting similarity matrix of the complement is no longer in the same regime, and requires a different approach to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' To capture the level of assortativity of our model, we define the following quan- tity, accordingly referred to as the assortativity ξ := M � m=1 d−1 � r=0 �d − 1 r � (d − 1 − 2r)α(m) (r,d−r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' (4) The summation over the different layers implies that the full hypergraph can be assortative even if individual layers W (m) are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Table 1 displays this formula for selected values of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='5, it is shown that ξ is a normalized expected difference between the number of hyperedges shared between two nodes when they are of the same community and when they are of different communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Therefore, a model is said to be assortative if ξ > 0, and disassortative if ξ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In order to state our main result concerning the performance of Algorithm 1, we will need the following information quantity I = sup λ∈R M � m=1 d−1 � r=0 2−(d−1) �d − 1 r � α(m) (r,d−r) � 1 − e−λ(d−1−2r)� (5) To be concise, we omit the dependence on model parameters in the definition of both ξ and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' We now state the main theorem that characterizes Algorithm 1 and provides a sufficient condition for exact recovery on the aggregate similarity matrix of a symmetric multilayer HSBM using the SDP approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 5 Table 1: Assortativity ξ in a d-uniform HSBM with M layers, where we denote αt = �M m=1 α(m) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' d ξ 2 α(0,2) − α(1,1) 3 2α(0,3) − 2α(1,2) 4 3α(0,4) − 3α(2,2) 5 4α(0,5) + 4α(1,4) − 8α(2,3) 6 5α(0,6) + 10α(1,5) − 5α(2,4) − 10α(3,3) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Suppose A ∼ HSBM(N, M, d, (α(m) t )), and let W be the aggregate similarity matrix of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' When I > 1, Algorithm 1 with s = sgn(ξ) achieves exact recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The proof of Theorem 1 is provided in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Taking M = 1 with parameters α(r,d−r) = � α for r = 0 and r = d β for 1 ≤ r ≤ d − 1 reduces to a homogeneous model that has been studied earlier in the assortative case with ξ = (d − 1)(α − β) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In this setting Kim, Bandeira, and Goemans [26] showed that the SDP algorithm does not achieve exact recovery when I < 1 and Gaudio and Joshi [12] proved that the SDP algorithm achieves exact recovery when I > 1, as conjectured in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 4 Numerical illustrations We perform numerical simulations to demonstrate the effect of the number of ob- served hypergraph layers on the classification accuracy of Algorithm 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The syn- thetic data is sampled from a 4-uniform HSBM(50, M, 4, (α(m) t )), with 1 ≤ M ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' We let the hypergraph layers be identically distributed giving α(m) t =: αt for all m ∈ [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Table 2 provides the parameter values used in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' These values are chosen such that the expected degree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' the number of hyperedges a node is incident to, is the same in both the homogeneous and the inhomogeneous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The associated hyperedge probabilities are computed from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Table 2: Four sets of parameter values (αt) used in simulation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Homogeneous Inhomogeneous t Assortat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Disassort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Assortat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Disassort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' (4, 0) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='7 (3, 1) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='4 (2, 2) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='3 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='7 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='8 To evaluate the accuracy of our estimate given σ, we use the Hubert-Arabie adjusted Rand index (AR) [17, 20], which is a measure of similarity between two 1Source code: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='com/kalaluusua/ 6 community assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The index is equal to 1 when the assignments are iden- tical, and 0 when they are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' For each simulated hypergraph, we also compute the classification error (CE), which we define as the fraction of misclassi- fied nodes N−1 min{Ham(ˆσ, σ), Ham(ˆσ, −σ)}, where Ham denotes the Hamming distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The results (averaged over five different random seed initializations) are depicted in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Table 3: Classification error (CE) and Adjusted Rand index (AR) of the community assignment estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Homogeneous Inhomogeneous Assortat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Disassort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Assortat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Disassort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' M |ξ| I CE AR CE AR |ξ| I CE AR CE AR 1 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='000 Based on the I-values, we expect that the community detection performance im- proves as M increases and is the same for the assortative and disassortative cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' This is precisely what Table 3 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Moreover, the larger I-values of the homo- geneous case lead us to expect an overall better performance in comparison to the homogeneous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Surprisingly, this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' An inspection of the ξ-values reveals a larger level of (dis)assortativity in the inhomogeneous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In small to moderate hypergraph sizes, we suspect that the level of assortativity may predict the detection performance of Algorithm 1 better than the information-theoretic quantity I, whose effect is more profound in the asymptotic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 5 Analysis of the algorithm In this section, we provide a detailed proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' We follow the procedure of [12, 19, 26] to analyze the SDP framework, and extend it to a more general model HSBM(N, M, d, (α(m) t )) that addresses multiple layers, disassortativity, and (symmetric) inhomogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' An outline of this section is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='1 constructs a dual certificate strategy to solve the SDP in (3) and specializes it to the assortative and disas- sortative cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Bounds on certain quantities that arise as part of this strategy are provided in Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='4 puts the parts together to com- plete the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Finally, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='5, we comment on the assorta- tive/disassortative nature of the model and its manifestation in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='1 SDP analysis To begin, we state a sufficient condition for optimality of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' This is a corollary of [12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='2] that asserts strong duality for (3) with s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Fix s ∈ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Suppose there is a diagonal matrix D ∈ RN×N and ν ∈ R such that the matrix S := D + ν11T −sW is positive semidefinite, its second 7 smallest eigenvalue λN−1(S) is strictly positive, and Sσ = 0, then X∗ = σσT is the unique optimal solution to (3) (with the same s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' For the HSBM(N, M, d, (α(m) t )) model with node communities σ and the aggre- gate similarity matrix W , define Dii := s � j Wijσiσj, (6) where s = sgn(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' With D = diag(Dii), it is easy to verify that Sσ = 0 and, therefore, it suffices to show that P � inf x⊥σ:∥x∥2=1 xTSx > 0 � = 1 − o(1) (7) for Lemma 1 to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Using a similar methodology as in [19, Theorem 2], we obtain the following complementary lemmas for the assortative and disassortative cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Let ξ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' With D defined via (6) with s = sgn(ξ) = +1 and S := D + 11T − W , for all x⊥σ such that ∥x∥2 = 1, we have xTSx ≥ min i Dii − ∥W − EW ∥2, where EW is the expected aggregate similarity matrix conditioned on σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The expected similarity matrix for a symmetric HSBM admits the following rank-2 decomposition: EW = �win + wout 2 � 11T + �win − wout 2 � σσT − winI, where win = E[Wij|σi = σj], wout = E[Wij|σi ̸= σj] and I is the N × N identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' We can then write xTSx = xTDx + � 1Tx �2 − xT(W − EW )x − xTEW x = xTDx + � 1Tx �2 − xT(W − EW )x − �win + wout 2 � � 1Tx �2 − �win − wout 2 � � σT x �2 + win||x||2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Because of the definition of the spectral norm and the facts that x⊥σ, and win, wout = Θ( log N N ) as shown in the proof of Proposition 1, we obtain xTSx ≥ Dii∥x∥2 2 + � 1Tx �2 � 1 − win + wout 2 � − xT(W − EW )x ≥ min i Dii − ∥W − EW ∥2, which proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Let ξ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' With D defined via (6) with s = sgn(ξ) = −1 and S := D + W , for all x⊥σ such that ∥x∥2 = 1, we have xTSx ≥ min i Dii − ∥W − EW ∥2 − win||x||2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The claim follows from applying the techniques from the proof of Lemma 2 on xTSx = xTDx − xT(EW − W )x + xTEW x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='2 Upper bound on ∥W − EW ∥2 Let E be the set of all node sets (hyperedges) e ⊂ [N] having size d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' We denote by H(d, N, (fe)) = ([N], (we, e ∈ E)) a weighted d-uniform hypergraph, where each hyperedge e ∈ E is independently assigned an fe distributed weight we.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Lemma 4 (Theorem 4, [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Let G ∼ H(d, N, (fe)) such that supp(fe) = [0, 1], and denote by W the similarity matrix of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Assume that maxe∈E Efe ≤ c0 log N (N−1 d−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Then there exists a constant C = C(d, c0) > 0 such that P � ∥W − EW ∥2 ≤ C � log N � ≥ 1 − O(N−11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' To apply Lemma 4, we note that the frequency of occurrences of hyperedge e in G ∼ HSBM(N, M, d, (α(m) t )) over the M layers follows the distribution fe = M−1 �M m=1 Ber(p(m) t ) with supp(fe) = [0, 1] and maxe∈E Efe ≤ α∗ log N (N−1 d−1) , where α∗ = arg maxt,m α(m) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The aggregate similarity matrix of G whose elements are normal- ized between zero and one, W/M, is equal in distribution to the similarity matrix of H ∼ H(d, N, (fe)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Finally, by Lemma 4, and by the definition of the spectral norm, P � ∥W − EW ∥2 ≤ CM � log N � ≥ 1 − O(N−11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='3 Lower bound on Dii Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Let I > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Then there exists a constant ǫ > 0 dependent on model parameters such that for all i ∈ [N], P(Dii ≤ ǫ log N) = o(N−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' We can write Dii in (6) as Dii = s � m � j:j̸=i � e∈E:e∋i,j A(m) e σiσj = s � m � e∈E:e∋i A(m) e � j∈e\\{i} σiσj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' We will split the sum on the right based on the community profile of the node set e \\ {i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Denote by Td−1 the set of vectors t = (t−1, t+1) with nonnegative integer- valued coordinates summing up to t−1 + t+1 = d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' For each t ∈ Td−1, denote by Ei,t the collection of node sets e of size d such that e ∋ i and such that the number of nodes j ∈ e\\{i} with community membership σj = k equals tk for k = {−1, +1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Then for any node i with block membership σi = k and any e ∈ Ei,t, � j∈e\\{i} σiσj = tk − t−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 9 Therefore, for any i with block membership σi = k, we find that Dii = s � m � t∈Td−1 � e∈Ei,t A(m) e (tk − t−k) = s � m � t∈Td−1 (tk − t−k)Y (m) i,t , (8) where Y (m) i,t = � e∈Ei,t A(m) e equals the number of hyperedges e in layer m that contain i and for which the i-excluded community profile equals t ∈ Td−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' For any such e, the full community profile equals t+ek, where ek is a basis vector for the coordinate k ∈ {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Furthermore, the size of the set Ei,t equals Rk,t := |Ei,t| = � N 2 −1 tk �� N 2 t−k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' It follows that the random variables Y (m) i,t are mutually independent and binomially distributed according to Y (m) i,t ∼ Bin(Rk,t, p(m) t+ek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Fix λ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' By independence and the inequality 1 − x ≤ ex, we find that the moment-generating function of Dii is bounded by E � e−λDii� = � m � t∈Td−1 E � e−sλ(tk−t−k)Y (m) i,t � = � m � t∈Td−1 � 1 − p(m) t+ek � 1 − e−sλ(tk−t−k)��Rk,t ≤ � t∈Td−1 � m exp � −Rk,tp(m) t+ek � 1 − e−sλ(tk−t−k)�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Using the bounds (1 − j−1 n ) nj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' ≤ �n j � ≤ nj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' and the scaling assumption (1), we find that Rk,tp(m) t+ek = (1 + o(1))2−(d−1) �d − 1 t−k � α(m) t+ek log N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' (9) We conclude that E � e−λDii� ≤ e−(1+o(1))ψk(sλ) log N, where, for x ∈ R, ψk(x) := 2−(d−1) � m � t∈Td−1 �d − 1 t−k � α(m) t+ek � 1 − e−x(tk−t−k)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' For the inner summation, taking t−k = r, we have that tk = d − 1 − r and αt+ek = α(r,d−r) = α(d−r,r), thus giving ψk(x) = 2−(d−1) � m d−1 � r=0 �d − 1 r � α(m) (r,d−r) � 1 − e−x(d−1−2r)� =: ψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Note that the above expression is independent of the community of node i owing to the symmetry inherent in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Markov’s inequality applied to the random variable e−λDii then implies that for any ǫ > 0, P(Dii ≤ ǫ log N) ≤ eλǫ log NEe−λDii ≤ Nλǫ−(1+o(1))ψ(sλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' (10) We note that ψ(x) is a concave function with ψ(0) = 0 and ψ′(0) = 2−(d−1) � m d−1 � r=0 �d − 1 r � (d − 1 − 2r)α(m) (r,d−r) = 2−(d−1)ξ, 10 where ξ is the assortativity defined by (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Letting s = sgn(ξ), it follows that I := sup x∈R ψ(x) = sup λ≥0 ψ(sλ), where I is the information quantity defined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Given d ≥ 2, we note that −(d − 1 − 2r) is positive for at least one 0 ≤ r ≤ d − 1, and the corresponding term in ψ(x) decreases to −∞ as x increases to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' On the other hand, −(d − 1 − 2r) is negative for at least one 0 ≤ r ≤ d − 1, and the corresponding term decreases to −∞ as x decreases to −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Moreover, all of the terms are bounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' It follows that ψ(x) attains its supremum on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' If we assume that I > 1 and choose a small enough ǫ > 0, then (10) implies that P(Dii ≤ ǫ log N) ≤ Nλ∗ǫ−(1+o(1))I = o(N−1), where λ∗ = arg maxλ≥0 ψ(sλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='4 Proof of Theorem 1 Lemma 5 shows that Dii ≤ ǫ log N with probability o(N−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Taking union bound over i, we obtain mini∈[N] Dii ≤ ǫ log N with probability o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' By Lemma 4, ∥W − EW ∥2 ≤ CM√log N with probability 1−O(N−11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Moreover, win = Θ(N−1 log N) as shown in the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' By Lemmas 2 and 3, we then have xTSx ≥ ǫ log N − CM√log N − N−1 log N > 0 with probability o(1) for all x⊥σ such that ∥x∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Application of Lemma 1 then concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content='5 Assortativity In this section, we provide an alternate interpretation of assortativity in terms of the aggregate similarity matrix W in the asymptotic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The following proposition shows that ξ is a normalized expected difference between the number of hyperedges shared between two nodes when they are of the same community and when they are of different communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Let win = E[Wij|σi = σj] and wout = E[Wij|σi ̸= σj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Then, for i ̸= j win − wout = log N 2d−2N ξ + o �log N N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' First, we note that for k ∈ {−1, 1} and i ̸= j win = E[Wij | σi = k, σj = k] = � m=1 � t∈Td−2 �N/2 − 2 tk ��N/2 t−k � p(m) t+ek+ek, and wout = E[Wij | σi = k, σj = −k] = � m=1 � t∈Td−2 �N/2 − 1 tk ��N/2 − 1 t−k � p(m) t+ek+e−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Applying the bounds (1 − j−1 n ) nj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' ≤ �n j � ≤ nj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' and the scaling assumption (1), win = log N N (1 + o(1))(d − 1) 2d−1 � m=1 � t∈Td−2 �d − 2 t−k � α(m) t+ek+ek = Θ �log N N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 11 Similarly, wout = Θ(log N/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' By (6), for two communities of equal size we have EDii = � j̸=i:σi=σj win − � j:σi̸=σj wout = N 2 (win − wout) − o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' (11) Using (8) and (9), the expected value of Dii can also be written as EDii = (1 + o(1))2−(d−1)ξ log N > 0 which combined with (11) implies the statement of the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 6 Conclusions In this work, we motivated and described the d-uniform multilayer inhomogeneous HSBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' We studied the problem of exact community recovery for the model using an SDP approach and the aggregate similarity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' For the symmetric case, our analysis provided a sufficient condition in terms of the information quantity I for community recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The generality of our model allows us to recover the sufficient conditions for some earlier models proposed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Our treatment of the problem brings to the fore numerous related questions which are listed below: The assumption of symmetry on the parameters could be relaxed to make the hyperedge probabilities depend on the community labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Additionally, it could be worthwhile to investigate asymmetry brought about by an imbalance in the community sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' This work provides sufficient conditions for exact-recovery based on the SDP approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Necessary conditions for the multilayer HSBM model with the knowledge of the similarity matrix can be obtained using a methodology sim- ilar to [26] which will be addressed in a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' In this paper, the number of layers, M, is taken to be a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' However, we expect that the analysis goes through when M grows slowly with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' The analysis of the SDP algorithm used here relies on the fact that there are just two communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Extensions to a larger number of communities is a question worthy of investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' 12 References [1] Abbé, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=': Community detection and stochastic block models: Recent develop- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Journal of Machine Learning Research 18, 1–86 (2018) [2] Ahn, K.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=': Learning with hypergraphs: Clustering, classification, and embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} +page_content=' Advances in Neural Information Processing Sys- tems (NeurIPS) (2006) 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdFJT4oBgHgl3EQf2y0y/content/2301.11657v1.pdf'} diff --git a/RdAyT4oBgHgl3EQft_l_/content/tmp_files/2301.00605v1.pdf.txt b/RdAyT4oBgHgl3EQft_l_/content/tmp_files/2301.00605v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e8bd9a7faf4d28633e65a6e639f222424e914adf --- /dev/null +++ b/RdAyT4oBgHgl3EQft_l_/content/tmp_files/2301.00605v1.pdf.txt @@ -0,0 +1,1551 @@ +arXiv:2301.00605v1 [math.AP] 2 Jan 2023 +Regularity of Time-Periodic Solutions to Autonomous +Semilinear Hyperbolic PDEs +Irina Kmit ∗ +Lutz Recke † +Abstract +This paper concerns autonomous boundary value problems for 1D semilinear hyper- +bolic PDEs. For time-periodic classical solutions, which satisfy a certain non-resonance +condition, we show the following: If the PDEs are continuous with respect to the space +variable x and C∞-smooth with respect to the unknown function u, then the solution is +C∞-smooth with respect to the time variable t, and if the PDEs are C∞-smooth with +respect to x and u, then the solution is C∞-smooth with respect to t and x. The same is +true for appropriate weak solutions. +Moreover, we show examples of time-periodic functions, which do not satisfy the non- +resonance condition, such that they are weak, but not classical solutions, and such that +they are classical solutions, but not C∞-smooth, neither with respect to t nor with respect +to x, even if the PDEs are C∞-smooth with respect to x and u. +For the proofs we use Fredholm solvability properties of linear time-periodic hyperbolic +PDEs and a result of E. N. Dancer about regularity of solutions to abstract equivariant +equations. +Keywords: 1D semilinear hyperbolic PDEs, autonomous boundary value problems, solution +regularity, non-resonance condition, Fredholm solvability +1 +Introduction +In this paper we consider time-periodic solutions to boundary value problems for 1D semi- +linear first-order hyperbolic systems of the type +∂tuj(t, x) + aj(x)∂xuj(t, x) = fj(x, u(t, x)) +and 1D semilinear second-order hyperbolic equations of the type +∂2 +t u(t, x) − a(x)2∂2 +xu(t, x) = f(x, u(t, x), ∂tu(t, x), ∂xu(t, x)). +Let us formulate our results concerning first-order systems. Specifically, we consider 2 × 2 +systems with reflection boundary conditions and time-periodic solutions with period one, i.e. +solutions u = (u1, u2) to problems of the type (for t ∈ R, x ∈ [0, 1]) +∂tuj(t, x) + aj(x)∂xuj(t, x) = fj(x, u(t, x)), j = 1, 2, +u1(t, 0) = r1u2(t, 0), u2(t, 1) = r2u1(t, 1), +u(t + 1, x) = u(t, x). +(1.1) +∗Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, D-10099 Berlin. On leave +from the Institute for Applied Problems of Mechanics and Mathematics, Ukrainian National Academy of +Sciences. E-mail: kmit@mathematik.hu-berlin.de +†Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, D-10099 Berlin. E-mail: +recke@mathematik.hu-berlin.de +1 + +We suppose that for j = 1, 2 +aj ∈ C([0, 1]), rj ∈ R, aj(x) ̸= 0 and a1(x) ̸= a2(x) for all x ∈ [0, 1], +∂k +u1∂l +u2fj exist and belong to C([0, 1] × R2) for all k, l ∈ N ∪ {0}. +(1.2) +Further, we write (for t ∈ R, x, y ∈ [0, 1], and j = 1, 2) +αj(x, y) := +� y +x +dz +aj(z). +Theorem 1.1 Suppose that (1.2) is fulfilled, and let u ∈ C(R × [0, 1]; R2) satisfy one of the +conditions +� 1 +0 +�∂u1f1(x, u(t − α1(x, 1), x)) +a1(x) +− ∂u2f2(x, u(t − α2(x, 1), x)) +a2(x) +� +dx +̸= ln |r1r2| for all t ∈ R +(1.3) +and +� 1 +0 +�∂u1f1(x, u(t + α1(0, x), x)) +a1(x) +− ∂u2f2(x, u(t + α2(0, x), x)) +a2(x) +� +dx +̸= ln |r1r2| for all t ∈ R. +(1.4) +Then the following is true: +(i) If u satisfies the boundary and the periodicity conditions in (1.1) and if there exists a +sequence u1, u2, . . . ∈ C1(R × [0, 1]; R2) such that, for j = 1, 2, +|un +j (t, x) − uj(t, x)| + |∂tun +j (t, x) + aj(x)∂xun +j (t, x) − fj(x, u(t, x))| → 0 for n → ∞ +uniformly with respect to (t, x) ∈ R×[0, 1], then u is a classical solution to (1.1), in particular, +u is C1-smooth. Moreover, all partial derivatives ∂k +t u, k ∈ N, exist and belong to C(R × +[0, 1]; R2). +(ii) If u is a classical solution to (1.1) and if the functions aj and fj, j = 1, 2, are C∞- +smooth, then u is C∞-smooth also. +Now we formulate our results concerning time-periodic solutions to second-order equations +subjected to one Dirichlet and one Neumann boundary conditions. More precisely, we consider +problems of the type (for t ∈ R and x ∈ [0, 1]) +∂2 +t u(t, x) − a(x)2∂2 +xu(t, x) = f(x, u(t, x), ∂tu(t, x), ∂xu(t, x)), +u(t, 0) = 0, ∂xu(t, 1) = 0, +u(t + 1, x) = u(t, x). +(1.5) +We assume that +a ∈ C1([0, 1]), a(x) ̸= 0 for all x ∈ [0, 1], +∂j +2∂k +3∂l +4f exist and belong to C([0, 1] × R3) for all j, k, l ∈ N ∪ {0}, +(1.6) +where ∂jf denotes the derivative of the function f with respect to its j-th argument. More +precisely, if f = f(x, u, v, w), then ∂2f is the derivative with respect to u, ∂3f is the derivative +2 + +with respect to v, and ∂4f is the derivative with respect to w. Further, we write (for t ∈ R, +x, y ∈ [0, 1], and u ∈ C([0, 1] × R2)) +α(x, y) := +� y +x +dz +a(z), +b+(t, x, u) := ∂3f(x, u(t, x), ∂tu(t, x), ∂xu(t, x)) + ∂4f(x, u(t, x), ∂tu(t, x), ∂xu(t, x)) +a(x) +, +b−(t, x, u) := ∂3f(x, u(t, x), ∂tu(t, x), ∂xu(t, x)) − ∂4f(x, u(t, x), ∂tu(t, x), ∂xu(t, x)) +a(x) +. +The weak formulation of the second-order problem (1.5) (see Theorem 1.1 (i)), which will +be used, is slightly more complicated than that for the first-order problem (1.1), and, in fact, +it is a technical tool only. We, therefore, will not include in Theorem 1.2 below a regularity +result for weak solutions to (1.5), but only regularity results for classical solutions to (1.5). +Theorem 1.2 Suppose that (1.6) is fulfilled. Let u ∈ C2(R × [0, 1]) be a classical solution to +(1.5), and suppose that it satisfies one of the conditions +� 1 +0 +b+(t + α(x, 1), x, u) − b−(t − α(x, 1), x, u) +a(x) +dx ̸= 0 for all t ∈ R +(1.7) +and +� 1 +0 +b+(t − α(0, x), x, u) − b−(t + α(0, x), x, u) +a(x) +dx ̸= 0 for all t ∈ R. +(1.8) +Then the following is true: +(i) All partial derivatives ∂k +t u, k ∈ N, exist and belong to C(R × [0, 1]). +(ii) If the functions a and f are C∞-smooth, then u is C∞-smooth also. +Remark 1.3 In most applications, solutions to problems of the type (1.1) and (1.5) are +found as a result of Hopf bifurcations from stationary solutions [7, 8, 12] and by continuation +of such solutions with respect to parameters [9]. +Remark 1.4 The paper [3] of J. K. Hale and J. Scheurle concerns smoothness with respect +to time of solutions to abstract autonomous semilinear evolution equations if those solutions +are bounded and close to be constant in time. The results are applied to slightly damped +nonlinear wave equations in 1D with constant coefficients, namely +∂2 +t u(t, x) − ∂2 +xu(t, x) + δ∂tu(t, x) − u(t, x) − λu(t, x) = f(u(t, x)), +(1.9) +subjected to homogeneous Dirichlet boundary conditions. The function f : R → R is smooth +and of order o(|u|) for u → 0, λ is small, δ is positive and small. It is shown that sufficiently +small bounded solutions are smooth with respect to time. +Let us compare this with Theorem 1.2: On one hand, the equation in our problem (1.5) +is more general than equation (1.9). Moreover, in Theorem 1.2 we do not suppose that the +solution is close to be constant in time. On the other hand, our Theorem 1.2 concerns time- +periodic solutions only, not general bounded ones. Anyway, if one applies definitions of the +functions b+ and b− to equation (1.9), then +b+(t, x, u) = b−(t, x, u) = −δ. +Hence, the assumption δ > 0 of [3] implies that the assumptions of Theorem 1.2 are fulfilled. +3 + +Remark 1.5 Let us consider Theorem 1.2 in the special case of a nonlinear wave equation +which is slightly more general than (1.9), namely +∂2 +t u(t, x) − a(x)2∂2 +xu(t, x) = β1(x)∂tu(t, x) + β2(x)∂xu(t, x) + f(x, u(t, x)). +If one applies definitions of b+ and b− to this equation, then +b+(t, x, u) = β1(x) + β2(x) +a(x) , b−(t, x, u) = β1(x) − β2(x) +a(x) . +Hence, the conditions (1.7) and (1.8) are identical, and they are satisfied for any u if and +only if +� 1 +0 +β1(x) +a(x) dx ̸= 0. +Remark 1.6 We do not know if Theorems 1.1 and 1.2 can be generalized to cases of more +than one space dimension. The reason is that linear autonomous hyperbolic partial differential +operators with one space dimension essentially differ from those with more than one space +dimension: They satisfy the spectral mapping property in Lp-spaces [16] and, which is more +important for applications to nonlinear problems, in C-spaces [14]. Moreover, they generate +Riesz bases (see, e.g. [2, 15]). This is not the case, in general, if the space dimension is +larger than one (see the counter-example of M. Renardy in [17]). Therefore, the question of +Fredholmness of those operators in appropriate spaces of time-periodic functions is highly +difficult. +Remark 1.7 Theorem 1.1 can be generalized to problems for n × n first-order hyperbolic +systems of the type (with natural numbers m < n) +∂tuj(t, x) + aj(x)∂xuj(t, x) = fj(x, u(t, x)), j ≤ n, +uj(t, 0) = +n +� +k=m+1 +rjkuk(t, 0), j ≤ m, +uj(t, 1) = +m +� +k=1 +rjkuk(t, 1), m < j ≤ n. +(1.10) +Here, instead of non-resonant conditions (1.3) and (1.4), one considers the following sufficient +conditions +max +s,t∈[0,1] max +j≤m +n +� +k=m+1 +m +� +l=1 +|rjkrkl| exp +� 1 +0 +�∂ukfk(x, u(t, x)) +ak(x) +− ∂ujfj(x, u(s, x)) +aj(x) +� +dx < 1 +and +max +s,t∈[0,1] max +m 1. +Then +|r1r2c1(t + α2(0, 1), 1, 0)c2(t, 0, 1)| ≥ 1 + c− +2 +≥ 1. +Equation (2.16) is equivalent to +v2(t, 0) += +v2(t − α1(1, 0) − α2(0, 1), 0) +r1r2c1(t − α1(1, 0), 1, 0)c2(t − α1(1, 0) − α2(0, 1), 0, 1) +− +f1(t − α1(1, 0), 0) +r1c1(t − α1(1, 0) − α2(0, 1), 1, 0) +− +f2(t − α1(1, 0) − α2(0, 1), 0) +r1r2c1(t − α1(1, 0), 1, 0)c2(t − α1(1, 0) − α2(0, 1), 0, 1) . +This equation is of the type (2.19) again, but now with +∥ �C∥L(Cper(R)) ≤ +2 +1 + c− +< 1. +Hence, we can proceed as above. +Similarly one deals with the case if condition (1.4) is satisfied. Then equation (2.18) is +uniquely solvable, and so is equation (2.17) and, hence, system (2.13)–(2.14). +11 + +Corollary 2.7 Let u ∈ C satisfy one of the conditions (1.3) and (1.4). Then the operator +¯A − B(u) is bijective from D( ¯A) ∩ Cbc (equipped with the operator norm) to C, and +( ¯A − B(u))−1v = (I − C(u))−1D(u)v for all v ∈ C. +(2.20) +Proof. To show the injectivity, suppose that ( ¯A − B(u))v = 0 for some v ∈ D( ¯A) ∩ Cbc. +Then Lemma 2.5 implies that (I − C(u))v = 0, and Lemma 2.6 yields v = 0. +To show the surjectivity and inversion formula (2.20), take f ∈ C. Because of Lemma 2.6, +there exists v ∈ C such that +v = C(u)v + D(u)f. +In particular, v ∈ Cbc (cf. the definitions of the operators C(u) and D(u)). Moreover, +Lemma 2.5 (i) and (ii) yields that v ∈ D( ¯A) and +( ¯A − B(u))v = ( ¯A − B(u))(C(u)v + D(u)f) = f. +Remark 2.8 Let us explain where the name “non-resonance condition” comes from. +Corollary 2.7 claims that, if u ∈ C satisfies one of the conditions (1.3) and (1.4), then for +any g ∈ C there exists exactly one solution v to the problem +∂tvj(t, x) + aj(x)∂xvj(t, x) − ∂ujfj(x, u(t, x))vj(t, x) = gj(t, x), j = 1, 2, +v1(t, 0) = r1v2(t, 0), v2(t, 1) = r2v1(t, 1), +v(t + 1, x) = v(t, x). +(2.21) +Suppose that the function u is independent of time, i.e. u(t, x) = u(x), and let bj(x) := +∂ujfj(x, u(x)). It is easy to calculate that the eigenvalues to the eigenvalue problem +aj(x)v′ +j(x) − bj(x)vj(x) = λvj(x), j = 1, 2, +v1(0) = r1v2(0), v2(1) = r2v1(1) +are +λk = +ln |r1r2| − +� 1 +0 +� b2(x) +a2(x) − b1(x) +a1(x) +� +dx +� 1 +0 +� +1 +a2(x) − +1 +a1(x) +� +dx ++ 2kπi, k ∈ Z. +Hence, all eigenvalues have non-vanishing real parts if and only if +ln |r1r2| ̸= +� 1 +0 +� b2(x) +a2(x) − b1(x) +a1(x) +� +dx, +and this is just condition (1.3) or condition (1.4) (in the case that the coefficients ∂ujfj(x, u(t, x)) +are independent of time, (1.3) and (1.4) are the same). In this case all ”internal frequencies” +λk/2πi, k ∈ Z, of system (2.21) are different to all ”external frequencies” k ∈ Z of the +right-hand sides gj, and one says that the external frequencies are not in resonance with the +internal frequencies. +12 + +2.2 +Proof of Lemma 2.4 +Suppose that u ∈ C satisfies one of the conditions (1.3) and (1.4). Write +�B(u) := F ′(u) − B(u). +We have to show that the operator ¯A − F ′(u) = ¯A − B(u) − �B(u) is Fredholm of index zero +from D( ¯A) ∩ Cbc (equipped with the graph norm) into C. Because of Corollary 2.7, this is the +case if and only if the operator +( ¯A − B(u))−1( ¯A − F ′(u)) = I − ( ¯A − B(u))−1 �B(u) = I − (I − C(u))−1D(u) �B(u) +is Fredholm of index zero from C into C. Hence, it suffices to show that +� +(I − C(u))−1D(u) �B(u) +�2 +is compact from C into C. +This is a consequence of the following Fredholmness criterion of S. M. Nikolskii (cf. e.g. [4, +Theorem XIII.5.2]): If U is a Banach space and K : U → U is a linear bounded operator such +that K2 is compact, then the operator I − K is Fredholm of index zero. +Since u is fixed, in what follows in this subsection we will not mention the dependence of +the operators B(u), �B(u), C(u) and D(u) on u, i.e. B := B(u), �B := �B(u), C := C(u), and +D := D(u). A straightforward calculation shows that +� +(I − C)−1D �B +�2 += (I − C)−1 � +(D �B)2 + D �BC(I − C)−1D �B +� +. +(2.22) +Then, on account of Lemma 2.6, it suffices to show that the operators D �BD and D �BC are +compact from C into C. +Let us show that D1 �BD (and similarly for D2 �BD) is compact from C into C. Take v ∈ C. +By definition, B and �B are the ”diagonal” and the ”non-diagonal” parts of F ′(u). Therefore, +[ �Bv](t, x) = +� +∂u2f1(x, u(t, x))v2(t, x), ∂u1f2(x, u(t, x))v1(t, x) +� +. +Hence, the first component of [D �Bv](t, x) is +[D1 �Bv](t, x) = +� x +0 +c1(t, x, y) +a1(y) +∂u2f1(y, u(t + α1(x, y), y))v2(t + α1(x, y), y) dy. +Therefore, +[D1 �BDv](t, x) = +� x +0 +� 1 +y +d(t, x, y, z)v2(t + α1(x, y) + α2(y, z), z) dzdy +(2.23) +with +d(t, x, y, z) := −c1(t, x, y)c2(t + α1(x, y), x, z) +a1(y)a2(z) +∂u2f1(y, u(t + α1(x, y), y)). +We change the order of integration in (2.23) according to +� x +0 +dy +� 1 +y +dz = +� x +0 +dz +� z +0 +dy + +� 1 +x +dz +� x +0 +dy. +(2.24) +13 + +Let us consider the first summand in the right-hand side of (2.24). It is the linear operator +[Kv](t, x) := +� x +0 +� z +0 +d(t, x, y, z)v(t + α1(x, y) + α2(y, z), z) dydz. +(2.25) +We have to show that K is compact from C([0, 1]2) (equipped with the maximum norm) into +itself. For that reason we replace in the inner integral in (2.25) the integration variable y with +a new integration variable η according to +η = �η(t, x, y, z) := t + α1(x, y) + α2(y, z) = t + +� y +x +dξ +a1(ξ) + +� z +y +dξ +a2(ξ). +Because of the assumption that a1(x) ̸= a2(x) for all x ∈ [0, 1] (cf. (1.2)), we have +∂y�η(t, x, y, z) = +1 +a1(y) − +1 +a2(y) ̸= 0 for all y ∈ [0, 1], +i.e. the function y �→ �η(t, x, y, z) is strictly monotone. Let us denote its inverse function by +η �→ �y(t, x, η, z). Then +[Kv2](t, x) = +� x +0 +� �η(t,x,z,z) +�η(t,x,0,z) +˜d(t, x, η, z)v2(η, z) dηdz +(2.26) +with +˜d(t, x, η, z) := +d(t, x, �y(t, x, η, z), z) +1 +a1(�y(t, x, η, z)) − +1 +a2(�y(t, x, η, z)) +. +Due to assumption (1.2), the function ˜d is continuous, and the function ˆη is C1-smooth. Hence, +the Arcela-Ascoli Theorem implies that the linear operator K is compact from C([0, 1]2), +equipped with the maximum norm, into itself. +Similarly one shows that also the second summand in the right-hand side of (2.24), which +is +� 1 +x +� x +0 +d(t, x, y, z)v2(t + α1(x, y) + α2(y, z), z) dydz, +generates a compact operator from C([0, 1]2) into itself. +Finally, let us show that the operator D �BC is compact from C into itself. We have (and +similarly for D2 �BC) +[D1 �BCv](t, x) = +� x +0 +d(t, x, y)v1(t + α1(x, y) + α2(y, 1), 1) dy +with +d(t, x, y) := r2 +c1(t, x, y)c2(t + α1(x, y), x, y) +a1(y) +∂u1f2(y, u(t + α1(x, y), y)). +Here we change the integration variable y to η = t + α1(x, y) + α2(y, 1), and then proceed as +above. +14 + +2.3 +Proof of Theorem 1.1 +Take a function u ∈ C(R × [0, 1]; R2) which satisfies the boundary and the periodicity condi- +tions as in (1.1) and such that there exists a sequence u1, u2, . . . ∈ C1(R×[0, 1]; R2) satisfying +the following convergence for j = 1, 2: +|un +j (t, x) − uj(t, x)| + |∂tun +j (t, x) + aj(x)∂xun +j (t, x) − fj(x, u(t, x))| → 0 for n → ∞, +uniformly in x ∈ [0, 1] and t ∈ R. Then u is a solution to (2.2), and Lemma 2.5 (iii) yields +u = C(u)u + D(u)(F(u) − B(u)u). +(2.27) +Further, we suppose that u satisfies one of the conditions (1.3) and (1.4). Then, due to +Lemma 2.4 and Corollary 4.2, all partial derivatives ∂k +t u, k ∈ N, exist and are continuous. +Therefore, all partial derivatives ∂k +t , k ∈ N, of the functions F(u) and B(u)u exist and are +continuous also. Hence, Lemma 2.5 (iv) and (2.27) yield that u ∈ C1 +bc and ¯Au = Au, i.e. u is +a classical solution to (1.1). Assertion (i) of Theorem 1.1 is therefore proved. +Similarly, if the functions aj and fj, j = 1, 2, are C∞-smooth, then Lemma 2.5 (v) and +(2.27) yield that u is C∞-smooth, i.e. assertion (ii) of Theorem 1.1 is proved also. +3 +Proofs for second-order equations +In this section we will prove Theorem 1.2. Hence, we suppose that assumption (1.6) is satisfied. +3.1 +Transformation of the second-order equation into a first-order system +In this subsection we show that any solution u to problem (1.5) for a second-order equation +creates a solution +v1(t, x) := ∂tu(t, x) + a(x)∂xu(t, x), +v2(t, x) := ∂tu(t, x) − a(x)∂xu(t, x) +(3.1) +to the following problem for a first-order system of integro-differential equations: +∂tv1(t, x) − a(x)∂xv1(t, x) = ∂tv2(t, x) + a(x)∂xv2(t, x) += f(x, [Jv](t, x), [Kv](t, x), [Lv](t, x)) + a′(x) +2 +(v1(t, x) − v2(t, x)), +v1(t, 0) + v2(t, 0) = v1(t, 1) − v2(t, 1) = 0, +v(t + 1, x) = v(t, x), +(3.2) +and vice versa. Here the partial integral operator J is defined by +[Jv](t, x) := 1 +2 +� x +0 +v1(t, y) − v2(t, y) +a(y) +dy, +and the ”local” operators K and L are defined by +[Kv](t, x) := v1(t, x) + v2(t, x) +2 +, +[Lv](t, x) := v1(t, x) − v2(t, x) +2a(x) +. +Lemma 3.1 (i) If u ∈ C2(R × [0, 1]) is a solution to (1.5), then the function v ∈ C1(R × +[0, 1]; R2) defined by (3.1) is a solution to (3.2). +(ii) Let v ∈ C1(R×[0, 1]; R2) be a solution to (3.2). Then the function u := Jv is C2-smooth +and is a solution to (1.5). +15 + +Proof. (i) Let u ∈ C2(R × [0, 1]; R2) be given, and let v ∈ C1 +per(R × [0, 1]; R2) be defined +by (3.1). Then +∂tu = v1 + v2 +2 += Kv, +∂xu = v1 − v2 +2a += Lv +(3.3) +and ∂tv1 = ∂2 +t u + a∂t∂xu, ∂xv1 = ∂t∂xu + a′∂xu + a∂2 +xu, ∂tv2 = ∂2 +t u − a∂t∂xu, and ∂xv2 = +∂t∂xu − a′∂xu − a∂2 +xu. Hence, +∂2 +t u − a2∂2 +xu − aa′∂xu = ∂tv1 − a∂xv1 = ∂tv2 + a∂xv2. +(3.4) +Further, let u be a solution to problem (1.5). Then (3.3) and the boundary conditions u(t, 0) = +0 and ∂xu(t, 1) + γu(t, 1) = 0 imply that v1(t, 0) + v2(t, 0) = 0 and v1(t, 1) − v2(t, 1) + +γa(1)[Lv](t, 1) = 0, i.e. the boundary conditions as in (3.2). Moreover, from u(t, 0) = 0 and +(3.3) follows that u(t, x) = [Jv](t, x). Hence, (3.3), (3.4), and the differential equation in (1.5) +yield the differential equations as in (3.2). +(ii) Let v ∈ C1(R × [0, 1]; R2) be a solution to (3.2). Set u := Jv. Then +∂tu(t, x) += +1 +2 +� x +0 +∂tv1(t, y) − ∂tv2(t, y) +a(y) +dy += +� x +0 +(∂yv1(t, y) + ∂yv2(t, y)) dy = v1(t, x) + v2(t, x). +Here we used the first boundary condition and the differential equations in (3.2). It follows +that ∂tu is C1-smooth, and +∂2 +t u = ∂tv1 + ∂tv2. +(3.5) +Further, the relation u = Jv yields that ∂xu = ∂xJv = Lv, i.e. ∂xu is C1-smooth also and, +hence u is C2-smooth. Moreover, 2(a′∂xu + a∂2 +xu) = ∂xv1 − ∂xv2, i.e. +a2∂2 +xu = a +2(∂xv1 − ∂xv2) − a′ +2 (v1 − v2). +(3.6) +But (3.2), (3.5), and (3.6) imply the differential equation as in (1.5). +The first boundary condition in (1.5) follows from u = Jv, and the second boundary +conditions in (1.5) follows from ∂xu = Lv and from the second boundary condition in (3.2). +Unfortunately, we cannot apply Theorem 1.1 directly to system (3.2) because there are +nonlocal terms in the equations in (3.2). Hence, we adapt the content of Section 2 to the +situation of system (3.2). +3.2 +Weak formulation of (3.2) +We use the notation of α(x, y), b+(t, x, u) and b−(t, x, u), which were introduced in Section 1, +as well as the function spaces C and C1, which were introduced in Section 2. Further, we +introduce a linear bounded operator A : C1 → C by +Av := (∂tv1 − ∂xv1, ∂tv2 + ∂xv2) +and a nonlinear C∞-smooth superposition operator F : C → C by +[Fj(v)](t, x) := f (x, [Jv](t, x), [Kv](t, x), [Lv](t, x)) + a′(x) +2 +(v1(t, x) − v2(t, x)), +j = 1, 2. +16 + +Any classical solution to (3.2) is a solution to the problem Av = F(v) and, hence, a solution +to the problem +v ∈ D( ¯A) ∩ Cbc : +¯Av = F(v). +(3.7) +In order to apply Corollary 4.2 to problem (3.7) (with U = D( ¯A) ∩ Cbc, V = C, and +F(v) = ¯Av − F(v)), we proceed as in Section 2. We write (for t ∈ R, x ∈ [0, 1], v ∈ C) +c+(t, x, v) +:= +b+(t, x, Jv) += +∂3f(x, [Jv](t, x), [Kv](t, x), [Lv](t, x)) + ∂4f(x, [Jv](t, x), [Kv](t, x), [Lv](t, x)) +a(x) +, +c−(t, x, v) +:= +b−(t, x, Jv) += +∂3f(x, [Jv](t, x), [Kv](t, x), [Lv](t, x)) − ∂4f(x, [Jv](t, x), [Kv](t, x), [Lv](t, x)) +a(x) +. +Note that, if a function u ∈ C1(R × [0, 1]) satisfies condition (1.7), then the function v ∈ +C(R × [0, 1]; R2), which is defined by (3.1), satisfies the condition +� 1 +0 +c+(t + α(x, 1), x, v) − c−(t − α(x, 1), x, v) +a(x) +dx ̸= 0 +for all t ∈ R. +(3.8) +Similarly, if u satisfies (1.8), then v satisfies the condition +� 1 +0 +c+(t − α(0, x), x, u) − c−(t + α(0, x), x, u) +a(x) +dx ̸= 0 +for all t ∈ R. +(3.9) +We divide the linearization F ′(v) into three parts, a “diagonal” one, a “non-diagonal” one, +and an “integral” one, as follows: +F ′(v) = B(v) + �B(v) + J (v) +(3.10) +with +[B(v)w](t, x) := 1 +2 +� (c+(t, x, v) + a′(x))w1(t, x) +(c−(t, x, v) − a′(x))w2(t, x) +� +, +[ �B(v)w](t, x) := 1 +2 +� (c−(t, x, v) − a′(x))w2(t, x) +(c+(t, x, v) + a′(x))w1(t, x) +� +, +and +[J1(v)w](t, x) = [J2(v)w](t, x) := ∂2f(x, [Jv](t, x), [Kv](t, x), [Lv](t, x))[Jw](t, x). +As in Subsection 2.1, for given v ∈ C, we introduce linear bounded operators C(v), D(v) : +C → C by +[C(v)w](t, x) := + + +−w2(t − α(x, 0), 0) +� +a(0) +a(x) exp +� +−1 +2 +� x +0 +c+(t − α(x, z), z, v) +a(z) +dz +� +w1(t + α(x, 1), 1) +� +a(1) +a(x) exp +� +−1 +2 +� 1 +x +c−(t + α(x, z), z, v) +a(z) +dz +� + + +and +[D(v)w](t, x) := + + +− +1 +� +a(x) +� x +0 +w1(t − α(x, y), y) +� +a(y) +exp +�1 +2 +� y +x +c+(t − α(x, z), z, v) +a(z) +dz +� +dy, +− +1 +� +a(x) +� 1 +x +w2(t + α(x, y), y) +� +a(y) +exp +�1 +2 +� x +y +c−(t + α(x, z), z, v) +a(z) +dz +� +dy + + . +17 + +Here we adapted the definitions of C(u)v and D(u)v from Subsection 2.1 as follows: We +replaced u by v, v by w, a1 by −a, a2 by a, r1 by minus one, r2 by one, ∂u1f1(x, u(t, x)) by +1 +2(c+(t, x, v) + a′(x)) and ∂u2f2(x, u(t, x)) by 1 +2(c−(t, x, v) − a′(x)). We used also the identity +exp +�1 +2 +� y +x +a′(z) +a(z) dz +� += +� +a(y) +a(x). +Remark that, if in (1.3) and in (1.4) we replace a1 by −a, a2 by a, r1 by minus one, r2 by +one, ∂u1f1(x, u(t, x)) by 1 +2(c+(t, x, v) + a′(x)), and ∂u2f2(x, u(t, x)) by 1 +2(c−(t, x, v) − a′(x)), +we immediately get (3.8) and (3.9). +Finally, the subspace of all functions, which satisfy the boundary conditions as in (3.2), is +Cbc := {v ∈ C : v1(t, 0) + v2(t, 0) = v1(t, 1) − v2(t, 1) = 0}. +Similar to Lemmas 2.5 and 2.6 and Corollary 2.7, we get the following: +Lemma 3.2 If v ∈ C satisfies one of the conditions (3.8) and (3.9), then I −C(v) is bijective +from C onto C, ¯A − B(v) is bijective from D( ¯A) ∩ Cbc onto C, and +( ¯A − B(v))−1w = (I − C(v))−1D(v)w +for all w ∈ C. +3.3 +Fredholmness +Lemma 3.3 Let a function v ∈ C satisfy one of the conditions (3.8) and (3.9). Then the +operator I − C(v) − D(v)( �B(v) + J (v)) is Fredholm of index zero from C to itself. +Proof. We proceed as in the proof of Lemma 2.4. We have to show that the operator I − +(I −C(v))−1 � +D(v)( �B(v) + J (v)) +� +is Fredholm of index zero from C into C. The compactness +criterion of Nikolskii implies that it suffices to show that +� +(I − C(v))−1 � +D(v)( �B(v) + J (v)) +� �2 +is compact from C into C. +(3.11) +As v is fixed, we will drop the dependence of the operators B(v), �B(v), C(v), D(u), E(v), +and J (v) on v, i.e. B := B(v), �B := �B(v), C := C(v), D := D(v) and J = J (v). As in +Subsection 2.2, we use the formula +� +(I − C)−1(D( �B + J )) +�2 += (I − C)−1 � +(D( �B + J ))2 + D( �B + J )C(I − C)−1D( �B + J ) +� +. +Similar to the proof of Lemma 2.2, to show (3.11) it suffices to prove that the operators +� +D( �B + J ) +�2 += D �BD �B + DJ DJ + D �BDJ + DJ D �B +and D( �B + J )C = D �BC + DJ C are compact from C into itself. Since in the proof of +Lemma 2.4 we already showed that the operators D �BD and D �BC are compact from C +into itself, it suffices to show that the operator DJ is compact from C into itself. The first +component of this operator (and similar for the second component) works as +[D1J w](t, x) = +� x +0 +c(t, x, y) +� y +0 +w1(t − α(x, y), z) − w2(t − α(x, y), z) +a(z) +dzdy +18 + +with +c(t, x, y) := − +� +∂2f(y, [Jv](s, y), [Kv](s, y), [Lv](s, y)) +2 +� +a(x)a(y) +exp +�1 +2 +� y +x +c+(z, v(s, z)) +a(z) +dz +�� +s=t−α(x,y) +. +We replace the integration variable y by η = �η(t, y, z) := t − α(x, y), hence dη = −dy/a(y). +If y = �y(t, η, z) is the inverse transformation, then we get +[D1J w](t, x) = +� t−α(0,x) +t +� �y(t,x,η) +0 +c(t, x, �y(t, x, η))a(�y(t, x, η))w1(η, z) − w2(η, z) +a(z) +dzdη. +Hence, the linear operator w ∈ C �→ D1J w ∈ C([0, 1]2) is compact because of the Arzela- +Ascoli Theorem. +3.4 +Proof of Theorem 1.2 +Let u ∈ C2(R × [0, 1]) be a classical solution to (1.5). Then, due to Lemma 3.1 (i), the +function v ∈ C1(R × [0, 1]; R2) defined by (3.1) is a classical solution to system (3.2) and, +hence, a solution to the abstract equation (3.7). If, moreover, u satisfies one of the conditions +(1.7) and (1.8), then v satisfies one of the conditions (3.8) and (3.9). Because of Lemma 3.3, +Corollary 4.2 can be applied to the solution v of the abstract equation (3.7). Hence, all partial +derivatives ∂k +t v, k ∈ N, exist and are continuous. Since u = Jv, all partial derivatives ∂k +t u, +k ∈ N, exist and are continuous also. Therefore, assertion (i) of Theorem 1.2 is proved. +In order to prove assertion (ii) of Theorem 1.2, suppose that the functions a and f are C∞- +smooth. Then, as in Subsection 2.3, we use Lemma 2.5 (v) and show that v is C∞-smooth. +Again, since u = Jv, u is C∞-smooth, as desired. +4 +Appendix +For given Banach spaces U and V , let Sϕ ∈ L(U) and Tϕ ∈ L(V ), with ϕ ∈ R, be one- +parameter C0-groups on U and V , respectively, i.e. +Sϕ ◦ Sψ = Sϕ+ψ for all ϕ, ψ ∈ R, S0 = I, +ϕ ∈ R �→ Sϕu ∈ U is continuous for all u ∈ U, +and similarly for Tϕ. Further, let F : U → V be a map such that +F(Sϕu) = TϕF(u) for all ϕ ∈ R and u ∈ U. +(4.1) +The following theorem is due to E. N. Dancer (see [1, Theorem 1]). Roughly speaking, it +claims the following: The map γ ∈ R �→ Sγu ∈ U is not C1-smooth, in general, but it is if u +solves an equivariant equation F(u) = 0 with a C1-Fredholm map F. +Theorem 4.1 Let U and V be Banach spaces. Let F be C1-smooth and u0 ∈ U be given +such that +F(u0) = 0, and F′(u0) is Fredholm of index zero from U into V . +(4.2) +If condition (4.1) is fulfilled, then the map γ ∈ R �→ Sγu0 ∈ U is C1-smooth. +19 + +This theorem can easily be generalized to the C∞ case as follows: +Corollary 4.2 Let U and V be Banach spaces. Let F be C∞-smooth and u0 ∈ U be given +such that (4.2) is satisfied. If condition (4.1) is fulfilled, then the map γ ∈ R �→ Sγu0 ∈ U is +C∞-smooth. +Proof. We have to show that for any k ∈ N the map ϕ ∈ R �→ Sϕu0 ∈ U is Ck-smooth. +To this end, we use the induction in k. +The assertion for k = 1 is true due to Theorem 4.1. +Doing the induction step, we suppose that, for a fixed k ∈ N, the map ϕ ∈ R �→ Sϕu0 ∈ U +is Ck-smooth and show that this map is Ck+1-smooth. +We denote by A : D(A) ⊆ U → U the infinitesimal generator of the C0-group Sϕ, i.e. +D(A) := {u ∈ U : ϕ ∈ R �→ Sϕu ∈ U is C1-smooth}, Au := d +dϕSϕu|ϕ=0 for u ∈ D(A). +Similarly we define D(Al) and Al with l ≥ 2. In particular, we have +d +dϕF(Sϕu)|ϕ=0 = F′(u)Au for u ∈ D(A), +d2 +dϕ2 F(Sϕu)|ϕ=0 = F′(u)A2u + F′′(u)(Au, Au) = 0 +for u ∈ D(A2) +More precisely, there exist C∞-maps Fl : U l → V , l ∈ N, such that +dl +dϕl F(Sϕu)|ϕ=0 = F′(u)Alu + Fl(u, Au, A2u, . . . , Al−1u) +for u ∈ D(Al). +On account of (4.1), for all ϕ ∈ R and u ∈ D(Al) it holds +Fl(Sϕu, SϕAu, SϕA2u, . . . , SϕAl−1u) = TϕFl(u, Au, A2u, . . . , Al−1u). +hence, F(Sϕu0) ≡ 0 yields that +F′(u0)Alu0 + Fl(u0, Au0, A2u0, . . . , Al−1u0) = 0 +for l ≤ k. +Now, let us consider the C∞-map G = (G0, G1, . . . , Gk) : U k+1 → V k+1 defined by +G0(u0, u1, . . . , uk) := F(u0), +Gj(u0, u1, . . . , uk) := F′(u0)uj + Fj(u0, u1, . . . , uj−1) +for j ≤ k. +In order to apply Theorem 4.1 to the equation G(u0, u1, . . . , uk) = 0 in its solution +(u0, u1, . . . , uk) = (u0, Au0, . . . , Aku0), +we have to show that the derivative G′(u0, Au0, . . . , Aku0) is Fredholm operator of index zero +from U k+1 into V k+1. We have +G′ +0(u0, Au0, . . . , Aku0)(u0, u1, . . . , uk) = F′(u0)u0 +and, for j ≤ k, +G′ +j(u0, Au0, . . . , Aku0)(u0, u1, . . . , uk) := F′(u0)uj + +j−1 +� +i=0 +∂iFj(u0, Au0, . . . , Aj−1u0)ui. +20 + +Hence, G′(u0, Au0, . . . , Aku0) is a triangular operator of the type +G′(u0, Au0, . . . , Aku0) = + + +F′(u0) +0 +0 +. . . +∂0F1(u0) +F′(u0) +0 +. . . +∂0F2(u0, Au0) +∂1F2(u0, Au0) +F′(u0) +. . . +. . . +. . . +. . . +. . . + + +By assumption, F′(u0) assumption is Fredholm operator of index zero from U into V . Hence, +there exist linear bounded operators J , K : U → V such that F′(u0) = J + K, that J is +bijective and that K is compact. Therefore +G′(u0, Au0, . . . , Aku0) = � +J + �K +with +� +J := + + +J +0 +0 +. . . +∂0F1(u0) +J +0 +. . . +∂0F2(u0, Au0) +∂1F2(u0, Au0) +J +. . . +. . . +. . . +. . . +. . . + + , �K := + + +K +0 +0 +. . . +0 +K +0 +. . . +0 +0 +K +. . . +. . . +. . . +. . . +. . . + + , +where � +J is a bijective operator from U k+1 to V k+1, and �K is a compact operator from U k+1 +into V k+1. Hence, G′(u0, Au0, . . . , Aku0) a is Fredholm operator of index zero from U k+1 to +V k+1. Now, Theorem 4.1 yields that +ϕ ∈ R �→ (Sϕu0, SϕAu0, . . . , SϕAku0) ∈ U k+1 is C1-smooth, +which means that ϕ ∈ R �→ Sϕu0 ∈ U is Ck+1-smooth. +Acknowledgments +Irina Kmit was supported by the VolkswagenStiftung Project “From Modeling and Analysis +to Approximation”. +References +[1] E. N. Dancer, The G-invariant implicit function theorem in infinite dimensions, Proc. +Royal Soc. Edinburgh 92A (1982), 13–30. +[2] Bao-Zhu Guo and Jun-Min Wang, Control of Wave and Beam PDEs. The Riesz +Basis Approach, Communications in Control Engineering, Springer 2019. +[3] J. K. Hale and J. Scheurle, Smoothness of bounded solutions of nonlinear evolution +equations, J. Differ. Equ. 56 (1985), 142-163. +[4] L. V. Kantorovich and G. P. Akilov, Functional Analysis, 2nd ed., Pergamon Press, +1982, Appl. Math. Sciences 156, Springer, 2004. +[5] I. Kmit, Smoothing effect and Fredholm property for first-order hyperbolic PDEs, Op- +erator Theory: Advances and Applications 231, Birkh¨auser, 2013, 219–238. +21 + +[6] I. Kmit and L. Recke, Fredholm alternative and solution regularity for time-periodic +hyperbolic systems. Differential and Integral Equations 29(11/12) (2016): 1049–1070. +[7] I. Kmit and L. Recke, Hopf bifurcation for semilinear dissipative hyperbolic systems, +J. Differ. Equ. 257 (2014), 246-309. +[8] I. Kmit and L. Recke, Hopf bifurcation for general 1D semilinear wave equations, J. +Dyn. Differ. Equ. (2021). +[9] I. Kmit and L. Recke, Solution regularity and smooth dependence for abstract equa- +tions and applications to hyperbolic PDEs, J. Differ. Equ. 259 (2015), 6287-6337. +[10] I. Kmit and L. Recke, Time-periodic second-order hyperbolic equations: Fredholm +solvability, regularity, and smooth dependence, in: Pseudodifferential Operators and Gen- +eralized Functions, Operator Theory: Advances and Applications 245, Birkh¨auser, 2015, +147-181. +[11] I. Kmit, L. Recke, V. Tkachenko, Bounded and almost periodic solvability of nonau- +tonomous quasilinear hyperbolic systems, J. Evol. Eq. 21(2021), 4171–4212. +[12] N. Kosovali´c and B. Pigott, Self-excited vibrations for damped and delayed 1- +dimensional wave equations, J. Dyn. Differ. Equ. 31 (2019), 129-152. +[13] N. Kosovali´c and B. Pigott, Self-excited vibrations for damped and delayed higher +dimensional wave equations, Discrete Contin. Dyn. Syst. 39 (2019), 2413-2435. +[14] M. Lichtner, A spectral mapping theorem for linear hyperbolic systems, Proc. Amer. +Math. Soc. 136 (6) (2008), 2091–2101. +[15] Z.-H. Luo, B.-Z. Guo and O. Mogul, Stability and Stabilization of Infinite Dimen- +sional Systems with Applications, Springer, 1999. +[16] A. F. Neves, H. de Souza Ribeiro and O. Lopes, On the spectrum of evolution +operators generated by hyperbolic systems, J. Funct. Anal. 670 (1986), 320-344. +[17] M. Renardy, On the linear stability of hyperbolic PDEs and viscoelastic flows, Z. +Angew. Math. Phys. (ZAMP) 45 (1994), 854-865. +22 + diff --git a/RdAyT4oBgHgl3EQft_l_/content/tmp_files/load_file.txt b/RdAyT4oBgHgl3EQft_l_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b81615d3c26f8d9350449ea189f3f67a31a4989 --- /dev/null +++ b/RdAyT4oBgHgl3EQft_l_/content/tmp_files/load_file.txt @@ -0,0 +1,1046 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf,len=1045 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='00605v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='AP] 2 Jan 2023 Regularity of Time-Periodic Solutions to Autonomous Semilinear Hyperbolic PDEs Irina Kmit ∗ Lutz Recke † Abstract This paper concerns autonomous boundary value problems for 1D semilinear hyper- bolic PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' For time-periodic classical solutions, which satisfy a certain non-resonance condition, we show the following: If the PDEs are continuous with respect to the space variable x and C∞-smooth with respect to the unknown function u, then the solution is C∞-smooth with respect to the time variable t, and if the PDEs are C∞-smooth with respect to x and u, then the solution is C∞-smooth with respect to t and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' The same is true for appropriate weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Moreover, we show examples of time-periodic functions, which do not satisfy the non- resonance condition, such that they are weak, but not classical solutions, and such that they are classical solutions, but not C∞-smooth, neither with respect to t nor with respect to x, even if the PDEs are C∞-smooth with respect to x and u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' For the proofs we use Fredholm solvability properties of linear time-periodic hyperbolic PDEs and a result of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Dancer about regularity of solutions to abstract equivariant equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Keywords: 1D semilinear hyperbolic PDEs, autonomous boundary value problems, solution regularity, non-resonance condition, Fredholm solvability 1 Introduction In this paper we consider time-periodic solutions to boundary value problems for 1D semi- linear first-order hyperbolic systems of the type ∂tuj(t, x) + aj(x)∂xuj(t, x) = fj(x, u(t, x)) and 1D semilinear second-order hyperbolic equations of the type ∂2 t u(t, x) − a(x)2∂2 xu(t, x) = f(x, u(t, x), ∂tu(t, x), ∂xu(t, x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Let us formulate our results concerning first-order systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Specifically, we consider 2 × 2 systems with reflection boundary conditions and time-periodic solutions with period one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' solutions u = (u1, u2) to problems of the type (for t ∈ R, x ∈ [0, 1]) ∂tuj(t, x) + aj(x)∂xuj(t, x) = fj(x, u(t, x)), j = 1, 2, u1(t, 0) = r1u2(t, 0), u2(t, 1) = r2u1(t, 1), u(t + 1, x) = u(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='1) ∗Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, D-10099 Berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' On leave from the Institute for Applied Problems of Mechanics and Mathematics, Ukrainian National Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' E-mail: kmit@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='hu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='de †Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, D-10099 Berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' E-mail: recke@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='hu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='de 1 We suppose that for j = 1, 2 aj ∈ C([0, 1]), rj ∈ R, aj(x) ̸= 0 and a1(x) ̸= a2(x) for all x ∈ [0, 1], ∂k u1∂l u2fj exist and belong to C([0, 1] × R2) for all k, l ∈ N ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='2) Further, we write (for t ∈ R, x, y ∈ [0, 1], and j = 1, 2) αj(x, y) := � y x dz aj(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='1 Suppose that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='2) is fulfilled, and let u ∈ C(R × [0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' R2) satisfy one of the conditions � 1 0 �∂u1f1(x, u(t − α1(x, 1), x)) a1(x) − ∂u2f2(x, u(t − α2(x, 1), x)) a2(x) � dx ̸= ln |r1r2| for all t ∈ R (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='3) and � 1 0 �∂u1f1(x, u(t + α1(0, x), x)) a1(x) − ∂u2f2(x, u(t + α2(0, x), x)) a2(x) � dx ̸= ln |r1r2| for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='4) Then the following is true: (i) If u satisfies the boundary and the periodicity conditions in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='1) and if there exists a sequence u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' ∈ C1(R × [0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' R2) such that, for j = 1, 2, |un j (t, x) − uj(t, x)| + |∂tun j (t, x) + aj(x)∂xun j (t, x) − fj(x, u(t, x))| → 0 for n → ∞ uniformly with respect to (t, x) ∈ R×[0, 1], then u is a classical solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='1), in particular, u is C1-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Moreover, all partial derivatives ∂k t u, k ∈ N, exist and belong to C(R × [0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' (ii) If u is a classical solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='1) and if the functions aj and fj, j = 1, 2, are C∞- smooth, then u is C∞-smooth also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Now we formulate our results concerning time-periodic solutions to second-order equations subjected to one Dirichlet and one Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' More precisely, we consider problems of the type (for t ∈ R and x ∈ [0, 1]) ∂2 t u(t, x) − a(x)2∂2 xu(t, x) = f(x, u(t, x), ∂tu(t, x), ∂xu(t, x)), u(t, 0) = 0, ∂xu(t, 1) = 0, u(t + 1, x) = u(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='5) We assume that a ∈ C1([0, 1]), a(x) ̸= 0 for all x ∈ [0, 1], ∂j 2∂k 3∂l 4f exist and belong to C([0, 1] × R3) for all j, k, l ∈ N ∪ {0}, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='6) where ∂jf denotes the derivative of the function f with respect to its j-th argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' More precisely, if f = f(x, u, v, w), then ∂2f is the derivative with respect to u, ∂3f is the derivative 2 with respect to v, and ∂4f is the derivative with respect to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Further, we write (for t ∈ R, x, y ∈ [0, 1], and u ∈ C([0, 1] × R2)) α(x, y) := � y x dz a(z), b+(t, x, u) := ∂3f(x, u(t, x), ∂tu(t, x), ∂xu(t, x)) + ∂4f(x, u(t, x), ∂tu(t, x), ∂xu(t, x)) a(x) , b−(t, x, u) := ∂3f(x, u(t, x), ∂tu(t, x), ∂xu(t, x)) − ∂4f(x, u(t, x), ∂tu(t, x), ∂xu(t, x)) a(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' The weak formulation of the second-order problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='5) (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='1 (i)), which will be used, is slightly more complicated than that for the first-order problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='1), and, in fact, it is a technical tool only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' We, therefore, will not include in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='2 below a regularity result for weak solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='5), but only regularity results for classical solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='2 Suppose that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='6) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Let u ∈ C2(R × [0, 1]) be a classical solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='5), and suppose that it satisfies one of the conditions � 1 0 b+(t + α(x, 1), x, u) − b−(t − α(x, 1), x, u) a(x) dx ̸= 0 for all t ∈ R (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='7) and � 1 0 b+(t − α(0, x), x, u) − b−(t + α(0, x), x, u) a(x) dx ̸= 0 for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='8) Then the following is true: (i) All partial derivatives ∂k t u, k ∈ N, exist and belong to C(R × [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' (ii) If the functions a and f are C∞-smooth, then u is C∞-smooth also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='3 In most applications, solutions to problems of the type (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='5) are found as a result of Hopf bifurcations from stationary solutions [7, 8, 12] and by continuation of such solutions with respect to parameters [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='4 The paper [3] of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Hale and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Scheurle concerns smoothness with respect to time of solutions to abstract autonomous semilinear evolution equations if those solutions are bounded and close to be constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' The results are applied to slightly damped nonlinear wave equations in 1D with constant coefficients, namely ∂2 t u(t, x) − ∂2 xu(t, x) + δ∂tu(t, x) − u(t, x) − λu(t, x) = f(u(t, x)), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='9) subjected to homogeneous Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' The function f : R → R is smooth and of order o(|u|) for u → 0, λ is small, δ is positive and small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' It is shown that sufficiently small bounded solutions are smooth with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Let us compare this with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='2: On one hand, the equation in our problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='5) is more general than equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Moreover, in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='2 we do not suppose that the solution is close to be constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' On the other hand, our Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='2 concerns time- periodic solutions only, not general bounded ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Anyway, if one applies definitions of the functions b+ and b− to equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='9), then b+(t, x, u) = b−(t, x, u) = −δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Hence, the assumption δ > 0 of [3] implies that the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='2 are fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' 3 Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='5 Let us consider Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='2 in the special case of a nonlinear wave equation which is slightly more general than (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='9), namely ∂2 t u(t, x) − a(x)2∂2 xu(t, x) = β1(x)∂tu(t, x) + β2(x)∂xu(t, x) + f(x, u(t, x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' If one applies definitions of b+ and b− to this equation, then b+(t, x, u) = β1(x) + β2(x) a(x) , b−(t, x, u) = β1(x) − β2(x) a(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Hence, the conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='8) are identical, and they are satisfied for any u if and only if � 1 0 β1(x) a(x) dx ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='6 We do not know if Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='2 can be generalized to cases of more than one space dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' The reason is that linear autonomous hyperbolic partial differential operators with one space dimension essentially differ from those with more than one space dimension: They satisfy the spectral mapping property in Lp-spaces [16] and, which is more important for applications to nonlinear problems, in C-spaces [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Moreover, they generate Riesz bases (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' [2, 15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' This is not the case, in general, if the space dimension is larger than one (see the counter-example of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Renardy in [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Therefore, the question of Fredholmness of those operators in appropriate spaces of time-periodic functions is highly difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='7 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='1 can be generalized to problems for n × n first-order hyperbolic systems of the type (with natural numbers m < n) ∂tuj(t, x) + aj(x)∂xuj(t, x) = fj(x, u(t, x)), j ≤ n, uj(t, 0) = n � k=m+1 rjkuk(t, 0), j ≤ m, uj(t, 1) = m � k=1 rjkuk(t, 1), m < j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='10) Here, instead of non-resonant conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdAyT4oBgHgl3EQft_l_/content/2301.00605v1.pdf'} +page_content='4), one considers the following sufficient conditions max s,t∈[0,1] max j≤m n � k=m+1 m � l=1 |rjkrkl| exp � 1 0 �∂ukfk(x, u(t, x)) ak(x) − ∂ujfj(x, u(s, x)) aj(x) � dx < 1 and max s,t∈[0,1] max m 0. Note that (·)2 : P → Q is bijective. Thus, if Q′ is another square of P, +then there is an obvious bijection Q → Q′ and it actually is an isomorphism of positive spaces. +Arrows are elements of a real vector space V 3 and the inner product is a function from V 3 × V 3 +to W 1, the extension of L2. Since W 1 is oriented, it has a natural ordering and the proposition +∀v ∈ V 3 : 0 ≤ ⟨v, v⟩ +makes sense. The codomain of the associated norm is not the positive space L, because the length of +a vector can be equal to zero. Thus, we have to introduce a new concept: Non-negative spaces, which +contain a neutral element of addition and whose elements can be multiplied by non-negative real +numbers. Given the definition of positive spaces, the definition of non-negative spaces is obvious. +Then the codomain of the inner product is the square root of the non-negative part of W 1, defined +as follows: +Definition 4. Let X be a non-negative space, then its square root consists of a non-negative space +Y together with a function +X ∋ x �→ √x ∈ Y +such that +√ +λx = +√ +λ +√ +x +for all x ∈ X and λ ≥ 0. Recall that two squares of the same positive space can be identified through +a natural isomorphism of positive spaces. A similar construction allows us to identify two square +roots of the same non-negative space through an isomorphism of non-negative spaces. +Lastly, we consider a speed c - a homomorphism of positive spaces from T to L - and the unique +inner product V 1 × V 1 → W 1 satisfying +∀u ∈ T : +� +⟨u, u⟩ = cu. +Definition 5. Consider some velocity v ∈ L(V 1, V 3), then the function +∥v∥: T → L +u �→ ∥vu∥ +is called its speed. +Remark 1. In the section on electromagnetism we consider a fixed set of units. Thus it is natural to +wonder about the invariance of the physical laws under a change of units - that is, if some equation +holds true for one particular choice of units, how do we know that it holds true for all possible +choices of units? To answer the question, we first reformulate it within a clear mathematical setting: +In general, we consider a list of positive spaces X1, . . . , Xn (e.g. the positive spaces associated to the +base dimensions of the International System of Quantities) and a physical quantity with values +in a real vector space V is a function +Q: +n +� +i=1 +Xi → V +5 + +with a well-defined dimension, meaning that there exists a list +α1, . . . , αn ∈ Q +such that +Q(λ1x1, . . . , λnxn) = λ1 +α1 · · · λn +αnQ(x1, . . . , xn) +for all x and positive real numbers λ1, . . . , λn. +That being said, the initial question can be rephrased as follows: If we consider two physical quan- +tities +Q, Q′: +n +� +i=1 +Xi → V +then what is a sufficient condition such that the following implication holds: +∃x : Q(x1, . . . , xn) = Q′(x1, . . . , xn) ⇒ ∀x : Q(x1, . . . , xn) = Q′(x1, . . . , xn) +A sufficient requirement that will always hold in practice is clearly that Q and Q′ have the same +dimension. +1.2 +Reference Frames +Definition 6. A reference frame on a set M consists of the following data: +• An affine space A1 with translation space V 1. +• An affine space A3 with translation space V 3. +• A function t: M → A1 and a function Π: M → A3 such that the induced function F : M → +A1 × A3 is bijective. +Definition 7. Suppose we have fixed a reference frame, a unit of time e0 and a unit of length. Then +for each choice of +• an origin of time 0 ∈ A1, +• an origin of space O ∈ A3 and +• an orthonormal basis e of V 3 (orthonormal w.r.t. the real valued inner product induced by +the unit of length) +the bijective function +A1 × A3 → R × R3 +(t, P) �→ (e0(t − 0), e(P − O)) +is called an orthonormal coordinate system. +Remark 2. In [3] a reference frame on M is a maximal atlas A such that +∀φ, φ′ ∈ A : ∃B ∈ O(3) : d(φ′ ◦ φ−1) = +�1 +0 +0 +B +� +(1.1) +6 + +holds true. This definition is compatible with our definition in the following sense: Given a reference +frame, a unit of time and a unit of length, then we can consider the composition of R with all +orthonormal coordinate systems in order to obtain such an atlas. +Conversely, suppose that the +following data is given: +• A maximal atlas A satisfying (1.1). +• An affine space A1 with translation space V 1. +• An affine space A3 with translation space V 3. +• A unit of time and a unit of length. +Let O be the set of all orthonormal coordinates, then we can easily construct a function R: M → +A1 × A3 such that +A = {κ ◦ R : κ ∈ O} : +(1.2) +We simply pick some κ ∈ O and a φ ∈ A and set R := κ−1 ◦ φ. In fact, each function R satisfying +(1.2) is obviously of this form. +1.3 +Generalized Manifolds and Tangent Bundles +Consider a set of reference frames on a set M such that F ′ ◦ F −1 continuously differentiable for +each pair of reference frames. Note that there is a unique topology such that all reference frames +are homeomorphisms. +Then one way to introduce the tangent bundle is to use coordinates to +define an atlas, but this is actually a detour: It is straightforward to generalize the definition of +a differentiable manifold and its tangent bundle such that the reference frames form the atlas of a +generalized manifold. +Definition 8. Suppose that we are given a topological space M and a positive integer n. +• A generalized n-dimensional reference frame (an n-frame for short) is a pair (A, F), where A +is an n-dimensional affine space and F : M → A is a homeomorphism.2 +• Let A be a set of n-frames on M. If the transition function +F ′ ◦ F −1 : A → A′ +is differentiable for all F, F ′ ∈ A, then (M, A) is called a n-dimensional differentiable space. +Remark 3. Let M be a differentiable space. We would like to emphasize that the differentials of the +transition functions are not assumed to be continuous. If they are, then M is called a continuously +differentiable space. +Definition 9. Let M be an n-dimensional differentiable space. A pre-tangent bundle consists of +the following data: +• A set T M. +• A function π: T M → M. +2More generally, F could be a bijective function between a subset of M and a subset of A, but this is sufficient +for our purposes. Given the usual definitions of differentiable manifolds with or without boundary, a generalization +should be straightforward. +7 + +• For each p ∈ M an n-dimensional real vector space structure on TpM := π−1({p}) (in partic- +ular, TpM is non-empty for all p ∈ M, i.e. π is surjective). +Definition 10. Let (M, A) be a differentiable space. A tangent bundle is a pair (T M, d), where +T M is a pre-tangent bundle and d is a function defined on A with the following properties: +• Let (F, A) be some frame in A and V the translation space of A, then dF : T M → A × V is +bijective, +∀x ∈ A : ∀v ∈ V : dF −1(x, v) ∈ TF −1(x)M +and for all p ∈ M the obvious function dFp : TpM → V is a vector space isomorphism. Note +that we made an abuse of notation by using the same letter for a frame and the associated +bijective function, i.e. F = (A, F). This will happen throughout the rest of the paper. +• If F and F ′ are two frames in A, then dF ′ ◦ dF −1 = d(F ′ ◦ F −1). +Remark 4. Let (M, A) be a continuously differentiable space. +• We can consider the unique topology on T M such that dF is a homeomorphism for all F ∈ A. +Then the differentials of the frames form a continuous atlas for the tangent bundle. +Fur- +thermore, each differential is a trivialization of the tangent bundle and we obtain a vector +bundle. +• The tangent bundle is defined up to a natural isomorphism: If (T M ′, d′) is a second tangent +bundle, then the vector bundle isomorphism +Φ := d′F ◦ dF −1 : T M → T M ′ +does not depend on F. +• To show the existence of a tangent bundle, we first note the existence of a pre-tangent bundle: +For example, we can choose an n-dimensional real vector space TpM for each p ∈ M and then +consider the disjoint union. That being said, let T M be a pre-tangent bundle. For each p ∈ M +we can pick a reference F, choose a vector space isomorphism dFp ∈ L(TpM, V ) (where V is +the translation space of the affine space associated to F) and set +dF ′ := d(F ′ ◦ F −1)F (p) ◦ dFp +for all F ′ ∈ A. We finally obtain a tangent bundle (T M, d). +8 + +Chapter 2 +Classical Mechanics vs. Special +Relativity +2.1 +Galilean Transformations +In view of our discussion of accelerated frames it is useful to introduce Galilean groups as a subgroup +of a larger group. Furthermore, it will play an important role that the differentials of Galilean +transformations are orientation-preserving (in the sense defined below), so we begin with a technical +lemma: +Lemma 1. Let V n and W n be two real vector spaces and suppose that A ∈ L(V, W) is invertible. +Then two bases v1 . . . , vn and w1 . . . , wn have the same orientation if and only if Av1, . . . , Avn and +Aw1, . . . , Awn have the same orientation. This has two implications: +• The function A defines a bijective function between the sets of orientations. +• If V = W, then A is either orientation-preserving or orientation-inverting. +Proof. Note that if e ∈ L(V, Rn) is the vector space isomorphism associated to the basis e1, . . . , en, +then e ◦ A−1 ∈ L(V, Rn) is the vector space isomorphism associated to the basis Ae1, . . . , Aen. That +being said, suppose that e and e′ are two bases of V with the same orientation, i.e. det(e′ ◦e−1) > 0. +Then e ◦ A−1 and e′ ◦ A−1 have the same orientation as well: +det(e′ ◦ A−1 ◦ (e ◦ A−1)−1) = det(e′ ◦ e−1) > 0. +Definition 11. Let F = (A1, T, A3, Π) and F ′ = (B1, T ′, B3, Π′) be two reference frames, then +F ′ ◦ F −1 is called an element of the general kinematic group if and only if +• T ′ = φ ◦ T , where φ: A1 → B1 is affine and dφ = 1. +9 + +• ∀t ∈ A1 the function +Σt : A3 → B3 +p �→ (Π′ ◦ F −1)(t, p) +is affine. +• The image of the function +R: A1 → L(V 3, V 3) +t �→ dΣt +is a subset of the rotation group (i.e. Rt is orientation-preserving and orthogonal). +Definition 12. Let F = (A1, T, A3, Π) and F ′ = (B1, T ′, B3, Π′) be two reference frames, then the +transition function T := F ′ ◦ F −1 is a Galilean transformation if and only if +• T is an element of the general kinematic group and +• ∀p ∈ A3 the function +A1 → B3 +t �→ (Π′ ◦ F −1)(t, p) = Σt(p) +is affine and the differential is independent of p. +Remark 5. Let (V1, V2, W1, W2) be a list of vector spaces over the same field. Then each +A ∈ L(V1 ⊕ V2, W1 ⊕ W2) +can be identified with the unique matrix satisfying +∀(v1, v2) ∈ V1 ⊕ V2 : A(v1, v2) = +�A11 +A12 +A21 +A22 +� �v1 +v2 +� += +�A11v1 + A12v2 +A21v1 + A22v2 +� +and the composition of two linear operators corresponds to the product of the matrices. Note that +Aij ∈ L(Vj, Wi). +Lemma 2. Let F and F ′ be two reference frames, then T := F ′ ◦ F −1 is a Galilean transformation +if and only if T is affine and +dT = +� +1 +0 +v +R +� +for some rotation R ∈ L(V 3, V 3). +Proof. If the transition function F ′ ◦ F −1 is assumed to be a Galilean transformation, then it is +straightforward to prove that it has the properties listed above. To prove the other direction we first +show that the function R: A1 → L(V 3, V 3) from definition 11 is constant: Given some v ∈ V 3 we +can choose p, q ∈ A3 such that v = q − p and then Rtv = d(Σt)(q − p) = Σt(q) − Σt(p) for all t ∈ A1. +This implies that +A1 ∋ t �→ Rtv ∈ V 3 +is constant for each v ∈ V 3. That being said, let R be a rotation in L(V 3, V 3) for the rest of the +proof. +10 + +Note that there exists a v ∈ L(V 1, V 3) such that +Σt+u(p) = Σt(p) + vu +for all t ∈ A1, u ∈ V 1, p ∈ A3: By assumption the function A1 ∋ t �→ Σt(p) =: p(t) ∈ B3 is affine +and its differential v := dp ∈ L(V 1, V 3) is independent of p. +That being said, consider t ∈ A1, u ∈ V 1, p ∈ A3, x ∈ V 3, then +(F ′ ◦ F −1)(t + u, p + x) = (φ(t + u), Σt+u(p + x)) = (F ′ ◦ F −1)(t, p) + (u, vu + Rx) +and this shows that F ′ ◦ F −1 is affine and that its differential has the desired form. +Remark 6. By our definition the differential of a Galilean transformation is orientation-preserving. +This will allow us to identify the orientations of the tangent bundle with the orientations of V 3 in +section 2.4, but most importantly this implies that the transformation of vectors defined in 2.6 is +orientation-preserving. +Lemma 3. Consider two frames of reference F and F ′. Fix some unit of time and length. Suppose +φ and φ′ are orthonormal coordinates for F and F ′. If the bases associated to φ and φ′ have the +same orientation, then F ′ ◦ F −1 is a Galilean transformation if and only if +d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) = +�1 +0 +v +R +� +for some R ∈ SO(3). +Proof. This follows from the following facts: Suppose that A and B are two affine functions, then +B ◦ A is affine and +d(B ◦ A) = dB ◦ dA. +Moreover, if A is invertible, then A−1 is affine and d(A−1) = (dA)−1. Now the key is to realize that +φ and φ′ are affine and to compute the differentials. +2.2 +Lorentz and Poincar´e Transformations +Definition 13. Consider the bilinear form η on V 1 ⊕ V 3 defined by +∀v, w ∈ V 1 : ∀x, y ∈ V 3 : η +�v +x +� �w +y +� += ⟨v, w⟩ − ⟨x, y⟩. +A vector space endomorphism Λ on V 1 ⊕ V 3 is called a Lorentz transformation if it preserves η, +i.e. +∀u, u′ ∈ V 1 ⊕ V 3 : η(Λu, Λu′) = η(u, u′). +Remark 7. Of course an endomorphism preserves η if and only if it preserves −η. But the signature +has not been chosen arbitrarily: The most important reason is explained in remark 14 and a more +aesthetic reason is that we do not need to consider the absolute value of the metric in the definition +of proper time. +11 + +Lemma 4. Let Λ be a Lorentz transformation. Note that Λ11 ∈ L(V 1, V 1) can be identified with a +real number: If A ∈ L(V 1, V 1) and m: R × V 1 → V 1 is the scalar multiplication associated to V 1, +then there is a unique x ∈ R such that A = m(x, ). That being said, 1 ≤ |Λ11|. +Proof. Let e0 be some unit of time and (e1, e2, e3) a basis of V 3 such that +∀i, j : ⟨ei, ej⟩ +⟨e0, e0⟩ = δij. +Then we obtain a basis (e0, e1, e2, e3) of V 1 ⊕ V 3. Lastly, we define �η: V → V ∗ through +∀v, w ∈ V : η(v, w) +⟨e0, e0⟩ =: (�ηv)w. +Then +1 = η(e0, e0) +⟨e0, e0⟩ = η(�η−1e0, �η−1e0) +⟨e0, e0⟩ += η(Λ−1�η−1e0, Λ−1�η−1e0) +⟨e0, e0⟩ += η(�η−1e0Λ, �η−1e0Λ) +⟨e0, e0⟩ += +3 +� +k=0 +3 +� +l=0 +e0Λeke0Λel +η(�η−1ek, �η−1el) +⟨e0, e0⟩ += Λ0 +0Λ0 +0 − +3 +� +α=1 +Λ0 +αΛ0 +α +and thus +Λ0 +0Λ0 +0 = 1 + +3 +� +α=1 +Λ0 +αΛ0 +α +(2.1) +which concludes the proof. +Definition 14. Let Λ be a Lorentz transformation. Lemma 4 shows that Λ11 is either positive or +negative. If Λ11 is positive, then Λ is called orthochronous. If Λ is additionally orientation-preserving, +then Λ is called proper orthochronous. +Definition 15. Let R and R′ be two reference frames such that T := R′ ◦ R−1 is affine. If dT +is a proper orthochronous Lorentz transformation, then T is called a Poincar´e transformation. We +already justified the requirement of orientation-preservation in our definition of Galilean transfor- +mations. +Lemma 5. Let F and F ′ be two frames of reference. Furthermore, fix a set of natural units and +let φ and φ′ be orthonormal coordinates for F and F ′ such that the associated bases of V 3 have the +same orientation. +• F ′ ◦ F −1 is affine if and only if φ′ ◦ F ′ ◦ F −1 ◦ φ−1 is affine. +• If F ′ ◦ F −1 is affine, then d(F ′ ◦ F −1) is a Lorentz transformation if and only if d(φ′ ◦ F ′ ◦ +F −1 ◦ φ−1) is a Lorentz transformation. +• Suppose that F ′ ◦F −1 is affine and d(F ′ ◦F −1) is a Lorentz transformation. If the bases of V 3 +associated to φ and φ′ have the same orientation, then d(F ′ ◦ F −1) is proper orthochronous if +and only if d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) is proper orthochronous. +Proof. Recall the proof of lemma 3 for the first item. To prove the second item, it helps to first +introduce some new terminology: +12 + +Definition 16. Let V be some vector space over the field F. If A: V × V → F is bilinear, then the +pair (V, A) is called a bilinear space. +Definition 17. Let (V, A) and (W, B) be two bilinear spaces over the same field. Then T ∈ L(V, W) +is called product-preserving if +∀u, v ∈ V : B(T u, T v) = A(u, v). +Lemma 6. Let U, V, W be bilinear spaces over the same field. If A ∈ L(U, V ) and B ∈ L(V, W) are +product-preserving, then and A−1 ∈ L(V, U) and B ◦ A ∈ L(U, W) are product-preserving as well. +Proof. The proof is left as an exercise. +Proof of lemma 5. Now the proof is straightforward: Since coordinate systems are affine and their +differentials are product-preserving (w.r.t. to the Minkowski metric), the claim follows from the last +lemma: If d(F ′ ◦ F −1) is a Lorentz transformation, then +d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) = d(φ′) ◦ d ◦ F ′ ◦ F −1) ◦ (dφ)−1 +is a Lorentz transformation and conversely, if d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) is a Lorentz transformation, +then +d(F ′ ◦ F −1) = (dφ′)−1 ◦ d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) ◦ dφ +is a Lorentz transformation. +Since we have already proven the second item, the third item boils down to the following fact: Since +the bases dφ and dφ′ have the same orientation, the determinant of d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) is positive +if and only if d(R′ ◦ R−1) is orientation-preserving. This can easily be verified. +2.3 +Representation of Lorentz transformations +Definition 18. We define an inner product on V 1 ⊕ V 3 as follows: +∀v, w ∈ V 1 : ∀x, y ∈ V 3 : +� +v +x +� +· +� +w +y +� += ⟨v, w⟩ + ⟨x, y⟩ +A Lorentz transformation is called a Lorentz boost if it is symmetric and positive w.r.t. this inner +product. +Lemma 7. Lorentz boosts are proper orthochronous. +Proof. Consider a basis like in the proof of lemma 4, then a Lorentz transformation Λ is boost +(proper orthochronous) if and only if the matrix (eiΛej)0≤i,j≤3 is a boost (proper orthochronous) +and thus the claim boils down to theorem 2 in [4]. +Definition 19. Suppose that v ∈ L(V 1, V 3) and ∥v∥ < c. We set +γ := +� +1 − ∥v∥ +c +∥v∥ +c +�−1/2 +∈ [1, ∞[ +13 + +and we define J ∈ L(V 3, V 1) trough +∀x ∈ V 3 : Jx = ⟨vu, x⟩ +⟨u, u⟩ u +where u is some basis of V 1 (but J does not depend on the choice of u). Lastly, let P ∈ L(V 3, V 3) +be the projection of V 3 onto the image of v. Then +� γ +γJ +γv +I + (γ − 1)P +� +=: Λ(v) +can be verified to be a Lorentz boost. +Corollary 1. Let Λ be a proper orthochronous Lorentz transformation, then there exist a unique +rotation R ∈ L(V 3, V 3) and a unique v ∈ L(V 1, V 3) with ∥v∥ < c such that +Λ = +�1 +0 +0 +R +� +Λ(v). +Similarly, there exist a unique rotation R′ and a unique v′ ∈ L(V 1, V 3) with ∥v′∥ < c such that +Λ(v′) +� +1 +0 +0 +R′ +� += Λ. +In fact R = R′ and v′ = R ◦ v. +Proof. This is an immediate consequence of the following two theorems: +Theorem 1. Let Λ be a Lorentz boost, then there is a unique v ∈ L(V 1, V 3) such that ∥v∥ < c and +Λ = Λ(v). +Proof. Consider the set X := {v ∈ L(V 1, V 3) : ∥v∥ < c} and let +Y ⊂ L(V 1 ⊕ V 3, V 1 ⊕ V 3) +be the set of all boosts. It can be verified that +Λ(v) = +� γ +γJ +γv +I + (γ − 1)P +� +∈ Y +for all v ∈ X, so we want to show that the function Λ: X → Y is bijective. We do so by considering +a basis e of V 1 ⊕ V 3 like the one in the proof of lemma 4 and showing that Λ is the composition of +bijective functions: +• Consider the bijective function +A: L(V 1, V 3) → R3 +v �→ +3 +� +i=1 +(ei ◦ v)(e0) +We have A(X) = B(0, 1) and hence we obtain a bijection �A: X → B(0, 1). +14 + +• The function +B : B(0, 1) → R3 +v �→ +v +√ +1 − vtv +is bijective (the function +v: R3 → B(0, 1) +B �→ +B +√ +1 + BtB +is its inverse.) +• Let Z ⊂ R4×4 be the set of all boosts, then +C : R3 → Z +B �→ +� +γ +Bt +B +I + BBt +1+γ +� +with γ(B) := +√ +1 + BtB is a bijective function (see [4] for a proof). +• The function +D: R4×4 → L(V 1 ⊕ V 3, V 1 ⊕ V 3) +A �→ e−1 ◦ A ◦ e +is bijective and D(Z) = Y , so we can consider the bijection �D: Z → Y . +It can be verified that Λ = �D ◦ C ◦ B ◦ �A. +It is well known that each Lorentz transformation on R4 can be decomposed into a boost and a +spatial rotation (see [5] for example). Furthermore it was observed in [4] that this decomposition is +nothing but the polar decomposition. The advantage is that the polar decomposition theorem can +just as well be applied to the Lorentz transformations from definition 13: +Theorem 2. Suppose that A ∈ L(V 1 ⊕ V 3, V 1 ⊕ V 3) is invertible. +• There exists a unique pair (Λ, Ω) such that: A = ΛΩ and +– Λ is a symmetric and positive +– Ω is orthogonal +w.r.t. the inner product from definition 18. +• There exists a unique pair (Λ′, Ω′) such that: A = Λ′Ω′ and +– Λ′ is a symmetric and positive +– Ω′ is orthogonal +w.r.t. the inner product from definition 18. +• Ω = Ω′ and Λ′ = ΩΛΩ† +15 + +• Let A be a Lorentz transformation, then Λ is Lorentz transformation and hence a boost. Since +Lorentz transformations form a group (a subgroup of the group of vector space automorphisms +on V 1 ⊕ V 3), this means that Ω = Λ−1A is a Lorentz transformation. Thus, +Λ′ = ΩΛΩ−1 +is a Lorentz transformation as well. +• Now suppose that A is a proper orthochronous Lorentz transformation. +Since proper or- +thochronous transformations form a subgroup of the Lorentz group, the last item shows that +Ω is a proper orthochronous Lorentz transformation. This together with the fact that Ω is +orthogonal means that there exists a rotation R ∈ L(V 3, V 3) such that +�1 +0 +0 +R +� += Ω. +Proof. The first three items are a special case of the polar decomposition theorem in the finite- +dimensional case. See [6] for a thorough discussion. We can proceed like in the proof of lemma 7 +and then our claim that Λ is a Lorentz transformation boils down to theorem 2 in [4]. +2.4 +Orientations +Consider a set of reference frames on a set M such that all transition functions are Poincar´e trans- +formations (Galilean transformations). Then there is a unique topology on M such that all reference +frames are homeomorphisms and proposition 15.9 in [7] tells us that there are precisely two con- +tinuous orientations of the tangent bundle. Furthermore, there is a natural bijection between the +orientations of V 3 and the continuous orientations of M: +Suppse that we have chosen an orientation of V 3. Since V 1 is oriented, the orientation determines an +orientation of V 1⊕V 3. Furthermore, if F is some reference frame, then the vector space isomorphism +dFp ∈ L(TpM, V 1 ⊕ V 3) +allows us to assign an orientation to TpM for all p ∈ M (see lemma 1). The assignment is independent +of F and equals one of the two continuous orientations of M. +2.5 +World Lines +We begin this section with a summary of the main results and highlight the differences between +special relativity and Newtonian mechanics: +Consider a reference frame F on a set M and a subset W of M. Our goal is to define what it +means that W is a world line w.r.t. F such that we can prove the following result: If W is a world +line w.r.t. F and F ′ is another reference frame, then W is also a world line w.r.t. F ′. Of course +the adequate definition will depend on the assumed relation between the reference frames. In the +context of Special Relativity it is natural to require that the speed of a world line w.r.t. F does not +exceed the speed of light and we will prove the covariance of this requirement (i.e. the theory is +compatible with the experimental data). In the simpler Galilean case this is not required. +16 + +Definition 20 (World lines in special relativity). Let W be a subset of M, i: W → M the inclusion +and F a reference frame. Suppose t: M → A1 and Π: M → A3 are the two projections associated +to F. W is called a world line w.r.t. F if +• the restriction of t: M → A1 to W is injective, +• its image is an interval I ⊂ A1, +• the function +P := Π ◦ i ◦ t−1 : I → A3 +is differentiable and ∥v∥ ≤ c with v := dP : I → L(V 1, V 3). +Theorem 3. If W ⊂ M is a world line w.r.t. to a reference frame F, then W is a world line w.r.t. +every reference frame. +Proof. Suppose that W ⊂ M is a world line w.r.t. F. We want to show that W is also a world +line w.r.t. F ′. The proof consists of two parts: In the first part, we show that the restriction of the +projection t′ : M → F 1 to W is injective and that its image is an interval. In the second part, we +show that ∥dP ′ < c∥. +Part 1: Let t: W → I ⊂ A1 be the obvious bijection. It suffices to show that t′ ◦ t−1 is strictly +increasing. To do so, consider the basis e from the proof of lemma 4. It suffices to show that +dt′ +dt := d(t′ ◦ t−1)e0 +e0 +> 0 +(the LHS is obviously independent of e). Firstly, we note that t′ ◦ t−1 = Π ◦ T ◦ X where X : I → +A1 × A3 is the representation of the world line w.r.t. F, +T := F ′ ◦ F −1 : A1 × A3 → B1 × B3 +and Π: B1 × B3 → B1 is the obvious projection. Thus, if Λ := dT , then: +dt′ +dt = d(Π ◦ T ◦ X)e0 +e0 += (e0 ◦ Λ ◦ dX)e0 = e0Λ +�3 +α=0 eα(dXe0)eα += e0Λe0 + e0Λ +�3 +α=1 eαv(e0)eα = e0Λe0 + +�3 +α=1 eαv(e0)e0Λeα +Note that 0 < e0Λe0 because Λ is orthochronous, so it suffices to show that +���� +�3 +α=1 eαv(e0)e0Λeα +���� < e0Λe0. +to conclude the proof. (2.1) implies that +��3 +i=1 Λ0iΛ0i < e0Λe0 +and the condition on the speed is +��3 +α=1 eαve0 = ∥ve0∥ +u += ∥ve0∥ +ce0 += ∥v∥ +c +< 1 +17 + +Now the Cauchy Schwarz inequality delivers the desired result: +���� +�3 +α=1 eαv(e0)e0Λeα +���� ≤ +��3 +α=1 eαv(e0)eαv(e0) +��3 +α=1 e0Λeαe0Λeα +≤ +��3 +α=1 e0Λeαe0Λeα < e0Λe0 +Part 2: The key is to realize that ∥v∥ < c and +0 < ⟨e0, e0⟩ − ⟨dPe0, dPe0⟩ +⟨e0, e0⟩ += η(dXe0, dXe0) +⟨e0, e0⟩ +are equivalent. Consider the obvious function t: I′ → I, then +X′ = T ◦ X ◦ t +and hence by the chain rule +η(dX′e0, dX′e0) +⟨e0, e0⟩ += +�η(dT dXe0, dT dXe0) +⟨e0, e0⟩ +◦ t +� dt +dt′ +dt +dt′ = +�η(dXe0, dXe0) +⟨e0, e0⟩ +◦ t +� dt +dt′ +dt +dt′ > 0. +Remark 8. In Newtonian mechanics, we do not require that the speed of world line does not exceed +the speed of light, i.e. we simply drop the last item in definition 20. Then the proof of theorem 3 is +similar, but much simpler: We only have to show that dt′/dt > 0 and it follows from our definition +of Galilean transformations that dt′/dt ≡ 1. +Corollary 2. . Let F be a reference frame on a set M, then each P ∈ A3 can be identified with +a constant function P : A1 → A3 and thus with a world line P ⊂ M (the preimage of the graph of +P : A1 → A3 under F). This holds true both in special relativity and Newtonian mechanics. +Proof. This is an immediate consequence of theorem 3. +2.6 +Transformation of vectors +Theorem 4. Let F and F ′ be two reference frames such that F ′ ◦ F −1 is a Galilean transformation +or a Poincar´e transformation and recall that each point in F corresponds to a world line by corollary +2. +1. Each point in F has a constant velocity in F ′. In addition, all points have the same velocity. +This velocity is called the velocity of F w.r.t. F ′ and is denoted by v(F|F ′). This allows us +to define a function +Φ: A3 × A3 → V 3 +where Φ(P, Q) is the (time-independent) vector from P to Q in F ′. +2. If P, Q, X, Y ∈ A3 and Q − P = Y − X, then +Φ(P, Q) = Φ(X, Y ) +and thus we can define a function T (F → F ′): V 3 → V 3. +18 + +3. If F ′ ◦ F −1 is a Galilean transformation, i.e. +d(F ′ ◦ F −1) = +� +1 +0 +0 +R +� � +1 +0 +v +1 +� +for some rotation R ∈ L(V 3, V 3) and v ∈ L(V 1, V 3), then +T (F → F ′) = R. +4. If F ′ ◦ F −1 is a Poincar´e transformation, i.e. +d(F ′ ◦ F −1) = +� +1 +0 +0 +R +� � +γ +γJ +γv +I + (γ − 1)P +� +(see corollary 1), then +T (F → F ′) = R + (γ − 1)(R ◦ P) − γ(R ◦ v ◦ J). +5. In both cases v(F|F ′) = Rv and v(F ′|F) = −v. +6. The items above show that T is an orientation-preserving vector space isomorphism, i.e. each +basis is mapped to another basis with the same orientation. This is a consequence of the +requirement that the differential of a Poincar´e transformation (a Galilean transformation) is +an orientation-preserving vector space isomorphism. +Proof. +Notation: +• Given P ∈ A3, the function +A1 ∋ t �→ (t, P) ∈ A1 × A3 +will be denoted by P as well. +• Given x ∈ V 3, the function +V 1 ∋ t �→ (t, x) ∈ V 1 × V 3 +will be denoted by x as well. +• T := F ′ ◦ F −1 : A1 × A3 → B1 × B3 +• Π1 : B1 × B3 → B1 and Π3 : B1 × B3 → B3 are the canonical projections. +1. +Suppose P ∈ A3, then +P ′ := (Π3 ◦ T ◦ P) ◦ (Π1 ◦ T ◦ P)−1 : B1 → B3 +is its path in F ′. P ′ is an affine function since the composition of affine functions is affine and the +inverse of an affine function is affine. This already shows that P has a constant velocity in F ′. Now +we show that each point has the same velocity: +Consider +A := +�1 +0 +� +∈ L(V 1, V 1 ⊕ V 3), +19 + +then d(P ′) = A and thus +d(P ′) = dΠ3 ◦ dT ◦ A +� +�� +� +=γRv +◦(dΠ1 ◦ dT ◦ A +� +�� +� +=γ +)−1 = R ◦ v =: v(F|F ′) ∈ L(V 1, V 3). +2. and 4. (3. is analogous) +Choose P, Q ∈ A3 such that Q − P = x ∈ V 3. Our goal is to prove that +∀t ∈ B1 : Q(t) − P(t) = Rx + (γ − 1)(R ◦ P)x − γ(R ◦ v ◦ J)x. +Firstly, note that +(Π3 ◦ T ◦ Q) ◦ (Π1 ◦ T ◦ Q)−1 − (Π3 ◦ T ◦ P) ◦ (Π1 ◦ T ◦ P)−1 += (dΠ ◦ dT ) +� +(Π1 ◦ T ◦ Q)−1 − (Π1 ◦ T ◦ P)−1 +Q − P +� +We now prove +∀t ∈ B1 : (Π1 ◦ T ◦ Q)−1(t) − (Π1 ◦ T ◦ P)−1(t) = −Jx +since this concludes the proof: +Choose some origin O ∈ M and let +F : A1 → V 1 +�F : B1 → V 1 +G: A3 → V 3 +�G: B3 → V 3 +H : A1 × A3 → V 1 × V 3 +�H : B1 × B3 → V 1 × V 3 +be the induced bijections. Note that +(Π1 ◦ T ◦ P)−1 = F −1 ◦ ( �F ◦ Π1 ◦ �H−1 +� +�� +� +=dΠ1 +◦ �H ◦ T ◦ H−1 +� +�� +� +dT +◦ H ◦ P ◦ F −1 +� +�� +� +=P −O +)−1 ◦ �F +and thus setting ⃗P := P − O for all P ∈ A3 yields +(Π1 ◦ T ◦ Q)−1 − (Π1 ◦ T ◦ P)−1 += F −1 ◦ (dΠ1 ◦ dT ◦ ⃗Q)−1 ◦ �F − F −1 ◦ (dΠ1 ◦ dT ◦ ⃗P)−1 ◦ �F += (dΠ1 ◦ dT ◦ ⃗Q)−1 ◦ �F − (dΠ1 ◦ dT ◦ ⃗P)−1 ◦ �F. +Since +∀x ∈ V 3 : ∀t ∈ V 1 : (dΠ1 ◦ dT ◦ x)−1(t) = t +γ − Jx +we finally obtain the desired result. +2.7 +Velocity Reciprocity +Theorem 5. +• If F and F ′ measure the speed of each other, then the measured speeds are equal: +∥v(F|F ′)∥ = ∥v(F ′|F)∥ +20 + +• If an observer in F ′ represents the direction of v(F|F ′) by an arrow, then the arrow and +v(F ′|F) have opposite directions from the point of view of an observer in F. In other words, +there exists a positive real number α such that +T (F → F ′) ◦ v(F ′|F) = −αv(F|F ′) ∈ L(V 1, V 3). +• If F ′◦F −1 is a Galilean transformation, then α = 1 and if F ′◦F −1 is a Poincar´e transformation, +then +α = 1 +γ +(this is an occurrence of length contraction). +Proof. We prove the Lorentzian case, because the Galilean case is analogous and simpler: +Theorem 4 tells us that v(F|F ′) = Rv and v(F ′|F) = −v and therefore ∥v(F|F ′)∥ = ∥v(F ′|F)∥. +Since P ◦ v = v and +v ◦ J ◦ v = ∥v∥ +c +∥v∥ +c v ∈ L(V 1, V 1) +we obtain the desired result: +T (F → F ′) ◦ v(F ′|F) = −T (F → F ′) ◦ v = −(R ◦ v) − (γ − 1)(R ◦ P ◦ v) + γ(R ◦ v ◦ J ◦ v) += −γRv + γRvJv = −γ +� +1 − ∥v∥ +c +∥v∥ +c +� +Rv = −Rv +γ += −v(F|F ′) +γ +Remark 9. In Newtonian Mechanics, we may assume that d(F ′ ◦ F −1) is a Galilean boost for each +pair of reference frames - the reason is that Galilean boosts form a group. Then the equations +T (F → F ′) ◦ v(F ′|F) = −v(F|F ′) +v(P|F ′) = T (F → F ′) ◦ v(P|F) + v(F|F ′) +(where P is a world line) simplify to +v(F ′|F) = −v(F|F ′) +v(P|F ′) = v(P|F) + v(F|F ′) +since T (F → F ′) = 1 for each pair of reference frames. But there is no physical motivation for this +assumption. In fact, the assumption can be misleading: We then get the impression that velocity +reciprocity means that +∀F, F ′ : v(F ′|F) = −v(F|F ′), +but since Lorentz boosts do not form a group, it then seems like velocity reciprocity does not hold +true in the context of Special Relativity. +21 + +2.8 +Interpretation of boosts +Theorem 6. Let F and F ′ be two reference frames on M such that F ′ ◦ F −1 is a Galilean trans- +formation, φ and φ′ are orthonormal coordinates for F and F ′. If e and e′ are the bases of V 3 +associated to φ and φ′, then the differential of +φ′ ◦ F ′ ◦ F −1 ◦ φ−1 : R4 → R4 +is a boost if and only if F observes that both bases are the same, i.e. +∀i : T (F ′ → F)ei′ = ei. +Proof. Let A ∈ L(V 3, V 3) be the isomorphism defined by ∀i : ei′ = Aei, then +d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) = dφ′ ◦ d(F ′ ◦ F −1) ◦ dφ−1 += +�1 +0 +0 +e ◦ A−1 +� � +1 +0 +R ◦ v +R +� �1 +0 +0 +e−1 +� += +� +1 +0 +e ◦ A−1 ◦ R ◦ v +e ◦ A−1 ◦ R ◦ e−1 +� +and e ◦ A−1 ◦ R ◦ e−1 = 1 ⇔ A = R ⇔ A = T (F ′ → F). +Remark 10. The last theorem does not hold if F ′ ◦ F −1 is a Poincar´e transformation: If d(φ′ ◦ R′ ◦ +R−1 ◦ φ−1) happens to be a boost, the basis of F ′ is not perceived as equal to the basis of F by an +observer in F: Set κ := φ ◦ R, then this boils down to the fact that the vector space isomorphism +T (κ → κ′) ∈ L(R3, R3) +defined in the obvious way does not map the standard basis to the standard basis. +2.9 +Inertial frames and accelerated frames +Frames accelerated with respect to another frame +Let F be a frame on a set M and P ⊂ M a world line w.r.t. F. It is natural to wonder about the +existence and uniqueness of a frame F ′ (e.g. uniqueness up to an affine transformation T with +dT = +�1 +0 +0 +R +� +for some rotation R on V 3) such that +1. P is a world line w.r.t. F ′, P is at rest in F ′ and +2. all points in F ′ are world lines w.r.t. F. +We consider two simple cases: +• If P has a constant velocity w.r.t. F and the speed of P is strictly smaller than c, then we +have at least two mathematical options: We can compose F with an appropriate Galilean or +a Poincar´e transformation to obtain a frame that even has a uniform velocity w.r.t. F. +22 + +• Suppose that P performs a uniform circular motion in F. We intuitively expect to find 1. a +frame F ′ such that all points in F ′ rotate around the same axis with the same angular velocity1 +and 2. a frame F ′′ such that all points in F ′′ have the same velocity w.r.t. F (namely the +velocity of P). In fact we can consider the composition of F with appropriate transformations +in the general kinematic group to construct such frames. +In summary, the general kinematics group is a natural extension of the Galilean group which allows us +to consider accelerated frames in Newtonian mechanics: Two frames can be defined to be accelerated +w.r.t. each other if the transition functions are in the general kinematic group, but not in the Galilean +group. However, an accelerated frame is usually meant to be accelerated w.r.t. the inertial frames, +which we haven’t introduced yet. +Strictly speaking the rest of this chapter does only apply to +Newtonian mechanics, since we lack a similar extension of the Lorentz group. +Transformation of velocities and accelerations +Let F and F ′ be two reference frames on a set M such that the transition functions are elements +of the general kinematic group. In the following we use the notation from definition 11. We will +assume that φ is the identity on A1 - i.e. A1 = B1 and T = T ′. (The differential of φ is the identity +on V 1 anyways, so the generalization - if ever necessary - is trivial.) +That being said, let w ⊂ M be a world line w.r.t. +F and P : I → A3 the position w.r.t. +F. +We assume that P is twice differentiable, i.e. the velocity v: I → L(V 1, V 3) and the acceleration +a: I → Q(V 1, V 3) exist. Note that w is also a world line w.r.t. F ′ and P ′ = ΣP is the position +w.r.t. F ′. We make the following two technical assumptions: +• R and R−1 are both differentiable w.r.t. the operator norm. +• For every O ∈ A3 the function ΣO: A1 → B3 is twice differentiable. +In this situation P ′ turns out to be twice differentiable and we now determine the relation between +v and v′ as well as a and a′. To do so, we consider the functions +˙Σ: A1 × A3 → L(V 1, V 3) +and +¨Σ: A1 × A3 → Q(V 1, V 3) +defined through the requirement that ˙ΣO = d(ΣO) and ¨ΣO = d( ˙ΣO) (i.e. +˙ΣO and ¨ΣO are the +velocity and the acceleration of O w.r.t. F ′). That being said, a first application of the product rule +to P ′ = ΣP yields +v′ = Rv + ˙ΣP. +(2.2) +For later purposes it is useful to introduce B := ˙RR−1 and differentiating (2.2) yields +a′ = Ra + 2 ˙Rv + ¨ΣP = Ra + 2BRv + ¨ΣP. +(2.3) +The choice of an origin allows us to further decompose the right-hand side of (2.2) and (2.3): Suppose +that O ∈ A3 and set x := P − O, then we have ΣP = ΣO + Rx and hence by the product rule: +v′ = Rv + BRx + ˙ΣO +(2.4) +a′ = Ra + 2BRv + BBRx + ˙BRx +(2.5) +1Note that the velocity of F ′ w.r.t. F is not bounded from above: The speed of the points goes to infinity as we +move away from the rotation axis. +23 + +We finally use the following lemma to introduce the angular velocity of F w.r.t. F ′ and to rewrite +these equations in a more common form. +Lemma 8. Let A1 be an affine space with translation space V 1 and +U : A1 → L(V 3, V 3) +a function with the following properties: +• The image of U is a subset of the orthogonal group. +• U and U −1 are both differentiable. +Let ˙U be the differential of U, i.e. ˙U = dU : A1 × V 1 → L(V 3, V 3), then ˙UU −1 is anti-symmetric. +Thus, if we fix an orientation of V 3 and a unit of length, then there is a unique +ω: A1 × V 1 → V 3 +such that +˙UU −1v = ω × v +for all functions v: A1 → V 3. +Proof. Let v and w be two differentiable vector-valued functions on A1, then +⟨v, w⟩ = ⟨Uv, Uw⟩ +and hence by the product rule +⟨dv, w⟩ + ⟨v, dw⟩ = d⟨v, w⟩ = d⟨Uv, Uw⟩ = ⟨ ˙Uv + Udv, Uw⟩ + ⟨Uv, ˙Uw + Udw⟩ += ⟨ ˙Uv, Uw⟩ + ⟨Udv, Uw⟩ + ⟨Uv, ˙Uw⟩ + ⟨Uv, Udw⟩ +Because U is orthogonal, this is equivalent to +0 = ⟨ ˙Uv, Uw⟩ + ⟨Uv, ˙Uw⟩. +Since U is invertible and U −1 : A1 → L(V 3, V 3) is differentiable (the differential of U −1 equals +U −1 ˙UU −1), we can consider the differentiable functions U −1v and U −1w and we obtain +0 = ⟨ ˙UU −1v, w⟩ + ⟨v, ˙UU −1w⟩. +The function +ω = ω(F|F ′): A1 → L(V 1, V 3) +associated to B through the last lemma is called the angular velocity of F w.r.t. F ′. We use it to +rewrite (2.4) and (2.5): +v′ = Rv + ω × Rx + ˙ΣO +(2.6) +a′ = Ra + 2ω × Rv + ω × (ω × Rx) + ˙ω × Rx + ¨ΣO +(2.7) +24 + +Inertial frames in Newtonian mechanics +To define inertial frames, we fix a set of reference frames on a set M such that all transition +functions are elements of the general kinematic group. Since the Galilean group is a subgroup, we +can introduce an equivalence relation through the definition that two frames are equivalent if and +only if the transition functions are Galilean transformations. +Now the purpose of Newton’s first law is to define inertial frames, i.e. a distinguished equivalence +class: +Roughly speaking, the laws of physics discussed in Newtonian mechanics are only invariant under +Galilean transformations, so the set of inertial frames can be defined to be precisely the equivalence +class where these laws hold true. We use an example to illustrate the idea and to show how our +formulation fits together with the original formulation of Newton’s first and second law in terms of +forces: +First of all, we postulate that a finite set of world lines is given.2 Next, we postulate the existence of +a frame with the property that we can find an assignment of time-independent masses to the world +lines such that the the representations of the world lines w.r.t. to the frame form a solution of the +ODE known as the n-body problem of Newtonian mechanics. Such a frame is called inertial. Since +(2.7) reduces to a′ = Ra for Galilean transformations, all frames in its equivalence class are inertial +as well and the masses are independent of the representative. Furthermore (2.7) suggests that we +can not find another equivalence class with inertial frames, i.e. the inertial frames form precisely +one equivalence class. +If we fix a frame, then we can assign two forces to each world line: The actual force - mass times +acceleration - and the force predicted by the ODE. The two forces are equal if the frame is inertial. +If we interpret the forces mentioned in Newton’s first and second law as the forces predicted by the +ODE, then these laws are nothing but a characterization of inertial frames (and consistent with our +definition): +1. Every body continues in its state of rest, or of uniform motion in a straight line, +unless it is compelled to change that state by forces impressed upon it. +2. The change of motion of an object is proportional to the force impressed; and is +made in the direction of the straight line in which the force is impressed. +2Since the transition functions are in the general kinematic group, it makes sense to talk about world lines without +referring to a reference frame +25 + +Chapter 3 +Special Relativity and +Electromagnetism +From now on we consider a set of reference frames on a set M such that all transition functions are +Poincar´e transformations. +3.1 +Proper Time +Remark 11. Let A1 be the affine space associated to some reference frame R. Since the translation +space V 1 is oriented, A1 has an obvious total order. Moreover, given x, y ∈ A1 with x < y we +can consider the interval I := [x, y] and its order topology T . Let Σ be the Borel σ-algebra (i.e. +the smallest σ-algebra containing T ), then there is a unique locally finite vector-valued measure +µ: Σ → V 1 such that µ([p, q]) = q − p for all p, q ∈ I with p ≤ q. If φ: I → R is continuous, then φ +is bounded (because (I, T ) is a compact space). Hence φ ∈ L1(I, Σ, µ) and +� y +x +φ ∈ V 1 +is our notation for its integral. +Definition 21. Let W ⊂ M be a world line, R a reference frame and t: M → A1 the projection +associated to R. According to our definition of world lines the image of W under t is an interval +I ⊂ A1 and t: W → I is bijective. Hence, W inherits an ordering which is independent of R since +the differentials of Poincar´e transformations are orthochronous. That being said, the proper time +associated to a world line is the function +W × W → V 1 +(x, y) �→ y − x +defined as follows: Suppose that x < y and let e0 be a basis of V 1. Then the integral +y − x := +� t(y) +t(x) +� +dXe0, dXe0 +⟨e0, e0⟩ +26 + +defined in remark 11 is independent of e0 and R. If y ≤ x, then y − x := −(x − y). +Proof. To be precise, the following calculation involves two measure spaces (I, Σ, µ) and (I′, Σ′, µ′). +In addition, we make an abuse of notation by considering the obvious bijections t: W → I and +t: I′ → I. As shown in the second part of the proof of theorem 3 we have that +� +dX′e0, dX′e0 +⟨e0, e0⟩ += +� +dXe0, dXe0 +⟨e0, e0⟩ +◦ t dt +dt′ +and hence the proof boils down to a change of variables: +� t′(y) +t′(x) +� +dX′e0, dX′e0 +⟨e0, e0⟩ += +� t′(y) +t′(x) +� +dXe0, dXe0 +⟨e0, e0⟩ +◦ t dt +dt′ = +� t(y) +t(x) +� +dXe0, dXe0 +⟨e0, e0⟩ +3.2 +The Riemannian Metric +Let F and R be two reference frames on M and consider the Lorentz transformation Λ := d(R◦F −1). +Furthermore, suppose that p ∈ M and v, w ∈ TpM. Then +η(dRpv, dRpw) = η(ΛdFpv, ΛdFpw) = η(dFpv, dFpw) +and hence +ηp(v, w) := η(dFpv, dFpw) +does not depend on F. +3.3 +4-vectors +Definition 22. Let W ⊂ M be a world line, i: W → M the obvious inclusion and F a reference +frame. Note that proper time allows us to differentiate functions from W to some affine space. +• For all p ∈ W the linear operator +Up := (dFp)−1 ◦ d(F ◦ i)p ∈ L(V 1, TpM) +is called the 4-velocity at p and is clearly independent of the reference frame by our definition +of the tangent bundle/by the chain rule. +• For all p ∈ W the quadratic function +Ap : V 1 → TpM +u �→ (dFp)−1d(d(F ◦ i)u)pu +is called the 4-acceleration at p. Since all transitions functions are affine, Ap is independent +of F: If R is another reference frame, then the differential of the transition function R ◦ F −1 +is constant and hence +d(d(R ◦ i)u) = d(d(R ◦ F −1 ◦ F ◦ i)u) = d(d(R ◦ F −1)d(F ◦ i)u) = d(R ◦ F −1)d(d(F ◦ i)u). +27 + +• Furthermore, if a mass m is associated to W, then f := mA is called the 4-force. +Definition 23. Suppose a world line W, a mass m and a reference frame F are given. Furthermore, +let X : I → A3 be the trajectory w.r.t. F. Then its differential +V := dX : I → L(V 1, V 3) +is called the velocity w.r.t. F and +γ := +� +1 − ∥V ∥ +c +∥V ∥ +c +�−1/2 +: I → [1, ∞[ +is called the Lorentz factor. Furthermore, +P := γmV : I → L(V 1, V 3) +is called the momentum w.r.t. F and +F := dP : I → Q(V 1, V 3) +is called the force w.r.t. F. +Lemma 9. Suppose a world line W, a mass and a reference frame F are given. Furthermore, let +t: M → A1 be the projection associated to F. According to the definition of world lines we obtain +a bijective function t: W → I onto some interval I ⊂ A1. That being said, we have the following +representation of the 4-velocity and the 4-force w.r.t. F: Let u be a basis of V 1, then +dF ◦ Uu ◦ t−1 = γ(u, V u) +(3.1) +and +dF ◦ fu ◦ t−1 = γ(u⟨F u, V u⟩ +⟨u, u⟩ +, F u) +(3.2) +where the sections Uu: W → T M and fu: W → T M are defined in the obvious way. +Proof. We use the following two facts: +• Set τ := t−1 : I → W. According to our definition of proper time and the fundamental theorem +of calculus we have that dτ/dt = 1/γ. Thus dt/dτ = γ ◦ t according to the inverse function +rule. +• Let X : I → A1 × A3 be the 4-position w.r.t. F, then X = F ◦ i ◦ τ. +Now the proof of (3.1) is straightforward: +dF ◦ Uu ◦ τ = dF ◦ (dF)−1 ◦ d(F ◦ i)u ◦ τ = d(F ◦ i)u ◦ τ = d(F ◦ i ◦ τ ◦ t)u ◦ τ += d(X ◦ t)u ◦ τ = (dX)u(dt)u +u +◦ τ = (dX)u dt +dτ ◦ τ = γ(dX)u = γ(u, V u) +Next, we want to prove (3.2). Set U := γdX, then the equation above implies that +dF ◦ mAu ◦ τ = dF ◦ (dF)−1 ◦ md(d(F ◦ i)u)u ◦ τ = md(d(F ◦ i)u)u ◦ τ += md(d(F ◦ i)u ◦ τ ◦ t)u ◦ τ = d(mUu ◦ t)u ◦ τ = γd(mUu)u. +28 + +Furthermore +d(mUu)u = d(γmu, γmV u)u = (d(γmu)u, d(γmV u)u) = (d(γmu)u, F u). +Set x := d(γmu)u: I → V 1, then all that remains to be shown is that +γ ⟨F u, V u⟩ +⟨u, u⟩ +u = x. +Note that +η(Uu, Uu) = ⟨u, u⟩ − ⟨V u, V u⟩ +1 − ⟨V u,V u⟩ +⟨u,u⟩ += ⟨u, u⟩ +i.e. the function η(Uu, Uu): I → W 1 is constant. By the product rule +0 = η(d(mUu)u, Uu) = ⟨x, γu⟩ − ⟨γF u, γV u⟩ +or equivalently ⟨x, u⟩ = γ⟨F u, V u⟩. This implies the desired result: +x = ⟨x, u⟩ +⟨u, u⟩u = γ ⟨F u, V u⟩ +⟨u, u⟩ +u +3.4 +Covariant Electromagnetism +We begin our reformulation of classical electromagnetism. The exposure in [8] has been an important +inspiration. +From now on we assume that a set of units has been fixed and all quantities are defined w.r.t. these +units. For example, for each p ∈ M the metric +ηp : TpM × TpM → W 1 +can be identified with a physical quantity +�ηp : L → L(TpM, TpM ∗) +since each unit of length defines a unit of area and hence a basis of W 1. See remark 1 for the precise +definition of physical quantities and a discussion of the invariance of the theory under a change of +units. +Definition 24. Let n be an integer, 0 < n < 4 and p ∈ M. Given a reference frame R and a unit +of length u, the vector space isomorphism +R = Rn : Λn(TpM ∗) → Λn−1(V ∗) ⊕ Λn(V ∗) +(with V = V 3) is defined as follows: +• Firstly, note that there is a unique unit of time e0 such that ce0 = l. In addition, the vector +space isomorphism +dRp ∈ L(TpM, V 1 ⊕ V 3) +allows to identify V 1 and V 3 with subspaces of TpM. +That being said, we simply define +e0 ∈ TpM ∗ through the requirement that the restriction to V 3 is equal to zero. +29 + +• Now consider some α ∈ Λk(TpM ∗) and let +i: Λ(V ∗) → Λ(TpM ∗) +be the canonical inclusion defined by the reference frame. Since +x := e0 ⌟ α +and +y := α − e0 ∧ x +are both inside the image of i, +Rα := (i−1x, i−1y) +is well-defined. +Remark 12. From now on we assume that we are given the following data: +• A reference frame R. +• Two real-valued and positive physical quantities1 k and α with arbitrary dimensions. +In +particular, k and α may be dimensionless, e.g. k = α = 1. +• A set of world lines W with a mass and a charge associated to each world line in W. +We define charge through the requirement that Coulomb’s law takes the form +∥F ∥ = k +4π +q +d +q′ +d +where d is the distance between q and q′. +Note that the dimension of charge depends on the +dimension of k. In order to introduce the electromagnetic field we make the idealized assumption +that there exist two unique vector fields E and B from M to V 3 such that +F = q(E + α +c v × B) +for all world lines in W. (The dimensions of E and B depend on the dimensions of k and α and +are only equal if α is a speed.) We can prove the covariance of this assumption, i.e. if R′ is another +reference frame, then there exist unique vector fields E′ and B′ such that +F ′ = q(E′ + α +c v′ × B′) +for all world lines in W. In fact this is an immediate consequence of the following theorem: +Corollary 3 (Covariance of the Lorentz force). TFAE in the situation of remark 12: +• There is a unique 2-form F such that +f ♭ = q α +c U ⌟ F +for all world lines in W. +1If a real-valued physical quantity is positive w.r.t. to one set of units, then it is positive for all sets of units. +30 + +• There is a unique pair of vector fields (E, B) such that +F = q(E + α +c v × B) +for all world lines in W. +In case of existence and uniqueness, +F = R−1(−E♭/α, ∗B♭). +Proof. Note that +V 3 ⊕ V 3 → Λ2(TpM ∗) +(E, B) �→ R−1(−E♭/α, ∗B♭) +is a vector space isomorphism for each p ∈ M. That being said, the following lemma completes the +proof: +Lemma 10. Consider the situation of remark 12. If E and B are two vector fields from M to V 3 +and F = R−1(−E♭/α, ∗B♭), then we have the following equivalence for each world line in W: +F = q(E + α +c v × B) ⇔ f ♭ = q α +c U ⌟ F +Proof. Set +(E , B) := (−E♭/α, ∗B♭) +and consider the following proposition: +P := +�v +c ⌟ F ♭ = −q α +c v ⌟ E and F ♭ = qα(−E − v +c ⌟ B) +� +We conclude the proof by showing the following equivalences (the last equivalence is obvious, since +R1 is bijective): +F = q(E + α +c v × B) ⇔ P ⇔ R1(f ♭) = R1(q α +c U ⌟ F) ⇔ f ♭ = q α +c U ⌟ F +Firstly, we prove that +(E + α +c v × B)♭ = α(−E − v +c ⌟ B) +in order two obtain the first equivalence: Let Ω ∈ Λ3(V ∗) be the volume form associated to the +oriented inner product space V 3, then X ⌟ Ω = ∗X♭ (see exercise 2-28 in [9]) and hence +(v × B)♭ = B ⌟ v ⌟ Ω = −v ⌟ B ⌟ Ω = −v ⌟ B. +The second equivalence is an immediate consequence of the following two equations: +R1(f ♭) = γ( v +c ⌟ F ♭, −F ♭) +(3.3) +R1(U ⌟ F) = γ(−v ⌟ E , cE + v ⌟ B) +(3.4) +Proof of (3.3): Firstly, note that if x ∈ R and X := (dR)−1(xe0, X), then +R1(X♭) = (x, −X♭). +31 + +Now the desired equation follows from +(dR)f = γ( F ·v +c e0, F ). +Proof of (3.4): Consider x := e0 ⌟ F and y := F − e0 ∧ x. We can use +U ⌟ F = U ⌟ (e0 ∧ x) + U ⌟ y = −e0 ∧ (U ⌟ x) + (U ⌟ e0) ∧ x + U ⌟ y +and +(dR)U = γ +�ce0 +v +� +to obtain the desired result. +Axiom 1. Consider the setting from remark 12. Furthermore, suppose that +• ρ is the charge density w.r.t. R, i.e. for all measurable V ⊂ A3 the integral of ρ over V +yields the charge inside V . +• J is the current density w.r.t. R, i.e. for all surfaces S in A3 the surface integral of J over +S yields the current through S. +Then the Maxwell equations hold true: +∇ · E = kρ +∇ · B = 0 +∇ × E +α = −Le0B +∇ × B = k +α +J +c + Le0 +E +α +Remark 13. The different forms of Maxwell’s equations that appear in the literature are due to +different choices of the quantities k and α: +k +α +SI +1/ǫ0 +c +Heaviside-Lorentz +1 +1 +Gaussian +4π +1 +A similar table can be found in [10]. We emphasize that the choice of k and α has nothing to do +with a choice of units. The units can still be chosen arbitrarily. +Theorem 7. If we consider the 2-form F := R−1(−E♭/α, ∗B♭) (as explained in corollary 3, F does +not depend on R) and the vector J := (dR)−1(cρe0, J), then we have the following equivalences: +dF = 0 ⇔ + + + +∇ × E +α = −Le0B +∇ · B = 0 +and +∗d∗F = k +α +J♭ +c ⇔ + + + +∇ · E = kρ +∇ × B = k +α +J +c + Le0 +E +α +Proof. We will prove this theorem after the following remark: +32 + +Remark 14. +• The last theorem proves the covariance of Maxwell’s equations: If they hold for one reference +frame, then they hold for all reference frames. +• In addition, this shows that J := (dR)−1(cρe0, J) does not depend on the R, i.e. 4-current is +indeed a 4-vector. +• If we consider the Riemannian metric −η and still define F through corollary 3, then theorem +7 only holds true with F replaced by −F. +• Throughout this section we assumed that a continuous orientation of M had been fixed (or +equivalently an orientation of V 3, see section 2.4). But the Maxwell equations are invariant +under a change of orientation: If we consider the Maxwell equations in terms of... +– ...F, then this follows from the fact that the composition of two Hodge stars (unlike a +single Hodge star) is invariant under a change of the orientation. +– ...E and B, then this can be seen as follows: If B is the magnetic field w.r.t. +one +orientation, then −B is the magnetic field w.r.t. the other orientation. Similarly, if X is +some vector field and ∇ × X is the rotation w.r.t. one orientation, then −∇ × X is the +rotation w.r.t. the other orientation. +Proof of theorem 7. Warning: In this proof we consider two different Riemannian manifolds, the +euclidean space E3 (the affine space A3 associated to the reference frame together with the inner +product on V 3 w.r.t. +the unit of length) and Minkowski space. We use bold symbols to avoid +confusion: If β is an exterior form on E3, then dβ is its exterior differential and ∗β is its Hodge +dual. +Firstly, we use the fact that ∇ · X = ∗d∗X♭ and ∇ × X = (∗dX♭)♯ for each vector field X (see +exercise 2-28 in [9]) to rewrite Maxwell’s equations: +∗dE♭ +α = −Le0B♭ +∗d∗B♭ = 0 +∗d∗E♭ +α = k +αρ +∗dB♭ = k +α +J♭ +c + Le0 +E♭ +α +Since ∗∗ = 1 on Λ3(V ∗), we can simplify two equations: +dE♭ +α + Le0∗B♭ = 0 +d∗B♭ = 0 +∗d∗E♭ +α = k +αρ +−Le0 +E♭ +α + ∗dB♭ = k +α +J♭ +c +Now we set +(E , B) := (−E♭ +α , ∗B♭) +and rewrite the equations one more time: +−dE + Le0B = 0 +dB = 0 +−∗d∗E = k +αρ +−Le0E − ∗d∗B = − k +α +J♭ +c +33 + +Thus, it remains to be shown: +R3(dF) = 0 ⇔ +� +−dE + Le0B = 0 +dB = 0 +and +R1(∗d∗F) = R1 +J♭ +c ⇔ + + + + + +−∗d∗E = k +αρ +−Le0E − ∗d∗B = − k +α +J♭ +c +Since +R1(J♭) = (cρ, −J♭) +(see the proof of lemma 10), the next lemma concludes the proof: +Lemma 11. Suppose F is a 2-form on M and F = R−1(E , B), then: +R3(dF) = (−dE + Le0B, dB) +(3.5) +R1(∗d∗F) = (−∗d∗E , −∗d∗B − Le0E ) +(3.6) +Proof. In the following, the isomorphisms i and R from definition 24 are mostly left implicit, e.g. +F = e0 ∧ E + B = (E , B). +We start by proving (3.5) and then we use this result to prove to (3.6): +Recall that +dω = +3 +� +i=0 +ei ∧ Leiω +for each exterior form ω. On the one hand, we can use +∀i : LXei = LXdxi = d(LXxi) = d(eiX) +(see equation 4.21 in [8]) to obtain +d(e0 ∧ E ) = +3 +� +i=0 +ei ∧ Lei(e0 ∧ E ) += +3 +� +i=0 +ei ∧ ( Leie0 ∧ E +� +�� +� +=d(e0ei)∧E =0 ++e0 ∧ LeiE ) = +3 +� +i=0 +ei ∧ e0 ∧ LeiE += +3 +� +i=1 +ei ∧ e0 ∧ LeiE = −e0 ∧ +3 +� +i=1 +ei ∧ LeiE = −e0 ∧ dE +and on the other hand +dB = e0 ∧ Le0B + +3 +� +i=1 +ei ∧ LeiB = e0 ∧ Le0B + dB. +In summary, +dF = d(e0 ∧ E + B) = d(e0 ∧ E ) + dB = e0 ∧ (−dE + Le0B) + dB = (−dE + Le0B, dB). +To prove (3.6), we need the following lemma. +34 + +Lemma 12. Let V be an n-dimensional oriented real vector space together with a non-degenerate +symmetric bilinear form g. If e1, . . . , en is a positively oriented orthonormal basis of V , then +∗ei1...ik = ei1g−1ei1 · · · gikg−1eikǫi1...ikik+1...ineik+1...in +where the RHS is not a sum: Suppose 0 < k < n and i1 < . . . < ik, then (ik+1, . . . , in) is the unique +tuple such that ik+1 < . . . < in and (i1, . . . , in) is a permutation of (1, . . . , n). +Proof. For a derivation of the coordinate representation of the Hodge dual based on the coordinate +invariant definition, see page 168 in [11]. +Proof of theorem 7. Let (ei)1≤i≤3 be a positively oriented orthonormal basis of V 3, then +(dRp +−1ei)0≤i≤3 +is a positively oriented orthonormal basis of TpM and we can use (3.5) and lemma 12 to obtain that +R∗F = R∗R−1(E , B) = (∗B, −∗E ). +Then (3.5) implies that +Rd∗F = RdR−1R∗F = R∗dR−1(∗B, −∗E ) = R∗dR−1(∗B, −∗E ) = (−d∗B − Le0∗E , −d∗E ). +Let α be a 3-form on M such that α = R−1(x, y), then we can use lemma 12 one more time to show +that R∗α = (∗y, ∗x) and this concludes the proof. +35 + +Bibliography +[1] +Valter Moretti. ANALYTICAL MECHANICS. Springer, forthcoming. +[2] +Josef Janyˇska, Marco Modugno, and Raffaele Vitolo. “An Algebraic Approach to Physical +Scales”. In: Acta Appl. Math. 110.3 (2010), pp. 1249–1276. doi: 10.1007/s10440-009-9505-6. +[3] +Valter Moretti. “Teoria della Relativit`a Speciale”. In: (2020). +[4] +Valter Moretti. “The interplay of the polar decomposition theorem and the Lorentz group”. 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Cambridge University Press, 2019. doi: 10.1017/9781108557917. +36 + diff --git a/VdFIT4oBgHgl3EQfgyug/content/tmp_files/load_file.txt b/VdFIT4oBgHgl3EQfgyug/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b948c21791763b564b390d6a4dbe9ba157c32b08 --- /dev/null +++ b/VdFIT4oBgHgl3EQfgyug/content/tmp_files/load_file.txt @@ -0,0 +1,1473 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf,len=1472 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='11285v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='class-ph] 21 Jan 2023 Filippo Saatkamp1 filippo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='saatkamp@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='com Reformulation of Special Relativity and Electromagnetism in terms of Reference Frames defined as maps from spacetime onto affine spaces 1Master’s student of mathematics at the LMU Munich 1 Abstract The starting point of this paper is one of the several definitions of reference frames (frames for short) introduced in [1]: In classical mechanics a frame can be defined as a triple (E, Π, t), where E is a 3-dimensional euclidean space, Π maps spacetime M onto E and t maps M onto R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The definition allows an intuitive and coordinate-free formulation of Newtonian mechanics in terms of frames instead of coordinates [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In particular, the postulate of a set of charts on M (an atlas) is replaced by the postulate of a set of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then the charts can be re-obtained through the choice of affine coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The main point of this paper is to continue the work and to reformulate special relativity and electromagnetism in terms of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In addition, we make some modifications to the definition of frames: In order to reflect the possibility to choose different origins of time, a frame is defined to be a quadruple - the additional item is a 1-dimensional affine space A and t maps M onto A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In addition, units enter the theory as elements of positive spaces and we obtain a geometric and manifestly unit-independent reformulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Each frame allows us to identify spacetime with a 4-dimensional product-space and we can generalize the definition of differentiable manifolds such that the frames turn out to form an atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then the only difference between Newtonian mechanics and special relativity is the assumed type of transition functions - Galilean transformations or Poincar´e transformations between affine spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This allows us to highlight the common features and differences in the second chapter of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' For example, we define velocity reciprocity in mathematical terms and prove the phenomenon for both kind of transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The chapter concludes with a discussion of inertial and accelerated frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In the third chapter we restrict our attention to the reformulation of special relativity and electro- magnetism with a strong emphasis on covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' World lines are introduced as particular subsets of spacetime, the proper time of world line is defined as an affine structure on the world line and the reformulation of electromagnetism is based on the representation of differential forms w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' a frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Acknowledgements A heartfelt thanks goes to Valter Moretti for carefully reviewing the manuscript and giving valuable feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' More generally, his didactic work - in particular [1] - has been a fundamental inspiration and I am thankful for his many elaborate answers to my questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2 Contents 1 Reference Frames 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1 Mathematical setup .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='2 Reference Frames .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='4 Covariant Electromagnetism .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 29 3 Chapter 1 Reference Frames 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1 Mathematical setup Units are elements of positive spaces - this statement simply summarizes the commonly accepted axioms for units [2]: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let R+ be the set of strictly positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' A positive space is a set P equipped with two operations +: P × P → P and R+ × P → P with the following properties: + is associative and commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' For all u ∈ P : 1u = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' For all x, y ∈ R+ and u ∈ P : (x · y)u = x(yu) and (x + y)u = xu + yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' For all x ∈ R+ and u, v ∈ P : x(u + v) = xu + xv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The operation R+ × P → P is a left free and transitive action of the group (R+, · ) on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' That being said, let T be the positive space associated to the units of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' T can be extended to a 1-dimensional oriented real vector space V 1: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let P be a positive space, then its extension consists of an oriented real vector space X1 and a function i: P → X such that the image of i is equal to the positive part of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the astriction1 of i onto the image is a homomorphism of positive spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Given two extensions of P, there is an obvious identification of the positive parts and this bijection extends to a unique vector space isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Units of length are elements of a positive space L and units of area are elements of its square L2, which is defined as follows: 1Let f : A → B be some function and f(A) ⊂ U ⊂ B, then the obvious function A → U is called an astriction of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 4 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let P be a positive space, then its square is a pair (Q, (·)2) consisting of a positive space Q and a function P ∋ l �→ l2 ∈ Q such that (λu)2 = λ2u2 for all u ∈ P and λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Note that (·)2 : P → Q is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Thus, if Q′ is another square of P, then there is an obvious bijection Q → Q′ and it actually is an isomorphism of positive spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Arrows are elements of a real vector space V 3 and the inner product is a function from V 3 × V 3 to W 1, the extension of L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Since W 1 is oriented, it has a natural ordering and the proposition ∀v ∈ V 3 : 0 ≤ ⟨v, v⟩ makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The codomain of the associated norm is not the positive space L, because the length of a vector can be equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Thus, we have to introduce a new concept: Non-negative spaces, which contain a neutral element of addition and whose elements can be multiplied by non-negative real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Given the definition of positive spaces, the definition of non-negative spaces is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then the codomain of the inner product is the square root of the non-negative part of W 1, defined as follows: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let X be a non-negative space, then its square root consists of a non-negative space Y together with a function X ∋ x �→ √x ∈ Y such that √ λx = √ λ √ x for all x ∈ X and λ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Recall that two squares of the same positive space can be identified through a natural isomorphism of positive spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' A similar construction allows us to identify two square roots of the same non-negative space through an isomorphism of non-negative spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Lastly, we consider a speed c - a homomorphism of positive spaces from T to L - and the unique inner product V 1 × V 1 → W 1 satisfying ∀u ∈ T : � ⟨u, u⟩ = cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Consider some velocity v ∈ L(V 1, V 3), then the function ∥v∥: T → L u �→ ∥vu∥ is called its speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In the section on electromagnetism we consider a fixed set of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Thus it is natural to wonder about the invariance of the physical laws under a change of units - that is, if some equation holds true for one particular choice of units, how do we know that it holds true for all possible choices of units?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' To answer the question, we first reformulate it within a clear mathematical setting: In general, we consider a list of positive spaces X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , Xn (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the positive spaces associated to the base dimensions of the International System of Quantities) and a physical quantity with values in a real vector space V is a function Q: n � i=1 Xi → V 5 with a well-defined dimension, meaning that there exists a list α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , αn ∈ Q such that Q(λ1x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , λnxn) = λ1 α1 · · · λn αnQ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , xn) for all x and positive real numbers λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' That being said, the initial question can be rephrased as follows: If we consider two physical quan- tities Q, Q′: n � i=1 Xi → V then what is a sufficient condition such that the following implication holds: ∃x : Q(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , xn) = Q′(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , xn) ⇒ ∀x : Q(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , xn) = Q′(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , xn) A sufficient requirement that will always hold in practice is clearly that Q and Q′ have the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='2 Reference Frames Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' A reference frame on a set M consists of the following data: An affine space A1 with translation space V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' An affine space A3 with translation space V 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' A function t: M → A1 and a function Π: M → A3 such that the induced function F : M → A1 × A3 is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Suppose we have fixed a reference frame, a unit of time e0 and a unit of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then for each choice of an origin of time 0 ∈ A1, an origin of space O ∈ A3 and an orthonormal basis e of V 3 (orthonormal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the real valued inner product induced by the unit of length) the bijective function A1 × A3 → R × R3 (t, P) �→ (e0(t − 0), e(P − O)) is called an orthonormal coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In [3] a reference frame on M is a maximal atlas A such that ∀φ, φ′ ∈ A : ∃B ∈ O(3) : d(φ′ ◦ φ−1) = �1 0 0 B � (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1) 6 holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This definition is compatible with our definition in the following sense: Given a reference frame, a unit of time and a unit of length, then we can consider the composition of R with all orthonormal coordinate systems in order to obtain such an atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Conversely, suppose that the following data is given: A maximal atlas A satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' An affine space A1 with translation space V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' An affine space A3 with translation space V 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' A unit of time and a unit of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let O be the set of all orthonormal coordinates, then we can easily construct a function R: M → A1 × A3 such that A = {κ ◦ R : κ ∈ O} : (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='2) We simply pick some κ ∈ O and a φ ∈ A and set R := κ−1 ◦ φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In fact, each function R satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='2) is obviously of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='3 Generalized Manifolds and Tangent Bundles Consider a set of reference frames on a set M such that F ′ ◦ F −1 continuously differentiable for each pair of reference frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Note that there is a unique topology such that all reference frames are homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then one way to introduce the tangent bundle is to use coordinates to define an atlas, but this is actually a detour: It is straightforward to generalize the definition of a differentiable manifold and its tangent bundle such that the reference frames form the atlas of a generalized manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Suppose that we are given a topological space M and a positive integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' A generalized n-dimensional reference frame (an n-frame for short) is a pair (A, F), where A is an n-dimensional affine space and F : M → A is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='2 Let A be a set of n-frames on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If the transition function F ′ ◦ F −1 : A → A′ is differentiable for all F, F ′ ∈ A, then (M, A) is called a n-dimensional differentiable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let M be a differentiable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We would like to emphasize that the differentials of the transition functions are not assumed to be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If they are, then M is called a continuously differentiable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let M be an n-dimensional differentiable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' A pre-tangent bundle consists of the following data: A set T M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' A function π: T M → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2More generally, F could be a bijective function between a subset of M and a subset of A, but this is sufficient for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Given the usual definitions of differentiable manifolds with or without boundary, a generalization should be straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 7 For each p ∈ M an n-dimensional real vector space structure on TpM := π−1({p}) (in partic- ular, TpM is non-empty for all p ∈ M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' π is surjective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let (M, A) be a differentiable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' A tangent bundle is a pair (T M, d), where T M is a pre-tangent bundle and d is a function defined on A with the following properties: Let (F, A) be some frame in A and V the translation space of A, then dF : T M → A × V is bijective, ∀x ∈ A : ∀v ∈ V : dF −1(x, v) ∈ TF −1(x)M and for all p ∈ M the obvious function dFp : TpM → V is a vector space isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Note that we made an abuse of notation by using the same letter for a frame and the associated bijective function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F = (A, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This will happen throughout the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If F and F ′ are two frames in A, then dF ′ ◦ dF −1 = d(F ′ ◦ F −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let (M, A) be a continuously differentiable space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We can consider the unique topology on T M such that dF is a homeomorphism for all F ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then the differentials of the frames form a continuous atlas for the tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Fur- thermore, each differential is a trivialization of the tangent bundle and we obtain a vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The tangent bundle is defined up to a natural isomorphism: If (T M ′, d′) is a second tangent bundle, then the vector bundle isomorphism Φ := d′F ◦ dF −1 : T M → T M ′ does not depend on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' To show the existence of a tangent bundle, we first note the existence of a pre-tangent bundle: For example, we can choose an n-dimensional real vector space TpM for each p ∈ M and then consider the disjoint union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' That being said, let T M be a pre-tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' For each p ∈ M we can pick a reference F, choose a vector space isomorphism dFp ∈ L(TpM, V ) (where V is the translation space of the affine space associated to F) and set dF ′ := d(F ′ ◦ F −1)F (p) ◦ dFp for all F ′ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We finally obtain a tangent bundle (T M, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 8 Chapter 2 Classical Mechanics vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Special Relativity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1 Galilean Transformations In view of our discussion of accelerated frames it is useful to introduce Galilean groups as a subgroup of a larger group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Furthermore, it will play an important role that the differentials of Galilean transformations are orientation-preserving (in the sense defined below), so we begin with a technical lemma: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let V n and W n be two real vector spaces and suppose that A ∈ L(V, W) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then two bases v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , vn and w1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , wn have the same orientation if and only if Av1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , Avn and Aw1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , Awn have the same orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This has two implications: The function A defines a bijective function between the sets of orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If V = W, then A is either orientation-preserving or orientation-inverting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Note that if e ∈ L(V, Rn) is the vector space isomorphism associated to the basis e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , en, then e ◦ A−1 ∈ L(V, Rn) is the vector space isomorphism associated to the basis Ae1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , Aen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' That being said, suppose that e and e′ are two bases of V with the same orientation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' det(e′ ◦e−1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then e ◦ A−1 and e′ ◦ A−1 have the same orientation as well: det(e′ ◦ A−1 ◦ (e ◦ A−1)−1) = det(e′ ◦ e−1) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let F = (A1, T, A3, Π) and F ′ = (B1, T ′, B3, Π′) be two reference frames, then F ′ ◦ F −1 is called an element of the general kinematic group if and only if T ′ = φ ◦ T , where φ: A1 → B1 is affine and dφ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 9 ∀t ∈ A1 the function Σt : A3 → B3 p �→ (Π′ ◦ F −1)(t, p) is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The image of the function R: A1 → L(V 3, V 3) t �→ dΣt is a subset of the rotation group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Rt is orientation-preserving and orthogonal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let F = (A1, T, A3, Π) and F ′ = (B1, T ′, B3, Π′) be two reference frames, then the transition function T := F ′ ◦ F −1 is a Galilean transformation if and only if T is an element of the general kinematic group and ∀p ∈ A3 the function A1 → B3 t �→ (Π′ ◦ F −1)(t, p) = Σt(p) is affine and the differential is independent of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let (V1, V2, W1, W2) be a list of vector spaces over the same field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then each A ∈ L(V1 ⊕ V2, W1 ⊕ W2) can be identified with the unique matrix satisfying ∀(v1, v2) ∈ V1 ⊕ V2 : A(v1, v2) = �A11 A12 A21 A22 � �v1 v2 � = �A11v1 + A12v2 A21v1 + A22v2 � and the composition of two linear operators corresponds to the product of the matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Note that Aij ∈ L(Vj, Wi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let F and F ′ be two reference frames, then T := F ′ ◦ F −1 is a Galilean transformation if and only if T is affine and dT = � 1 0 v R � for some rotation R ∈ L(V 3, V 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If the transition function F ′ ◦ F −1 is assumed to be a Galilean transformation, then it is straightforward to prove that it has the properties listed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' To prove the other direction we first show that the function R: A1 → L(V 3, V 3) from definition 11 is constant: Given some v ∈ V 3 we can choose p, q ∈ A3 such that v = q − p and then Rtv = d(Σt)(q − p) = Σt(q) − Σt(p) for all t ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This implies that A1 ∋ t �→ Rtv ∈ V 3 is constant for each v ∈ V 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' That being said, let R be a rotation in L(V 3, V 3) for the rest of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 10 Note that there exists a v ∈ L(V 1, V 3) such that Σt+u(p) = Σt(p) + vu for all t ∈ A1, u ∈ V 1, p ∈ A3: By assumption the function A1 ∋ t �→ Σt(p) =: p(t) ∈ B3 is affine and its differential v := dp ∈ L(V 1, V 3) is independent of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' That being said, consider t ∈ A1, u ∈ V 1, p ∈ A3, x ∈ V 3, then (F ′ ◦ F −1)(t + u, p + x) = (φ(t + u), Σt+u(p + x)) = (F ′ ◦ F −1)(t, p) + (u, vu + Rx) and this shows that F ′ ◦ F −1 is affine and that its differential has the desired form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' By our definition the differential of a Galilean transformation is orientation-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This will allow us to identify the orientations of the tangent bundle with the orientations of V 3 in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='4, but most importantly this implies that the transformation of vectors defined in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='6 is orientation-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Consider two frames of reference F and F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Fix some unit of time and length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Suppose φ and φ′ are orthonormal coordinates for F and F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If the bases associated to φ and φ′ have the same orientation, then F ′ ◦ F −1 is a Galilean transformation if and only if d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) = �1 0 v R � for some R ∈ SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This follows from the following facts: Suppose that A and B are two affine functions, then B ◦ A is affine and d(B ◦ A) = dB ◦ dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Moreover, if A is invertible, then A−1 is affine and d(A−1) = (dA)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Now the key is to realize that φ and φ′ are affine and to compute the differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='2 Lorentz and Poincar´e Transformations Definition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Consider the bilinear form η on V 1 ⊕ V 3 defined by ∀v, w ∈ V 1 : ∀x, y ∈ V 3 : η �v x � �w y � = ⟨v, w⟩ − ⟨x, y⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' A vector space endomorphism Λ on V 1 ⊕ V 3 is called a Lorentz transformation if it preserves η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' ∀u, u′ ∈ V 1 ⊕ V 3 : η(Λu, Λu′) = η(u, u′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Of course an endomorphism preserves η if and only if it preserves −η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' But the signature has not been chosen arbitrarily: The most important reason is explained in remark 14 and a more aesthetic reason is that we do not need to consider the absolute value of the metric in the definition of proper time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 11 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let Λ be a Lorentz transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Note that Λ11 ∈ L(V 1, V 1) can be identified with a real number: If A ∈ L(V 1, V 1) and m: R × V 1 → V 1 is the scalar multiplication associated to V 1, then there is a unique x ∈ R such that A = m(x, ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' That being said, 1 ≤ |Λ11|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let e0 be some unit of time and (e1, e2, e3) a basis of V 3 such that ∀i, j : ⟨ei, ej⟩ ⟨e0, e0⟩ = δij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then we obtain a basis (e0, e1, e2, e3) of V 1 ⊕ V 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Lastly, we define �η: V → V ∗ through ∀v, w ∈ V : η(v, w) ⟨e0, e0⟩ =: (�ηv)w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then 1 = η(e0, e0) ⟨e0, e0⟩ = η(�η−1e0, �η−1e0) ⟨e0, e0⟩ = η(Λ−1�η−1e0, Λ−1�η−1e0) ⟨e0, e0⟩ = η(�η−1e0Λ, �η−1e0Λ) ⟨e0, e0⟩ = 3 � k=0 3 � l=0 e0Λeke0Λel η(�η−1ek, �η−1el) ⟨e0, e0⟩ = Λ0 0Λ0 0 − 3 � α=1 Λ0 αΛ0 α and thus Λ0 0Λ0 0 = 1 + 3 � α=1 Λ0 αΛ0 α (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1) which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Definition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let Λ be a Lorentz transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Lemma 4 shows that Λ11 is either positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If Λ11 is positive, then Λ is called orthochronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If Λ is additionally orientation-preserving, then Λ is called proper orthochronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Definition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let R and R′ be two reference frames such that T := R′ ◦ R−1 is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If dT is a proper orthochronous Lorentz transformation, then T is called a Poincar´e transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We already justified the requirement of orientation-preservation in our definition of Galilean transfor- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let F and F ′ be two frames of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Furthermore, fix a set of natural units and let φ and φ′ be orthonormal coordinates for F and F ′ such that the associated bases of V 3 have the same orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F ′ ◦ F −1 is affine if and only if φ′ ◦ F ′ ◦ F −1 ◦ φ−1 is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If F ′ ◦ F −1 is affine, then d(F ′ ◦ F −1) is a Lorentz transformation if and only if d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) is a Lorentz transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Suppose that F ′ ◦F −1 is affine and d(F ′ ◦F −1) is a Lorentz transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If the bases of V 3 associated to φ and φ′ have the same orientation, then d(F ′ ◦ F −1) is proper orthochronous if and only if d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) is proper orthochronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Recall the proof of lemma 3 for the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' To prove the second item, it helps to first introduce some new terminology: 12 Definition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let V be some vector space over the field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If A: V × V → F is bilinear, then the pair (V, A) is called a bilinear space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let (V, A) and (W, B) be two bilinear spaces over the same field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then T ∈ L(V, W) is called product-preserving if ∀u, v ∈ V : B(T u, T v) = A(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let U, V, W be bilinear spaces over the same field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If A ∈ L(U, V ) and B ∈ L(V, W) are product-preserving, then and A−1 ∈ L(V, U) and B ◦ A ∈ L(U, W) are product-preserving as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The proof is left as an exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof of lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Now the proof is straightforward: Since coordinate systems are affine and their differentials are product-preserving (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' to the Minkowski metric), the claim follows from the last lemma: If d(F ′ ◦ F −1) is a Lorentz transformation, then d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) = d(φ′) ◦ d ◦ F ′ ◦ F −1) ◦ (dφ)−1 is a Lorentz transformation and conversely, if d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) is a Lorentz transformation, then d(F ′ ◦ F −1) = (dφ′)−1 ◦ d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) ◦ dφ is a Lorentz transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Since we have already proven the second item, the third item boils down to the following fact: Since the bases dφ and dφ′ have the same orientation, the determinant of d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) is positive if and only if d(R′ ◦ R−1) is orientation-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This can easily be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='3 Representation of Lorentz transformations Definition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We define an inner product on V 1 ⊕ V 3 as follows: ∀v, w ∈ V 1 : ∀x, y ∈ V 3 : � v x � � w y � = ⟨v, w⟩ + ⟨x, y⟩ A Lorentz transformation is called a Lorentz boost if it is symmetric and positive w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' this inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Lorentz boosts are proper orthochronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Consider a basis like in the proof of lemma 4, then a Lorentz transformation Λ is boost (proper orthochronous) if and only if the matrix (eiΛej)0≤i,j≤3 is a boost (proper orthochronous) and thus the claim boils down to theorem 2 in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Definition 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Suppose that v ∈ L(V 1, V 3) and ∥v∥ < c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We set γ := � 1 − ∥v∥ c ∥v∥ c �−1/2 ∈ [1, ∞[ 13 and we define J ∈ L(V 3, V 1) trough ∀x ∈ V 3 : Jx = ⟨vu, x⟩ ⟨u, u⟩ u where u is some basis of V 1 (but J does not depend on the choice of u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Lastly, let P ∈ L(V 3, V 3) be the projection of V 3 onto the image of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then � γ γJ γv I + (γ − 1)P � =: Λ(v) can be verified to be a Lorentz boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let Λ be a proper orthochronous Lorentz transformation, then there exist a unique rotation R ∈ L(V 3, V 3) and a unique v ∈ L(V 1, V 3) with ∥v∥ < c such that Λ = �1 0 0 R � Λ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Similarly, there exist a unique rotation R′ and a unique v′ ∈ L(V 1, V 3) with ∥v′∥ < c such that Λ(v′) � 1 0 0 R′ � = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In fact R = R′ and v′ = R ◦ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This is an immediate consequence of the following two theorems: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let Λ be a Lorentz boost, then there is a unique v ∈ L(V 1, V 3) such that ∥v∥ < c and Λ = Λ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Consider the set X := {v ∈ L(V 1, V 3) : ∥v∥ < c} and let Y ⊂ L(V 1 ⊕ V 3, V 1 ⊕ V 3) be the set of all boosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' It can be verified that Λ(v) = � γ γJ γv I + (γ − 1)P � ∈ Y for all v ∈ X, so we want to show that the function Λ: X → Y is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We do so by considering a basis e of V 1 ⊕ V 3 like the one in the proof of lemma 4 and showing that Λ is the composition of bijective functions: Consider the bijective function A: L(V 1, V 3) → R3 v �→ 3 � i=1 (ei ◦ v)(e0) We have A(X) = B(0, 1) and hence we obtain a bijection �A: X → B(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 14 The function B : B(0, 1) → R3 v �→ v √ 1 − vtv is bijective (the function v: R3 → B(0, 1) B �→ B √ 1 + BtB is its inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=') Let Z ⊂ R4×4 be the set of all boosts, then C : R3 → Z B �→ � γ Bt B I + BBt 1+γ � with γ(B) := √ 1 + BtB is a bijective function (see [4] for a proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The function D: R4×4 → L(V 1 ⊕ V 3, V 1 ⊕ V 3) A �→ e−1 ◦ A ◦ e is bijective and D(Z) = Y , so we can consider the bijection �D: Z → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' It can be verified that Λ = �D ◦ C ◦ B ◦ �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' It is well known that each Lorentz transformation on R4 can be decomposed into a boost and a spatial rotation (see [5] for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Furthermore it was observed in [4] that this decomposition is nothing but the polar decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The advantage is that the polar decomposition theorem can just as well be applied to the Lorentz transformations from definition 13: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Suppose that A ∈ L(V 1 ⊕ V 3, V 1 ⊕ V 3) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' There exists a unique pair (Λ, Ω) such that: A = ΛΩ and – Λ is a symmetric and positive – Ω is orthogonal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the inner product from definition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' There exists a unique pair (Λ′, Ω′) such that: A = Λ′Ω′ and – Λ′ is a symmetric and positive – Ω′ is orthogonal w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the inner product from definition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Ω = Ω′ and Λ′ = ΩΛΩ† 15 Let A be a Lorentz transformation, then Λ is Lorentz transformation and hence a boost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Since Lorentz transformations form a group (a subgroup of the group of vector space automorphisms on V 1 ⊕ V 3), this means that Ω = Λ−1A is a Lorentz transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Thus, Λ′ = ΩΛΩ−1 is a Lorentz transformation as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Now suppose that A is a proper orthochronous Lorentz transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Since proper or- thochronous transformations form a subgroup of the Lorentz group, the last item shows that Ω is a proper orthochronous Lorentz transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This together with the fact that Ω is orthogonal means that there exists a rotation R ∈ L(V 3, V 3) such that �1 0 0 R � = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The first three items are a special case of the polar decomposition theorem in the finite- dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' See [6] for a thorough discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We can proceed like in the proof of lemma 7 and then our claim that Λ is a Lorentz transformation boils down to theorem 2 in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='4 Orientations Consider a set of reference frames on a set M such that all transition functions are Poincar´e trans- formations (Galilean transformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then there is a unique topology on M such that all reference frames are homeomorphisms and proposition 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='9 in [7] tells us that there are precisely two con- tinuous orientations of the tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Furthermore, there is a natural bijection between the orientations of V 3 and the continuous orientations of M: Suppse that we have chosen an orientation of V 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Since V 1 is oriented, the orientation determines an orientation of V 1⊕V 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Furthermore, if F is some reference frame, then the vector space isomorphism dFp ∈ L(TpM, V 1 ⊕ V 3) allows us to assign an orientation to TpM for all p ∈ M (see lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The assignment is independent of F and equals one of the two continuous orientations of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='5 World Lines We begin this section with a summary of the main results and highlight the differences between special relativity and Newtonian mechanics: Consider a reference frame F on a set M and a subset W of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Our goal is to define what it means that W is a world line w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F such that we can prove the following result: If W is a world line w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F and F ′ is another reference frame, then W is also a world line w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Of course the adequate definition will depend on the assumed relation between the reference frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In the context of Special Relativity it is natural to require that the speed of a world line w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F does not exceed the speed of light and we will prove the covariance of this requirement (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the theory is compatible with the experimental data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In the simpler Galilean case this is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 16 Definition 20 (World lines in special relativity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let W be a subset of M, i: W → M the inclusion and F a reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Suppose t: M → A1 and Π: M → A3 are the two projections associated to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' W is called a world line w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F if the restriction of t: M → A1 to W is injective, its image is an interval I ⊂ A1, the function P := Π ◦ i ◦ t−1 : I → A3 is differentiable and ∥v∥ ≤ c with v := dP : I → L(V 1, V 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If W ⊂ M is a world line w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' to a reference frame F, then W is a world line w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' every reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Suppose that W ⊂ M is a world line w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We want to show that W is also a world line w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The proof consists of two parts: In the first part, we show that the restriction of the projection t′ : M → F 1 to W is injective and that its image is an interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In the second part, we show that ∥dP ′ < c∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Part 1: Let t: W → I ⊂ A1 be the obvious bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' It suffices to show that t′ ◦ t−1 is strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' To do so, consider the basis e from the proof of lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' It suffices to show that dt′ dt := d(t′ ◦ t−1)e0 e0 > 0 (the LHS is obviously independent of e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Firstly, we note that t′ ◦ t−1 = Π ◦ T ◦ X where X : I → A1 × A3 is the representation of the world line w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F, T := F ′ ◦ F −1 : A1 × A3 → B1 × B3 and Π: B1 × B3 → B1 is the obvious projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Thus, if Λ := dT , then: dt′ dt = d(Π ◦ T ◦ X)e0 e0 = (e0 ◦ Λ ◦ dX)e0 = e0Λ �3 α=0 eα(dXe0)eα = e0Λe0 + e0Λ �3 α=1 eαv(e0)eα = e0Λe0 + �3 α=1 eαv(e0)e0Λeα Note that 0 < e0Λe0 because Λ is orthochronous, so it suffices to show that ���� �3 α=1 eαv(e0)e0Λeα ���� < e0Λe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' to conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1) implies that ��3 i=1 Λ0iΛ0i < e0Λe0 and the condition on the speed is ��3 α=1 eαve0 = ∥ve0∥ u = ∥ve0∥ ce0 = ∥v∥ c < 1 17 Now the Cauchy Schwarz inequality delivers the desired result: ���� �3 α=1 eαv(e0)e0Λeα ���� ≤ ��3 α=1 eαv(e0)eαv(e0) ��3 α=1 e0Λeαe0Λeα ≤ ��3 α=1 e0Λeαe0Λeα < e0Λe0 Part 2: The key is to realize that ∥v∥ < c and 0 < ⟨e0, e0⟩ − ⟨dPe0, dPe0⟩ ⟨e0, e0⟩ = η(dXe0, dXe0) ⟨e0, e0⟩ are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Consider the obvious function t: I′ → I, then X′ = T ◦ X ◦ t and hence by the chain rule η(dX′e0, dX′e0) ⟨e0, e0⟩ = �η(dT dXe0, dT dXe0) ⟨e0, e0⟩ t � dt dt′ dt dt′ = �η(dXe0, dXe0) ⟨e0, e0⟩ t � dt dt′ dt dt′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In Newtonian mechanics, we do not require that the speed of world line does not exceed the speed of light, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' we simply drop the last item in definition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then the proof of theorem 3 is similar, but much simpler: We only have to show that dt′/dt > 0 and it follows from our definition of Galilean transformations that dt′/dt ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let F be a reference frame on a set M, then each P ∈ A3 can be identified with a constant function P : A1 → A3 and thus with a world line P ⊂ M (the preimage of the graph of P : A1 → A3 under F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This holds true both in special relativity and Newtonian mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This is an immediate consequence of theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='6 Transformation of vectors Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let F and F ′ be two reference frames such that F ′ ◦ F −1 is a Galilean transformation or a Poincar´e transformation and recall that each point in F corresponds to a world line by corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Each point in F has a constant velocity in F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In addition, all points have the same velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This velocity is called the velocity of F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F ′ and is denoted by v(F|F ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This allows us to define a function Φ: A3 × A3 → V 3 where Φ(P, Q) is the (time-independent) vector from P to Q in F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If P, Q, X, Y ∈ A3 and Q − P = Y − X, then Φ(P, Q) = Φ(X, Y ) and thus we can define a function T (F → F ′): V 3 → V 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If F ′ ◦ F −1 is a Galilean transformation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' d(F ′ ◦ F −1) = � 1 0 0 R � � 1 0 v 1 � for some rotation R ∈ L(V 3, V 3) and v ∈ L(V 1, V 3), then T (F → F ′) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If F ′ ◦ F −1 is a Poincar´e transformation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' d(F ′ ◦ F −1) = � 1 0 0 R � � γ γJ γv I + (γ − 1)P � (see corollary 1), then T (F → F ′) = R + (γ − 1)(R ◦ P) − γ(R ◦ v ◦ J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In both cases v(F|F ′) = Rv and v(F ′|F) = −v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The items above show that T is an orientation-preserving vector space isomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' each basis is mapped to another basis with the same orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This is a consequence of the requirement that the differential of a Poincar´e transformation (a Galilean transformation) is an orientation-preserving vector space isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Notation: Given P ∈ A3, the function A1 ∋ t �→ (t, P) ∈ A1 × A3 will be denoted by P as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Given x ∈ V 3, the function V 1 ∋ t �→ (t, x) ∈ V 1 × V 3 will be denoted by x as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' T := F ′ ◦ F −1 : A1 × A3 → B1 × B3 Π1 : B1 × B3 → B1 and Π3 : B1 × B3 → B3 are the canonical projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Suppose P ∈ A3, then P ′ := (Π3 ◦ T ◦ P) ◦ (Π1 ◦ T ◦ P)−1 : B1 → B3 is its path in F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' P ′ is an affine function since the composition of affine functions is affine and the inverse of an affine function is affine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This already shows that P has a constant velocity in F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Now we show that each point has the same velocity: Consider A := �1 0 � ∈ L(V 1, V 1 ⊕ V 3), 19 then d(P ′) = A and thus d(P ′) = dΠ3 ◦ dT ◦ A � �� � =γRv (dΠ1 ◦ dT ◦ A � �� � =γ )−1 = R ◦ v =: v(F|F ′) ∈ L(V 1, V 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' is analogous) Choose P, Q ∈ A3 such that Q − P = x ∈ V 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Our goal is to prove that ∀t ∈ B1 : Q(t) − P(t) = Rx + (γ − 1)(R ◦ P)x − γ(R ◦ v ◦ J)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Firstly, note that (Π3 ◦ T ◦ Q) ◦ (Π1 ◦ T ◦ Q)−1 − (Π3 ◦ T ◦ P) ◦ (Π1 ◦ T ◦ P)−1 = (dΠ ◦ dT ) � (Π1 ◦ T ◦ Q)−1 − (Π1 ◦ T ◦ P)−1 Q − P � We now prove ∀t ∈ B1 : (Π1 ◦ T ◦ Q)−1(t) − (Π1 ◦ T ◦ P)−1(t) = −Jx since this concludes the proof: Choose some origin O ∈ M and let F : A1 → V 1 �F : B1 → V 1 G: A3 → V 3 �G: B3 → V 3 H : A1 × A3 → V 1 × V 3 �H : B1 × B3 → V 1 × V 3 be the induced bijections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Note that (Π1 ◦ T ◦ P)−1 = F −1 ◦ ( �F ◦ Π1 ◦ �H−1 � �� � =dΠ1 �H ◦ T ◦ H−1 � �� � dT H ◦ P ◦ F −1 � �� � =P −O )−1 ◦ �F and thus setting ⃗P := P − O for all P ∈ A3 yields (Π1 ◦ T ◦ Q)−1 − (Π1 ◦ T ◦ P)−1 = F −1 ◦ (dΠ1 ◦ dT ◦ ⃗Q)−1 ◦ �F − F −1 ◦ (dΠ1 ◦ dT ◦ ⃗P)−1 ◦ �F = (dΠ1 ◦ dT ◦ ⃗Q)−1 ◦ �F − (dΠ1 ◦ dT ◦ ⃗P)−1 ◦ �F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Since ∀x ∈ V 3 : ∀t ∈ V 1 : (dΠ1 ◦ dT ◦ x)−1(t) = t γ − Jx we finally obtain the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='7 Velocity Reciprocity Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If F and F ′ measure the speed of each other, then the measured speeds are equal: ∥v(F|F ′)∥ = ∥v(F ′|F)∥ 20 If an observer in F ′ represents the direction of v(F|F ′) by an arrow, then the arrow and v(F ′|F) have opposite directions from the point of view of an observer in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In other words, there exists a positive real number α such that T (F → F ′) ◦ v(F ′|F) = −αv(F|F ′) ∈ L(V 1, V 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If F ′◦F −1 is a Galilean transformation, then α = 1 and if F ′◦F −1 is a Poincar´e transformation, then α = 1 γ (this is an occurrence of length contraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We prove the Lorentzian case, because the Galilean case is analogous and simpler: Theorem 4 tells us that v(F|F ′) = Rv and v(F ′|F) = −v and therefore ∥v(F|F ′)∥ = ∥v(F ′|F)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Since P ◦ v = v and v ◦ J ◦ v = ∥v∥ c ∥v∥ c v ∈ L(V 1, V 1) we obtain the desired result: T (F → F ′) ◦ v(F ′|F) = −T (F → F ′) ◦ v = −(R ◦ v) − (γ − 1)(R ◦ P ◦ v) + γ(R ◦ v ◦ J ◦ v) = −γRv + γRvJv = −γ � 1 − ∥v∥ c ∥v∥ c � Rv = −Rv γ = −v(F|F ′) γ Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In Newtonian Mechanics, we may assume that d(F ′ ◦ F −1) is a Galilean boost for each pair of reference frames - the reason is that Galilean boosts form a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then the equations T (F → F ′) ◦ v(F ′|F) = −v(F|F ′) v(P|F ′) = T (F → F ′) ◦ v(P|F) + v(F|F ′) (where P is a world line) simplify to v(F ′|F) = −v(F|F ′) v(P|F ′) = v(P|F) + v(F|F ′) since T (F → F ′) = 1 for each pair of reference frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' But there is no physical motivation for this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In fact, the assumption can be misleading: We then get the impression that velocity reciprocity means that ∀F, F ′ : v(F ′|F) = −v(F|F ′), but since Lorentz boosts do not form a group, it then seems like velocity reciprocity does not hold true in the context of Special Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='8 Interpretation of boosts Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let F and F ′ be two reference frames on M such that F ′ ◦ F −1 is a Galilean trans- formation, φ and φ′ are orthonormal coordinates for F and F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If e and e′ are the bases of V 3 associated to φ and φ′, then the differential of φ′ ◦ F ′ ◦ F −1 ◦ φ−1 : R4 → R4 is a boost if and only if F observes that both bases are the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' ∀i : T (F ′ → F)ei′ = ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let A ∈ L(V 3, V 3) be the isomorphism defined by ∀i : ei′ = Aei, then d(φ′ ◦ F ′ ◦ F −1 ◦ φ−1) = dφ′ ◦ d(F ′ ◦ F −1) ◦ dφ−1 = �1 0 0 e ◦ A−1 � � 1 0 R ◦ v R � �1 0 0 e−1 � = � 1 0 e ◦ A−1 ◦ R ◦ v e ◦ A−1 ◦ R ◦ e−1 � and e ◦ A−1 ◦ R ◦ e−1 = 1 ⇔ A = R ⇔ A = T (F ′ → F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The last theorem does not hold if F ′ ◦ F −1 is a Poincar´e transformation: If d(φ′ ◦ R′ ◦ R−1 ◦ φ−1) happens to be a boost, the basis of F ′ is not perceived as equal to the basis of F by an observer in F: Set κ := φ ◦ R, then this boils down to the fact that the vector space isomorphism T (κ → κ′) ∈ L(R3, R3) defined in the obvious way does not map the standard basis to the standard basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='9 Inertial frames and accelerated frames Frames accelerated with respect to another frame Let F be a frame on a set M and P ⊂ M a world line w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' It is natural to wonder about the existence and uniqueness of a frame F ′ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' uniqueness up to an affine transformation T with dT = �1 0 0 R � for some rotation R on V 3) such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' P is a world line w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F ′, P is at rest in F ′ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' all points in F ′ are world lines w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We consider two simple cases: If P has a constant velocity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F and the speed of P is strictly smaller than c, then we have at least two mathematical options: We can compose F with an appropriate Galilean or a Poincar´e transformation to obtain a frame that even has a uniform velocity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 22 Suppose that P performs a uniform circular motion in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We intuitively expect to find 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' a frame F ′ such that all points in F ′ rotate around the same axis with the same angular velocity1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' a frame F ′′ such that all points in F ′′ have the same velocity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F (namely the velocity of P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In fact we can consider the composition of F with appropriate transformations in the general kinematic group to construct such frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In summary, the general kinematics group is a natural extension of the Galilean group which allows us to consider accelerated frames in Newtonian mechanics: Two frames can be defined to be accelerated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' each other if the transition functions are in the general kinematic group, but not in the Galilean group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' However, an accelerated frame is usually meant to be accelerated w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the inertial frames, which we haven’t introduced yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Strictly speaking the rest of this chapter does only apply to Newtonian mechanics, since we lack a similar extension of the Lorentz group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Transformation of velocities and accelerations Let F and F ′ be two reference frames on a set M such that the transition functions are elements of the general kinematic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In the following we use the notation from definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We will assume that φ is the identity on A1 - i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' A1 = B1 and T = T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' (The differential of φ is the identity on V 1 anyways, so the generalization - if ever necessary - is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=') That being said, let w ⊂ M be a world line w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F and P : I → A3 the position w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We assume that P is twice differentiable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the velocity v: I → L(V 1, V 3) and the acceleration a: I → Q(V 1, V 3) exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Note that w is also a world line w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F ′ and P ′ = ΣP is the position w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We make the following two technical assumptions: R and R−1 are both differentiable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the operator norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' For every O ∈ A3 the function ΣO: A1 → B3 is twice differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In this situation P ′ turns out to be twice differentiable and we now determine the relation between v and v′ as well as a and a′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' To do so, we consider the functions ˙Σ: A1 × A3 → L(V 1, V 3) and ¨Σ: A1 × A3 → Q(V 1, V 3) defined through the requirement that ˙ΣO = d(ΣO) and ¨ΣO = d( ˙ΣO) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' ˙ΣO and ¨ΣO are the velocity and the acceleration of O w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' That being said, a first application of the product rule to P ′ = ΣP yields v′ = Rv + ˙ΣP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='2) For later purposes it is useful to introduce B := ˙RR−1 and differentiating (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='2) yields a′ = Ra + 2 ˙Rv + ¨ΣP = Ra + 2BRv + ¨ΣP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='3) The choice of an origin allows us to further decompose the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='3): Suppose that O ∈ A3 and set x := P − O, then we have ΣP = ΣO + Rx and hence by the product rule: v′ = Rv + BRx + ˙ΣO (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='4) a′ = Ra + 2BRv + BBRx + ˙BRx (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='5) 1Note that the velocity of F ′ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F is not bounded from above: The speed of the points goes to infinity as we move away from the rotation axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 23 We finally use the following lemma to introduce the angular velocity of F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F ′ and to rewrite these equations in a more common form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let A1 be an affine space with translation space V 1 and U : A1 → L(V 3, V 3) a function with the following properties: The image of U is a subset of the orthogonal group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' U and U −1 are both differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let ˙U be the differential of U, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' ˙U = dU : A1 × V 1 → L(V 3, V 3), then ˙UU −1 is anti-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Thus, if we fix an orientation of V 3 and a unit of length, then there is a unique ω: A1 × V 1 → V 3 such that ˙UU −1v = ω × v for all functions v: A1 → V 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let v and w be two differentiable vector-valued functions on A1, then ⟨v, w⟩ = ⟨Uv, Uw⟩ and hence by the product rule ⟨dv, w⟩ + ⟨v, dw⟩ = d⟨v, w⟩ = d⟨Uv, Uw⟩ = ⟨ ˙Uv + Udv, Uw⟩ + ⟨Uv, ˙Uw + Udw⟩ = ⟨ ˙Uv, Uw⟩ + ⟨Udv, Uw⟩ + ⟨Uv, ˙Uw⟩ + ⟨Uv, Udw⟩ Because U is orthogonal, this is equivalent to 0 = ⟨ ˙Uv, Uw⟩ + ⟨Uv, ˙Uw⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Since U is invertible and U −1 : A1 → L(V 3, V 3) is differentiable (the differential of U −1 equals U −1 ˙UU −1), we can consider the differentiable functions U −1v and U −1w and we obtain 0 = ⟨ ˙UU −1v, w⟩ + ⟨v, ˙UU −1w⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The function ω = ω(F|F ′): A1 → L(V 1, V 3) associated to B through the last lemma is called the angular velocity of F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We use it to rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='5): v′ = Rv + ω × Rx + ˙ΣO (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='6) a′ = Ra + 2ω × Rv + ω × (ω × Rx) + ˙ω × Rx + ¨ΣO (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='7) 24 Inertial frames in Newtonian mechanics To define inertial frames, we fix a set of reference frames on a set M such that all transition functions are elements of the general kinematic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Since the Galilean group is a subgroup, we can introduce an equivalence relation through the definition that two frames are equivalent if and only if the transition functions are Galilean transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Now the purpose of Newton’s first law is to define inertial frames, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' a distinguished equivalence class: Roughly speaking, the laws of physics discussed in Newtonian mechanics are only invariant under Galilean transformations, so the set of inertial frames can be defined to be precisely the equivalence class where these laws hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We use an example to illustrate the idea and to show how our formulation fits together with the original formulation of Newton’s first and second law in terms of forces: First of all, we postulate that a finite set of world lines is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='2 Next, we postulate the existence of a frame with the property that we can find an assignment of time-independent masses to the world lines such that the the representations of the world lines w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' to the frame form a solution of the ODE known as the n-body problem of Newtonian mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Such a frame is called inertial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Since (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='7) reduces to a′ = Ra for Galilean transformations, all frames in its equivalence class are inertial as well and the masses are independent of the representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Furthermore (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='7) suggests that we can not find another equivalence class with inertial frames, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the inertial frames form precisely one equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If we fix a frame, then we can assign two forces to each world line: The actual force - mass times acceleration - and the force predicted by the ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The two forces are equal if the frame is inertial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If we interpret the forces mentioned in Newton’s first and second law as the forces predicted by the ODE, then these laws are nothing but a characterization of inertial frames (and consistent with our definition): 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The change of motion of an object is proportional to the force impressed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' and is made in the direction of the straight line in which the force is impressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2Since the transition functions are in the general kinematic group, it makes sense to talk about world lines without referring to a reference frame 25 Chapter 3 Special Relativity and Electromagnetism From now on we consider a set of reference frames on a set M such that all transition functions are Poincar´e transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1 Proper Time Remark 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let A1 be the affine space associated to some reference frame R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Since the translation space V 1 is oriented, A1 has an obvious total order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Moreover, given x, y ∈ A1 with x < y we can consider the interval I := [x, y] and its order topology T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let Σ be the Borel σ-algebra (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the smallest σ-algebra containing T ), then there is a unique locally finite vector-valued measure µ: Σ → V 1 such that µ([p, q]) = q − p for all p, q ∈ I with p ≤ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If φ: I → R is continuous, then φ is bounded (because (I, T ) is a compact space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Hence φ ∈ L1(I, Σ, µ) and � y x φ ∈ V 1 is our notation for its integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Definition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let W ⊂ M be a world line, R a reference frame and t: M → A1 the projection associated to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' According to our definition of world lines the image of W under t is an interval I ⊂ A1 and t: W → I is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Hence, W inherits an ordering which is independent of R since the differentials of Poincar´e transformations are orthochronous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' That being said, the proper time associated to a world line is the function W × W → V 1 (x, y) �→ y − x defined as follows: Suppose that x < y and let e0 be a basis of V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then the integral y − x := � t(y) t(x) � dXe0, dXe0 ⟨e0, e0⟩ 26 defined in remark 11 is independent of e0 and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If y ≤ x, then y − x := −(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' To be precise, the following calculation involves two measure spaces (I, Σ, µ) and (I′, Σ′, µ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In addition, we make an abuse of notation by considering the obvious bijections t: W → I and t: I′ → I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' As shown in the second part of the proof of theorem 3 we have that � dX′e0, dX′e0 ⟨e0, e0⟩ = � dXe0, dXe0 ⟨e0, e0⟩ t dt dt′ and hence the proof boils down to a change of variables: � t′(y) t′(x) � dX′e0, dX′e0 ⟨e0, e0⟩ = � t′(y) t′(x) � dXe0, dXe0 ⟨e0, e0⟩ t dt dt′ = � t(y) t(x) � dXe0, dXe0 ⟨e0, e0⟩ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='2 The Riemannian Metric Let F and R be two reference frames on M and consider the Lorentz transformation Λ := d(R◦F −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Furthermore, suppose that p ∈ M and v, w ∈ TpM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then η(dRpv, dRpw) = η(ΛdFpv, ΛdFpw) = η(dFpv, dFpw) and hence ηp(v, w) := η(dFpv, dFpw) does not depend on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='3 4-vectors Definition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let W ⊂ M be a world line, i: W → M the obvious inclusion and F a reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Note that proper time allows us to differentiate functions from W to some affine space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' For all p ∈ W the linear operator Up := (dFp)−1 ◦ d(F ◦ i)p ∈ L(V 1, TpM) is called the 4-velocity at p and is clearly independent of the reference frame by our definition of the tangent bundle/by the chain rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' For all p ∈ W the quadratic function Ap : V 1 → TpM u �→ (dFp)−1d(d(F ◦ i)u)pu is called the 4-acceleration at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Since all transitions functions are affine, Ap is independent of F: If R is another reference frame, then the differential of the transition function R ◦ F −1 is constant and hence d(d(R ◦ i)u) = d(d(R ◦ F −1 ◦ F ◦ i)u) = d(d(R ◦ F −1)d(F ◦ i)u) = d(R ◦ F −1)d(d(F ◦ i)u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 27 Furthermore, if a mass m is associated to W, then f := mA is called the 4-force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Definition 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Suppose a world line W, a mass m and a reference frame F are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Furthermore, let X : I → A3 be the trajectory w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then its differential V := dX : I → L(V 1, V 3) is called the velocity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F and γ := � 1 − ∥V ∥ c ∥V ∥ c �−1/2 : I → [1, ∞[ is called the Lorentz factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Furthermore, P := γmV : I → L(V 1, V 3) is called the momentum w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F and F := dP : I → Q(V 1, V 3) is called the force w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Suppose a world line W, a mass and a reference frame F are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Furthermore, let t: M → A1 be the projection associated to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' According to the definition of world lines we obtain a bijective function t: W → I onto some interval I ⊂ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' That being said, we have the following representation of the 4-velocity and the 4-force w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F: Let u be a basis of V 1, then dF ◦ Uu ◦ t−1 = γ(u, V u) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1) and dF ◦ fu ◦ t−1 = γ(u⟨F u, V u⟩ ⟨u, u⟩ , F u) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='2) where the sections Uu: W → T M and fu: W → T M are defined in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We use the following two facts: Set τ := t−1 : I → W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' According to our definition of proper time and the fundamental theorem of calculus we have that dτ/dt = 1/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Thus dt/dτ = γ ◦ t according to the inverse function rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let X : I → A1 × A3 be the 4-position w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F, then X = F ◦ i ◦ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Now the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1) is straightforward: dF ◦ Uu ◦ τ = dF ◦ (dF)−1 ◦ d(F ◦ i)u ◦ τ = d(F ◦ i)u ◦ τ = d(F ◦ i ◦ τ ◦ t)u ◦ τ = d(X ◦ t)u ◦ τ = (dX)u(dt)u u τ = (dX)u dt dτ ◦ τ = γ(dX)u = γ(u, V u) Next, we want to prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Set U := γdX, then the equation above implies that dF ◦ mAu ◦ τ = dF ◦ (dF)−1 ◦ md(d(F ◦ i)u)u ◦ τ = md(d(F ◦ i)u)u ◦ τ = md(d(F ◦ i)u ◦ τ ◦ t)u ◦ τ = d(mUu ◦ t)u ◦ τ = γd(mUu)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 28 Furthermore d(mUu)u = d(γmu, γmV u)u = (d(γmu)u, d(γmV u)u) = (d(γmu)u, F u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Set x := d(γmu)u: I → V 1, then all that remains to be shown is that γ ⟨F u, V u⟩ ⟨u, u⟩ u = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Note that η(Uu, Uu) = ⟨u, u⟩ − ⟨V u, V u⟩ 1 − ⟨V u,V u⟩ ⟨u,u⟩ = ⟨u, u⟩ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the function η(Uu, Uu): I → W 1 is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' By the product rule 0 = η(d(mUu)u, Uu) = ⟨x, γu⟩ − ⟨γF u, γV u⟩ or equivalently ⟨x, u⟩ = γ⟨F u, V u⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' This implies the desired result: x = ⟨x, u⟩ ⟨u, u⟩u = γ ⟨F u, V u⟩ ⟨u, u⟩ u 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='4 Covariant Electromagnetism We begin our reformulation of classical electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The exposure in [8] has been an important inspiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' From now on we assume that a set of units has been fixed and all quantities are defined w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' these units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' For example, for each p ∈ M the metric ηp : TpM × TpM → W 1 can be identified with a physical quantity �ηp : L → L(TpM, TpM ∗) since each unit of length defines a unit of area and hence a basis of W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' See remark 1 for the precise definition of physical quantities and a discussion of the invariance of the theory under a change of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Definition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let n be an integer, 0 < n < 4 and p ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Given a reference frame R and a unit of length u, the vector space isomorphism R = Rn : Λn(TpM ∗) → Λn−1(V ∗) ⊕ Λn(V ∗) (with V = V 3) is defined as follows: Firstly, note that there is a unique unit of time e0 such that ce0 = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In addition, the vector space isomorphism dRp ∈ L(TpM, V 1 ⊕ V 3) allows to identify V 1 and V 3 with subspaces of TpM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' That being said, we simply define e0 ∈ TpM ∗ through the requirement that the restriction to V 3 is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 29 Now consider some α ∈ Λk(TpM ∗) and let i: Λ(V ∗) → Λ(TpM ∗) be the canonical inclusion defined by the reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Since x := e0 ⌟ α and y := α − e0 ∧ x are both inside the image of i, Rα := (i−1x, i−1y) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' From now on we assume that we are given the following data: A reference frame R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Two real-valued and positive physical quantities1 k and α with arbitrary dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In particular, k and α may be dimensionless, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' k = α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' A set of world lines W with a mass and a charge associated to each world line in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We define charge through the requirement that Coulomb’s law takes the form ∥F ∥ = k 4π q d q′ d where d is the distance between q and q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Note that the dimension of charge depends on the dimension of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In order to introduce the electromagnetic field we make the idealized assumption that there exist two unique vector fields E and B from M to V 3 such that F = q(E + α c v × B) for all world lines in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' (The dimensions of E and B depend on the dimensions of k and α and are only equal if α is a speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=') We can prove the covariance of this assumption, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' if R′ is another reference frame, then there exist unique vector fields E′ and B′ such that F ′ = q(E′ + α c v′ × B′) for all world lines in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In fact this is an immediate consequence of the following theorem: Corollary 3 (Covariance of the Lorentz force).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' TFAE in the situation of remark 12: There is a unique 2-form F such that f ♭ = q α c U ⌟ F for all world lines in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 1If a real-valued physical quantity is positive w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' to one set of units, then it is positive for all sets of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 30 There is a unique pair of vector fields (E, B) such that F = q(E + α c v × B) for all world lines in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In case of existence and uniqueness, F = R−1(−E♭/α, ∗B♭).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Note that V 3 ⊕ V 3 → Λ2(TpM ∗) (E, B) �→ R−1(−E♭/α, ∗B♭) is a vector space isomorphism for each p ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' That being said, the following lemma completes the proof: Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Consider the situation of remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If E and B are two vector fields from M to V 3 and F = R−1(−E♭/α, ∗B♭), then we have the following equivalence for each world line in W: F = q(E + α c v × B) ⇔ f ♭ = q α c U ⌟ F Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Set (E ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' B) := (−E♭/α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' ∗B♭) and consider the following proposition: P := �v c ⌟ F ♭ = −q α c v ⌟ E and F ♭ = qα(−E − v c ⌟ B) � We conclude the proof by showing the following equivalences (the last equivalence is obvious,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' since R1 is bijective): F = q(E + α c v × B) ⇔ P ⇔ R1(f ♭) = R1(q α c U ⌟ F) ⇔ f ♭ = q α c U ⌟ F Firstly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' we prove that (E + α c v × B)♭ = α(−E − v c ⌟ B) in order two obtain the first equivalence: Let Ω ∈ Λ3(V ∗) be the volume form associated to the oriented inner product space V 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' then X ⌟ Ω = ∗X♭ (see exercise 2-28 in [9]) and hence (v × B)♭ = B ⌟ v ⌟ Ω = −v ⌟ B ⌟ Ω = −v ⌟ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The second equivalence is an immediate consequence of the following two equations: R1(f ♭) = γ( v c ⌟ F ♭, −F ♭) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='3) R1(U ⌟ F) = γ(−v ⌟ E , cE + v ⌟ B) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='4) Proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='3): Firstly, note that if x ∈ R and X := (dR)−1(xe0, X), then R1(X♭) = (x, −X♭).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 31 Now the desired equation follows from (dR)f = γ( F ·v c e0, F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='4): Consider x := e0 ⌟ F and y := F − e0 ∧ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We can use U ⌟ F = U ⌟ (e0 ∧ x) + U ⌟ y = −e0 ∧ (U ⌟ x) + (U ⌟ e0) ∧ x + U ⌟ y and (dR)U = γ �ce0 v � to obtain the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Axiom 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Consider the setting from remark 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Furthermore, suppose that ρ is the charge density w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' for all measurable V ⊂ A3 the integral of ρ over V yields the charge inside V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' J is the current density w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' for all surfaces S in A3 the surface integral of J over S yields the current through S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then the Maxwell equations hold true: ∇ · E = kρ ∇ · B = 0 ∇ × E α = −Le0B ∇ × B = k α J c + Le0 E α Remark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The different forms of Maxwell’s equations that appear in the literature are due to different choices of the quantities k and α: k α SI 1/ǫ0 c Heaviside-Lorentz 1 1 Gaussian 4π 1 A similar table can be found in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We emphasize that the choice of k and α has nothing to do with a choice of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The units can still be chosen arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If we consider the 2-form F := R−1(−E♭/α, ∗B♭) (as explained in corollary 3, F does not depend on R) and the vector J := (dR)−1(cρe0, J), then we have the following equivalences: dF = 0 ⇔ \uf8f1 \uf8f2 \uf8f3 ∇ × E α = −Le0B ∇ · B = 0 and ∗d∗F = k α J♭ c ⇔ \uf8f1 \uf8f2 \uf8f3 ∇ · E = kρ ∇ × B = k α J c + Le0 E α Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We will prove this theorem after the following remark: 32 Remark 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The last theorem proves the covariance of Maxwell’s equations: If they hold for one reference frame, then they hold for all reference frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In addition, this shows that J := (dR)−1(cρe0, J) does not depend on the R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 4-current is indeed a 4-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If we consider the Riemannian metric −η and still define F through corollary 3, then theorem 7 only holds true with F replaced by −F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Throughout this section we assumed that a continuous orientation of M had been fixed (or equivalently an orientation of V 3, see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' But the Maxwell equations are invariant under a change of orientation: If we consider the Maxwell equations in terms of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' – .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='F, then this follows from the fact that the composition of two Hodge stars (unlike a single Hodge star) is invariant under a change of the orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' – .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='E and B, then this can be seen as follows: If B is the magnetic field w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' one orientation, then −B is the magnetic field w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the other orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Similarly, if X is some vector field and ∇ × X is the rotation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' one orientation, then −∇ × X is the rotation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the other orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof of theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Warning: In this proof we consider two different Riemannian manifolds, the euclidean space E3 (the affine space A3 associated to the reference frame together with the inner product on V 3 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the unit of length) and Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We use bold symbols to avoid confusion: If β is an exterior form on E3, then dβ is its exterior differential and ∗β is its Hodge dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Firstly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' we use the fact that ∇ · X = ∗d∗X♭ and ∇ × X = (∗dX♭)♯ for each vector field X (see exercise 2-28 in [9]) to rewrite Maxwell’s equations: ∗dE♭ α = −Le0B♭ ∗d∗B♭ = 0 ∗d∗E♭ α = k αρ ∗dB♭ = k α J♭ c + Le0 E♭ α Since ∗∗ = 1 on Λ3(V ∗),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' we can simplify two equations: dE♭ α + Le0∗B♭ = 0 d∗B♭ = 0 ∗d∗E♭ α = k αρ −Le0 E♭ α + ∗dB♭ = k α J♭ c Now we set (E ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' B) := (−E♭ α ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' ∗B♭) and rewrite the equations one more time: −dE + Le0B = 0 dB = 0 −∗d∗E = k αρ −Le0E − ∗d∗B = − k α J♭ c 33 Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' it remains to be shown: R3(dF) = 0 ⇔ � −dE + Le0B = 0 dB = 0 and R1(∗d∗F) = R1 J♭ c ⇔ \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∗d∗E = k αρ −Le0E − ∗d∗B = − k α J♭ c Since R1(J♭) = (cρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' −J♭) (see the proof of lemma 10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' the next lemma concludes the proof: Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Suppose F is a 2-form on M and F = R−1(E , B), then: R3(dF) = (−dE + Le0B, dB) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='5) R1(∗d∗F) = (−∗d∗E , −∗d∗B − Le0E ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In the following, the isomorphisms i and R from definition 24 are mostly left implicit, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' F = e0 ∧ E + B = (E , B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' We start by proving (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='5) and then we use this result to prove to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='6): Recall that dω = 3 � i=0 ei ∧ Leiω for each exterior form ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' On the one hand, we can use ∀i : LXei = LXdxi = d(LXxi) = d(eiX) (see equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='21 in [8]) to obtain d(e0 ∧ E ) = 3 � i=0 ei ∧ Lei(e0 ∧ E ) = 3 � i=0 ei ∧ ( Leie0 ∧ E � �� � =d(e0ei)∧E =0 +e0 ∧ LeiE ) = 3 � i=0 ei ∧ e0 ∧ LeiE = 3 � i=1 ei ∧ e0 ∧ LeiE = −e0 ∧ 3 � i=1 ei ∧ LeiE = −e0 ∧ dE and on the other hand dB = e0 ∧ Le0B + 3 � i=1 ei ∧ LeiB = e0 ∧ Le0B + dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In summary, dF = d(e0 ∧ E + B) = d(e0 ∧ E ) + dB = e0 ∧ (−dE + Le0B) + dB = (−dE + Le0B, dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' To prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='6), we need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 34 Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let V be an n-dimensional oriented real vector space together with a non-degenerate symmetric bilinear form g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' If e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , en is a positively oriented orthonormal basis of V , then ∗ei1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='ik = ei1g−1ei1 · · · gikg−1eikǫi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='ikik+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='ineik+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='in where the RHS is not a sum: Suppose 0 < k < n and i1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' < ik, then (ik+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , in) is the unique tuple such that ik+1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' < in and (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , in) is a permutation of (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' , n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' For a derivation of the coordinate representation of the Hodge dual based on the coordinate invariant definition, see page 168 in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Proof of theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let (ei)1≤i≤3 be a positively oriented orthonormal basis of V 3, then (dRp −1ei)0≤i≤3 is a positively oriented orthonormal basis of TpM and we can use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='5) and lemma 12 to obtain that R∗F = R∗R−1(E , B) = (∗B, −∗E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='5) implies that Rd∗F = RdR−1R∗F = R∗dR−1(∗B, −∗E ) = R∗dR−1(∗B, −∗E ) = (−d∗B − Le0∗E , −d∗E ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Let α be a 3-form on M such that α = R−1(x, y), then we can use lemma 12 one more time to show that R∗α = (∗y, ∗x) and this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 35 Bibliography [1] Valter Moretti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' ANALYTICAL MECHANICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Springer, forthcoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' [2] Josef Janyˇska, Marco Modugno, and Raffaele Vitolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' “An Algebraic Approach to Physical Scales”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In: Acta Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='3 (2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 1249–1276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1007/s10440-009-9505-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' [3] Valter Moretti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' “Teoria della Relativit`a Speciale”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In: (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' [4] Valter Moretti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' “The interplay of the polar decomposition theorem and the Lorentz group”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In: (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='MATH-PH/0211047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='org/abs/math-ph/0211047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' [5] Helmuth K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Urbantke Roman U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Sexl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Relativity, Groups, Particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Springer, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' isbn: 978-3-211-83443-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' [6] Valter Moretti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' “Geometric Methods in Mathematical Physics I”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' In: (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' [7] John M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Introduction to Smooth Manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Springer, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1007/978-1-4419-9982-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' [8] Theodore Frankel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' The Geometry of Physics: An Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Cambridge University Press, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1017/CBO9781139061377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' [9] John M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Introduction to Riemannian Manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1007/978-3-319-91755-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' [10] John David Jackson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Classical Electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Wiley, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' isbn: 978-0-471-30932-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' [11] Alexander Altland and Jan von Delft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Mathematics for Physicists: Introductory Concepts and Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' Cambridge University Press, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content='1017/9781108557917.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFIT4oBgHgl3EQfgyug/content/2301.11285v1.pdf'} +page_content=' 36' metadata={'source': 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The classification of isomorphism classes of group grad- +ings on a given algebra is an interesting problem. However, not much +has been done in the context of non-simple algebras and the infinite- +dimensional ones. In this work, we focus our attention on an algebra +that is neither simple nor finite dimensional. More precisely, we classify +the group gradings on the infinite upper triangular matrix algebra. The +definition of an infinite-dimensional upper triangular matrix algebra is +a particular case of a triangularizable algebra, in the sense defined by +Mesyan [26]. The topology on the infinite-dimensional algebras play a +fundamental role in our main results. +1. Introduction +The classification of group gradings on algebras has been attracting the +attention of researchers since the paper by Patera and Zassenhauss [28]. The +authors started a systematic study of group gradings, focused on simple Lie +algebras. +Afterwards, several ground-breaking papers have appeared, for +instance [6, 7, 8, 10, 5, 14, 16]. The state-of-the-art may be found in the +monograph [17]. +Almost all the efforts on the classification problem aim to study the finite- +dimensional algebras. +Nonetheless, there are a lot of important infinite- +dimensional graded algebras, for instance, the polynomial algebras, the +graded algebra obtained from a filtration, Leviatt path algebras (see, for +instance, [18]), just to mention a few examples. The problem of classifying +the group gradings on a given infinite-dimensional algebra is far from trivial +and not much is known. +One of the first papers dealing directly with such a problem is [9], where +the authors classified the gradings on infinite matrices with finitely many +nonzero entries on an algebraically closed field of characteristic zero. +In +another direction, using Functional Identities techniques, the authors were +able to obtain interesting consequences to the structure of certain group +gradings on Lie and Jordan algebras in [2, 4]. +Next, as a generalization +of the previous work, in [3], the authors classified the group gradings on +2010 Mathematics Subject Classification. 16W50. +The second named author acknowledges the support of the S˜ao Paulo Research Foun- +dation (FAPESP), grant no. 2018/23690-6. +1 + +2 +WALDECK SCH¨UTZER AND FELIPE YUKIHIDE YASUMURA +the infinite-dimensional primitive algebras with minimal ideals. As a con- +sequence, they obtain a classification of the abelian group gradings on the +infinite-dimensional finitary simple Lie algebras. +All works mentioned deal with algebras which are closely related to the +simple ones. On another direction, there are some papers dedicated to clas- +sify the group gradings on a non-simple algebra. +Bahturin classified the +group gradings on a free nilpotent Lie algebra [1]. On the class of finite- +dimensional associative algebras, the isomorphism classes of group gradings +on the upper triangular matrices UTnF over an arbitrary field F were clas- +sified in [13, 31, 32] and it was shown that every group grading on UTnF is +elementary. Moreover, a classification of the graded isomorphism classes was +obtained. Next, a classification of group gradings on upper-block triangular +matrices was obtained in [12, 33, 21, 34]. The works [11, 20, 27, 29, 30] +investigated the group gradings on the incidence algebras. It seems that +following this trend the next natural step would be to consider the upper +triangular matrices in the infinite-dimensional setting. However, there is no +unique way to define an infinite upper triangular matrix. One such way, as +done in [9], would be to consider the direct limit lim +→ UTn. A much more +interesting possibility has recently been opened up by Mesyan [25], alowing +one to construct an algebra of infinite upper triangular matrices that not +only can have inifitely many nonzero columns but even uncountably many, +yet retaining only finitely many nonzero entries in each column. See also +[15, 26]. +Following these works, we let V be an (infinite-dimensional) vector space +over a field F and (β, ≤) be a well-ordered basis of V. +For each v ∈ β, +let V(v) = Span{w ∈ β | w ≤ v}. Mesyan defines an element t ∈ EndF V +as being upper triangular with respect to (β, ≤) if t(v) ∈ V(v) for all v ∈ +β. +Here we define UTβ V as the subset of all upper triangular elements +in EndF V with respect to the basis β (cf [25, Definition 4]). +This is a +triangularizable algebra in the sense of [26, Definition 2.1]. +For each pair of nonzero vectors u, v ∈ β we let euv be the unique trans- +formation in EndF V defined by +euv(w) = +� +u, +if w = v, +0, +otherwise, +for all w ∈ β, called matrix units relative to β. In particular, evv is the +projection of V onto the subspace spanned by v with kernel Span(β \ {v}), +and every t ∈ EndF V satisfies euutevv = auveuv for some auv ∈ F. We find +it convenient to define t(u, v) = auv, so that euutevv = t(u, v)euv for all +u, v ∈ β. +Other examples of triangularizable algebras which are of interest to us +are UTfin +β V = {t ∈ UTβ V | dim t(V) < +∞} and UT→β V = Span{euv | +u ≤ v ∈ β}. The latter can be identified with direct limit lim +→ UTn F, when +β = {vn | n ∈ N} is countable. +The algebra UTβ V is clearly a unital + +GROUP GRADINGS ON INFINITE UPPER TRIANGULAR MATRICES +3 +subalgebra of EndF V and UTfin +β V is a (non-unital) subalgebra of UTβ V. +Furthermore, from [26, Lemma 2.5], the closure of UTβ V in the function +topology of EndF V is still triangular with respect to β, hence UTβ V is +topologically closed in EndF V. +Let G be any group, and recall that an F-algebra A is G-graded if there +exist subspaces Ag, for g ∈ G, satisfying A = � +g∈G Ag and AgAh ⊆ Agh, +for all g, h ∈ G. +In direct analogy with the finite dimensional case, we +consider good and elementary gradings as follows: +Definition 1. A G-grading on A, where UT→β V ⊆ A ⊆ UTβ V is a good +grading if every matrix unit euv (u, v ∈ β) is homogeneous in the grading. +The grading is elementary if there is a function γ : β → G such that euv is +homogeneous of degree γ(u)−1γ(v). +We notice that if each euv is homogeneous, then the grading is completely +determined (see Corollary 21). This fact is far from obvious in the present +context. In fact, it is not at all obvious why A should have any nontrivial +gradings. +Our first main result is the following (cf. section 2.2 for specific details): +Theorem 2. Let β = {ui | i ∈ N}. Then, every G-grading on A, where +A = UT→β V or A = UTβ V, is isomorphic to an elementary grading. +Furthermore, any grading on UTβ V has a finite support. +We shall prove this result by constructing certain sets of primitive homoge- +neous pairwise orthogonal idempotents. We shall see that such a set defines +a grading on a subalgebra given by the direct limit of finite-dimensional ones. +Then, with the aid of the natural topology, we shall extend this grading to +the whole algebra. +Next, we shall investigate the isomorphism classes of elementary gradings +on the triangular algebras (Theorem 34 and Corollary 35). It turns out that +each good grading is uniquely determined by a map β \ {1} → G (where +1 ∈ β is the first element). +Finally, we remark that, if β is uncountably infinite, then UTβ V is iso- +morphic to the finitary incidence algebra of the non locally-finite partially +ordered set β (cf. [22]), and if β is finite or N, then UTβ V is isomorphic to +the incidence algebra of the locally-finite partially ordered set β. +2. Preliminaries +2.1. Topological algebras. A topological vector space is a vector space +V endowed with a topology where the sum V × V → V and the scalar +multiplication F × V → V are continuous maps. The Cartesian products +have the product topology and F is endowed with the discrete topology. A +topological algebra is a topological vector space A endowed with a continuous +bilinear map A × A → A. All the topological algebras in this paper will be +Hausdorff. + +4 +WALDECK SCH¨UTZER AND FELIPE YUKIHIDE YASUMURA +Let X be a topological space and Y a set. Then, the space XY = {f : Y → +X map} has a natural topology given by the product topology if we identify +XY with � +Y X. This topology on XY is called the topology of pointwise +convergence. In the particular case where X has the discrete topology, then +the topology on XY is called the finite topology. +In this case, a basis of +neighbourhoods of f ∈ XY is given by +N (f, S) = {g ∈ XY | g(x) = f(x), ∀x ∈ S}, +where S ⊆ Y is a finite set. +Now, let V be a vector space endowed with the discrete topology, so +that VV is endowed with the Hausdorff finite topology. Then, the closed +subset EndF V ⊆ VV inherits the finite topology, and thus it is a Hausdorff +topological algebra. A basis of neighbourhoods of 0, in this case, is given by +N (W) = {T ∈ EndF V | T|W = 0}, +where W ⊆ V is finite-dimensional. Indeed, let S be a finite set that spans +W. Then N (W) = N (0, S), in the above notation. +Let E = {ei | i ∈ J} ⊆ V, where V is a Hausdorff topological vector +space, so that every convergent net has a unique limit in V. We say that E +is summable if the net +�� +i∈S +ei | S ⊆ J is finite +� +converges. +In this case, it is usual to denote the limit by � +i∈J ei or by +� +e∈E e. +Let (J, ≤) be a directed set. Recall that a net (xα)α∈J in a topological +vector space V is called a Cauchy net if for any neighbourhood U of 0, there +exists γ ∈ J such that α, α′ ≥ γ implies xα − xα′ ∈ U. A topological vector +space is called complete if every Cauchy net converges. Moreover, if V is +complete and W ⊆ V is a closed subspace, then W is complete as well. Since +V is Hausdoff and complete, so are VV and EndF V. +The following extension result will be very useful for our purposes: +Proposition 3. Let V, W be Hausdorff topological vector spaces, where W +is complete, and let V0 ⊆ V be a dense subspace. If f : V0 → W is linear and +continuous, then there exists a unique extension ¯f : V → W which is linear +and continuous. +Proof. Let f : V0 → W be a continuous linear map. +Claim 1. If (xα)α∈J is a Cauchy net in V0, then (f(xα))α∈J is a Cauchy +net in W. +Indeed, let U ⊆ W be a neighbourhood of 0. Then, f −1(U) is a neigh- +bourhood of 0 in V0, so there exists γ ∈ J such that α, α′ ≥ γ implies +xα − xα′ ∈ f −1(U). Hence, U ∋ f(xα − xα′) = f(xα) − f(xα′), for all α, +α′ ≥ γ. +Claim 2. There exists a linear extension ¯f : V → W of f. + +GROUP GRADINGS ON INFINITE UPPER TRIANGULAR MATRICES +5 +Indeed, if x ∈ V, let (xα)α∈J be a net in V0 converging to x. Then, the +net is Cauchy, and so is (f(xα))α∈J. Since W is Hausdorff and complete, it +converges to a unique limit. So, we define +¯f(x) = lim +α f(xα). +If (x′ +α′)α′∈J′ is another net converging to the same x, then (xα−x′ +α′)(α,α′)∈J×J′ +is a net in V0 converging to 0. Using continuity and linearity of f, we obtain +that limα f(xα) = limα′ f(x′ +α′), so ¯f is well-defined. Clearly if x ∈ V0, then +¯f(x) = f(x), so ¯f is an extension of f. Finally, from the continuity of the +sum and scalar multiplication of V and W, we see that ¯f is linear. +Claim 3. ¯f is continuous. +Indeed, let U ⊆ W be a neighbourhood of 0, and let U0 be an open set +such that 0 ∈ U0 ⊆ U0 ⊆ U. Then f −1(U0) is an open set of V0. Thus, +f −1(U0) = V0 ∩ Z, for some open set Z ⊆ V. Given x ∈ Z, up to taking a +subnet, we may find a net (xα)α∈J in V0 ∩Z converging to x. So (f(xα))α∈J +is a convergent net in U0. Thus, +¯f(x) = lim +α f(xα) ∈ U0 ⊆ U. +Hence, ¯f(Z) ⊆ U, so ¯f is continuous. +□ +2.2. Graded algebras. Let A be an arbitrary algebra over a field F, and +let G be any group. We use multiplicative notation for G, and denote its +identity by 1. We say that A is G-graded if A is endowed with a fixed vector +space decomposition, +Γ: A = +� +g∈G +Ag, +where some of the subspaces Ag may be 0, and such that AgAh ⊆ Agh, +for all g, h ∈ G. The subspace Ag is called the homogeneous component +of degree g, and the non-zero elements x ∈ Ag are said to be homogeneous +of degree g. We write deg x = g for these elements. We also write degG +or degΓ to make it clear that we consider the degree with respect to the +grading Γ. Not all elements of G are required to participate in the grading +with corresponding nonzero homogeneous subspace, hence it is often useful +to define the support of the grading, SuppΓ = {g ∈ G | Ag ̸= 0}, to single +out those elements which do participate. A G-grading is called finite if its +support is finite. +A subspace S ⊆ A is homogeneous of degree g if S ⊆ Ag. For an arbitrary +subspace S ⊆ A, the subspace Sg = S ∩ Ag is the (possibly zero) homoge- +neous component of S of degree g. We say that a subspace S is graded if it +is the sum of all its homogeneous components, namely S = � +g∈G(S ∩ Ag). +Let Γ′ : B = � +g∈G Bg be another G-graded algebra. A map f : A → B +is called a homomorphism of G-graded algebras if f is a homomorphism of +algebras and f(Ag) ⊆ Bg, for all g ∈ G. If, moreover, f is an isomorphism, +we call f a G-graded isomorphism (or an isomorphism of G-graded algebras), +and we say that (A, Γ) and (B, Γ′) are G-graded isomorphic (or isomorphic + +6 +WALDECK SCH¨UTZER AND FELIPE YUKIHIDE YASUMURA +as G-graded algebras). Two G-gradings, Γ and Γ′, on the same algebra A +are isomorphic if (A, Γ) and (A, Γ′) are isomorphic as graded algebras. In +this case, we write Γ ∼= Γ′. +Now, if f : A → A is an algebra automorphism, then f induces a new +G-grading on A via A = � +g∈G A′ +g, where A′ +g = f(Ag). Note that f will be +an isomorphism between these two gradings on A. +3. Complete set of idempotents +Recall that an element t ∈ EndF V is topologically nilpotent in the fi- +nite topology if the sequence (ti)+∞ +i=1 converges to 0. Accordingly, a subset +X ⊆ EndF V is topologically nilpotent whenever the sequence (ti · · · t2t1)∞ +i=1 +converges to 0 in the finite topology for every infinite list t1, t2, . . . ∈ X. +Following [26, Definition 5.1] for any subring A of EndF V, we let TNil(A) +denote the set of all topologically nilpotent elements of A. If A is triangu- +larizable, it follows from [25, Proposition 5.4] that TNil(A) is an ideal of +A consisting of all strictly triangular elements in A. Moreover, TNil(A) is +topologically nilpotent, and J(A) ⊆ TNil(A). If A is closed in the finite +topology, then J(A) = TNil(A). +If β is a basis of V, we define the β-support of t ∈ EndF V as the set +Suppβ(t) = {v ∈ β | evvtevv ̸= 0}. +If β = {v1, v2, . . . , vn} is finite, this is the subset of indices corresponding to +the nonzero diagonal entries of the matrix [t]β of t. When there is no risk +of confusion, for instance when β is clear from the context, we shall simply +write Supp(t) for Suppβ(t) and call it the support of t. We obviously have +v ∈ Supp(t) if, and only if, t(v, v) ̸= 0. +Remark 4. When s, t are triangular with respect to the well-ordered basis +(β, ≤), it is straightforward to show that evvstevv = (evvsevv)(evvtevv), hence +(st)(v, v) = s(v, v)t(v, v) for all v ∈ β, and thus Supp(st) = Supp(s) ∩ +Supp(t). +We need the following results, the first of which is inspired by the finite +dimensional Jordan-Chevalley decomposition of an endomorphism. +Lemma 5. Let t ∈ EndF V be triangularizable and such that dim t(V) < ++∞. Then there exist polynomials p(λ), f(λ) ∈ F[λ] without constant term +such that p(t) = 0 and e = f(t) is an idempotent with Supp(e) = Supp(t). +Consequently, t is algebraic over F, e commutes with every endomorphism +commuting with t and stabilizes every subspace of V stabilized by t. +Proof. By [25, Proposition 6], since t is triangularizable and dim t(V) < +∞, +there is a finite dimensional t-invariant subspace W ⊂ V such that t(V) ⊂ W. +Further, by [25, Theorem 8(2)], there is a polynomial p0(λ) ∈ F[λ] that +factors into linear terms and such that p0(t) annihilates W. It follows that +p(λ) = p0(λ)λ is such that p(t) annihilates V. We may assume that p(λ) = + +GROUP GRADINGS ON INFINITE UPPER TRIANGULAR MATRICES +7 +�k +j=1(λ − αj)mj for distinct scalars α1, . . . , αk, with αk = 0, and natural +numbers m1, . . . , mk. Then, by [25, Lemma 2], V = �k +j=1 ker((t − αjI)mj). +Now, by the Chinese Remainder Theorem, there exists a polynomial +f(λ) ∈ F[λ] satisfying the congruences +f(λ) +≡ +δj +mod ((λ − αj)mj) +(1 ≤ j ≤ k), +f(λ) +≡ +0 +mod (λ), +where δj = 1 if αj ̸= 0 and δj = 0 otherwise (in which case the last con- +gruence is superfluous). Let e = f(t). In view of the last congruence, f +has no constant term, hence the last assertions about e are obviously true. +The j-th congruence above means that e acts diagonally on the subspace +ker((t − αjI)mj) of V as the scalar δj ∈ {0, 1}, hence this element is an +idempotent. Lastly, as e stabilizes every subspace stabilized by t, of course +Supp(e) = Supp(t), and this completes the proof. +□ +The next lemma generalizes well-known properties of idempotents in the +finite dimensional algebra UTn F. Recall that two idempotents f, g in a ring +R are isomorphic, written f ∼= g, if fR ∼= gR as right R-modules. +Lemma 6. Let f, g ∈ EndF V be idempotents which are triangular with +respect to a common well-ordered basis (β, ≤). The following are equivalent: +(1) Suppβ(f) = Suppβ(g); +(2) There is an invertible element h ∈ EndF V also triangular with re- +spect to (β, ≤) and such that f = hgh−1. +(3) There exist a, b ∈ EndF V such that f = ab and g = ba. +(4) f ∼= g in EndF V. +Proof. (1) =⇒ (2): Define h = fg + (1 − f)(1 − g). This transformation is +still triangular with respect to (β, ≤) and satisfies fh = fg = hg. Now, each +v ∈ β is either in Suppβ(f) = Suppβ(g) or in Suppβ(1 − f) = Suppβ(1 − g), +hence h(v, v) ̸= 0 for all v ∈ β. It follows from [25, Proposition 16] that h is +invertible and that h−1 is also triangular with respect to (β, ≤). +(2) =⇒ (3): Take a = h and b = gh−1. +(3) =⇒ (1): Obvious, since Suppβ(ab) = Suppβ(ba). +(3) ⇐⇒ (4) This is [23, Proposition 21.20]. +□ +Recall the usual partial order on idempotents f, g in a ring R: f ≤ g +if, and only if, fg = f = gf (equivalently, fRf ⊆ gRg). The next result +tells that this partial order is somewhat determined by the supports of the +idempotents. +Corollary 7. Let A be a closed triangularizable subalgebra of EndF V. If +f ≤ g are idempotents in A, then Supp(f) ⊆ Supp(g). +If f < g, then +Supp(f) ⊊ Supp(g). +Proof. Since f = fg, it is clear that Supp(f) ⊆ Supp(g). +If Supp(f) = +Supp(g), then from Lemma 6, we have f ∼= g. Since f and g commute, we +get f = g. Therefore, f < g implies Supp(f) ⊊ Supp(g). +□ + +8 +WALDECK SCH¨UTZER AND FELIPE YUKIHIDE YASUMURA +If f is an idempotent in a triangularizable algebra A, then of course the +corner subalgebra fAf is still a triangularizable algebra with respect to the +same well-ordered basis as A. The next proposition shows that this is so in +a stronger sense. +Proposition 8. Let A be a triangularizable subalgebra of EndF V with re- +spect to the well-ordered basis (β, ≤), and let f ∈ A be an idempotent. Also, +let β′ = Suppβ(f) and V′ = Span β′. Then, the corner subalgebra D = fAf +of A is isomorphic as a topological F-algebra to a triangularizable subalgebra +A′ of EndF V′. Moreover, if A is the set of all transformations in EndF V +which are triangular with respect to (β, ≤), then so is A′ in EndF V′ with +respect to (β′, ≤). +Proof. Let g be the unique idempotent element of EndF V such that g(v) = v +for all v ∈ β′ and with kernel the Span(β \ β′). Since Suppβ(f) = Suppβ(g), +Lemma 6 provides an invertible element h ∈ EndF V such that g = hfh−1, +with both h, h−1 still triangular with respect to (β, ≤). Let ϕ: EndF V → +EndF V be the continuous inner automorphism such that ϕ(t) = hth−1. +For every t ∈ D, we have t = ft = tf = ftf, hence ϕ(t)(V) = ϕ(ft)(V) = +(gϕ(t))(V) ⊆ V′, while ϕ(t)(v) = ϕ(tf)(v) = (ϕ(t)g)(v) = 0 for all v ∈ β\β′, +hence ϕ(t) ∈ EndF V′ and this transformation is triangular with respect +to (β, ≤). Here we identify EndF V′ with the subset of transformations in +EndF V which keep V′ invariant and vanish on β \ β′. Under this identifi- +cation, the element g becomes the identity transformation of V′. It follows +that A′ = ϕ(D) is a triangularizable subalgebra of EndF V with respect to +(β, ≤) as well as a triangularizable subalgebra of EndF V′ with respect to +(β′, ≤). +Now assume that A is the set of all transformations that are triangular +with respect to (β, ≤) and let t ∈ EndF V′ be triangular with respect to +(β′, ≤). Since t(v) ∈ Span{u ∈ β′ | u ≤ v} for all v ∈ β′ and t(v) = 0 +for all v ∈ β \ β′, it is clear that the element s = ϕ−1(t) = h−1th satisfies +s(v) ∈ Span{u ∈ β | u ≤ v} for all v ∈ β, so it is triangular with respect to +(β, ≤), and thus it lies in A. Also sf = h−1thfh−1h = h−1tgh = h−1th = s, +since g is the identity in EndF V′, and similarly fs = s, hence s ∈ D. But +then t = ϕ(s) ∈ A′, as required. +□ +Remark 9. There is another interesting/useful point of view to look at the +previous proposition. Assume that A is the set of all transformations which +are triangular with respect to β and that h ∈ A is an invertible element. +Define β′ = h(β) with the induced ordering, namely u′ ≼ v′ in (β′, ≼) ⇐⇒ +h−1(u′) ≤ h−1(v′) in (β, ≤). For each v ∈ β and t ∈ A, we have v = h−1(v′), +and hence by triangularity th−1(v′) = t(v) ∈ Span{u ∈ β | u ≤ v} = +Span{h−1(u′) | h−1(u′) ≤ h−1(v′)} =⇒ hth−1(v′) ∈ Span{u′ ∈ β′ | u′ ≼ v′} +for all v′ ∈ β′. +Since hth−1 is still in A, it follows that A = hAh−1 is +triangular with respect to β′. +The main advantage of this point of view is that, for f, g, h with f = hgh−1 +as above, f is diagonal and so is 1−f with respect to the new basis β′ = h(β). + +GROUP GRADINGS ON INFINITE UPPER TRIANGULAR MATRICES +9 +Indeed, for each v′ ∈ β′, v′ = h(v) for v ∈ β, so h−1f(v′) = h−1fh(v) = g(v), +hence +f(v′) = hg(v) = +� +hv, +v ∈ Suppβ(f) +0, +v /∈ Suppβ(f) = +� +v′, +v′ ∈ Suppβ′(f) +0, +v′ /∈ Suppβ′(f) , +where we noticed that Suppβ′(f) = h +� +Suppβ(f) +� +. In particular, it follows +that dim f(V) = | Suppβ(f)| for every idempotent f ∈ EndF V which is +triangular with respect to (β, ≤). +In view of the previous proposition, it follows that: +Corollary 10. Let f ∈ UTβ V be an idempotent, β′ = Suppβ(f) and +V ′ = Span β′. Then f(UTβ V)f ∼= UTβ′ V ′ as topological F-algebras. In +particular, if | Suppβ(f)| = n < +∞, then f(UTβ V)f ∼= UTn F. +□ +Note that, if we consider an algebra A, where UT→β V ⊆ A ⊆ UTβ V, +then an indempotent f ∈ A is an element of UTβ V. Hence, we may repeat +the proof of Proposition 8, and obtain: +Corollary 11. Let A be a subalgebra of UTβ V containing UT→β V, and let +f ∈ A be an idempotent. Then fAf ∼= A′, where UT→β′ V ⊆ A′ ⊆ UTβ′ V′ +as topological F-algebras, and where β′ = Suppβ(f) and V′ = Span β′. In +particular, if | Suppβ(f)| = n < +∞, then fAf ∼= UTn F. +□ +4. Gradings on topological algebras +Let A be a topological algebra. It means that A is a topological Hausdorff +vector space endowed with a continuous bilinear product (a, b) �→ ab, where +A × A is endowed with the product topology. +Definition 12. Let Γ: A = � +g∈G Ag be a G-grading on the topological +algebra A, and denote by πg : A → A the respective projection of A over +Ag. +(1) We say that Γ is closed if each subspace Ag is a (topologically) closed +subspace. +(2) We say that Γ is continuous if each πg is a continuous map. +When A is finite dimensional, it is natural to consider the discrete topol- +ogy, in which case every G-grading is obviously continuous and closed. The +next lemma shows that every continuous grading is closed, however the con- +verse need not hold. See Example 5. +Lemma 13. A continuous G-grading is a closed one. +Proof. Indeed, since the topology on A is Hausdorff, {0} is closed. +So +π−1 +g (0) = � +h̸=g Ah is also closed. Thus, Ag = � +h̸=g π−1 +h (0) is closed as +well. +□ + +10 +WALDECK SCH¨UTZER AND FELIPE YUKIHIDE YASUMURA +Let A0 ⊆ A be a dense subalgebra of a complete topological algebra, +namely that A0 is a subalgebra and ¯ +A0 = A (the topological closure). If +Γ is a continuous G-grading on A0, then the projections πg : A0 → A0 are +continuous. Hence, from Proposition 3, there exists a unique continuous +extension ¯πg : A → A. +Lemma 14. In the above notation, if the grading is finite, then {¯πg | g ∈ G} +is a complete set of projections of A. Moreover, A = � +g∈G ¯πg (A) is a +continous G-grading on A, and ¯πg(A) = πg(A0). +Proof. Let x ∈ A and (xα)α∈J be a net on A0 converging to x. Then, by +definition, ¯πg(x) = lim πg(xα). First, from the continuity of ¯πg, we have +¯π2 +g(x) = lim π2 +g(xα) = lim πg(xα) = ¯πg(x), +so, ¯π2 +g = ¯πg, for each g ∈ G. Also, whenever g ̸= h, we have +¯πg¯πh(x) = lim πgπh(xα) = lim 0 = 0, +hence ¯πg¯πh = 0. Finally, since the grading is finite, +� +g∈G +¯πg(x) = lim +� +g∈G +πg(xα) = lim xα = x, +thus, � +g∈G ¯πg = 1. Hence, {¯πg | g ∈ G} is a complete set of projections of +A. Let us show that the corresponding vector space decomposition, namely +A = � +g∈G ¯πg(A), is a G-grading on A. +Let g, h ∈ G, x, y ∈ A, and +{(xα, yα)}α∈J be a net on A0 × A0 converging to (x, y) (with respect to the +product topology). We have +¯πg(x)¯πh(y) = πg(lim xα)πh(lim yα) = (lim πg(xα))(lim πh(yα)) += lim πg(xα)πh(yα) = lim πgh(xαyα) = ¯πgh(lim xα · yα) += ¯πgh(xy). +By construction, each ¯πg is a continuous map, so the G-grading is a +continuous one. By construction, ¯πg(A) is obtained from limits of nets in +πg(A0), so ¯πg(A) ⊆ πg(A0). On the other hand, clearly πg(A0) ⊆ ¯πg(A). +Since the grading defined by the {¯πg} is continuous, it is also closed (Lemma +13). Thus, πg(A0) ⊂ ¯πg(A), so equality holds. +□ +Remark 15. Another way to prove that A = � +g∈G ¯πg(A) defines a grad- +ing is as follows. First, let f1, . . . , fm : A0 → A0 be continuous linear maps, +¯f1, . . . , ¯fm : A → A their respective linear extensions, and p = p(x1, . . . , xm) ∈ +F⟨x1, . . . , xm⟩ be such that p(f1, . . . , fm) = 0, where F⟨x1, . . . , xm⟩ is the free +associative algebra of rank m. Given x ∈ A, let (xα)α∈J be a net in A0 con- +verging to x. Then +p( ¯f1, . . . , ¯fm)x = lim +α p(f1, . . . , fm)xα = 0. + +GROUP GRADINGS ON INFINITE UPPER TRIANGULAR MATRICES +11 +Hence, p( ¯f1, . . . , ¯fm) = 0. Now, assume that A0 = �m +i=1 πgi(A0) is a finite +G-grading on A0. Then {πgi | i = 1, . . . , m} satisfies the polynomials: +pij(xi, xj) = xixj − δijxi, +p(x1, . . . , xm) = x1 + x2 + · · · + xm − 1. +Thus, the set of extensions {¯πgi | i = 1, . . . , m} satisfies the same polynomi- +als, so it constitutes a (vector space) G-grading on A. +As mentioned above, EndF V is a complete topological space and UTβ V ⊆ +EndF V is a closed subalgebra. Thus, UTβ V is complete as well. Hence, +every finite continous grading on UT→β V extends to a continuous grading +on UTβ V. The next example shows that we cannot always do without the +finiteness condition. +Example 1. Let β = {vi | i ∈ N} and, for simplicity, denote evivj just by eij. +Let G = Z be the free abelian group with 1 generator, and let γ : β → Z be +defined by +γ(vi) = i(i + 1) +2 +. +Then, γ defines a good Z-grading on UT→β V if we set deg e1i = γ(vi) (see +Subsection 5.1), which is equivalent to setting deg ei,i+1 = i + 1, ∀i ∈ N. +Denote, as before, πi : UT→β V → UT→β V the respective projections, and +let ¯πi : UTβ V → UTβ V be the continuous extensions. Then {¯πi | i ∈ N} is +a set of pairwise orthogonal idempotents. However, it is not true that their +image sums to the whole space UTβ V. Indeed, let a = � +i∈N ei,i+1. Then, +¯πj(a) = lim +i πj( +i +� +k=1 +ek,k+1). +The term inside the limit equals ej,j+1 for i ≥ j and 0 otherwise. Hence, +¯πj(a) = ej,j+1 ̸= 0, for all j ∈ N. Thus, +a /∈ +� +j∈N +¯πj(UTβ V). +Therefore, the grading defined by γ on UT→β V does not admit an extension +to a Z-grading on UTβ V. +Notation. Let A0 ⊆ A be a dense subalgebra, where A is complete. If Γ is +a continuous G-grading on A0, we denote by Γ its extension to A. +Remark 16. If Γ ∼= Γ′ by a continuous isomorphism then Γ ∼= Γ′. +Now, let us study a special kind of group grading. Let V be a vector +space with a basis β. We equip EndF V with the finite topology. We let +A = A(I) ⊆ EndF V be a subalgebra such that there exists I ⊆ β × β +satisfying: +(i) (v, v) ∈ I, for all v ∈ β, +(ii) A(I) = Span{euv | (u, v) ∈ I}. + +12 +WALDECK SCH¨UTZER AND FELIPE YUKIHIDE YASUMURA +It is worth noticing that, since A(I) is an algebra, the relation defined by I +is in fact a preorder on β. For instance, if V is finite-dimensional, then ex- +amples of such algebra include the full matrix algebras, the upper triangular +ones, and more generally the incidence algebras of a finite set equipped with +a pre-ordering. If β is well-ordered and I = {(x, y) ∈ β × β | x ≤ y}, then +we obtain UT→β V. +The algebra A = A(I) is known in the literature as a structural matrix +algebra when I is finite, and the notion of good grading applies to it as well: +Definition 17. A G-grading on A is called good if every exy, (x, y) ∈ I, is +homogeneous in the grading. +First, we shall prove that every good grading is a continuous one. +Theorem 18. Let G be a group and Γ a good G-grading on A. Then Γ is +continuous. +Proof. Denote by Γ: A = � +g∈G Ag and πg the projections. Let {rα}α∈I be +a net on A converging to r ∈ A. Let S = {x1, . . . , xm} ⊆ β. Write +I = Ig ˙∪I̸=g ˙∪Ir, +where +Ig = {(x, y) ∈ I | exy ∈ Ag, y ∈ S}, +I̸=g = {(x, y) ∈ I \ Ig | y ∈ S}, +and the remaining Ir = I \ (Ig ∪ I̸=g). Then, if +r = +� +(x,y)∈Ig +λxyexy + +� +(x,y)∈I̸=g +λxyexy + +� +(x,y)∈Ir +λxyexy, +then +πg(r) = +� +(x,y)∈Ig +λxyexy + +� +(x,y)∈Ir +λ′ +xyexy. +Denote N (r, S) = {z ∈ End A | z(x) = r(x), ∀x ∈ S}, the elements consti- +tuting a basis of the topology of End V. Then, note that +z ∈ N (r, S) ∩ A ⇐⇒ z ≡ +� +(x,y)∈Ig +λxyexy + +� +(x,y)∈I̸=g +λxyexy +(mod Span Ir), +and +z ∈ N (πg(r), S) ∩ A ⇐⇒ z ≡ +� +(x,y)∈Ig +λxyexy +(mod Span Ir). +Hence, πg(N (r, S)) ⊆ N (πg(r), S). Thus, πg is continuous. +□ +Now, let U be an algebra, where A(I) ⊆ U ⊆ A(I), where A(I) is the +topological closure of A(I) in End V. +Definition 19. A G-grading on U is called good if every exy is homogeneous +in the grading, for all (x, y) ∈ I. + +GROUP GRADINGS ON INFINITE UPPER TRIANGULAR MATRICES +13 +Let U = � +g∈G Ug be a finite good G-grading. Then Ag := A∩Ug defines a +good G-grading on A; which extends to a G-grading on A via � +g∈G A ∩ Ug. +It is not entirely obvious that Ag ∩ U = Ug. However, as the next result +proves, every Ug is a closed subspace. +Theorem 20. A good G-grading on U is closed. +Proof. Denote U = � +g∈G Ug the good G-grading. First, note that if x ∈ U +is homogeneous and eiixejj ̸= 0, then degG x = deg eij. So, let g ∈ G and +(xα)α∈J ⊆ Ug be a net converging to an x ∈ U. Write x = � +h∈G xh. Let +(i, j) ∈ I be such that eiixhejj ̸= 0, for some h. Since the multiplication +is continuous, (eiixαejj)α∈J is a net converging to eiixejj = � +h∈G eiixhejj. +However, (eiixαejj)α∈J is eventually constant (since eiiUejj = Feij, whose +topology is discrete). Thus, for some α ∈ J, eiixαejj = � +h∈G eiixhejj. Since +this equation involves only homogeneous elements, we should have +g = degG eiixαejj = degG(eiixhejj) = h. +Hence, x ∈ Ug. +□ +Corollary 21. If G is finite, then every good grading on A is induced from +a good grading on A. +Proof. Let Γ: A = � +g∈G Ug be a finite good G-grading on A. Then, Ag = +Ug ∩ A defines a good G-grading Γ0 on A. Since each Ug is closed, one has +Ug ⊇ Ag. However, by Theorem 18, A = � +g∈G Ag. Thus, Γ = Γ0. +□ +We proved that every good grading on A is continuous (Theorem 18). +Also, an extension of a continuous grading is continuous (Lemma 14). Hence, +as a consequence of Corollary 21, we also prove: +Corollary 22. Every good grading with finite support on A is continuous. +□ +Specializing the previous corollaries for our interest, we can state: +Corollary 23. Every finite good grading on UTβ V is continuous, induced +from a finite good grading on UT→β V. +□ +5. Classification of gradings on triangular algebras +5.1. Elementary grading on UT→β V. We shall prove that there is a +correspondence between elementary G-gradings on UT→β V and maps ¯β → +G, where ¯β is obtained from β by removing its first element (here, we denote +it by 1). As mentioned before, if Γ is an elementary G-grading on UT→β V, +then we define γΓ(i) = degΓ e1x, x ∈ ¯β. +Conversely, let γ : ¯β → G be a map. For convenience, define γ(1) = 1. +Given i ≤ j, let +degG eij = γ(i)−1γ(j). + +14 +WALDECK SCH¨UTZER AND FELIPE YUKIHIDE YASUMURA +Then, we obtain a well-defined elementary G-grading on UT→β V. Indeed, +on one hand, UT→β V = Span{eij | i ≤ j}, thus we obtain a vector space +G-grading on UT→β V. +Given i ≤ j ≤ k, the equation eik = eijejk is +compatible with the vector space grading since +degG eij degG ejk = γ(i)−1γ(j)γ(j)−1γ(k) = γ(i)−1γ(k) = degG eik. +Then we obtain a G-grading on UT→β V, denoted by Γ(γ). +Clearly, by +construction, the associated map γΓ(γ) = γ. Thus, we obtain a bijective +correspondence between the elementary G-gradings and maps ¯β → G. +We close this subsection with the following elementary remarks. First, +a G-grading on UT→β V is good if and only if it is elementary. +Indeed, +an elementary grading is good by definition. Conversely, given a good G- +grading on UT→β V, we may define γ : β → G via +γ(i) = degΓ e1i. +Since e1j = e1ieij, we see that degΓ eij = γ(i)−1γ(j). Thus, a good grading +may be described in terms of an elementary one. +Finally, if a G-grading on UT→β V is such that every eii is homogeneous +in the grading, then Γ is a good grading. Indeed, for i ≤ j, Span eij = +eii UT→β Vejj, thus each eij is homogeneous. +5.2. Primitive Homogeneous Idempotents. We fix a G-grading Γ on +UTβ V (respec. UTfin +β V). So, whenever we refer to “homogeneous elements” +or “graded subspaces” of UTβ V (respec. UTfin +β V), we understand as “ho- +mogeneous” or “graded” with respect to the grading Γ. +The purpose of this subsection is to establish the existence of primitive +homogeneous idempotents in UTβ V (respec. UTfin +β V) when these algebras +are graded by a group G. Moreover, it is shown that a homogeneous idem- +potent is primitive if, and only if, its support is a singleton. +Theorem 24. Let (β, ≤) be a well-ordered basis of V and assume that UTβ V +is graded by a group G. If v is a minimal or a maximal element in β, then +there exists a primitive idempotent fv in (UTβ V)1 with Supp(fv) = {v}. +Proof. It is enough to consider the case when v ∈ β is minimal. +Step 1: Decompose evv. Let evv = γ1 + · · · + γn, with γi ∈ (UTβ V)hi, +be the homogeneous decomposition of evv, with the h1, . . . , hn ∈ G distinct. +Since evv ̸= 0, we may assume that, for some p ≥ 1, evvγievv ̸= 0 for +i = 1, 2, . . . , p and evvγievv = 0 for i = p + 1, . . . , n (if any). +Step 2: H = {h1, . . . , hp} is a subgroup of G. For t ∈ UTβ V arbitrary, +we have tevv = t(v, v)evv, by the minimality of v, hence +tγ1 + · · · + tγn = t(v, v)γ1 + · · · t(v, v)γn. +For t homogeneous, this equation gives either tγj = 0 for all j or tγj = +t(v, v)γσ(j) ̸= 0 for all j and some permutation σ ∈ Sn. Applied to (v, v) +this identity gives (tγj)(v, v) = t(v, v)γj(v, v) = t(v, v)γσ(j)(v, v). Hence, for +t(v, v) ̸= 0, we have 1 ≤ j ≤ p if, and only if, 1 ≤ σ(j) ≤ p. + +GROUP GRADINGS ON INFINITE UPPER TRIANGULAR MATRICES +15 +In particular, for t = γi (1 ≤ i ≤ p), we obtain γiγj = γi(v, v)γσi(j) ̸= 0 +for some permutation σi ∈ Sn for all j, and thus hihj = hσi(j) ∈ H for +1 ≤ j ≤ p. +It follows that H is a finite subgroup of G. +Relabeling if +necessary, we may assume that h1 = 1. +Step 3: The element fv = γ1(v, v)−1γ1 is a homogeneous idem- +potent. Since γ1(v, v) ̸= 0, the element γ1 satisfies γ1γj = γ1(v, v)γj for +all j. In particular, γ2 +1 = γ1(v, v)γ1. It follows that fv = γ1(v, v)−1γ1 is a +homogeneous idempotent, acting as the identity on the left of γj for all j. +Step 4: The corner subalgebra D = fv(UTβ V)fv is a finite dimen- +sional graded subalgebra spanned by the elements γ1, . . . , γp. Con- +sider the corner subalgebra D = fv(UTβ V)fv, which is obviously graded. As +before, for 1 ≤ j ≤ p, we have γjγ1 = γj(v, v)γj and γ1γj = γ1(v, v)γj, hence +γj = γj(v, v)−1γjγ1 = γj(v, v)−1γ1(v, v)fvγjfv ∈ D. Thus Span{γ1, . . . , γp} ⊆ +D. +On the other hand, let t ∈ Dg = fv(UTβ V)gfv be a homogeneous element +for some g ∈ G. From Step 2, if t(v, v) = 0, then t = tfv = 0, otherwise +t = tfv = t(v, v)γ1(v, v)−1γσ(1) for some permutation σ ∈ Sn with σ(1) ≤ p. +It follows that D ⊆ Span{γ1, . . . , γp}. +Step 5: D is one-dimensional It follows from Step 4 and Corollary 10 +that D ∼= UTm F for m = | Supp(fv)| and this algebra is still graded. More- +over, 1 (= the image of fv) is the only nonzero homogeneous idempotent in +UTm F. From the classification of the group gradings on UTm F in [32], this +is possible only if D is one-dimensional. +Step 6: Conclusion. Since D is one-dimensional, it follows that m = 1 +and Supp(fv) = {v} is a singleton. +Then fv is an idempotent with the +sought properties which finishes the proof. +□ +It follows from the previous theorem that the support of every primitive +homogeneous idempotent in UTβ V is a singleton: +Corollary 25. Let (β, ≤) be a well-ordered basis of V and assume that +UTβ V is graded by a group G. +Let f be a homogeneous idempotent in +(UTβ V)1. Then f is primitive if, and only if, Supp(f) is a singleton. +Proof. Assume that f is a primitive idempotent in (UTβ V)1. From Corol- +lary 10, we have that f(UTβ V)f is isomorphic as a topological F-algebra +to UTβ′ V′ for β′ = Supp(f) and V′ = Span β′. Since f is homogeneous, +we have that UTβ′ V′ is still graded, and 1 (the image of f under the said +isomorphism) is the only nonzero homogeneous idempotent. For v the least +element in β′, it follows from Theorem 24 that there exists in (UTβ′ V′)1 a +primitive homogeneous idempotent g whose support is {v}. Since g ̸= 0, we +must have g = 1 (and UTβ′ V′ ∼= UT1 F). It follows that Supp(f) = {v} is a +singleton. The converse is obvious. +□ +Definition 26. A set E of idempotents in a ring A is complete if for any +idempotent f ∈ A, there exists e ∈ E such that fe ̸= 0. + +16 +WALDECK SCH¨UTZER AND FELIPE YUKIHIDE YASUMURA +Remark 27. Note that we may repeat word by word the proof of Theo- +rem 24 and Corollary 25 for any algebra A, where UT→β V ⊆ A ⊆ UTβ. +We shall only replace the use of Corollary 10 to Corollary 11. +The following result is interesting by its own. It may be used to obtain a +proof of Theorem 2 that works for algebras A with UT→β V ⊆ A ⊆ UTfin +(see Remark 31). +Lemma 28. Let A be an algebra such that UT→β V ⊆ A ⊆ UTfin +β V, and +f ∈ EndF V be an idempotent such that fA is a graded subspace. For each +v ∈ β such that evv ∈ fA, there exists a homogeneous idempotent e ∈ fA +with Supp(e) = {v}. +Proof. Let t ∈ fA be homogeneous and such that evvtevv ̸= 0 (say t is one +of the components in the homogeneous decomposition of evv with respect to +the group grading on fA). +By Lemma 5, t is algebraic over F and the powers tn (n ∈ N) are nonzero, +homogeneous, and span a finite dimensional subspace of fA, hence degG t is +a torsion element of G. Then, replacing t by one of its powers, if necessary, +we may assume that degG t = 1. +It follows from Lemma 5 that there exists an idempotent element ǫ in +the (finite-dimensional) subspace of EndF V spanned by {t, t2, . . .} such that +evvǫevv ̸= 0. Obviously ǫ is also homogeneous of degree 1 and lies in fA. +Finally, ǫAǫ is a G-graded algebra isomorphic to UTs F by Corollary 11, +where s = | Supp(ǫ)|. Since every G-grading on UTs F is elementary (see the +main result of [32]), we find a (complete) set of s orthogonal homogeneous +idempotents in UTs F whose supports are singletons among which we are +sure to find an idempotent e whose support is {v}. By construction, such an +idempotent is homogeneous and belongs to fA. The proof is complete. +□ +Now, we shall specialize to the case where the basis β is indexed by N. +In this case, we shall find a summable set of primitive pairwise idempotents +that sums to the identity map. +Lemma 29. Let β = {vi | i ∈ N}. +Then, there exists a complete set +of primitive pairwise orthogonal homogeneous idempotents E ⊆ A, where +UT→β V ⊆ A ⊆ UTβ, such that � +e∈E Supp e = β. +Proof. First, we may use Theorem 24 (see also Remark 27) to find a primitive +homogeneous idempotent e1 ∈ A1 and define E1 = {e1}. Next, we assume +that the set of primitive pairwise orthogonal homogeneous idempotents Em +with m elements has been constructed for some integer m > 1, and consider +the element f = 1 − �m +i=1 ei, which is a homogeneous idempotent in A1. +By Corollary 11, the corner subalgebra fAf is isomorphic to a subalgebra +A′ with UT→βm Vm ⊆ A′ ⊆ UTβm Vm, where βm = {vi | i ≥ m + 1} and +Vm = Span βm, and this subalgebra is still graded. A further application +of Theorem 24 to A′ yields a primitive homogeneous idempotent em+1 ∈ +fA1f, which is orthogonal to every element in Em by construction. We then + +GROUP GRADINGS ON INFINITE UPPER TRIANGULAR MATRICES +17 +define Em+1 = Em ∪ {em+1}. To finish the proof, we define E = � +m≥1 Em. +Since every v ∈ β belongs to the support of exactly one element in E (see +Corollary 25) and, since V = � +i≥1 ei(V), this set is summable to 1 (see [19, +Corollary 2.12]), it follows that it is also complete, and we are done. +□ +Example 2. Let β = {vi | i ∈ N}, and E = {eii + e1i | i > 1}. Then E +is a complete set of pairwise orthogonal idempotents in UT→β V, but it is +not complete in UTfin +β V or UTβ V. Hence, it is not true that any complete +set E satisfies � +e∈E Suppe = β. However, note that E is summable and, +(1 − E) UT→β V(1 − E) is an 1-dimensional algebra, isomorphic to UT1(F). +It is not true that the (unique) non-zero idempotent of (1−E) UT→β V(1−E) +can be moved to UT→β V in an attempt at “completing” the set E. This +happens because E /∈ UT→β V. +Example 3. Let ω = N∪{N} be a well-ordered set containing a limit ordinal, +and β = {vi | i ∈ ω}. Let E = {eii + eiN | i ∈ N}. Then E is a complete set +of pairwise orthogonal idempotents in UTβ V. However, it is not true that +� +e∈E Supp e = ω. It is not even true that E is summable. Indeed, the set +{e ∈ E | e(vN) ̸= 0} is infinite, so E is not summable by [19, Lemma 2.11]. +Thus, it seems a hard problem to extend Lemma 29 for an ordinal larger +than N. +Example 4. This example shows that it is not impossible to extend Lemma 29 +to a limit ordinal. Let ω = N ∪ {N} be an ordinal and β as in the previous +example and assume that UTβ V is graded by a group G. By Theorem 24 +there exists a primitive homogeneous idempotent eN ∈ (UTβ V)1 whose sup- +port is {vN}. For f = 1−eN, the corner subalgebra f(UTβ V)f is isomorphic +to UTβ′ V′ by Corollary 10, where β′ = β \ {vN} and V′ = Span β′, and this +algebra is still graded. By Lemma 29, f(UTβ V)f contains a complete set +of primitive pairwise orthogonal idempotents E′ = {ei | i ∈ N} satisfying +Supp(ei) = {vi} for all i ∈ N. Then E = E′ ∪ {eN} is a complete set of +primitive pairwise orthogonal idempotents in (UTβ V)1. Furthermore, since +� +i∈ω ei(V) = V, it follows from [19, Corollary 2.12] that E is summable and +sums 1. +Finally, we shall prove that a complete set of pairwise orthogonal elements +indexed by N is simultaneously diagonalizable. +Lemma 30. Let β = {vi | i ∈ N} and E = {ei | i ∈ N} be a complete +set of primitive pairwise orthogonal homogeneous idempotents as in Lemma +29. +Then there exists a basis {wi | i ∈ N} such that, for each i ∈ N, +Span{wj | j ≤ i} = Span{vj | j ≤ i} =: Vi, and eiwj = δijwj, for each +j ∈ N. +Proof. We define wi = eivi, i ∈ N. Then, by construction, for each i ∈ N, +wi ∈ Vi but wi /∈ � +j +⇒ +> +[35]. Finally, RankSVM is used to +calculate out d, as in Equation (3) and Equation (4). +( +) +( +) +( +) +* +1 +2 +d +, +, +, +; +argmin +T +F F +F +E D +η +σ += += + + (3) +( +) +( +) +( +) +( +) +{ +} +2 +t +2 +max 0,1 +| +| +2 +1 +l +d +E D +d +l d +t d +T T +ψ +ψ +> ++ +× +− ++ +− +∑ += + (4) +Equation (3) defines a function +( +) +1 +2 +, +, +, +; +T +F +F +F +η +σ + + that converts the video frames into a +single vector d*. Thus, d* combines the details of all frames and is often used as a video descriptor. +Equation (4) is the solution of two key functions: a quadratic regularize implemented in SVMs, +and a hinge-loss soft-counting function that indicates how many pairs ( +) +l +t +> + are misaligned by +the rank function [36]. +The dynamic imaging of micro-expressions is shown in Figure 2. The figure shows that the +generated dynamic image successfully preserves the consistent and inconsistent information of +different categories of expressions within a single frame. + + +Figure 1 Overall framework of the algorithm + + +raw clips +preprocessing +feature extract +Multi-scale +backbone +dynamic imaging +concatenate +class score +contrastive +learning +FC + FC +Multi-scale +backbone +optical flow+os3.2 OF component +The concept of OF refers to the movement of target pixels in an image due to the movement +of objects in the image or the movement of the camera in two consecutive frames. It encodes the +motion of the object in vector notation, representing the flow direction and intensity of each image +pixel. There have been many feature extraction algorithms on optical flow in existing +micro-expression studies, and it has been confirmed that OF can extract subtle facial change +features of micro-expressions [1,38]. The mathematical definition of optical flow is as follows. +0 +t +I p +I +→ +∇ ++ += + + (5) +where +( +) +, , +I x y t denotes the pixel intensity of the space ( +) +,x y at time t. +( +) +, +x +y +I +I I +∇ = +refers +to the spatial gradient and +tI denotes the temporal gradient representing the intensity function. +p +→ is the horizontal and vertical components of the optical flow, denoted as follows. +, +T +dx +dy +p +p +q +dt +dt +→ + + += += += + + + + + (6) +which dx , dy refers to the change in the two-dimensional position and +td represents the +change in time. This constraint on luminance constancy assumes that the pixel intensities of the +two images are constant over time. We adopt TV-L1 [39] for optical flow approximation because +it has two main advantages: better noise robustness and the ability to preserve the discontinuity of +the flow. +To obtain essential features of optical flow features in different directions, we compute +optical strain(OS) for feature learning. Each video can be represented by a single OS image +showing facial deformation [40]. The two-dimensional displacement vector of the moving object +can be expressed as +[ , ]T +u +u v += +. Assuming that the moving object is in a small motion, the strain +magnitude can be defined as Equation (7). +( +) +1 +2 +T +u +u +ε + + += +∇ + ∇ + + (7) +It can be further expanded as + + + + + + +Figure 2 Dynamic images of different micro-expression expressions, from left to right: happiness, disgust, repression, +surprise, anger, sadness. + +1 +2 +1 +2 +xx +xy +yx +yy +u +u +v +x +y +x +v +u +v +x +y +y +ε +ε +ε +ε +ε + + + + +∂ +∂ +∂ += += ++ + + + + +∂ +∂ +∂ + + + + +=  + + + +∂ +∂ +∂ + + += ++ += + + +∂ +∂ +∂ + + + + + + + (8) +The diagonal strain component ( +) +, +xx +yy +ε +ε +is the normal strain component, which ( +) +, +xy +yx +ε +ε + is +shear strain component. Specifically, normal strain measures the change in length along a +particular direction. And shear strain measures the change in two angles. +The optical strain for each pixel can be obtained by calculating the sum of the squares of the +normal and shear strain components, as shown in equation (9). +2 +2 +2 +2 +, +2 +2 +2 +1 +2 +x y +xx +yy +xy +yx +u +v +u +u +x +x +x +x +ε +ε +ε +ε +ε += ++ ++ ++ +∂ +∂ +∂ +∂ + + += ++ ++ ++ + + +∂ +∂ +∂ +∂ + + + (9) +To obtain the first- and second-order optical flow variation features, referring to [22,41], we +use the horizontal optical flow, vertical optical flow, and optical strain as three channels of the +image for feature learning, as shown in Figure 3. +3.3 Multi-scale multi-modal transformer +To obtain multi-scale features of images, this paper converts images into multiple scales, +then learns the spatial relationship between different patches through the network self-attention +mechanism to get local discriminative features of the same modal at different scales and improve +the recognition ability of the model. The multi-scale module and the specific patch feature +weighted module are shown in figure 4 and figure 5. +3.3.1 Multi-scale patch +The original ViT model has a fixed size on the patch, which tends to ignore finer-grained +micro-expression features, despite the results achieved. To acquire images at different scales, as +shown in figure 4, the image is converted to the specified size( +) +, +m n for patches at the first scale, +which can be divided into ( +) ( +) +16 +16 +m +m +× +patches. At the second scale, the image is reduced to + +Figure 3 Optical flow-optical strain image + +onset trame +of os +apex frame( +) +2, +2 +m +n +size, still patched embedding according to the pixel size of 16 16 +× +, and at the third +scale, the image is reduced to ( +) +4, +4 +m +n +size and patched embedding in the same way. Since ViT +is used to learn the spatial relationships of different patch regions, we do not consider 1 1 +× patch. +We proposed a new method of patch selection in which the intensity of attention can be +intuitively used as an indicator to characterize the importance of symbols. As shown in Figure 5, +in this module, we integrate all the original attention weights of the first +1 +L − layers of the +transformer into an attention graph to guide the network to efficiently and accurately select the +distinguished image patches and compute their relationships. The embedding lacks token +identifiability, and the original attention weights do not necessarily correspond to the relative +importance of the input token, especially for the higher levels of the model [42]. To effectively +utilize the attention mechanism of ViT, we improved the input of the last layer of the network. As +shown in Figure 6, the weights of the first +1 +L − layers of the network are first extracted. + + +Figure 5 patch feature weighting module + + +Figure 4 Multi-scale feature selection module + +patch feature weighted module +TransformerEncoder +position embedding +class_token +MLP +(1-7)x +Transformer +patch feature weighting +Transformer Encoder +Norm +linear projection +TransformerEncoder +Multi-Head +Encoder +... +Attention +.. +... +... +Norm +Embedded +Patchesclass score +fc2 +1 +fc1 +1 +concatenate +cls_token +cls_token +cls_token +patch feature +patch feature +patch feature +weighted module +weighted module +weighted module +个 +↑1 +2 +, +, +, +H +l +l +l +L +w +w w +w + + +=  + + +, +1,2, +, +1 +l +L +∈ +− + + (10) +1 +2 +, +, +, +i +D +l +l +l +l +w +w w +w + + +=  + + +, +1,2, +, +i +H +∈ + + (11) +where H is the number of heads, L is the number of layers, and D is the embedded feature +dimension. +We consider that the critical attention features are gradually accumulated and amplified in +each layer, so we first regularized the weights of multiple attentions, that is, we sum up the +multiple attention values of each layer weight and take the mean value, and then regularize them +in each embedding dimension (Regularization is mainly used to avoid the generation of +over-fitting and reduce network errors): + + + + + + +h 1 +1 +1 +1 +1 +1 +H +h +l +l +D +H +h +l +h +d +w +H +G +w +D +H += += += += + + + + + + +∑ +∑ +∑ + +(12) + +The attention weights of all previous layers are then integrated and the matrix multiplication +operation is performed recursively on the modified attention weights of all layers as follows: + +Figure 6 patch feature weighting + + +output, +Output efficient patch features +W +final Layer +outputr- +WL-1 +0.15 +0.91 +0.58 +0.41 +L-1 Layers +0.25 +0.83 +0.69 +0.55 +0.34 +0.46 +0.51 +0.28 +W2 +0.24 +0.21 +0.85 +0.17 +LayerAttention +MatrixProcessing +inptt( +) +1 +1 +L +l +L +G +G +− += +=∏ + (13) +Since the elements in G are the attention-related values of each patch with respect to other +patches, to obtain the weight value of each patch, we average the results by row and normalize +them: +( +) +( +) +( +) +,0 / max +,0 +G +mean G +mean G +− += + (14) +Finally, the output features of the L-1th layer are multiplied with the adjusted attention +weights to obtain the weighted patch features, which are then concatenated with the cls_token and +input to the last Transformer Layer. +1 +1 +_ +, +L +L +L +input +cls token +G Z +− +− +− + + +=  + + + (15) +The output of the last layer is shown in equation (16): +( +) +( +) +( +) +( +) +L +L +L +Z +FFN LN MSM LN input +input += ++ + (16) + Since facial expressions occur in some local regions of the face, other regional features +contribute less to the expressions, but some features will be lost if the regions with lower weights +are dropped, so these features are still kept. In this way, we not only keep the global information, +but focus on the nuances of micro-expressions in the last Transformer Layer. +3.3.3 Multi-feature fusion +Since optical flow and dynamic imaging have different representations of micro-expression +and have complementary roles, therefore, to capture the key features of different modalities in +multi-scale mode, we concatenate the cls_token of the features learned at different scales as the +final features of individual modal. The two features are then concatenated into two fully connected +layers for expression classification, as shown in equations (17) and (18). To avoid over-fitting, we +add dropout and ReLU after the first fully connected layer. +( +) +( +) +2 +1 +class_ +( +_ +, +_ +) +final +final +score +FC +FC Concat dy imaging +flow os += + (17) +( +) +1 +2 +3 +( +_ +, +_ +, +_ +), +_ +, +_ +final +F +Concat cls token cls token cls token +F +dy imaging flow os += +∈ + (18) +3.3.4 Network optimization +Contrast learning generates anchor samples, positive and negative samples from an unlabeled +database and learns the similarity of the samples. The common features of the samples are +extracted by encoding the positive samples similarly and encoding the negative samples +differently through an encoder. Since the dynamic and optical flow maps used in this paper belong +to two different modalities, the similarity of the two modal features on the same category is +maximized and the similarity of different categories is minimized by using contrast learning [43]. + +The micro-expression database i consists of two modalities, the sample set +i +dy and +_ +i +flow os , and positive and negative sample pairs are constructed according to whether the two +modalities come from the same micro-expression sample. For the positive sample pair, fixed +i +dy +and the loss function is: +( +) +( +) +( +) +( +) +( +) +( +) +, +, +_ +1, +1 +exp +, +_ +/ +log +exp +, +_ +/ +exp +, +_ +/ +i +i +i dy flow +os +B +B +i +i +i +k +k +k i +k +s dy flow os +L +s dy flow os +s dy flow os +τ +τ +τ += +≠ += += − ++ +∑ +∑ + (19) +where B is the small batch size, τ is the temperature coefficient, and s(.) denotes the cosine +similarity function. +Fixed +_ +i +flow os ,the loss can be obtained, and the comparative loss function +con +L + for the two +modes is then denoted as: +, +, +_ +, +_ +, +1 +1 +[ +] +2 +B +con +i dy flow +os +i flow +os dy +i +L +L +L +B += += ++ +∑ + (20) +The total loss of the network is: +( +) +_ +1 +loss +con +cross +entroy +L +L +L +α +α += +− +∗ ++ +∗ + (21) +4 Experiments +4.1 Datasets +Three spontaneous micro-expression datasets are involved in the experimental validation of +the algorithm in this paper, i.e., SMIC, CASMEII, SAMM. To be able to compare fairly with other +state-of-the-art algorithms, we not only test the three datasets individually, but also refer to the +combined data set proposed in MEG2019 [44] to measure the algorithm in this paper. +A. SMIC +The SMIC database [45] is the first database containing spontaneous micro-expressions +obtained through an emotion elicitation experiment. The experimental test data is an HS subset +with a frame rate of 100 frames and a resolution of 640 +480 +× +. The data set includes a total of 164 +micro-expression clips from 16 participants and does not contain action unit labels, and the data +were labeled by two coders into three categories of expression clips based on participants' +statements about their subjective emotions while watching the video: positive (51), negative (70), +and surprised (43). +B. CASMEII +CASME II [46] is an improved version of the CASME database with a higher frame rate +(200pfs) and a resolution of 640 +480 +× +. The CASME II database contains 255 micro-expression +clips from 26 subjects. Each micro-expression clip is classified by the coders according to the +facial AU, the subject's self-reports and the video content, and the database contains seven +categories, namely disgust(60), happiness(32), repression(27), surprise(25), sadness(7),fear(2) and + +others(102). Because there are few samples of sadness and fear, only the first five classes of data +are selected in single data set experiment. In addition, the database provides the onset frame, apex +frame, offset frame and AU labels of the micro-expression. +C. SAMM +SAMM [47] contains a total of 159 micro-expression sequences from 32 participants from +13 ethnic groups, a database with a high resolution of 2040 1088 +× + and a frame rate of 200. The +coders objectively classified expressions into seven basic expression categories based on FACS: +anger (57), happiness (26), surprise (15), contempt (12), disgust (9), fear (8), sadness (6), and +other (26). Similarly, in the single database test, the data participating in the experimental analysis +only contains the five categories with the largest number of categories. SAMM also provides onset +frames, apex frames, offset frames, and AU labels of expression clips. +The combined database included 442 micro-expression samples from 68 participants from +three datasets. Of these, 164 samples were from SMIC, 145 were from CASMEII, and 133 were +from SAMM. These samples were reclassified into three categories: positive, negative, and +surprise [44]. Since the transformer network needs enough data for network learning, we enhance +the data by rotating [-10,10], flipping horizontally, scale change, etc. The SMIC dataset does not +provide the onset and apex frames of expressions, so we take the intermediate frames as the apex +frames for feature extraction. +4.2 Evaluation metrics +Due to the imbalanced distribution of the number of categories in the database, we used +three metrics to reduce bias: accuracy (Acc), unweighted average recall (UAR), and unweighted +F1 score (UF1). The unweighted F1 score (UF1) is obtained by summing the F1 value of each +class and then calculating the average value according to the number of classes. The unweighted +average recall (UAR) is obtained by summing the accuracy of each class and then averaging over +the number of classes. +1 +1 +C +c +c +C +c +c +TP +Acc +N += += += ∑ +∑ + (22) +1 +1 +C +c +c +c +TP +UAR +C +N += += ∑ + (23) +' +1 +1 +2 +1 +2 +C +c +c +c +c +c +TP +UF +C +TP +FP +FN += += ++ ++ +∑ + (24) +where C is the number of classes and +c +N is the number of samples per class. TP , FN and FP are +true positives, false negatives and false positives, respectively. +In the experiments, we used the LOSO cross-validation method. All the micro-expression +data of one subject were used for testing, and the samples of the other subjects were used for + +training until each subject completed the test as a test set. +4.3 Pre-processing +Since the SAMM database does not provide cropped data, we processed the three datasets in +a unified manner to reduce interference terms in the micro-expression recognition. As shown in +Figure 7, we used the Dlib library to locate 68 feature points of the face. Since the duration of +micro-expressions is relatively short, some tight facial movements can be ignored. Therefore, we +only compute the affine transformation for the first frame image with two inner eye angle +coordinates, align the subsequent frames with the same transformation, and then perform facial +cropping with the landmark points of the aligned face. +4.3.1 Eulerian video magnification +The Eulerian video amplification (EVM) technique [48] can be used to amplify subtle +motion changes in video that are difficult to see with the naked eye. Motion amplification has +been widely used in micro-expression recognition tasks due to the increased micro-motion +amplitude, which is better for feature extraction and analysis [49,50]. In this paper, through +experimental comparison, the optimal results can be obtained when the amplification factor is set +as 10. +4.4 Implementation details +For multi-scale feature extraction process, we used the ViT-base with 12 layers of Encoder +and hidden layers of size 768, with 12 heads. For initialization, we used the pre-trained official +ViT-B model on ImageNet[32]. In the training phase, we set the batch size to 16 and trained the +network for 50 epochs. We initialized the learning rate to 6e-5 and weight decay to 0.05, updating +the network weights using AdamW. AdamW is an optimization algorithm that uses adaptive +learning rate gradient descent to make the network converge faster. All our experiments are +windows systems and Nvidia GeForce RTX 1080Ti GPU. +4.5 Results and discussion +We compare the proposed approach with commonly used manual feature extraction methods +and recently outstanding deep learning methods in single database experiment (SDE) and +combined database experiment(CDE) settings based on widely used micro-expression databases: + + +(a) (b) +Figure 7 (a)68 landmark points on the face (b)cropped face + + + +SMIC-HS, CASME II, SAMM. +4.5.1 SDE +To ensure consistency and fairness of comparison, SDE results for all methods were +obtained under the same conditions, i.e., for the same number of samples, number of labels +(classes), and using the same cross-validation method. +As can be easily seen from Table 1, among the experimental results under SDE settings, the +proposed method in this paper performs the best on the SMIC database with an accuracy of 78.3%， +which is 5.13% higher than the next best, and UF1 reaches 0.7216, which is at the next best level. +The accuracy on the SAMM database reached 73.83%, which is second to the best result and at an +advanced level. However, the proposed algorithm is not outstanding in CASMEII database, while +the results of the Sparse transformer algorithm are more outstanding. We analyzed the reasons for +this result and found that the CASMEII database has many categories and small differences +between categories, there are samples with similar or the same facial motion regions belong to +different categories, which has a great interference to the classification. The Sparse transformer +algorithm is able to achieve outstanding results on the database mainly due to the use of CNN and +Table1 Micro-expression recognition results of the proposed MSMMT algorithm and the state-of-the-art +method tested singularly on the three datasets + +SMIC-HS +CASMEII +SAMM + +Acc +F1 +Acc +F1 +Acc +F1 +LBP-TOP[10*] +53.66 +0.5384 +46.46 +0.4241 +- +- +MDMO[6](2016) +61.5 +0.406 +51.0 +0.418 +- +- +DiSTLBP-RIP[14](2017) +63.41 +- +64.78 +- +- +- +Bi-WOOF[1](2018) +59.3 +0.620 +58.9 +0.610 +59.8 +0.591 +DSSN[54](2019) +63.41 +0.6462 +70.78 +0.7297 +57.35 +0.4644 +STRCN[24](2019) +53.1 +0.514 +56.0 +0.542 +54.5 +0.492 +MER-GCN[30](2020) +- +- +42.71 +- +- +- +SLSTT[10](2021) +73.17 +0.724 +75.806 +0.753 +72.388 +0.640 +GEME[35](2021) +64.63 +0.6158 +75.2 +0.7354 +55.88 +0.4538 +Later[11](2022) +73.17 +0.7447 +70.68 +0.7106 +- +- +FDCN[38](2022) +- +- +73.09 +0.72 +58.07 +0.57 +KTGSL[51](2022) +72.58 +0.6820 +75.64 +0.6917 +- +- +Sparse Transformer[13](2022) +- +- +76.11 +0.7192 +80.15 +0.7547 +MSMMT(Ours) +78.30 +0.7216 +68.00 +0.6681 +73.83 +0.5881 + + +transformer for spatio-temporal feature extraction. The lower-level features of the video are first +extracted from the CNN, and then more advanced features are extracted using the transformer, and +the algorithm is able to combine the advantages of both networks. The algorithm in this paper is +based on a pure ViT network, although slightly inferior to the Sparse Transformer algorithm, but +both algorithms have their own advantages. Different from the methods based on the transformer +algorithm [10,11,13], the multi-scale transformer in this paper focuses more on extracting features +of different sizes of the face, making full use of the attention weights of multiple encoder layers of +the transformer to select key region features, which are more expressive for features of different +granularity and can obtain comparable results even when using only a single frame feature image. +4.5.2 CDE +Some similar and the latest micro-expression recognition algorithms are compared in Table 2, +and it can be easily seen that our method achieves the best level with UF1 of 0.8160 and UAR of +0.8191 on the combined database. The performance on the SMIC and CASMEII datasets is +outstanding, both outperforming the other algorithms, with UF1 reaching 0.7651 and UAR +reaching 0.7780 for the SMIC database, UF1 reaching 0.9071 and UAR reaching 0.8878 for the +CASMEII database. For the SAMM database, we can see that the results are also comparable but +not at the optimal level. The above results show an important conclusion that our proposed method +is effective and can obtain better representation of features in the CDE. The reason for being able +to achieve this result, we believe that benefiting from multi-scale multi-modal, the network is able +to learn the local features of the samples with sufficient samples than the single data set. + 4.5.3 Impact of α +Table2 Testing the micro-expression recognition results of the proposed MSMMT and the state-of-the-art +method on the combined three datasets + +FULL +SMIC-HS +CASMEII +SAMM + +UF1 +UAR +UF1 +UAR +UF1 +UAR +UF1 +UAR +LBP-TOP[47*] +0.5882 +0.5785 +0.2 +0.528 +0.7026 +0.7429 +0.3954 +0.4102 +Bi-WOOF[47*] +0.6296 +0.6227 +0.5727 +0.5829 +0.7805 +0.8026 +0.5211 +0.5139 +OFF-ApexNet[53](2019) +0.7196 +0.7096 +0.6817 +0.6695 +0.8764 +0.8681 +0.5409 +0.5392 +STSTNet[22](2019) +0.7353 +0.7605 +0.6801 +0.7013 +0.8382 +0.8686 +0.6588 +0.681 +EMR[49](2019) +0.7885 +0.7824 +0.7461 +0.753 +0.8293 +0.8209 +0.7754 +0.7152 +STA-GCN[26](2021) +- +- +- +- +0.7608 +0.7096 +- +- +SLSTT[10](2021) +0.816 +0.790 +0.724 +0.707 +0.901 +0.885 +0.715 +0.642 +AUGCN[52](2021) +0.7914 +0.7933 +0.7192 +0.7215 +0.8798 +0.871 +0.7715 +0.7890 +MSMMT(Ours) +0.8160 +0.8191 +0.7651 +0.7780 +0.9071 +0.8878 +0.7392 +0.7163 + + +This subsection discusses in detail the effect of the loss weight factor on the network, and we +show the results for the three data sets from CDE's experiments as well as the combined set. As +can be seen in Figure 8, the accuracy increases first and then decreases as the weighting factor α +gradually increases. Although the parameters of α are different when the optimal results are +obtained on the three datasets, comprehensive comparison shows that when +0.1 +α = +, the two +modes have the best performance in the three datasets. Similarly, we selected the optimal results +under the α parameter for the three data sets in the SDE experiment. The α parameter was +different for the three datasets, with 0.1 for the SMIC and CASMEII database, 0.2 for the SAMM +database. +5 Conclusion +In this work, we proposed a micro-expression recognition method based on multi-scale +learning of bimodal features. The multi-scale features are learned by using ViT for dynamic +imaging and optical flow features, and the patch features are weighted by using multi-headed +self-attention weights in the network to obtain the most expressive facial region features and +reduce the influence of irrelevant facial patches on the results. By combining cross-modal +unsupervised contrastive learning, the information of texture and motion modal features are +processed in the same category close to and different categories far from each other, enabling the +network to fully use both features for expression learning. In this paper, a large number of +experiments are conducted on three datasets, and the results obtained have good recognition rates, +which fully demonstrate the effectiveness of the multi-scale algorithm proposed in this paper. In +the future, we will further study the local feature expressions of micro-expressions on this basis +and explore more meaningful features of different categories of micro-expressions. +References +[1] S.T. Liong, J. See, K. S. Wong, R. Phan, Less is more: Micro-expression recognition from video using apex +frame [J]. Signal Processing: Image Communication, 2018, 62:82-92. doi:10.1016/j.image.2017.11.006 + +(a) (b) +Figure 8 Variation curves of UF1, UAR values with α for three categories (a) Variation of UF1 values (b) +Variation of UAR values + +Variation of UF1 value +1 +6'0 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +FULL +SMIC +CASMEL +SAMMVariation of UAR value +1 +6'0 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +FULI +SMIC +CASMEI +SAMM[2] W. -J. Yan, Q. Wu, J. Liang, Y. -H. Chen, X. Fu, How Fast are the leaked facial expressions: The duration of +micro-expressions, J. 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Lin, Dual-stream Shallow Networks for Facial +Micro-expression +Recognition,2019 +IEEE +International +Conference +on +Image +Processing +(ICIP), +doi:10.1109/ICIP.2019.8802965. +Acknowledgments + + + diff --git a/b9E1T4oBgHgl3EQfKwO_/content/tmp_files/load_file.txt b/b9E1T4oBgHgl3EQfKwO_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c77caba00e728ad49d7dbf93b604aebdacdbce44 --- /dev/null +++ b/b9E1T4oBgHgl3EQfKwO_/content/tmp_files/load_file.txt @@ -0,0 +1,1095 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf,len=1094 +page_content='Multi-scale multi-modal micro-expression recognition algorithm based on transformer Fengping Wang, Jie Li, Chun Qi, Lin Wang, Pan Wang Abstract: A micro-expression is a spontaneous unconscious facial muscle movement that can reveal the true emotions people attempt to hide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Although manual methods have made good progress and deep learning is gaining prominence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Due to the short duration of micro-expression occurrence and different scales of expressing in facial regions, existing algorithms cannot extract multi-modal multi-scale facial region features while taking into account contextual information to learn underlying features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Therefore, in order to solve the above problems, a multi-modal multi-scale algorithm based on transformer network is proposed in this paper, aiming to fully learn local multi-grained features of micro-expressions through two modal features of micro-expressions - motion features and texture features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' To obtain local area features of the face at different scales, we learned patch features at different scales for both modalities, and then fused multi-layer multi-headed attention weights to obtain effective features by weighting the patch features, and combined cross-modal contrastive learning for model optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' We conducted comprehensive experiments on three spontaneous datasets, and the results show the accuracy of the proposed algorithm in single measurement SMIC database is up to 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='73% and the F1 value on CASMEII of the combined database is up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='9071, which is at the leading level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=" Keywords: multi-scale, multi-modal, transformer, micro-expression, recognition 1 Introduction Micro-expressions, as a kind of human emotion, can express people's inner real feelings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Therefore, the detection and recognition of micro-expressions have been widely used in clinical diagnosis, criminal investigation analysis, and national defense and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The short duration of micro-expressions, no more than 1/2s, and occurring only in localized areas of the face [1,2], makes detecting and recognizing micro-expressions difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Early micro-expression studies were based on manual features and there are two main categories: texture feature algorithms based on Local Binary Patterns-Three Orthogonal Planes(LBP-TOP) and motion feature algorithms based on optical flow (OF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' In recent years, deep learning algorithms have been the main trend in micro-expression recognition research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Convolutional neural network(CNN)-based algorithms are used to extract spatiotemporal motion features of micro-expressions, while also combining them with traditional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' In terms of feature extraction, since facial expressions are triggered by facial muscle unit movements, mining micro-features from local spatial regions is one of the ways to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' For example, LBP-TOP features with multi-scale activation patches [3,4,5], Main Directional Mean Optical Flow (MDMO) [6,7], Local region feature learning [8,9], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' However, although these efforts emphasize the task of local region feature extraction, there are still some drawbacks that need to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' For instance, 1) multi-scale optical flow features cannot fully represent the changing features of expressions, or different modal features cannot effectively capture local features even by using fixed patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 2)treating all key regions equally ignores the validity of the representation of local features and the connection relationship between local blocks and expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' In order to extract local features of micro-expressions more rationally and to learn the relationship between local features, transformer has been introduced into the study of micro-expressions [10,11,12,13], and good results have been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' However, these algorithms will all utilize fixed-size patches and do not take into account local area features at different scales of micro-expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Therefore, to solve the above problem, we applied the multi-headed self-attention mechanism transformer to build our network framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' This network attempts to perform multi-scale patches on the inputs of two different modalities and learn meaningful facial features by transformer self-attention property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The results confirm that this method can learn the subtle features of micro-expressions with better performance than the current level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The main contributions of this paper are summarized as follows: 1: We proposed a multi-scale multi-modality transformer network-MSMMT for learning micro-expression features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' To learn facial expression features at different scales, we first acquire images of different scales and then input them to the transformer for patch embedding to extract multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 2: Considering the different degrees of the contribution of features to micro-expressions at different scales, the patches attention mechanism is proposed after multi-scale feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The core idea is to multiply the weights learned in the first N-1 layers of the Transformer and then weight the patch features in the last layer to obtain the most effective region features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 3: We apply unsupervised contrastive learning loss to make similar features closer and dissimilar features away in two multi-scale features, and use cross-entropy loss for joint features to optimize the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The effectiveness of this algorithm is experimentally demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The rest of the article is organized as follows: Section 2 presents the related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Section 3 is the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The experimental results are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Finally, we conclude the paper in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 2 Related works 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='1 Micro-expression recognition Traditional manual feature extraction algorithms based on local regions are the main tools in the preliminary stage of micro-expression research, and local blocking of images or sequences using different methods is the main way to extract local features of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [14] first extracted the fine texture motion features of the sparse part of the micro-expression using robust principal component analysis (RPCA), then extracted the local texture orientation features of the 16 ROIs using a local spatiotemporal algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [15] proposed a spatiotemporal LBP-TOP descriptor for multi-scale patch fusion, which considers the active contribution of different regional areas of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' To better capture the low-intensity image features corresponding to small local areas, Zong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [16] used a multi-scale spatial segmentation grid to segment video clips into multi-level local blocks to extract spatiotemporal descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' A kernelized group sparse learning (KGSL) model is then used to learn more efficient multi-level spatiotemporal descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [17] proposed the MDMO, a normalized statistical feature based on the region of interest (ROI), and extracted optical flow statistical features for the 36 ROI regions of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [18] proposed sparse MDMO, whose core idea is to introduce classical graph regularized sparse encoding in the MDMO feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The article captures this sparsity with a new distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Allaert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [19] proposed a new feature extraction algorithm-Local Motion Pattern (LMP)-which performs a local analysis of the motion distribution to separate consistent motion patterns from the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Since LMP extracts local features based on 25 ROI regions divided by the motion pattern of facial expressions, the method is applicable to handle all expressions that cause facial skin deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Liong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [20] proposed an optical strain (OS) weighted feature extraction method for subtle expression recognition of human faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' OS has better recognition of deformation results than OF and can better represent the motion characteristics of micro-expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' In the article, the micro-expression sequences are locally blocked to extract local LBP-TOP features, and then the contribution values of different regions are weighted by optical strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' To reduce redundancy, Liong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [1] extracted double-weighted directional OF features for the Apex frames of micro-expression sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Similarly, the algorithm still uses OS for local region feature weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Micro-expression deep learning algorithms have become more popular in these years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' In addition to end-to-end learning using original images or sequences, many scholars also input expression feature maps into networks for learning research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' And these algorithms utilize different ways to extract the local features of micro-expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [21] used optical flow to capture small changes in micro-expressions as they occur and then designed convolution kernels of different scales to extract local features of micro-expressions of different intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Liong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [22] also proposed a shallow three-stream 3D CNN (STSTNet) to extract discriminative high-level and detailed micro-expressions features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The network learns three optical flow features (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=', OS, horizontal optical flow field, and vertical optical flow field) based on the onset and apex frames computed from each video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' This also solves the problem of insufficient samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [23] proposed a three-stream CNN (TSCNN) to fuse temporal, spatial, and local region cues of micro-expression videos to learn micro-expression salient features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [24] proposed to capture the spatiotemporal deformation features of expression sequences in a deep recurrent convolutional network (STRCN) based on the local area features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' To obtain useful local area features, the temporal differences of video frames on the whole database were first accumulated to calculate a differential heat map based on the entire database, and the local perceptual area of micro-expressions was obtained after the thresholding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [25] proposed an end-to-end marker point-assisted dual-stream graph-attentive convolutional network for micro-expression feature recognition, where one stream uses the coordinate positions of facial marker points to construct graph nodes and edges to learn facial muscle movements, and one stream uses the local area optical flow features of marker points to learn facial features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Similarly, graph convolutional network-based methods are also available in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [27] proposed a micro-expression recognition method based on key facial components guidance (KF-MER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The idea is to divide the face into semantic regions, obtain division probability maps, learn the relationship between parts, and then use shallow residual networks for micro-expression learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [28] fed optical flow images into a multi-scale joint network for feature extraction and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The proposed joint feature module integrates features at different levels and facilitates capturing micro-expression features of various magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' In addition, some works have used attention mechanisms to obtain local ROI features of faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [29] proposed that MERTA integrates three types of attentional mechanisms - general attention, motivation attention, and channel attention - to extract landmark areas, motion regions, and expression-related semantic features, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [30] designed spatial attention, channel attention, and self-attentive ternary attention modules in a network to learn meaningful optical flow features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [31] proposed a micro-attention mechanism in concert with the residual network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Micro-attention enables the neural network to learn to attend to regions of interest of faces covering different action units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The self-output of each residual block at different scales is used to compute the attention map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='2 Vision Transformer Transformer is a very successful model proposed in NLP in 2017 and subsequently used in computer vision with notable success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Visual transformer (ViT) can transform images into sequences by dividing them into sub-images and sorting them consistently, so that spatial correlations can be learned like temporal features, and then image classification can be performed [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The most significant difference between Transformer and CNN is that it uses a self-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' It allows each token to represent contextual information in the group it belongs to rather than representing a single meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Since self-attention models the relationship of patches in an image, it is more expressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Recently, researchers have used the transformer for various computer vision tasks and obtained remarkable results, and micro-expression algorithms related to the transformer have also been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [10] proposed a long and short-term correlation based on a spatiotemporal transformer for micro-expression recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The architecture includes a spatial encoder for learning spatial patterns, a temporal aggregator for temporal dimensional analysis, and a classification head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [11] proposed a late-fusion-based transformer for expression recognition algorithm for video, where OF and grayscale sequences are fused after learning by the transformer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Both methods obtained good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [12] proposed two branching modules for micro-expression recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The transformer-based module was used for position calibration, and the continuous attention-based module was used for learning motion features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' [13] proposed a spatio-temporal feature learning method based on sparse transformation to obtain effective features of facial micro-expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Its core is to extract strongly correlated spatiotemporal features of micro-expression categories using spatial and temporal attention mechanisms, reducing influence of the irrelevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 3 The proposed approach We proposed a multi-scale transformer-based feature extraction algorithm to learn more discriminative local region features of micro-expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The algorithm takes the dynamic imaging and the optical flow-optical strain images as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' To select important local region features, inspired by the RAMS-Trans[33] algorithm, the attention weights learned in all previous layers except the last layer of ViT are processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Then the features are weighted and input to the last encoder layer for learning and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The overall process is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='1 dynamic imaging Micro-expressions are fast and fleeting, appearing in only a few frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' These temporary offsets can be obtained from the video using dynamic imaging methods [34,35,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' A dynamic imaging is a standard RGB image that preserves the spatial and temporal information of an entire video sequence into a single image [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' To construct a dynamic imaging, a sorting function is applied to the video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The RGB feature vector of each frame T F is expressed as ( ) T F σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The temporal average of the available feature vectors ( tϕ ) is calculated using equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' ( ) t 1 1 t T T F t ϕ σ = = ∑ (1) The score related to time t is then calculated by the sorting function, as in Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' ( ) t | , t d d ψ φ = (2) Where d d R ∈ represents a vector to calculate the fraction of frames in the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Higher ranks are assigned to frames of time l, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' ( ) ( ) ( ) | | l t l d t d ψ ψ > ⇒ > [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Finally, RankSVM is used to calculate out d, as in Equation (3) and Equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' ( ) ( ) ( ) 1 2 d , , , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' argmin T F F F E D η σ = = \uf04c (3) ( ) ( ) ( ) ( ) { } 2 t 2 max 0,1 | | 2 1 l d E D d l d t d T T ψ ψ > + × − + − ∑ = (4) Equation (3) defines a function ( ) 1 2 , , , ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' T F F F η σ \uf04c that converts the video frames into a single vector d*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Thus, d* combines the details of all frames and is often used as a video descriptor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Equation (4) is the solution of two key functions: a quadratic regularize implemented in SVMs, and a hinge-loss soft-counting function that indicates how many pairs ( ) l t > are misaligned by the rank function [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The dynamic imaging of micro-expressions is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The figure shows that the generated dynamic image successfully preserves the consistent and inconsistent information of different categories of expressions within a single frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Figure 1 Overall framework of the algorithm raw clips preprocessing feature extract Multi-scale backbone dynamic imaging concatenate class score contrastive learning FC FC Multi-scale backbone optical flow+os3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='2 OF component The concept of OF refers to the movement of target pixels in an image due to the movement of objects in the image or the movement of the camera in two consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' It encodes the motion of the object in vector notation, representing the flow direction and intensity of each image pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' There have been many feature extraction algorithms on optical flow in existing micro-expression studies, and it has been confirmed that OF can extract subtle facial change features of micro-expressions [1,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The mathematical definition of optical flow is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 0 t I p I → ∇ + = \uf067 (5) where ( ) , , I x y t denotes the pixel intensity of the space ( ) ,x y at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' ( ) , x y I I I ∇ = refers to the spatial gradient and tI denotes the temporal gradient representing the intensity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' p → is the horizontal and vertical components of the optical flow, denoted as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' , T dx dy p p q dt dt → \uf8ee \uf8f9 = = = \uf8ef \uf8fa \uf8f0 \uf8fb (6) which dx , dy refers to the change in the two-dimensional position and td represents the change in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' This constraint on luminance constancy assumes that the pixel intensities of the two images are constant over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' We adopt TV-L1 [39] for optical flow approximation because it has two main advantages: better noise robustness and the ability to preserve the discontinuity of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' To obtain essential features of optical flow features in different directions, we compute optical strain(OS) for feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Each video can be represented by a single OS image showing facial deformation [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The two-dimensional displacement vector of the moving object can be expressed as [ , ]T u u v = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Assuming that the moving object is in a small motion, the strain magnitude can be defined as Equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' ( ) 1 2 T u u ε \uf8ee \uf8f9 = ∇ + ∇ \uf8f0 \uf8fb (7) It can be further expanded as Figure 2 Dynamic images of different micro-expression expressions, from left to right: happiness, disgust, repression, surprise, anger, sadness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 1 2 1 2 xx xy yx yy u u v x y x v u v x y y ε ε ε ε ε \uf8ee \uf8f9 \uf8eb \uf8f6 ∂ ∂ ∂ = = + \uf8ef \uf8fa \uf8ec \uf8f7 ∂ ∂ ∂ \uf8ed \uf8f8 \uf8ef \uf8fa = \uf8ef \uf8fa \uf8eb \uf8f6 ∂ ∂ ∂ \uf8ef \uf8fa = + = \uf8ec \uf8f7 ∂ ∂ ∂ \uf8ef \uf8fa \uf8ed \uf8f8 \uf8f0 \uf8fb (8) The diagonal strain component ( ) , xx yy ε ε is the normal strain component, which ( ) , xy yx ε ε is shear strain component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Specifically, normal strain measures the change in length along a particular direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' And shear strain measures the change in two angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The optical strain for each pixel can be obtained by calculating the sum of the squares of the normal and shear strain components, as shown in equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 2 2 2 2 , 2 2 2 1 2 x y xx yy xy yx u v u u x x x x ε ε ε ε ε = + + + ∂ ∂ ∂ ∂ \uf8eb \uf8f6 = + + + \uf8ec \uf8f7 ∂ ∂ ∂ ∂ \uf8ed \uf8f8 (9) To obtain the first- and second-order optical flow variation features, referring to [22,41], we use the horizontal optical flow, vertical optical flow, and optical strain as three channels of the image for feature learning, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='3 Multi-scale multi-modal transformer To obtain multi-scale features of images, this paper converts images into multiple scales, then learns the spatial relationship between different patches through the network self-attention mechanism to get local discriminative features of the same modal at different scales and improve the recognition ability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The multi-scale module and the specific patch feature weighted module are shown in figure 4 and figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='1 Multi-scale patch The original ViT model has a fixed size on the patch, which tends to ignore finer-grained micro-expression features, despite the results achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' To acquire images at different scales, as shown in figure 4, the image is converted to the specified size( ) , m n for patches at the first scale, which can be divided into ( ) ( ) 16 16 m m × patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' At the second scale, the image is reduced to Figure 3 Optical flow-optical strain image onset trame of os apex frame( ) 2, 2 m n size, still patched embedding according to the pixel size of 16 16 × , and at the third scale, the image is reduced to ( ) 4, 4 m n size and patched embedding in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Since ViT is used to learn the spatial relationships of different patch regions, we do not consider 1 1 × patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' We proposed a new method of patch selection in which the intensity of attention can be intuitively used as an indicator to characterize the importance of symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' As shown in Figure 5, in this module, we integrate all the original attention weights of the first 1 L − layers of the transformer into an attention graph to guide the network to efficiently and accurately select the distinguished image patches and compute their relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The embedding lacks token identifiability, and the original attention weights do not necessarily correspond to the relative importance of the input token, especially for the higher levels of the model [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' To effectively utilize the attention mechanism of ViT, we improved the input of the last layer of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' As shown in Figure 6, the weights of the first 1 L − layers of the network are first extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Figure 5 patch feature weighting module Figure 4 Multi-scale feature selection module patch feature weighted module TransformerEncoder position embedding class_token MLP (1-7)x Transformer patch feature weighting Transformer Encoder Norm linear projection TransformerEncoder Multi-Head Encoder .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Attention .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Norm Embedded Patchesclass score fc2 1 fc1 1 concatenate cls_token cls_token cls_token patch feature patch feature patch feature weighted module weighted module weighted module 个 ↑1 2 , , , H l l l L w w w w \uf8ee \uf8f9 = \uf8f0 \uf8fb \uf04c , 1,2, , 1 l L ∈ − \uf04c (10) 1 2 , , , i D l l l l w w w w \uf8ee \uf8f9 = \uf8f0 \uf8fb \uf04c , 1,2, , i H ∈ \uf04c (11) where H is the number of heads, L is the number of layers, and D is the embedded feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' We consider that the critical attention features are gradually accumulated and amplified in each layer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' so we first regularized the weights of multiple attentions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' we sum up the multiple attention values of each layer weight and take the mean value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' and then regularize them ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='in each embedding dimension (Regularization is mainly used to avoid the generation of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='over-fitting and reduce network errors): ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='h 1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='The attention weights of all previous layers are then integrated and the matrix multiplication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='operation is performed recursively on the modified attention weights of all layers as follows: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='Figure 6 patch feature weighting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='output,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Output efficient patch features W final Layer outputr- WL-1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='41 L-1 Layers 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='28 W2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='17 LayerAttention MatrixProcessing inptt( ) 1 1 L l L G G − = =∏ (13) Since the elements in G are the attention-related values of each patch with respect to other patches,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' to obtain the weight value of each patch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' we average the results by row and normalize them: ( ) ( ) ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='0 / max ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='0 G mean G mean G − = (14) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' the output features of the L-1th layer are multiplied with the adjusted attention weights to obtain the weighted patch features,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' which are then concatenated with the cls_token and input to the last Transformer Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 1 1 _ , L L L input cls token G Z − − − \uf8ee \uf8f9 = \uf8ef \uf8fa \uf8f0 \uf8fb (15) The output of the last layer is shown in equation (16): ( ) ( ) ( ) ( ) L L L Z FFN LN MSM LN input input = + (16) Since facial expressions occur in some local regions of the face, other regional features contribute less to the expressions, but some features will be lost if the regions with lower weights are dropped, so these features are still kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' In this way, we not only keep the global information, but focus on the nuances of micro-expressions in the last Transformer Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='3 Multi-feature fusion Since optical flow and dynamic imaging have different representations of micro-expression and have complementary roles, therefore, to capture the key features of different modalities in multi-scale mode, we concatenate the cls_token of the features learned at different scales as the final features of individual modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The two features are then concatenated into two fully connected layers for expression classification, as shown in equations (17) and (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' To avoid over-fitting, we add dropout and ReLU after the first fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' ( ) ( ) 2 1 class_ ( _ , _ ) final final score FC FC Concat dy imaging flow os = (17) ( ) 1 2 3 ( _ , _ , _ ), _ , _ final F Concat cls token cls token cls token F dy imaging flow os = ∈ (18) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='4 Network optimization Contrast learning generates anchor samples, positive and negative samples from an unlabeled database and learns the similarity of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The common features of the samples are extracted by encoding the positive samples similarly and encoding the negative samples differently through an encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Since the dynamic and optical flow maps used in this paper belong to two different modalities, the similarity of the two modal features on the same category is maximized and the similarity of different categories is minimized by using contrast learning [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The micro-expression database i consists of two modalities, the sample set i dy and _ i flow os , and positive and negative sample pairs are constructed according to whether the two modalities come from the same micro-expression sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' For the positive sample pair, fixed i dy and the loss function is: ( ) ( ) ( ) ( ) ( ) ( ) , , _ 1, 1 exp , _ / log exp , _ / exp , _ / i i i dy flow os B B i i i k k k i k s dy flow os L s dy flow os s dy flow os τ τ τ = ≠ = = − + ∑ ∑ (19) where B is the small batch size, τ is the temperature coefficient, and s(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=') denotes the cosine similarity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Fixed _ i flow os ,the loss can be obtained, and the comparative loss function con L for the two modes is then denoted as: , , _ , _ , 1 1 [ ] 2 B con i dy flow os i flow os dy i L L L B = = + ∑ (20) The total loss of the network is: ( ) _ 1 loss con cross entroy L L L α α = − ∗ + ∗ (21) 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='1 Datasets Three spontaneous micro-expression datasets are involved in the experimental validation of the algorithm in this paper, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=', SMIC, CASMEII, SAMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' To be able to compare fairly with other state-of-the-art algorithms, we not only test the three datasets individually, but also refer to the combined data set proposed in MEG2019 [44] to measure the algorithm in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' SMIC The SMIC database [45] is the first database containing spontaneous micro-expressions obtained through an emotion elicitation experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The experimental test data is an HS subset with a frame rate of 100 frames and a resolution of 640 480 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=" The data set includes a total of 164 micro-expression clips from 16 participants and does not contain action unit labels, and the data were labeled by two coders into three categories of expression clips based on participants' statements about their subjective emotions while watching the video: positive (51), negative (70), and surprised (43)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' CASMEII CASME II [46] is an improved version of the CASME database with a higher frame rate (200pfs) and a resolution of 640 480 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The CASME II database contains 255 micro-expression clips from 26 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=" Each micro-expression clip is classified by the coders according to the facial AU, the subject's self-reports and the video content, and the database contains seven categories, namely disgust(60), happiness(32), repression(27), surprise(25), sadness(7),fear(2) and others(102)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Because there are few samples of sadness and fear, only the first five classes of data are selected in single data set experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' In addition, the database provides the onset frame, apex frame, offset frame and AU labels of the micro-expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' SAMM SAMM [47] contains a total of 159 micro-expression sequences from 32 participants from 13 ethnic groups, a database with a high resolution of 2040 1088 × and a frame rate of 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The coders objectively classified expressions into seven basic expression categories based on FACS: anger (57), happiness (26), surprise (15), contempt (12), disgust (9), fear (8), sadness (6), and other (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Similarly, in the single database test, the data participating in the experimental analysis only contains the five categories with the largest number of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' SAMM also provides onset frames, apex frames, offset frames, and AU labels of expression clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The combined database included 442 micro-expression samples from 68 participants from three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Of these, 164 samples were from SMIC, 145 were from CASMEII, and 133 were from SAMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' These samples were reclassified into three categories: positive, negative, and surprise [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Since the transformer network needs enough data for network learning, we enhance the data by rotating [-10,10], flipping horizontally, scale change, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The SMIC dataset does not provide the onset and apex frames of expressions, so we take the intermediate frames as the apex frames for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='2 Evaluation metrics Due to the imbalanced distribution of the number of categories in the database, we used three metrics to reduce bias: accuracy (Acc), unweighted average recall (UAR), and unweighted F1 score (UF1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The unweighted F1 score (UF1) is obtained by summing the F1 value of each class and then calculating the average value according to the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The unweighted average recall (UAR) is obtained by summing the accuracy of each class and then averaging over the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=" 1 1 C c c C c c TP Acc N = = = ∑ ∑ (22) 1 1 C c c c TP UAR C N = = ∑ (23) ' 1 1 2 1 2 C c c c c c TP UF C TP FP FN = = + + ∑ (24) where C is the number of classes and c N is the number of samples per class." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' TP , FN and FP are true positives, false negatives and false positives, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' In the experiments, we used the LOSO cross-validation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' All the micro-expression data of one subject were used for testing, and the samples of the other subjects were used for training until each subject completed the test as a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='3 Pre-processing Since the SAMM database does not provide cropped data, we processed the three datasets in a unified manner to reduce interference terms in the micro-expression recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' As shown in Figure 7, we used the Dlib library to locate 68 feature points of the face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Since the duration of micro-expressions is relatively short, some tight facial movements can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Therefore, we only compute the affine transformation for the first frame image with two inner eye angle coordinates, align the subsequent frames with the same transformation, and then perform facial cropping with the landmark points of the aligned face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='1 Eulerian video magnification The Eulerian video amplification (EVM) technique [48] can be used to amplify subtle motion changes in video that are difficult to see with the naked eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Motion amplification has been widely used in micro-expression recognition tasks due to the increased micro-motion amplitude, which is better for feature extraction and analysis [49,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' In this paper, through experimental comparison, the optimal results can be obtained when the amplification factor is set as 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='4 Implementation details For multi-scale feature extraction process, we used the ViT-base with 12 layers of Encoder and hidden layers of size 768, with 12 heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' For initialization, we used the pre-trained official ViT-B model on ImageNet[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' In the training phase, we set the batch size to 16 and trained the network for 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' We initialized the learning rate to 6e-5 and weight decay to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='05, updating the network weights using AdamW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' AdamW is an optimization algorithm that uses adaptive learning rate gradient descent to make the network converge faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' All our experiments are windows systems and Nvidia GeForce RTX 1080Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='5 Results and discussion We compare the proposed approach with commonly used manual feature extraction methods and recently outstanding deep learning methods in single database experiment (SDE) and combined database experiment(CDE) settings based on widely used micro-expression databases: (a) (b) Figure 7 (a)68 landmark points on the face (b)cropped face SMIC-HS, CASME II, SAMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='1 SDE To ensure consistency and fairness of comparison, SDE results for all methods were obtained under the same conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=', for the same number of samples, number of labels (classes), and using the same cross-validation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' As can be easily seen from Table 1, among the experimental results under SDE settings, the proposed method in this paper performs the best on the SMIC database with an accuracy of 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='3%， which is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='13% higher than the next best, and UF1 reaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='7216, which is at the next best level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The accuracy on the SAMM database reached 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='83%, which is second to the best result and at an advanced level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' However, the proposed algorithm is not outstanding in CASMEII database, while the results of the Sparse transformer algorithm are more outstanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' We analyzed the reasons for this result and found that the CASMEII database has many categories and small differences between categories, there are samples with similar or the same facial motion regions belong to different categories, which has a great interference to the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The Sparse transformer algorithm is able to achieve outstanding results on the database mainly due to the use of CNN and Table1 Micro-expression recognition results of the proposed MSMMT algorithm and the state-of-the-art method tested singularly on the three datasets SMIC-HS CASMEII SAMM Acc F1 Acc F1 Acc F1 LBP-TOP[10*] 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='5384 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='4241 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+page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='6917 Sparse Transformer[13](2022) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='7192 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='7547 MSMMT(Ours) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='30 0.' metadata={'source': 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transformer, and the algorithm is able to combine the advantages of both networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The algorithm in this paper is based on a pure ViT network, although slightly inferior to the Sparse Transformer algorithm, but both algorithms have their own advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Different from the methods based on the transformer algorithm [10,11,13], the multi-scale transformer in this paper focuses more on extracting features of different sizes of the face, making full use of the attention weights of multiple encoder layers of the transformer to select key region features, which are more expressive for features of different granularity and can obtain comparable results even when using only a single frame feature image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='2 CDE Some similar and the latest micro-expression recognition algorithms are compared in Table 2, and it can be easily seen that our method achieves the best level with UF1 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='8160 and UAR of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='8191 on the combined database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The performance on the SMIC and CASMEII datasets is outstanding, both outperforming the other algorithms, with UF1 reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='7651 and UAR reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='7780 for the SMIC database, UF1 reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='9071 and UAR reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='8878 for the CASMEII database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' For the SAMM database, we can see that the results are also comparable but not at the optimal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The above results show an important conclusion that our proposed method is effective and can obtain better representation of features in the CDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The reason for being able to achieve this result, we believe that benefiting from multi-scale multi-modal, the network is able to learn the local features of the samples with sufficient samples than the single data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='5.' metadata={'source': 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+page_content='7192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='7215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='8798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='7715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='7890 MSMMT(Ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='8160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='8191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='7651 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='7780 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='9071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='8878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='7392 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content="7163 This subsection discusses in detail the effect of the loss weight factor on the network, and we show the results for the three data sets from CDE's experiments as well as the combined set." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' As can be seen in Figure 8, the accuracy increases first and then decreases as the weighting factor α gradually increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Although the parameters of α are different when the optimal results are obtained on the three datasets, comprehensive comparison shows that when 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='1 α = , the two modes have the best performance in the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Similarly, we selected the optimal results under the α parameter for the three data sets in the SDE experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The α parameter was different for the three datasets, with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='1 for the SMIC and CASMEII database, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='2 for the SAMM database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' 5 Conclusion In this work, we proposed a micro-expression recognition method based on multi-scale learning of bimodal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' The multi-scale features are learned by using ViT for dynamic imaging and optical flow features, and the patch features are weighted by using multi-headed self-attention weights in the network to obtain the most expressive facial region features and reduce the influence of irrelevant facial patches on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' By combining cross-modal unsupervised contrastive learning, the information of texture and motion modal features are processed in the same category close to and different categories far from each other, enabling the network to fully use both features for expression learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' In this paper, a large number of experiments are conducted on three datasets, and the results obtained have good recognition rates, which fully demonstrate the effectiveness of the multi-scale algorithm proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' In the future, we will further study the local feature expressions of micro-expressions on this basis and explore more meaningful features of different categories of micro-expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' Liong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content=' See, K.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content="006 (a) (b) Figure 8 Variation curves of UF1, UAR values with α for three categories (a) Variation of UF1 values (b) Variation of UAR values Variation of UF1 value 1 6'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} +page_content='7 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9E1T4oBgHgl3EQfKwO_/content/2301.02969v1.pdf'} diff --git a/c9FKT4oBgHgl3EQfqS4B/content/tmp_files/2301.11873v1.pdf.txt b/c9FKT4oBgHgl3EQfqS4B/content/tmp_files/2301.11873v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8bb2e49efaf17310771cd3979f1e2c7beaba18fc --- /dev/null +++ b/c9FKT4oBgHgl3EQfqS4B/content/tmp_files/2301.11873v1.pdf.txt @@ -0,0 +1,3040 @@ +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN +HIERARCHICAL MODELS +Lasse Elsem¨uller +Department of Psychology +University of Mannheim +lasse.elsemueller@gmail.com +Martin Schnuerch +Department of Psychology +University of Mannheim +martin.schnuerch@gmail.com +Paul-Christian B¨urkner +Cluster of Excellence SimTech +University of Stuttgart +paul.buerkner@gmail.com +Stefan T. Radev +Cluster of Excellence STRUCTURES +Heidelberg University +stefan.radev93@gmail.com +ABSTRACT +Bayesian model comparison (BMC) offers a princi- +pled approach for assessing the relative merits of com- +peting computational models and propagating uncer- +tainty into model selection decisions. However, BMC +is often intractable for the popular class of hierarchical +models due to their high-dimensional nested parame- +ter structure. To address this intractability, we propose +a deep learning method for performing BMC on any +set of hierarchical models which can be instantiated +as probabilistic programs. Since our method enables +amortized inference, it allows efficient re-estimation +of posterior model probabilities and fast performance +validation prior to any real-data application. In a se- +ries of extensive validation studies, we benchmark the +performance of our method against the state-of-the- +art bridge sampling method and demonstrate excel- +lent amortized inference across all BMC settings. We +then use our method to compare four hierarchical evi- +dence accumulation models that have previously been +deemed intractable for BMC due to partly implicit +likelihoods. In this application, we corroborate evi- +dence for the recently proposed L´evy flight model of +decision-making and show how transfer learning can +be leveraged to enhance training efficiency. Repro- +ducible code for all analyses is provided. +1 +Introduction +Hierarchical or multilevel models (HMs) play an increas- +ingly important methodological role in the social and cog- +nitive sciences (Farrell & Lewandowsky, 2018; Rouder et +al., 2017). HMs embody probabilistic and structural in- +formation about nested data occurring frequently in vari- +ous settings, such as educational research (Ulitzsch et al., +2020), experimental psychology (Vandekerckhove et al., +2011), epidemiology (Jalilian & Mateu, 2021) or astro- +physics (Hinton et al., 2019), to name just a few. Cru- +cially, HMs can often extract more information from rich +data structures than their non-hierarchical counterparts +(e.g., aggregate analyses), while retaining a relatively high +intrinsic interpretability of their structural components +(i.e., parameters). Moreover, viewed as formal instanti- +ations of scientific hypotheses, HMs can be employed to +systematically assign preferences to these hypotheses by +means of formal model comparison. For example, Haaf +and Rouder (2017) proposed a powerful framework based +on Bayesian HMs for formulating and testing competing +theoretical positions on quantitative vs. qualitative indi- +vidual differences. +We consider Bayesian model comparison (BMC) as a +principled framework for comparing and ranking compet- +ing HMs via Occam’s razor (Kass & Raftery, 1995; Lotfi +et al., 2022; MacKay, 2003). However, standard BMC is +analytically intractable for non-trivial HMs, as it requires +marginalization over high-dimensional parameter spaces. +Moreover, BMC for complex HMs without explicit like- +lihoods (i.e., HMs available only as randomized simula- +tors) becomes increasingly hopeless and precludes many +interesting applications in the rapidly expanding field of +simulation-based inference (Cranmer et al., 2020). +In this work, we propose to tackle the problem of BMC +for arbitrarily complex HMs from a simulation-based per- +spective using deep learning. In particular, we build on +the BayesFlow framework (Radev, D’Alessandro, et al., +2021; Radev et al., 2020) for simulation-based Bayesian +inference and propose a novel hierarchical neural network +architecture for approximating Bayes factors (BFs) and +arXiv:2301.11873v1 [stat.ML] 27 Jan 2023 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +posterior model probabilities (PMPs) for any collection +of HMs. +Our neural approach circumvents the steps of explicitly +fitting all models and marginalizing over each model’s pa- +rameter space. Thus, it is applicable to both HMs with +explicit likelihood functions and HMs accessible only +through Monte Carlo simulations (i.e., with implicit like- +lihood functions). Moreover, our neural networks come +with an efficient way to compute their calibration error +(Guo et al., 2017), which provides an important diagnos- +tic for self-consistency. Lastly, trained networks can be +adapted to related tasks, substantially reducing the com- +putational burden when dealing with demanding simula- +tors. +The remainder of this paper is organized as follows. In +Section 2, we introduce the theoretical background and +related work on (hierarchical) BMC. We then present the +rationale and details of our deep learning method in Sec- +tion 3. In Sections 4.1 and 4.2, we present two valida- +tion studies of the proposed method: One that includes +toy models for illustrative purposes and one that includes +two popular classes of models from the field of cognitive +psychology. In Section 4.3, we then apply our method +to compare hierarchical diffusion decision models with +partly intractable likelihoods on a real data set. Finally, +Section 5 summarizes our contributions and discusses fu- +ture perspectives. +2 +Theoretical Background +2.1 +Bayesian Hierarchical Modeling +In order to streamline statistical analyses, researchers rely +on assumptions about the probabilistic structure or sym- +metry of the assumed data-generating process. For in- +stance, the canonical IID assumption in psychological +modeling states that (multivariate) observations are in- +dependent of each other and sampled from the same +latent probability distribution (Nicenboim et al., 2022; +Singmann & Kellen, 2019). +However, more complex dependencies may arise in a +variety of contexts. For instance, if there are repeated +measurements per participant or participants belong to +different natural groups (e.g., school classes, working +groups), the respective observations exhibit higher cor- +relations within those clusters than across them. Ignor- +ing this nested structure in statistical analyses may re- +sult in biased conclusions (Singmann & Kellen, 2019). +Bayesian HMs formalize this structural knowledge by as- +suming that observations are sampled from a multilevel +generative process (Gelman, 2006). +For instance, +the generative recipe for a two-level +Bayesian HM can be written as: +η ∼ p(η) +(1) +θm ∼ p(θ | η) for m = 1, . . . , M +(2) +xmn ∼ p(x | θm) for n = 1, . . . , Nm, +(3) +where η denotes the group-level parameters, θm denotes +the individual parameters in group m and xmn represents +the n-th observation in group m. Such a model suggests +the following (non-unique) factorization of the joint dis- +tribution: +p(η, {θm}, {xmn}) = +p(η) +M +� +m=1 +p (θm η) +Nm +� +n=1 +p (xmn | θ) . +(4) +The set notation {θm} and {xmn} implies that the num- +ber of groups and observations in each group can vary +across simulations, data sets and experiments and that +these quantities are exchangeable. +HMs can be considered as a compromise between a sepa- +rate analysis of each group (no-pooling) that neglects the +information contained in the rest of the data and an aggre- +gate analysis of the data (complete pooling) that loses the +distinction between intra-group and inter-group variabil- +ity (Hox et al., 2017). The partial pooling of information +induced by HMs leads to more stable and accurate indi- +vidual estimates through the shrinkage properties of mul- +tilevel priors, whereby single estimates inform each other +(B¨urkner, 2017; Gelman, 2006). +Despite having desirable properties, hierarchical model- +ing comes at the cost of increased complexity and compu- +tational demands. These increased demands make it hard +or even impossible to compare competing HMs within +the probabilistic framework of BMC. Before we highlight +these challenges, we first describe the basics of BMC for +non-hierarchical models. +2.2 +Bayesian Model Comparison +The starting point of BMC is a collection of J compet- +ing generative models M = {M1, M2, . . . , MJ}. Each +Mj is associated with a prior p (θj | Mj) on the pa- +rameters θj and a generative mechanism, which is either +defined analytically through a (tractable) likelihood den- +sity function p (x | θj, Mj) or realized as a Monte Carlo +simulation program gj(θ, z) with random states z. To- +gether, the prior and the likelihood define the Bayesian +joint model +p (θj, x | Mj) = p (θj | Mj) p (x | θj, Mj) , +(5) +2 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +x0 +x1 +p(x|M1) +p(x|M2) +(a) Marginal Likelihoods +M1 +M2 +0.2 +0.4 +0.6 +0.8 +1 +Probability +p(M) +M1 +M2 +p(M|x0) +M1 +M2 +p(M|x1) +(b) Posterior Model Probabilities +Figure 1: Hypothetical BMC setting with a simple model M1 and a more complex model M2. (a) The complex +model which accounts for a broader range of observations needs to spread its marginal likelihood to cover its larger +generative scope. It does so at the cost of diminished sharpness. Thus, even though observation x1 is well within its +generative scope, the simpler model M1 yields a higher marginal likelihood and is therefore preferred. In contrast, +observation x0 has a higher marginal likelihood under model M2, as it is very unlikely to be generated by the simpler +model M1. (b) The corresponding posterior model probabilities (PMPs) given a uniform model prior. +which is also tacitly defined for simulator-based models +by marginalizing the joint distribution p (x, z | θj, Mj) +over all possible execution paths (i.e., random states) of +the simulation program to obtain the implicit likelihood +p (x | θj, Mj) = +� +p (x, z | θj, Mj) dz. +(6) +This integral is typically intractable for complex simula- +tors (Cranmer et al., 2020), which makes it impossible to +evaluate the likelihood and use standard Bayesian meth- +ods for parameter inference or model comparison. +The likelihood function, be it explicit or implicit, is a key +object in Bayesian inference. When the parameters θ are +systematically varied and the data x held constant, the +likelihood quantifies the relative fit of each model instan- +tiation (defined by a fixed configuration θ) to the observed +data. +When we marginalize the Bayesian joint model (Eq. 5) +over its parameter space, we obtain the marginal likeli- +hood or Bayesian evidence (see MacKay, 2003, Chap- +ter 28): +p (x | Mj) = +� +p (x | θj, Mj) p (θj | Mj) dθj. +(7) +The marginal likelihood can be interpreted as the prob- +ability that we would generate data x from model Mj +when we randomly sample from the model’s parame- +ter prior p (θj | Mj). Moreover, the marginal likelihood +is a central quantity for prior predictive hypothesis test- +ing or model selection (Kass & Raftery, 1995; O’Hagan, +1995; Rouder & Morey, 2012). It is well-known that the +marginal likelihood encodes a notion of Occam’s razor +arising from the basic principles of probability (Kass & +Raftery, 1995, see also Figure 1). Thus, the marginal like- +lihood provides a foundation for the widespread use of +Bayes factors (BFs; Heck et al., 2022) or posterior model +probabilities (PMPs; Congdon, 2006) for BMC. +The relative evidence for a pair of models can be com- +puted through the ratio of marginal likelihoods for the two +competing models Mj and Mk, +BFjk = p (x | Mj) +p (x | Mk). +(8) +This ratio is called Bayes factor (BF) and is widely used +for quantifying pairwise model preference in Bayesian +settings (Heck et al., 2022; Kass & Raftery, 1995). Ac- +cordingly, a BFjk > 1 indicates preference for model j +over model k given available data x. Alternatively, one +can directly focus on the (marginal) posterior probability +of a model Mj, +p (Mj | x) = +p(x | Mj) p(Mj) +�J +j=1 p (x | Mj) p(Mj) +, +(9) +where p(Mj) is a categorical (typically uniform) prior +distribution encoding a researcher’s prior beliefs regard- +ing the plausibility of each considered model. This prior +distribution is then updated with the information con- +tained in the marginal likelihood p(x | Mj) to obtain +the corresponding posterior model probability (PMP), +p(Mj | x). Occasionally in the text, we will refer to the +vector of PMPs for all J models as π and to the individual +PMPs as πj. The ratio of two PMPs, known as posterior +odds, is in turn connected to the Bayes factor via the cor- +responding model priors: +p(Mj | x) +p(Mk | x) = p (x | Mj) +p (x | Mk) × p(Mj) +p(Mk). +(10) +Despite its intuitive appeal, the marginal likelihood (and +thus BFs and PMPs) represents a well-known and widely +appreciated source of intractability in Bayesian work- +flows, since it typically involves a multi-dimensional inte- +gral (Eq. 7) over potentially unbounded parameter spaces +(Gronau, Sarafoglou, et al., 2017; Lotfi et al., 2022). +3 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +Furthermore, the marginal likelihood becomes doubly in- +tractable when the likelihood function is itself not avail- +able (e.g., in simulation-based settings), thereby making +the comparison of such models a challenging and some- +times, up to this point, hopeless endeavor. +Unsurprisingly, estimating the marginal likelihood (Eq. 7) +in the context of hierarchical models becomes even more +challenging, since the number of parameters over which +we need to perform marginalization grows dramatically +(i.e., parameters at all hierarchical levels enter the compu- +tation). These computational demands render probabilis- +tic comparison of HMs based on BFs or PMPs analyti- +cally intractable even for relatively simple models with +explicit (analytical) likelihoods. Therefore, researchers +need to resort to costly, approximate methods which typi- +cally only work for models with explicit likelihoods (Gel- +man & Meng, 1998; Gronau, Sarafoglou, et al., 2017; +Meng & Schilling, 2002). +2.3 +Approximate Bayesian Model Comparison +2.3.1 +Explicit Likelihoods +The most efficient approximate methods to date require +all candidate models to possess explicitly available like- +lihood functions. For the most simple scenario in which +two HMs are nested (e.g., through an equality constraint +on a parameter), the Savage-Dickey density ratio (Dickey +& Lientz, 1970) provides a convenient approximation of +the BF (Wagenmakers et al., 2010). Typically, however, +the candidate models are not nested but exhibit notable +structural differences. Thus, a general-purpose method is +needed to encompass the entire plethora of model com- +parison scenarios arising in practical applications. +A more general method, and the current state-of-the-art +for comparing HMs in psychological and cognitive mod- +eling (Gronau et al., 2020; Gronau et al., 2019; Schad +et al., 2022), is given by bridge sampling (Bennett, 1976; +Meng & Wong, 1996). Bridge sampling has enabled com- +parisons within families of complex process models, such +as multinomial processing trees (MPTs; Gronau et al., +2019) or evidence accumulation models (EAMs; Gronau +et al., 2020), and serves as a simple add-on for Markov +chain Monte Carlo (MCMC) based Bayesian workflows. +Crucially, bridge sampling relies on the posterior draws +generated by an MCMC sampler (e.g., Stan; Carpenter +et al., 2017) to efficiently approximate the marginal likeli- +hood of each respective model (Gronau, Sarafoglou, et al., +2017). Note, however, that bridge sampling requires con- +siderably more random draws for stable results than stan- +dard parameter estimation (usually about an order of mag- +nitude more; Gronau, Singmann, et al., 2017). Moreover, +the approximation quality of bridge sampling is depen- +dent on the convergence of the MCMC chains (Gronau et +al., 2020). Finally, there are no strong theoretical guaran- +tees that the approximations are unbiased and accurately +reflect the true marginal likelihoods (Schad et al., 2022). +2.3.2 +Implicit Likelihoods +With the rise of complex, high-resolution models, in- +tractable likelihood functions (i.e., functions that do not +admit a closed form or are too costly to evaluate) be- +come more and more common in statistical modeling. +Such models are not limited to psychology and cogni- +tive science (Nicenboim et al., 2022; Van Rooij et al., +2019), but are also common in fields such as neuro- +science (Gonc¸alves et al., 2020), epidemiology (Radev, +Graw, et al., 2021), population genetics (Pudlo et al., +2016) or astrophysics (Hermans et al., 2021) (see Cran- +mer et al., 2020). Despite the common term likelihood- +free, simulator-based models still possess an implicitly +defined likelihood (see Section 2.2) from which we can +obtain random draws through Monte Carlo simulations. +This enables model comparison through simulation-based +methods, usually by means of approximate Bayesian com- +putation (ABC; Marin et al., 2018; Mertens et al., 2018; +Pudlo et al., 2016). +Traditional (rejection-based) ABC methods for BMC re- +peatedly simulate data sets from the specified generative +models, retaining only those simulations that are suffi- +ciently similar to the empirical data. To enable the cal- +culation of this (dis-)similarity even in high-dimensional +cases, the information contained in the simulated data sets +is reduced by computing hand-crafted summary statis- +tics, such as the mean and variance (Csill´ery et al., 2010; +Sunn˚aker et al., 2013). +The resulting acceptance rates +of the candidate models represent the approximations of +their PMPs (Marin et al., 2018; Mertens et al., 2018). +Even for non-hierarchical models, ABC methods are +known to be notoriously inefficient and highly dependent +on the concrete choice of summary statistics (Cranmer et +al., 2020; Marin et al., 2018). This choice is even more +challenging for HMs, as modelers now have to retain an +optimal amount of information on multiple levels. More- +over, the rapidly growing number of summary statistics +reduces the probability that a simulated data set is similar +enough to the empirical data, which vastly increases the +number of required simulations (Beaumont, 2010; Marin +et al., 2018). +Regardless of the number of summary statistics, their +manual computation carries the danger of insufficiently +summarizing the simulations and thereby producing bi- +ased approximations (a phenomenon known as curse of +insufficiency; Marin et al., 2018). While many improve- +ments of rejection-based ABC have been proposed, most +4 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +notably ABC-MCMC (Marjoram et al., 2003; Turner & +Sederberg, 2014), ABC-SMC (Sisson et al., 2007), as well +as Gibbs ABC (Turner & Van Zandt, 2014) for Bayesian +hierarchical modeling in particular (see also Clart´e et al., +2021; Fengler et al., 2021), these advancements are still +limited by their dependence on hand-crafted summary +statistics or kernel density estimation methods. +Recent developments, such as ABC-RF (Pudlo et al., +2016), combine ABC with machine learning methods to +build more expressive approximators for BMC problems. +Accordingly, model comparison is treated as a supervised +learning problem – the simulated data encompasses a +training set for a machine learning algorithm that learns to +recognize the true generative model from which the data +set was simulated. The machine learning approach re- +duces the inefficiency problem that haunts rejection-based +ABC methods, but does not alleviate the curse of insuffi- +ciency (Marin et al., 2018). +2.4 +Bayesian Model Comparison with Neural +Networks +Recently, Radev, D’Alessandro, et al. (2021) explored a +method for simulation-based BMC using specialized neu- +ral networks. The authors proposed to jointly train two +specialized neural networks using Monte Carlo simula- +tions from each candidate model in M: a summary net- +work and an evidential network. The goal of the summary +network is to extract maximally informative (in the opti- +mal case, sufficient) summary statistics from complex data +sets. The goal of the evidential network is to approximate +PMPs as accurately as possible and, optionally, to quan- +tify their epistemic uncertainty. +Importantly, simulation-based training of neural networks +enables amortized inference for both implicit and ex- +plicit likelihood models. Amortization is a property that +ensures rapid inference for an arbitrary amount of data +sets after a potentially high computational investment for +simulation and training (Mestdagh et al., 2019; Radev, +D’Alessandro, et al., 2021; Radev et al., 2020). +As a +consequence, the calibration (Guo et al., 2017; Talts et +al., 2018) or the inferential adequacy (Schad et al., 2021; +Schad et al., 2022) of an amortized Bayesian method are +embarrassingly easy to validate in practice. +In contrast, non-amortized methods, such as ABC- +MCMC (Turner & Sederberg, 2014) or ABC-SMC (Sis- +son et al., 2007) need to repeat all computations from +scratch for each observed data set. Thereby, it is often in- +feasible to assess their calibration or inferential adequacy +in the pre-data phase of a Bayesian workflow (Gelman et +al., 2020). +Unfortunately, the evidential method proposed by Radev, +D’Alessandro, et al. (2021) is not applicable to HMs due +to their nested probabilistic structure which cannot be +tackled via previous summary networks. This severely +limits the applicability of the method in quantitative re- +search, where hierarchical models have been advocated +as a default choice (Lee, 2011; McElreath, 2020; Rouder +et al., 2017). In the following, we describe how to extend +the original method to enable amortized BMC for HMs. +3 +Method +At its core, our method involves a multilevel permutation +invariant neural network which is aligned to the proba- +bilistic symmetry of the underlying HMs (see Figure 2 for +a visualization). We hold that any method which does not +rely on ad hoc summary statistics should take this prob- +abilistic symmetry (e.g., exchangeability) into account in +order to ensure the structural faithfulness of its approx- +imations. +Moreover, respecting the probabilistic sym- +metry implied by a generative model cannot only make +simulation-based training easier but also suggests a par- +ticular architecture for building neural Bayesian approxi- +mators. +3.1 +Permutation Invariance +Permutation invariance is the functional equivalent of the +probabilistic notion of exchangeability (Bloem-Reddy & +Teh, 2020; Gelman, 2006), which roughly states that the +order of random variables should not influence their joint +probability. +To illustrate this point, consider the model in Eq. 4, +which has two exchangeable levels by design, indexed by +m ∈ {1 . . . , M} and n ∈ {1, . . . , Nm}. In a setting fa- +miliar to social scientists, we might have M individuals, +each of whom provides Nm (multivariate) responses on +some scale or in repeated trials of an experiment. Now, +suppose that we want to compare a set of HMs M = +{M1, . . . , MJ} of the form given by Eq. 4 that might +differ in various ways (e.g., different prior/hyperprior as- +sumptions or disparate likelihoods). Due to the structure +of the models, the PMPs p(M | {xmn}) depend on nei- +ther the ordering of the individuals nor the ordering of +their responses (which also holds true for the correspond- +ing BFs). +More precisely, if S(·) is an arbitrary permutation of an +index set, then +p(M | {xmn}) = p(M | S({xmn})) +(11) +for any S(·) acting on {1 . . . , M} × {1, . . . , N1} × · · · × +{1, . . . , NM} where × denotes the Cartesian product of +5 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +Figure 2: Our proposed hierarchical neural network architecture for encoding permutation invariance in the trans- +formation of nested, two-level data into posterior model probabilities. A first invariant module Σ(1) +I +reduces all Nm +observations within each of the M groups to a single intermediary embedding vector �xm. For readability of the fig- +ure, we display Nm = N as constant across each group. A second invariant module Σ(2) +I +reduces all intermediary +embedding vectors to a hierarchical embedding vector z, which gets passed through an inference network to arrive at +the final vector �π of approximated posterior model probabilities. +two (index) sets. Note that this notation implies that only +permuting each m and permuting each n within, but not +across each group m is allowed. The property of permu- +tation invariance is immediately obvious from the right- +hand side of Eq. 4 that involves two nested products (prod- +ucts being permutation invariant transformations when +seen as functions operating on sets). Naturally, learning +permutation invariance directly from data or simulations +is hardly feasible with standard neural networks, even for +non-nested data. Indeed, for non-hierarchical generative +models, Radev, D’Alessandro, et al. (2021) propose to use +composite permutation invariant networks as employed +by Zaheer et al. (2017). In the following section, we gen- +eralize this architectural concept to the hierarchical set- +ting. +3.2 +Hierarchical Invariant Neural Network +Architecture +Permutation invariant networks differ from standard feed- +forward networks in that they can process inputs of dif- +ferent sizes and encode the probabilistic symmetry of the +data directly (i.e., remove the need to learn the symmetry +implicitly during training by supervised learning alone). +For the purpose of BMC with HMs, we realize a hi- +erarchical permutation invariant function via a stack of +invariant modules Σ(l) +I +for each hierarchical level l = +1, . . . , L of the Bayesian model (see Figure 2). Each in- +variant module performs an equivariant non-linear trans- +formation h(l) +1 +acting on the individual data points, fol- +lowed by a pooling operator (e.g., sum or max) and a fur- +ther non-linear transformation h(l) +2 +acting on the pooled +data. +In order to preserve hierarchical symmetry, we apply each +Σ(l) +I independently to each nested sequence of data points. +To make this point concrete, consider the two-level model +given by Eq. 4 and let data point xmn denote the multi- +variate response of person m in trial n of some data col- +lection experiment. Accordingly, the first invariant mod- +ule Σ(1) +I +operates by reducing the trial data {xn}m of each +person m to a single person-vector �xm of fixed size: +�xm = Σ(1) +I +({xn}m) = h(1) +2 +� Nm +� +n=1 +h(1) +1 (xmn) +� +, +(12) +where h1 and h2 are implemented as simple feedforward +neural networks with trainable parameters suppressed for +clarity. +The second invariant module Σ(2) +I +then com- +6 + +X11 +Invariant +Module Z(1) +X1N +Invariant +Module Z(1) +Invariant +Inference +Module Z(2) +Network +ZN +Hierarchical +M1 +M2 +Embedding +Posterior Model +Probabilities π +Invariant +Module Z(1) +XMN +Intermediary +Embeddings +Nested DataA DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +presses all person vectors to a final vector z of fixed size: +z = Σ(2) +I +({xm}) = h(2) +2 +� M +� +m=1 +h(2) +1 (�xm) +� +. +(13) +In this way, the architecture becomes completely indepen- +dent of the number of persons M or number of trials per +person Nm, which could vary arbitrarily across persons. +The vector z, whose dimensionality represents a tunable +hyperparameter, can be interpreted as encoding learned +summary statistics for the BMC task at hand (to be dis- +cussed shortly). Moreover, it is easy to see that z is inde- +pendent of the ordering of persons or the ordering of trials +within persons, as necessitated by the model formulation +in Eq. 4. Thus, the composition Σ(2) +I +◦ Σ(1) +I ({xmn}) re- +duces a hierarchical data set with two levels to a single +vector z which respects the probabilistic symmetry im- +plied by the particular hierarchical model formulation. +3.3 +Increasing the Capacity of Invariant Networks +Encoding an entire hierarchical data set {xmn} into a sin- +gle vector z forces the composite neural network to per- +form massive data compression, creating a potential in- +formation bottleneck. +For complex generative models, +this task can become rather challenging and will depend +highly on the representational capacity of the neural net- +work (i.e., its ability to extract informative data set em- +beddings). Fortunately, we can enhance the simple archi- +tecture described in the preceding paragraph by using in- +sights from Zaheer et al. (2017) and Bloem-Reddy and +Teh (2020). +In order to increase the capacity of the previously +introduced invariant transformation, we can stack to- +gether multiple equivariant modules Σ(l) +E . Each equivari- +ant module implements a combination of equivariant and +invariant transformations. For instance, focusing on our +two-level model example (Eq. 4), the transformations at +level 1 for each person m are now given by: +�xm = h(1) +2 +� Nm +� +n=1 +h(1) +1 (xmn) +� +(14) +�xmn = h(1) +3 ([xmn, �xm]) +for +n = 1, . . . , Nm, (15) +where h3 is also implemented as a simple feedforward +neural network. +In this way, each intermediary output +�xmn of the equivariant module now contains information +from all data points, so the network can learn considerably +more flexible transformations. Moreover, we can stack K +equivariant modules followed by an invariant module, in +order to obtain a deep invariant module, which for the first +hierarchical level (l = 1) takes the following form: +�xm = (Σ(1) +I +◦ Σ(K,1) +E +◦ · · · ◦ Σ(1,1) +E +)({xn}m). +(16) +Compared to the simple invariant module from Eq. 12, the +deep invariant module involves a larger number of com- +putations but allows the network to learn more expressive +representations. Accordingly, the transformation for the +second hierarchical level (l = 2), which yields the final +summary representation z, is given by: +z = (Σ(2) +I +◦ Σ(K′,2) +E +◦ · · · ◦ Σ(1,2) +E +)({�xm}), +(17) +where the number of equivariant modules K′ for level 2 +can differ from the number of equivariant modules K for +level 1. In our experiments, reported in Section 4, we ob- +serve a clear advantage of using deep invariant networks +over their simple counterparts. Furthermore, for two-level +models, we find that the performance of the networks is +largely insensitive to the choice of K or K′. +3.4 +Learning the Model Comparison Problem +In order to get from the learned summary representation +z to an approximation of the analytic PMPs �π, we ap- +ply a final neural classifier (i.e., the inference network) +I(z) = �π, as visualized in Figure 2. We deviate from +the Dirichlet-based setting in Radev, D’Alessandro, et al. +(2021), since we found that implementing the inference +network as a standard softmax classifier (Grathwohl et +al., 2019) provides slightly better calibration and leads to +more stable training in the specific context of HMs. +Denoting the entire hierarchical neural network as +fφ({x}) = �π and an arbitrary hierarchical data set as +{x}, we aim to minimize the expected logarithmic loss +min +φ Ep(M,{x}) +� +�− +J +� +j=1 +IMj · log fφ({x})j +� +� , +(18) +where φ represents the vector of trainable neural network +parameters (e.g., weights and biases), IMj is the indicator +function for the “true” model. The expectation runs over +the joint generative (mixture) distribution of all models +p(M, {x}), which we access through Monte Carlo sim- +ulations. Since the logarithmic loss is a strictly proper +loss (Gneiting & Raftery, 2007), it drives the outputs of +fφ({x}) to estimate the actual PMPs p(M | {x}) as best +as possible. +In practice, we approximate Eq. 18 over a training set of +B simulations from the competing HMs. Each entry b +for b = 1, ..., B in this training set represents a hierar- +chical data set {x(b)} itself along with a corresponding +one-hot encoded vector for the “true” model index M(b) +j . +The latter denotes the model from which the data set was +generated and serves as the “ground truth” for supervised +learning. +7 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +Similarly to Radev, D’Alessandro, et al. (2021), our +neural method encodes an implicit preference for sim- +pler HMs (i.e., Occam’s razor) inherent in all marginal +likelihood-based methods (see MacKay, 2003, Chap- +ter 28). Since our simulation-based training approximates +an expectation over the marginal likelihoods of all HMs +p(M) p(x | M), data sets generated by a simpler HM will +tend to be more similar compared to those generated by a +more complex one (cf. Figure 1). Thus, data sets that are +plausible under both HMs will be generated more often +by the simpler model than by the more complex model. A +sufficiently expressive neural network will capture this be- +havior by assigning a higher PMP for the simpler model1, +thereby capturing complexity differences arising directly +from the generative behavior of the HMs. +Finally, to increase training efficiency when working un- +der a limited simulation budget, we also explore a novel +pre-training method inspired by transfer learning (Ben- +gio et al., 2009; Torrey & Shavlik, 2010). First, we train +the networks on data sets with a reduced number of ex- +changeable units (e.g., reducing the number of observa- +tions at level l = 1). This procedure accelerates training +since it uses fewer simulator calls and the forward pass +through the networks becomes cheaper. In a second step, +we generate data with a realistic number of exchangeable +units. Crucially, since we can use the pre-trained network +from step one as a better-than-random initialization, we +need a much smaller amount of simulations than if we +trained the network from scratch. Indeed, Experiment 4.3 +demonstrates the utility of this training method. +4 +Experiments +In this section, we first conduct two simulation studies +in which we extensively test the approximation perfor- +mance of our hierarchical neural method. We start with +a comparison of two nested toy HMs in Section 4.1, fol- +lowed by a comparison of two complex non-nested HMs +of cognition in Section 4.2. For both validation studies, +we test our method internally by examining the calibra- +tion of the approximated PMPs. Additionally, we vali- +date our method externally by benchmarking its perfor- +mance against the current state-of-the-art for comparing +HMs, namely, bridge sampling (Gelman & Meng, 1998; +Gronau, Singmann, et al., 2017). +To enable this chal- +lenging benchmark, we limit our validation studies to the +comparison of models with explicit likelihoods to which +bridge sampling is applicable. +Finally, in a real-data application, we use our deep learn- +ing method to compare four hierarchical evidence accu- +mulation models (EAMs) of response times data in Sec- +1Assuming equal prior model probabilities. +tion 4.3. Two of these models have no analytic likeli- +hood, which makes the entire BMC setup intractable with +current state-of-the-art methods (e.g., bridge sampling). +Moreover, with this example, we also address the utility +of a novel EAM, the L´evy flight model (Voss et al., 2019), +that has previously been impossible to investigate directly +using Bayesian HMs. +For all experiments, we assume uniform model pri- +ors p(Mj) += +1/J. +All computations are con- +ducted on a single-GPU machine with an NVIDIA +RTX 3070 graphics card. +The reported computation +times are measured as wall-clock times. +Details on +the implementation of our neural networks and the em- +ployed training procedures are provided in Appendix A. +Code for reproducing all results from this paper is +freely available at https://github.com/elseml/ +DeepHierarchicalModelComparison. +4.1 +Validation study 1. Hierarchical Normal Models +In this first experiment, we examine a simple and control- +lable model comparison setup to examine the behavior of +our method under various conditions, before moving on +to more complex scenarios. Inspired by Gronau (2021), +we compare two hierarchical normal models M1 and M2 +that share the same hierarchical structure +τ 2 ∼ Normal+(0, 1) +(19) +σ2 ∼ Normal+(0, 1) +(20) +θm ∼ Normal(µ, +√ +τ 2) for m = 1, . . . , M +(21) +xmn ∼ Normal(θm, +√ +σ2) for n = 1, . . . , Nm, +(22) +with Normal+(·) denoting a zero-truncated normal distri- +bution. The models differ with respect to the parameter µ +that describes the location of the individual-level param- +eters θm: Whereas M1 assumes the location of θm to be +fixed at 0, the more flexible M2 allows for µ to vary +M1: µ = 0 +(23) +M2: µ ∼ Normal(0, 1). +(24) +4.1.1 +Calibration +The most important properties of an approximate infer- +ence method are the trustworthiness of its results and, +more pragmatically, whether we can diagnose the lack of +trustworthiness in a given application. A useful proxy for +trustworthiness is the calibration of a probabilistic classi- +fier, which measures how closely the predicted probabili- +ties of outcomes match their true underlying probabilities +(Guo et al., 2017; Schad et al., 2022). +However, computing the calibration of a BMC procedure +is hardly feasible in a non-amortized setting, since it in- +8 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +volves applying the method to a large number of simu- +lated data sets. For bridge sampling, for example, that +would imply re-fitting the models via MCMC and running +bridge sampling on at least hundreds, if not thousands +of simulated data sets. The calibration of our networks, +on the other hand, can be determined almost immediately +after training due to their amortization property (Radev, +D’Alessandro, et al., 2021). +In the following experiments, we assess the calibration +of our networks visually (via calibration curves) and nu- +merically (via a measure of calibration error). For gen- +erating a calibration curve (DeGroot & Fienberg, 1983; +Niculescu-Mizil & Caruana, 2005), we first sort the pre- +dicted PMPs �π(s) +j +on S simulated data sets s = 1, . . . , S, +which we then partition into I equally spaced probability +bins i = 1, . . . , I (we use I = 15 bins for all validation +experiments). For each model j and each bin i contain- +ing a set Bij of predicted model indices, we compute the +mean prediction for the model (predicted probability, PP) +and the actual fraction of this model being true (true prob- +ability, TP) as follows: +PP(Bij) := +1 +|Bij| +� +b∈Bij +�π(b) +j , +(25) +TP(Bij) := +1 +|Bij| +� +b∈Bij +IM(b) +j , +(26) +where I again denotes the indicator function for the “true +model”. +These two quantities varying over the bins +form the X- and Y -axis of a calibration curve. A well- +calibrated model comparison method with an agreement +in each bin (as indicated by a diagonal line) thus yields ap- +proximations that reflect the true probabilities of the com- +pared models (Guo et al., 2017). We further summarize +this information via the Expected Calibration Error (ECE; +Naeini et al., 2015) as a single number bounded between +0 and 1, which we estimate by averaging the individual +deviations between predicted and true probability in each +bin: +� +ECEj := +I +� +i=1 +|Bij| +S +����PP(Bij) − TP(Bij) +����. +(27) +If follows from Eq. 27 that a perfect ECE can be achieved +by always predicting indifferent probabilities (e.g., �π1 = +�π2 = .5 when comparing two models). +We therefore +complement our calibration assessment by measuring the +accuracy of recovery, for which we dichotomize the pre- +dicted PMPs �π(s) +j +on S simulated data sets into one-vs-rest +model predictions � +M(s) +j : +Accj := 1 +S I � +M(s) +j +=M(s) +j . +(28) +Thus, in our BMC context, accuracy roughly is to ECE +what sharpness is to posterior calibration in Bayesian pa- +rameter estimation (B¨urkner et al., 2022).2 +Fixed data set sizes +In the first calibration experiment, +we examine the performance of our method for the most +simple application case of learning a model comparison +problem on a specific (fixed) data set size. Here, all data +sets simulated for training and validating the network con- +sist of M = 50 groups and Nm = 50 observations for +each group m = 1, . . . , M. +We train the network for 10, 000 backpropagation steps, +taking 12 minutes. Subsequently, we calculate its calibra- +tion on 5, 000 held-out validation data sets and repeat this +process 25 times to obtain stable results with uncertainty +quantification. +Figure 3a depicts the resulting median +calibration curve. Its close alignment to the dashed di- +agonal line representing perfect calibration indicates that +the PMP approximations are very well-calibrated (median +ECE over all repetitions of � +ECE = 0.011). The curve’s +coverage of the full range of predicted probabilities and +the median accuracy of � +Acc = .88 confirm that the ex- +cellent calibration does not stem from indifferent predic- +tions. The subsequent comparison of our method to bridge +sampling suggests that this accuracy is indeed close to the +upper bound imposed by the aleatoric uncertainty in the +model-implied data. +Data sets with varying numbers of observations +We +now train our hierarchical network to approximate BMC +over a range of hierarchical data sets with varying num- +bers of observations within groups Nm. This amortiza- +tion over observation sizes would provide a substantial +efficiency gain if a researcher desires to compare HMs +on multiple data sets with differing Nm, as only a sin- +gle network would have to be trained for all data sets.3 In +our validation setup, each simulated data set still consist +of M = 50 groups, but now the number of observations +within those groups varies in Nm = 1, . . . , 100. +We train the network for 20, 000 training steps, taking 25 +minutes. At each training step, we draw the number of +observations for the current batch of simulations from a +discrete uniform distribution Nm ∼ UniformD(1, 100). +For each Nm used during training, we evaluate the cali- +bration 25 times on 5, 000 held-out simulated validation +2We focus on the accuracy, since we use a uniform model +prior p(M), but other metrics of predictive performance, such +as the logarithmic scoring rule, would have been expedient as +well. +3Note that we refer to variability between data sets. We de- +scribe an approach for handling within data set variability of +nested trials in Section 4.3. +9 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted probability +0.2 +0.4 +0.6 +0.8 +1.0 +True probability +Median +50% CI +95% CI +(a) Median calibration curve and confidence intervals (CIs) +for data sets of M = 50 groups with Nm = 50 observations +within each group. +20 +40 +60 +80 +100 +Number of observations (Nm) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +d +ECE +Median +50% CI +95% CI +(b) Median expected calibration errors (ECEs) and confidence +intervals for data sets of M = 50 groups with differing numbers +of observations Nm within each group. +Figure 3: Validation study 1: Calibration results for (a) the neural network trained on fixed data set sizes and (b) the +neural network trained on data sets with varying numbers of observations. Medians and confidence intervals (CIs) are +computed over 25 repetitions. +data sets. This repetition procedure allows us to quantify +the uncertainty of our ECE estimates. +Figure 3b plots the median ECE values for each observa- +tion size. The neural network achieves high calibration +with a median ECE over all observation sizes (and rep- +etitions) of � +ECE = 0.012. Moreover, the unsystematic +pattern of the median curve and the homoskedastic vari- +ation between the observation sizes indicate that the net- +work has learned the model comparison task equally well +for all settings. Together, the low calibration error and the +accurate model predictions (median accuracy � +Acc = .88) +indicate that our method incurs no trade-off between cali- +bration and accuracy. +Data sets with varying numbers of groups and obser- +vations +In the third calibration experiment, we test the +ability of the network to learn a model comparison prob- +lem over a range of data sets with varying numbers of +groups M and varying observations per group Nm. This +training scheme allows for amortized model comparison +on multiple data sets with different sizes, which can be +especially useful for a priori sample size determination on +simulated data. Additionally, the trained network can be +stored and reused on future data sets with yet-unknown +sample sizes. For this experiment, training and valida- +tion data sets are simulated with M = 1, . . . , 100 groups +and Nm = 1, . . . , 100 observations, resulting in a vast +variability of data set sizes between 1 up to 10, 000 data +points. +Given the complexity of the learning task, we now train +the network for 40, 000 training steps, taking 49 minutes. +At each training step, we draw the number of groups and +observations from discrete uniform distributions M ∼ +UniformD(1, 100) and Nm ∼ UniformD(1, 100). We es- +timate calibration on 5, 000 held-out simulations for each +combination of M and Nm. As this implies simulating +50, 000, 000 data sets, we forego the repetition procedure +employed in the previous experiments. +Figure 4 depicts the calibration and accuracy results for +all combinations of M and Nm. We observe low ECEs +for the vast majority of settings in Figure 4a (median ECE +over all settings of � +ECE = 0.012). In other words, the +trained network is capable of generating highly calibrated +PMPs over a broad range of data set sizes. Moreover, the +BMC results are sensitive to the number of nested obser- +vations Nm, but not to the number of groups M, in our +experimental setups. The only systematic drop in cali- +bration occurs for data sets containing just a few nested +observations (Nm ≤ 5). Considering that we observed +high calibration even for this low number of observations +in a network trained on data sets with varying Nm (see +Figure 3b), we surmise that the drop in the edge areas in +Figure 4a arises from the challenging learning task over +vastly different data set sizes (a phenomenon known as +10 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +Number of groups (M) +20 +40 +60 +80 +100 +Number of observations (Nm) +20 +40 +60 +80 +100 +d +ECE +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +(a) Expected Calibration Error (ECE). +Number of groups (M) +20 +40 +60 +80 +100 +Number of observations (Nm) +20 +40 +60 +80 +100 +Accuracy +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +(b) Accuracy of recovery. +Figure 4: Validation study 1: Results for the neural network trained and tested over variable data set sizes. +amortization gap; Cremer et al., 2018). The overall low +(i.e., good) ECEs for all cases but the poorly identifiable +Nm = 1 setting suggest that the networks’ approxima- +tions are generally trustworthy. This is further confirmed +by Figure 4b, where the observable accuracy pattern as- +sures that this high calibration does not arise from a trade- +off with predictive performance. Despite the demanding +amortization setting, the network achieves an excellent +median accuracy of � +Acc = .88, similar to the earlier ex- +periments. +4.1.2 +Bridge Sampling Comparison +After validating the general trustworthiness of our +method, we now benchmark it against the current gold +standard for comparing HMs, namely, bridge sampling, +as implemented by Gronau, Singmann, et al., 2017. As +the non-amortized nature of bridge sampling restricts the +feasible number of test sets, we conduct the benchmarking +on 100 test sets which are simulated equally from M1 and +M2. All simulated data sets consist of M = 50 groups +and Nm = 50 observations per group. The fixed sample +sizes of the test sets allow us to compare the two most dis- +tinct networks from Section 4.1.1 to bridge sampling: The +fixed network that is trained for this specific sample size +and the more complex variable network that is trained for +amortized model comparison over variable sample sizes +between M = 1, . . . , 100 groups and Nm = 1, . . . , 100 +observations per group. +For bridge sampling, we first run four parallel MCMC +chains with a warm-up period of 1, 000 draws and 49, 000 +post-warmup posterior draws per chain in Stan (Carpenter +et al., 2017; Stan Development Team, 2019). We assess +convergence through a visual inspection of the MCMC +chains and an assessment of the �R, bulk ESS and tail +ESS metrics (Vehtari et al., 2021). Afterwards, we use +the posterior draws to approximate PMPs and BFs with +the bridgesampling R package (Gronau, Singmann, et al., +2017). We confirm the sufficiency of the total of 196, 000 +posterior draws by assessing the variability between mul- +tiple runs as in Schad et al. (2022), which yields highly +similar results. +Further insights via our calibration di- +agnostics are precluded by bridge sampling being a non- +amortized method. +Approximation performance +As we compare approxi- +mate PMPs, we can use a number of complementary met- +rics commonly employed to evaluate the quality of prob- +abilistic predictions. +First, we quantify the fraction of +times the correct model M(s) +j +underlying a simulated data +set s was detected, that is, the accuracy of recovery (see +Equation 28). Second, we assess the Mean Absolute Error +(MAE) to investigate the average deviation of the approx- +imated model probabilities �π(s) +j +from a perfect classifica- +tion: +MAEj := 1 +S +S +� +s=1 +����π(s) +j +− I(s) +Mj +���. +(29) +Third, we measure the Root Mean Squared Error (RMSE), +which places particular emphasis on large prediction er- +rors, to detect whether one method produces highly incor- +11 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +Table 1: Validation study 1: Performance metrics. +Accuracy +MAE +RMSE +Log-Score +SBC +Bridge sampling +0.86 (0.03) +0.19 (0.03) +0.32 (0.03) +0.32 (0.06) +-0.02 (0.04) +Fixed network +0.85 (0.04) +0.2 (0.03) +0.33 (0.03) +0.33 (0.06) +-0.02 (0.04) +Variable network +0.86 (0.03) +0.19 (0.03) +0.32 (0.03) +0.32 (0.06) +-0.01 (0.04) +Note. Bootstrapped mean values and standard errors (in parentheses) are presented. We use 1000 bootstrap versions of the test +data sets and estimate the standard errors from the bootstrap standard deviations of the metrics. +rect approximations more frequently than the other: +RMSEj := +� +� +� +� 1 +S +S +� +s=1 +� +�π(s) +j +− I(s) +Mj +�2 +. +(30) +Fourth, we calculate the Log-Score following the logarith- +mic scoring rule: +LogScorej := − 1 +S +S +� +s=1 +� +I(s) +Mj · log�π(s) +j +� +. +(31) +Its property as a strictly proper scoring rule implies that +it is asymptotically minimized if and only if the approxi- +mate probabilities equal the true probabilities (Gneiting & +Raftery, 2007). Lastly, we measure simulation-based cal- +ibration (SBC; Talts et al., 2018) as adapted by Schad et +al. (2022) for model inference by the difference between +the prior probability for a model and its average posterior +probability in the test sets: +SBCj := p(Mj) − 1 +S +S +� +s=1 +�π(s) +j . +(32) +We evaluate all metrics for M2, so that a bias towards M1 +is indicated by positive SBC values and a bias towards +M2 by negative SBC values. +Table 1 depicts the comparison results for our experi- +mental setting. All metrics show equal performances for +bridge sampling and the two neural network variants, with +any differences being well within the range of the standard +errors. +Approximation convergence +In the following, we ana- +lyze the degree of convergence between the two methods +at the level of individual data sets. We explore this vi- +sually by contrasting the PMP and (natural logarithmic) +BF approximations of bridge sampling with the two neu- +ral network variants in Figure 5. +We observe that the +two methods’ PMP approximations agree for the easy +cases where the true underlying model is clearly clas- +sifiable. Thus, discrepancies between the two methods +arise mainly for data sets with predicted PMPs close to +�π = 0.5. Even for the data sets with the largest discrepan- +cies, the two methods do not map to qualitatively different +decisions: �π(bridge) +2 += .67 and �π(neural) +2 += .83 for the fixed +network, �π(bridge) +2 += .32 and �π(neural) +2 += .25 for the vari- +able network. Most importantly, we detect no systematic +pattern in these deviations. +As BFs represent the ratio of marginal likelihoods, they al- +low for a closer inspection of the degree of agreement be- +tween the methods in those edge cases with PMPs close to +0 or 1. We observe a close convergence for data sets clas- +sified as stemming from M1. Considering the predictions +favoring M2, there are discrepancies for data sets with log +BFs > 9.49. 4 Since this corresponds to BFs > 13, 000 +and PMPs > .9999, it is not visible in the PMP approx- +imation plots. We obtain such extreme results only for +M2, as this model allows for deviations of the group level +parameters’ location from 0 and enables the occurrence of +extreme evidence in its favor. The divergence in this area +of extreme evidence emerges most likely from the loss +function employed for training the neural networks: The +logarithmic loss obtained from a minuscule deviation of +the PMP from 1 is near 0, which results in a negligible in- +centive for further optimization of the network’s weights. +We could reject a competing explanation based on limited +floating-point precision, since training with an increased +floating-point precision from 32-bit to 64-bit resulted in +identical patterns. +The divergence we encountered provides insights into the +technical nature of our method, but only arises in cases of +extreme evidence. Thus, it is far from altering the substan- +tive conclusions derived from the simulated BMC setting. +Considering the convergence between the two methods in +the realm of practical relevance, we can conclude that our +method produces highly similar approximations to bridge +sampling in this scenario. +4For visibility purposes, we exclude the 27 data sets for +which bridge sampling approximated a BF > 1, 000, 000 for +the BF plots, all continuing the observed plateau pattern. Plots +with all 100 data sets are provided in Appendix B. +12 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p(M2 |x) - bridge sampling +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p(M2 |x) - fixed network +Simulated from M1 +Simulated from M2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p(M2 |x) - bridge sampling +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p(M2 |x) - variable network +Simulated from M1 +Simulated from M2 +Posterior Model Probabilities +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +log(BF21) - bridge sampling +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +log(BF21) - fixed network +Simulated from M1 +Simulated from M2 +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +log(BF21) - bridge sampling +2.5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +log(BF21) - variable network +Simulated from M1 +Simulated from M2 +Log Bayes Factors +Figure 5: Validation study 1: Comparison of approximation results obtained via bridge sampling vs. the neural +network trained on fixed data set sizes (left) and the neural network trained on variable data set sizes (right). Note that +the Bayes factor plots contain only those 73 data sets for which bridge sampling approximated a BF21 < 1, 000, 000. +13 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +Approximation time +Both bridge sampling and our +deep learning method can be divided into two computa- +tional phases. For bridge sampling, the first phase con- +sists of drawing from the posterior parameter distributions +(taking 52 seconds per data set on average). Bridge sam- +pling itself takes place in the second phase (taking 38 sec- +onds on average). Notably, in contrast to amortized in- +ference with neural networks, both phases need to be re- +peated for each (simulated or observed) data set. Taking +the initial compilation time of 42 seconds into account, +bridge sampling consequently took 152 minutes for BMC +on our 100 test data sets. +For the neural networks, the first phase (training) is +resource-intensive (taking 12 minutes for the fixed net- +work and 49 minutes for the variable network). The sec- +ond phase (inference) is then performed in near real-time +(taking 0.003 seconds for both networks on all 100 test +data sets) and thus amortizes the training cost over mul- +tiple applications. For the simple HMs compared here, +the amortization gains of our networks over bridge sam- +pling come into effect after performing BMC on 8 (fixed +network) or 39 (variable network) data sets. +We acknowledge our likely suboptimal choices of com- +putational steps for the bridge sampling workflow or the +neural networks and hence wish to stress the general pat- +terns of non-amortized vs. amortized methods demon- +strated here. In general, we expect an advantage of bridge +sampling in terms of efficiency in situations where only +one or a few data sets are available and obtaining a large +number of posterior draws is feasible. The demonstrated +amortization property of our method might not be so rel- +evant for inference on a single hierarchical data set, but +it becomes crucial for performing calibration or recov- +ery studies, which necessitate multiple re-fits of the same +model (Schad et al., 2022). +4.2 +Validation study 2: Hierarchical SDT vs. MPT +Models +We now extend our validation experiments from the sim- +ple setup with nested HMs to the comparison of non- +nested HMs of cognition. In this simulation study, we +examine the ability of our method to distinguish between +data sets generated either from an HM based on signal de- +tection theory (SDT model; Green, Swets, et al., 1966) or +a hierarchical multinomial processing tree model (MPT +model; Riefer & Batchelder, 1988). For illustrative pur- +poses, we embed our simulation study within an old-new +recognition scenario, where participants indicate whether +or not a stimulus was previously presented to them. +We ensure a challenging model comparison setting via +three design aspects: First, we specify both models to +possess a similar generative behavior, that is, hardly dis- +tinguishable prior predictive distributions of hit rates and +false alarm rates (prior predictive plots are provided in +Appendix C). Second, data sets of old-new recognition +typically contain low information as they only consist +of binary variables indicating the stimulus type and re- +sponse, respectively. Third, we further amplify the infor- +mation sparsity of the data sets by choosing a particularly +small size for all data sets of M = 25 simulated partici- +pants and Nm = 50 observations per participant. +A major difference between the compared cognitive +model classes lies in the assumption of a continuous la- +tent process by the SDT model and discrete processes (or +states) by the MPT model. Our specification of the SDT +model follows the hierarchical formulation of the standard +equal-variance model by Rouder and Lu (2005). As the +competing MPT model, we specify a hierarchical latent- +trait two-high-threshold model (Klauer, 2010), which, in +contrast to the SDT model, explicitly models correlations +between its parameters. We follow the convention of re- +stricting the parameters that describe the probability of +recognizing a previously presented stimulus as old and +a distractor stimulus as new to be equal, DO = DN, to +render the MPT model identifiable (Erdfelder et al., 2009; +Singmann & Kellen, 2013). Our prior choices for the pa- +rameters of both models are described in Appendix C. +We train the neural network for 30, 000 training steps. As +in Section 4.1, we first leverage the amortization prop- +erty of our method to inspect its calibration for the cur- +rent model comparison task. Figure 6a shows that the +trained neural network generates well-calibrated PMP ap- +proximations (median ECE over 25 repetitions of � +ECE = +0.009). +Next, we assess whether the observed calibration of the +network translates into a competitive performance relative +to bridge sampling. The benchmarking setup (50 simu- +lated data sets from each model) and the implementation +of the bridge sampling workflow follow the procedure de- +scribed in Section 4.1.2. +The classification metrics depicted in Table 2 reveal the +excellent performance of both methods, despite the chal- +lenging BMC scenario. We further observe a high degree +of convergence between approximate PMPs derived by +the two methods (cf. Figure 6b). As depicted in Figure 6c, +obtaining PMP approximations for the 100 test data sets +took more than 6 hours for bridge sampling and 36 min- +utes for the neural network. For this comparison of more +complex cognitive models than in Section 4.1.2, the amor- +tization advantage of our method emerges when analyz- +ing 10 or more data sets. Note that this advantage would +quickly show up in validation studies involving multiple +14 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted probability +0.2 +0.4 +0.6 +0.8 +1.0 +True probability +Median +50% CI +95% CI +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p(MPT|x) - bridge sampling +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p(MPT|x) - neural network +Simulated from SDT +Simulated from MPT +20 +40 +60 +80 +100 +Number of data sets +0 +50 +100 +150 +200 +250 +300 +350 +Computation time in minutes +Bridge sampling +Neural network +(a) Calibration of the neural network over +25 repetitions with 5, 000 data sets each. +(b) Convergence of approximate PMPs. +(c) Computation times. +Figure 6: Validation study 2: Results for the comparison between hierarchical SDT and MPT models. +Table 2: Validation study 2: Performance metrics for the comparison between hierarchical SDT and MPT models. +Accuracy +MAE +RMSE +Log Score +SBC +Bridge sampling +0.95 (0.02) +0.1 (0.02) +0.22 (0.03) +0.16 (0.04) +-0.01 (0.04) +Neural network +0.95 (0.02) +0.1 (0.02) +0.21 (0.03) +0.16 (0.04) +0.00 (0.04) +Note. Bootstrapped mean values and standard errors (in parentheses) are presented. We use 1000 bootstrap versions of the test +data sets and estimate the standard errors from the bootstrap standard deviations of the metrics. +model re-fits (e.g., bootstrap, sensitivity analysis or cross- +validation). +The converging results from the two validation stud- +ies demonstrate that our neural method generates well- +calibrated and accurate PMP approximations. Despite our +method only accessing the likelihood function indirectly +via simulations, it can successfully compete with bridge +sampling, which has direct access to the likelihood func- +tion. +4.3 +Application: Hierarchical Evidence +Accumulation Models +In the following, we showcase the utility of our method by +comparing complex hierarchical evidence accumulation +models (EAMs) in a real-data situation where likelihood- +based methods such as bridge sampling would not be ap- +plicable. +More precisely, we seek to test the explana- +tory power of different stochastic diffusion model formu- +lations proposed by Voss et al. (2019) for experimental +response times data. +The so-called L´evy flight model increases the flexibility +of the standard Wiener diffusion model (Ratcliff et al., +2016) but renders its likelihood function intractable with +standard numerical approximations (Voss & Voss, 2007). +The complete incorporation of all information through hi- +erarchical modeling and the realization of BMC has con- +sequently been infeasible so far. Thus, in a recent study, +Wieschen et al. (2020) had to resort to a separate com- +putation of the Bayesian Information Criterion (BIC) for +each participant with subsequent aggregation. We aim to +extend the study of Wieschen et al. (2020) by comparing +fully hierarchical EAMs through PMPs and BFs. More- +over, we intend to answer the question formulated by Wi- +eschen et al., 2020 as to whether the superior performance +of the more complex models in their study stems from an +insufficient punishment of model flexibility by the BIC. In +addition to addressing a substantive research question in +this application, we also demonstrate multiple advantages +of our deep learning method on empirical data: +• Compare HMs with intractable likelihoods: As our +method is simulation-based, including models with +intractable likelihood functions in the comparison set +does not alter its feasibility. +• Adequately model nested data: Our method allevi- +ates computational challenges that prevent modelers +from adequately capturing the information contained +in nested data structures through HMs. +15 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +• Re-use trained networks via fine-tuning: We acceler- +ate the training of our neural network by pre-training +it on less complex simulated data and subsequently +fine-tuning it on simulated data resembling the ac- +tual experimental setting. +• Handle missing data: We train a neural network that +can handle varying amounts of missing data by ran- +domly masking simulated data during the training +process. +• Validate a trained network on simulated data: The +amortized nature of our method allows for extensive +validation of a trained network prior to its application +to empirical data. +4.3.1 +Model specification +For this application, we consider a L´evy flight model with +non-Gaussian noise (Voss et al., 2019). The L´evy flight +process is driven by the following stochastic ordinary dif- +ferential equation (SDE): +dx = v dt + σdξ +(33) +ξ ∼ AlphaStable(α, µ = 0, σ, β = 0), +(34) +which represents a L´evy walk characterized by a fat-tailed +stable noise distribution. +In the above equation, x de- +notes the accumulated (perceptual) evidence, v denotes +the rate of accumulation and α controls the tail exponent +of the noise variate ξ. Voss et al. (2019) and Wieschen +et al. (2020) argue that the more abrupt changes in the +information accumulation process that this model allows +for could provide a better description of human decision- +making than a Gaussian noise. The addition of L´evy noise +renders the standard numerical approximation of the dif- +fusion model likelihood intractable (Voss & Voss, 2007). +Consequently, neither standard MCMC nor bridge sam- +pling are applicable for Bayesian parameter estimation +and BMC, respectively. +There is an ongoing debate about the inclusion of addi- +tional parameters that account for inter-trial variability in +the diffusion model parameters: While they can provide +a better model fit, the estimation of inter-trial variabil- +ity parameters is often difficult and can result in unsta- +ble results (Boehm et al., 2018; Lerche & Voss, 2016). +Thus, Wieschen et al. (2020) also compared basic (with- +out inter-trial variability parameters) to full (with inter- +trial variability parameters) versions of the drift-diffusion +and L´evy flight model. +Consequently, the set of candidate models considered here +consists of four EAMs with increasing flexibility (i.e., the +scope of possible data patterns that they can generate): +• M1, the most parsimonious basic diffusion model +with the parameter v describing the mean rate of +information uptake, the parameter a describing the +threshold at which a decision is made, the parameter +zr describing a bias of the starting point towards one +decision alternative and the parameter t0 describing +the non-decision time, that is, the time spent encod- +ing the stimulus and executing the decision. +• M2, the basic L´evy flight model in which the as- +sumption of a Wiener diffusion process with Gaus- +sian noise is replaced by the above introduced L´evy +flight process. The additional free parameter α de- +notes the heaviness of the noise distribution’s tails. +Note that α = 2 equals a Gaussian distribution, +while α = 1 describes a Cauchy distribution. +• M3, the full diffusion model, which extends M1 +with the parameters sv, sz and st that denote the vari- +ability (i.e., standard deviations) of drift rate, starting +point bias and non-decision time, respectively, be- +tween trials. +• M4, the full L´evy flight model that possesses the +largest flexibility by including inter-trial variability +parameters as well as the flexible L´evy noise distri- +bution controlled by α. +Parameter priors and prior predictive checks are provided +in Appendix D.1. +4.3.2 +Data +The reanalyzed data set by Wieschen et al. (2020) con- +tains 40 participants who completed a total of 900 trials +of binary decision tasks (color discrimination and lexical +decision) each. On average, 3.17% of trials per partici- +pant were excluded due to extremely short or long reac- +tion times. +4.3.3 +Simulation-based training +Since simulating data from EAMs can be challenging, es- +pecially when they include non-Gaussian noise, we lever- +age the advantage that neural networks are capable of +transfer learning as described in Section 3.4. +Transfer +learning describes the utilization of representations that +had been previously learned by a neural network in a par- +ticular task for a new, related task (e.g., Ng et al., 2015). +In this way, neural networks can be applied in small data +settings by re-using the training knowledge encoded from +similar (possibly big data) settings. +For the purpose of model comparison, we first pre-train +the network for 20 epochs (passes over the whole training +data) on 10, 000 simulated data sets per model. These data +16 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.4 +0.6 +0.8 +1.0 +True probability +d +ECE = 0.003 +M1 +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.4 +0.6 +0.8 +1.0 +d +ECE = 0.003 +M2 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted probability +0.2 +0.4 +0.6 +0.8 +1.0 +True probability +d +ECE = 0.010 +M3 +0.2 +0.4 +0.6 +0.8 +1.0 +Predicted probability +0.2 +0.4 +0.6 +0.8 +1.0 +d +ECE = 0.009 +M4 +M1 +M2 +M3 +M4 +Predicted model +M1 +M2 +M3 +M4 +True model +0.97 +0.03 +0.00 +0.00 +0.04 +0.96 +0.00 +0.00 +0.00 +0.00 +0.89 +0.10 +0.00 +0.00 +0.14 +0.86 +0.0 +0.2 +0.4 +0.6 +0.8 +(a) Calibration curves. +(b) Confusion matrix. +Figure 7: Application experiment: Validation results for the evidence accumulation models on 2, 000 simulated data +sets per model. +sets resemble the empirical data in that they consist of 40 +simulated participants, but differ in that the number of tri- +als is reduced by a factor of 9 (100 instead of 900 trials per +participant). Afterwards, we fine-tune the network for ad- +ditional 30 epochs on 2, 000 simulated data sets per model +that match the empirical data set with 40 simulated partic- +ipants and 900 trials per participant. Thereby, we consid- +erably reduce the computational demand of the training +process. We further speed up the training phase by simu- +lating all data prior to the training of the network in the +high-performance programming language Julia (Bezan- +son et al., 2017). Pre-training took 60 minutes for the +simulations and 39 minutes for training of the networks. +Fine-tuning took 110 minutes for the simulations and 74 +minutes for training of the networks, resulting in a total of +4 hours and 43 minutes for the training phase. +To fully adapt the network to the characteristics of the em- +pirical data, we also simulate missing data during fine- +tuning. In each training epoch, we generate a random bi- +nary mask f coding the simulated missing values. We +sample the number of masked trials from a (discretized) +normal distribution truncated between 1 and the number +of trials, 900. The distributions’ mean and standard de- +viation match the amount and variability of missing trials +in the empirical data. We then perform an element-wise +multiplication ˜x = x ⊗ f and feed the “contaminated” +data ˜x to the network. This procedure results in a robust +network that can process various proportions of missing +data. We show the stability of our results even in the pres- +ence of up to 25% missing data in Appendix D.2. +4.3.4 +Results +Before applying our trained network to the empirical data, +we validate it on 2, 000 simulated data sets per model. +First, the individual calibration curves in Figure 7a show +excellent calibration for all models with � +ECEs close to 0. +The calibration curves now consist of 10 instead of 15 in- +tervals to obtain stable results despite the smaller amount +of validation data sets per model. Second, we evaluate +the accuracy of recovery and patterns of misclassification +through the confusion matrix depicted in Figure 7b. The +confusion matrix confirms that the excellent calibration +of the network does not stem from chance performance. +It also reveals that the selection of the “true” model be- +comes more difficult with increasing model complexity, +which is a direct consequence of the Occam’s razor prop- +erty inherent in BMC (cf. Figure 1). +Table 3 presents the model comparison results on the em- +pirical data set. Additionally, Figure 8 displays the model +posteriors under different data perturbations. We find lit- +tle evidence for both the basic diffusion model M1 and +the basic L´evy flight model M2, meaning that the addi- +tional complexity of allowing parameters to vary between +trials in M3 and M4 is outweighed by better model fit. +Consistent with the results of Wieschen et al. (2020), the +17 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +M1 +M2 +M3 +M4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Posterior model probability +Bootstrapped trials +M1 +M2 +M3 +M4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Bootstrapped participants +M1 +M2 +M3 +M4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Leave-one-participant-out +Figure 8: Application experiment: Model posteriors on the empirical data set with uncertainty under different data +perturbations. We use 100 bootstrap samples for the bootstrapped results. +Table 3: Application experiment: Bayes factors (BFs) +and posterior model probabilities (PMPs) on the empirical +data set. The preferred model is indicated by an asterisk. +M1 +M2 +M3 +M4 +BFj4 +1.46e-07 +2.76e-04 +0.03 +* +BF4j +6.85e+06 +3.62e+03 +33.96 +* +PMP +1.42e-07 +2.68e-04 +0.03 +0.97 +full L´evy flight model M4 explains the experimental data +best. While the magnitude and robustness of this advan- +tage were unclear in Wieschen et al. (2020) due to the +indirect approach of calculating separate BIC values, we +now obtain clear evidence of BF43 = 33.96 for the full +L´evy flight model M4 over the second best performing +full diffusion model M3. Remarkably, the strong evi- +dence for the most complex model M4 occurs despite the +strict penalization of prior-predictive flexibility in BMC +and proves to be robust to multiple data perturbations. +5 +Discussion +Nested data are ubiquitous in the quantitative sciences, +including psychological and cognitive research (Farrell +& Lewandowsky, 2018). +Yet, to avoid dealing with +the complex dependencies resulting from these data, re- +searchers often resort to simpler analyses, ignoring poten- +tially important structural information. Hierarchical mod- +els (HMs) provide a flexible way to represent the multi- +level structure of nested data, but this flexibility can make +Bayesian model comparison a daunting undertaking. +In this work, we proposed a powerful remedy to this prob- +lem: Building on the BayesFlow framework (Radev et al., +2020), we developed a neural network architecture that +enables approximate BMC for arbitrarily complex HMs. +In two simulation studies, we showed that our deep learn- +ing method is well-calibrated and performs as accurately +as bridge sampling, which is the current state-of-the-art +for comparing HMs with simple likelihoods. Moreover, +in a subsequent real-data application, we compared the +relatively new L´evy flight model with existing evidence +accumulation models. Thus, we argue that our method is +well-suited to enhance the applicability of (complex) HMs +in psychological research. Below, we summarize the key +properties and limitations of our method while also out- +lining future research directions. +5.1 +Amortized Inference +Our method offloads the computational demands for com- +paring HMs onto the training phase of a custom neural +network, allowing for near real-time model comparison +using the trained network. The resulting amortization of- +fers several advantages over non-amortized methods. +First, it enables thorough validation of a trained network +on thousands of simulated data sets, allowing large-scale +simulation-based diagnostics to become an integral part +of the BMC workflow (Gelman et al., 2020; Schad et al., +2022). Second, the trained and validated network can be +used not only for point estimates of BFs or PMPs on em- +pirical data but also for exploring the robustness of the +results against multiple data perturbations, as showcased +in our real-data application. +Third, we demonstrated the feasibility of amortizing over +variable data set sizes in our first validation study. This +is particularly advantageous in the context of HMs since +nested data sets often contain multiple exchangeable lev- +els with variable sizes (e.g., different numbers of clusters, +participants and observations). Analyzing multiple hier- +18 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +archical data sets with variable sizes only requires a sin- +gle network that has seen different data set sizes during +training. The same network could also be used for var- +ious simulation studies, such as the challenging task of +designing maximally informative experiments in a hierar- +chical BMC setting (Heck & Erdfelder, 2019; Myung & +Pitt, 2009). +Lastly, we showed that researchers do not even need to +consider all possible shapes of future data sets when train- +ing such a network, as they can use transfer learning to +efficiently adapt a trained network to a related setting. Be- +yond allowing more flexibility in reusing networks across +experiments, researchers or even fields, transfer learning +can also considerably reduce the computational demands +associated with comparing complex HMs. +As demon- +strated in our real-data application, a network can be pre- +trained on simulated data sets with reduced size and fine- +tuned afterwards on sizes matching the empirical data. +5.2 +Independence From Explicit Likelihoods +Unlike other popular methods for performing BMC on +HMs, such as the Savage-Dickey density ratio or bridge +sampling, our method is not constrained by the availabil- +ity of an explicit likelihood function for all competing +models. As long as the models in question can be imple- +mented as simulators, the neural network can be trained +to perform BMC on these models. The value of such a +method is evident, as it decouples the substantive task of +model specification from concerns about the feasibility of +estimation methods. +Statistical models are instantiations of substantive knowl- +edge or hypotheses. As such, we argue that model speci- +fication should not be unduly restricted by considerations +of computational tractability – a sentiment that is closely +related to what Haaf et al. (2021) call the “specification- +first-principle”. Our proposed deep learning method sat- +isfies this principle, as model specification may be guided +exclusively by substantive arguments with few concerns +about tractability. +Thus, we believe that our method +makes a contribution to the recent upsurge of innova- +tive psychological models (Collins & Shenhav, 2022; +Ghaderi-Kangavari et al., 2022; Heathcote & Matzke, +2022) by allowing for an efficient assessment of their in- +cremental value in a hierarchical setting. +5.3 +Limitations and Outlook +One of the main challenges of approximate methods and, +more broadly, statistical inference is ensuring the faithful- +ness of the obtained results. The outlined possibilities for +validating the network and examining the robustness of +the results are important contributions of our method but +come with open questions. Concerning the validation of +the network, framing model comparison as a supervised +learning problem allows us to draw from the rich litera- +ture on classification performance metrics. Nevertheless, +determining a “good-enough” score for an approximate +BMC method remains challenging, as the optimally pos- +sible performance is application-specific and usually un- +known. +Concerning the application of the network to empirical +data, our robustness checks are a practical proxy for the +stability of BMC results in a closed-world setting. How- +ever, these checks cannot possibly capture the (lack of) +absolute evidence for an HM: As a relative method, BMC +may indicate that one model fits the data better than a set +of competing models, but it does not provide any measure +of how well (or poorly) the model itself approximates the +underlying data-generating process. A promising direc- +tion to address this limitation could be the combination of +our method with the recently proposed meta-uncertainty +framework for BMC (Schmitt et al., 2022), which can be +greatly accelerated with amortized methods. This combi- +nation could provide a principled delineation of different +uncertainty sources, enabling the detection of model mis- +specification cases where none of the competing HMs can +explain the observed data. +Since BMC is a marginal likelihood (i.e., prior predictive) +approach, the priors should be informed by scientific the- +ory and will thus have a decisive influence on the results +(Vanpaemel, 2010). We do not intend to re-iterate the +ongoing discussion about this property of BMC (Gronau +& Wagenmakers, 2019a, 2019b; Haaf et al., 2021; Ve- +htari et al., 2019), but want to highlight a specific dif- +ficulty that arises for HMs: Parameter priors of an HM +are connected via multilevel dependencies, increasing the +risk that poor prior choices may dominate the final re- +sults (for a recent discussion of this problem in cognitive +modeling, see Sarafoglou et al., 2022). Therefore, prior +predictive checks and prior sensitivity analyses become +especially important when conducting BMC on compet- +ing HMs. While transfer learning reduces the computa- +tional demands of retraining a neural network for sensi- +tivity analyses, another avenue for future research would +be the amortization over different prior choices, enabling +immediate prior sensitivity assessment. +Finally, it should be noted that the version of our +method explored here can only compare HMs assum- +ing exchangeable data at each hierarchical level. +Al- +though the majority of HMs in social science research +follow this probabilistic symmetry, some researchers may +want to compare non-exchangeable HMs, for example, to +study within-person dynamics (Driver & Voelkle, 2018; +Lodewyckx et al., 2011; Schumacher et al., 2022). For- +19 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +tunately, the modularity of our method allows easy adap- +tation of the neural network architecture to handle non- +exchangeable HMs. +To compare hierarchical time se- +ries models with temporal dependencies at the lowest +level, for instance, the first invariant module could be ex- +changed for a recurrent network, as proposed in Radev, +D’Alessandro, et al. (2021). Thus, future research could +extend and validate our method in BMC settings involving +non-exchangeable HMs. +Acknowledgments +The authors thank L. Schumacher for reading this article +and providing helpful feedback. +LE and MS were supported by a grant from the Deutsche +Forschungsgemeinschaft (DFG, German Research Foun- +dation; GRK 2277) to the research training group Statisti- +cal Modeling in Psychology (SMiP). LE was additionally +supported by the Google Cloud Research Credits program +with the award GCP19980904. 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We employ the +Adam optimizer (Kingma & Ba, 2015) with an initial +learning rate of 5e-4 and a cosine decay schedule. For +all validation studies, we use online training, i.e. simulate +new training data sets flexibly right before each training +step. For the application study, we simulate all data sets +efficiently a priori in the Julia programming language and +therefore use offline training, i.e. training with a predeter- +mined amount of data sets. +We use the following neural network architectures for all +experiments: The hierarchical summary network is com- +posed of two deep invariant modules, each consisting of +K = 2 equivariant modules followed by an invariant +module. The first deep invariant module reduces the infor- +mation within each group to vectors �xm of size 32, while +the second deep invariant module reduces the information +between the groups to a single vector z of size 128. The +inference network is realized through a standard feedfor- +ward network with two hidden layers and the number of +output units equaling the number of competing HMs, J. +We did not conduct a thorough search for optimal hyper- +parameter settings of the neural networks and the training +process. +B +Validation study 1 details +Figure 9 displays the log BFs approximated by bridge +sampling and the neural network variants for all 100 test +data sets, including those 27 data sets for which bridge +sampling approximated a BF > 1, 000, 000 and that were +therefore excluded in Figure 5 for visibility purposes. +C +Validation study 2 details +Here, we provide details on our model specifications and +prior choices. We reformulate the observation-level struc- +ture of the MPT model as a binomial instead of a multino- +mial process to obtain identical response generation im- +plementations for both models +xh +mn ∼ Bernoulli(hm) for n = 1, . . . , Nm +(35) +xf +mn ∼ Bernoulli(fm) for n = 1, . . . , Nm, +(36) +where hm denotes the probability of detecting an old item +as old (”hit”) and fm denotes the probability of detecting a +new item as old (”false alarm”). The generating processes +of these probabilities with our distributional choices are +described in Tables 4 and 5 for the SDT model and Tables +6 and 7 for the MPT models. Figure 10 shows the prior +predictive patterns of hit rates and false alarm rates arising +from 5, 000 simulated data sets for each model. +D +Application study details +D.1 +Application experiment: Parameter priors and +prior predictive checks. +We base our priors upon the comprehensive collection of +diffusion model parameter estimates by Tran et al., 2021. +For the L´evy flight models, M2 and M4, we inform the +prior on the additional α parameter by the estimates for +comparable tasks (those completed under speed instruc- +tions) in Voss et al., 2019. For the inter-trial variability +parameters included in M3 and M4, we follow the non- +hierarchical priors that Wiecki et al., 2013 suggest to use +in hierarchical drift-diffusion models. Table 8 contains the +hyperprior choices and table 9 the group-level priors. +To ensure that the informed priors for our HMs accurately +reflect prior knowledge at both levels, we conduct prior +predictive checks based on 10, 000 simulations (displayed +in Figures 11, 12 and 13). +D.2 +Robustness against artificial noise +Here, we inspect the stability of our neural network +against additional noise injection. Figure 14 displays the +model comparison results as increasing percentages of tri- +als per participant are artificially masked as missing. We +repeat the random masking of trials 100 times per per- +centage step to assess the sensitivity of the results to spe- +cific parts of the empirical data. The low variability indi- +cated by the thin-shaded areas means that our model com- +parison results do not depend on a specific subset of the +data. Further, consistent with our model comparison re- +sults, the PMPs of M1 and M2 are indistinguishable in +Figure 14. Despite our network being trained on the em- +pirical amount of missing data, 3.17% over both tasks, we +observe robust model comparison results for up to 25% +missing data per participant. +25 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +0 +20 +40 +60 +80 +log(BF21) - bridge sampling +0 +20 +40 +60 +80 +log(BF21) - fixed network +Simulated from M1 +Simulated from M2 +0 +20 +40 +60 +80 +log(BF21) - bridge sampling +0 +20 +40 +60 +80 +log(BF21) - variable network +Simulated from M1 +Simulated from M2 +Figure 9: Validation study 1: Full comparison results for the log Bayes factors (all 100 test data sets). +Table 4: Validation study 2: Hyperprior distributions of the SDT model. +Parameter +Symbol +Prior distribution +Probit-transformed hit probability +µh′ +Normal(1, 0.5) +σh′ +Gamma(1, 1) +Probit-transformed false alarm probability +µf ′ +Normal(−1, 0.5) +σf ′ +Gamma(1, 1) +Table 5: Validation study 2: Group-level prior distributions and transformations of the SDT model. +Parameter +Symbol +Prior distribution / transformation +Probit-transformed hit probability +h′ +m +Normal(µh′, σh′) +Probit-transformed false alarm probability +f ′ +m +Normal(µf ′, σf ′) +Hit probability +hm +Φ(h′ +m) +False alarm probability +fm +Φ(f ′ +m) +Table 6: Validation study 2: Hyperprior distributions and transformations of the MPT model. +Parameter +Symbol +Prior distribution / transformation +Probit-transformed recognition probability +hd′ +Normal(0, 0.25) +Probit-transformed guessing probability +hg′ +Normal(0, 0.25) +Covariance matrix +λd′ +Uniform(0, 2) +λg′ +Uniform(0, 2) +Q +InvWishart(3, I) +Σ +Diag(λd′, λg′) Q Diag(λd′, λg′) +26 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +Table 7: Validation study 2: Group-level prior distributions and transformations of the MPT model. +Parameter +Symbol +Prior distribution / transformation +Probit-transformed recognition probability +d′ +m +Normal +��µd′ +µg′ +� +, Σ +� +Probit-transformed guessing probability +g′ +m +Recognition probability +dm +Φ(d′ +m) +Guessing probability +gm +Φ(g′ +m) +Hit probability +hm +dm + (1 − dm) ∗ gm +False alarm probability +fm +(1 − dm) ∗ gm +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +10000 +20000 +30000 +40000 +50000 +Hit rates +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +10000 +20000 +30000 +40000 +50000 +False alarm rates +Hierarchical SDT Model +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +5000 +10000 +15000 +20000 +25000 +30000 +35000 +40000 +Hit rates +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +5000 +10000 +15000 +20000 +25000 +30000 +35000 +40000 +False alarm rates +Hierarchical MPT Model +Figure 10: Validation study 2: Prior predictive checks for the SDT and the MPT model. The green vertical lines +indicate the mean. +27 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +Table 8: Application experiment: Hyperprior distributions of the evidence accumulation models. +Parameter +Symbol +Prior distribution +Threshold separation +µa +Normal(5, 1) +σa +Normal+(0.4, 0.15) +Relative starting point +µzr +Normal(0, 0.25) +σzr +Normal+(0, 0.05) +Drift rate for blue/non-word stimuli +µv0 +Normal(5, 1) +σv0 +Normal+(0.5, 0.25) +Drift rate for orange/word stimuli +µv1 +Normal(5, 1) +σv1 +Normal+(0.5, 0.25) +Non-decision time +µt0 +Normal(5, 1) +σt0 +Normal+(0.1, 0.05) +Stability parameter of the noise distribution +µα +Normal(1.65, 0.15) +σα +Normal+(0.3, 0.1) +Table 9: Application experiment: Group-level prior distributions of the evidence accumulation models. +Parameter +Symbol +Prior distribution +Threshold separation +am +Gamma(µa, σa) +Relative starting point +zrm +invlogit(Normal(µzr, σzr)) +Drift rate for blue/non-word stimuli +v0m +-Gamma(µv0, σv0) +Drift rate for orange/word stimuli +v1m +Gamma(µv1, σv1) +Non-decision time +t0m +Gamma(µt0, σt0) +Stability parameter of the noise distribution +αm +TruncatedNormal(µα, σα, 1, 2) +Inter-trial variability of starting point +sz +Beta(1, 3) +Inter-trial variability of drift +sv +Normal+(0, 2) +Inter-trial variability of non-decision time +st +Normal+(0, 0.3) +28 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +Figure 11: Application experiment: Prior predictive checks for the hyperpriors in the comparison of evidence accu- +mulation models. The green vertical lines indicate the mean. +Figure 12: Application experiment: Prior predictive checks for the hierarchical group-level priors in the comparison +of evidence accumulation models. The green vertical lines indicate the mean. +Figure 13: Application experiment: Prior predictive checks for the non-hierarchical group-level priors in the compar- +ison of evidence accumulation models. The green vertical lines indicate the mean. +29 + +A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Percentage of missing data +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Posterior model probability +M1 +M2 +M3 +M4 +Figure 14: Application experiment: Robustness of the model comparison results against artificially injected random +noise. The lines represent the average probabilities of 100 repetitions per percentage step (in each repetition masking +a random subset of the data), while the shaded areas indicate the standard deviation between these repetitions. +30 + diff --git a/c9FKT4oBgHgl3EQfqS4B/content/tmp_files/load_file.txt b/c9FKT4oBgHgl3EQfqS4B/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2cf7c5b040a7eca786ef9a7a02147f5c696e4b7b --- /dev/null +++ b/c9FKT4oBgHgl3EQfqS4B/content/tmp_files/load_file.txt @@ -0,0 +1,2096 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf,len=2095 +page_content='A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS Lasse Elsem¨uller Department of Psychology University of Mannheim lasse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='elsemueller@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='com Martin Schnuerch Department of Psychology University of Mannheim martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='schnuerch@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='com Paul-Christian B¨urkner Cluster of Excellence SimTech University of Stuttgart paul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='buerkner@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='com Stefan T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Radev Cluster of Excellence STRUCTURES Heidelberg University stefan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='radev93@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='com ABSTRACT Bayesian model comparison (BMC) offers a princi- pled approach for assessing the relative merits of com- peting computational models and propagating uncer- tainty into model selection decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' However, BMC is often intractable for the popular class of hierarchical models due to their high-dimensional nested parame- ter structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' To address this intractability, we propose a deep learning method for performing BMC on any set of hierarchical models which can be instantiated as probabilistic programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Since our method enables amortized inference, it allows efficient re-estimation of posterior model probabilities and fast performance validation prior to any real-data application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In a se- ries of extensive validation studies, we benchmark the performance of our method against the state-of-the- art bridge sampling method and demonstrate excel- lent amortized inference across all BMC settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We then use our method to compare four hierarchical evi- dence accumulation models that have previously been deemed intractable for BMC due to partly implicit likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In this application, we corroborate evi- dence for the recently proposed L´evy flight model of decision-making and show how transfer learning can be leveraged to enhance training efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Repro- ducible code for all analyses is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 1 Introduction Hierarchical or multilevel models (HMs) play an increas- ingly important methodological role in the social and cog- nitive sciences (Farrell & Lewandowsky, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Rouder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' HMs embody probabilistic and structural in- formation about nested data occurring frequently in vari- ous settings, such as educational research (Ulitzsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020), experimental psychology (Vandekerckhove et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2011), epidemiology (Jalilian & Mateu, 2021) or astro- physics (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2019), to name just a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Cru- cially, HMs can often extract more information from rich data structures than their non-hierarchical counterparts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', aggregate analyses), while retaining a relatively high intrinsic interpretability of their structural components (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Moreover, viewed as formal instanti- ations of scientific hypotheses, HMs can be employed to systematically assign preferences to these hypotheses by means of formal model comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For example, Haaf and Rouder (2017) proposed a powerful framework based on Bayesian HMs for formulating and testing competing theoretical positions on quantitative vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' qualitative indi- vidual differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We consider Bayesian model comparison (BMC) as a principled framework for comparing and ranking compet- ing HMs via Occam’s razor (Kass & Raftery, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Lotfi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' MacKay, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' However, standard BMC is analytically intractable for non-trivial HMs, as it requires marginalization over high-dimensional parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Moreover, BMC for complex HMs without explicit like- lihoods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', HMs available only as randomized simula- tors) becomes increasingly hopeless and precludes many interesting applications in the rapidly expanding field of simulation-based inference (Cranmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In this work, we propose to tackle the problem of BMC for arbitrarily complex HMs from a simulation-based per- spective using deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In particular, we build on the BayesFlow framework (Radev, D’Alessandro, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Radev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020) for simulation-based Bayesian inference and propose a novel hierarchical neural network architecture for approximating Bayes factors (BFs) and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='11873v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='ML] 27 Jan 2023 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS posterior model probabilities (PMPs) for any collection of HMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Our neural approach circumvents the steps of explicitly fitting all models and marginalizing over each model’s pa- rameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thus, it is applicable to both HMs with explicit likelihood functions and HMs accessible only through Monte Carlo simulations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', with implicit like- lihood functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Moreover, our neural networks come with an efficient way to compute their calibration error (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017), which provides an important diagnos- tic for self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Lastly, trained networks can be adapted to related tasks, substantially reducing the com- putational burden when dealing with demanding simula- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In Section 2, we introduce the theoretical background and related work on (hierarchical) BMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We then present the rationale and details of our deep learning method in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2, we present two valida- tion studies of the proposed method: One that includes toy models for illustrative purposes and one that includes two popular classes of models from the field of cognitive psychology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3, we then apply our method to compare hierarchical diffusion decision models with partly intractable likelihoods on a real data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Finally, Section 5 summarizes our contributions and discusses fu- ture perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 2 Theoretical Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1 Bayesian Hierarchical Modeling In order to streamline statistical analyses, researchers rely on assumptions about the probabilistic structure or sym- metry of the assumed data-generating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For in- stance, the canonical IID assumption in psychological modeling states that (multivariate) observations are in- dependent of each other and sampled from the same latent probability distribution (Nicenboim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Singmann & Kellen, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' However, more complex dependencies may arise in a variety of contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For instance, if there are repeated measurements per participant or participants belong to different natural groups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', school classes, working groups), the respective observations exhibit higher cor- relations within those clusters than across them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Ignor- ing this nested structure in statistical analyses may re- sult in biased conclusions (Singmann & Kellen, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Bayesian HMs formalize this structural knowledge by as- suming that observations are sampled from a multilevel generative process (Gelman, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For instance, the generative recipe for a two-level Bayesian HM can be written as: η ∼ p(η) (1) θm ∼ p(θ | η) for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , M (2) xmn ∼ p(x | θm) for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , Nm, (3) where η denotes the group-level parameters, θm denotes the individual parameters in group m and xmn represents the n-th observation in group m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Such a model suggests the following (non-unique) factorization of the joint dis- tribution: p(η, {θm}, {xmn}) = p(η) M � m=1 p (θm η) Nm � n=1 p (xmn | θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (4) The set notation {θm} and {xmn} implies that the num- ber of groups and observations in each group can vary across simulations, data sets and experiments and that these quantities are exchangeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' HMs can be considered as a compromise between a sepa- rate analysis of each group (no-pooling) that neglects the information contained in the rest of the data and an aggre- gate analysis of the data (complete pooling) that loses the distinction between intra-group and inter-group variabil- ity (Hox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The partial pooling of information induced by HMs leads to more stable and accurate indi- vidual estimates through the shrinkage properties of mul- tilevel priors, whereby single estimates inform each other (B¨urkner, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Gelman, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Despite having desirable properties, hierarchical model- ing comes at the cost of increased complexity and compu- tational demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' These increased demands make it hard or even impossible to compare competing HMs within the probabilistic framework of BMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Before we highlight these challenges, we first describe the basics of BMC for non-hierarchical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 Bayesian Model Comparison The starting point of BMC is a collection of J compet- ing generative models M = {M1, M2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , MJ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Each Mj is associated with a prior p (θj | Mj) on the pa- rameters θj and a generative mechanism, which is either defined analytically through a (tractable) likelihood den- sity function p (x | θj, Mj) or realized as a Monte Carlo simulation program gj(θ, z) with random states z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' To- gether, the prior and the likelihood define the Bayesian joint model p (θj, x | Mj) = p (θj | Mj) p (x | θj, Mj) , (5) 2 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS x0 x1 p(x|M1) p(x|M2) (a) Marginal Likelihoods M1 M2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1 Probability p(M) M1 M2 p(M|x0) M1 M2 p(M|x1) (b) Posterior Model Probabilities Figure 1: Hypothetical BMC setting with a simple model M1 and a more complex model M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (a) The complex model which accounts for a broader range of observations needs to spread its marginal likelihood to cover its larger generative scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' It does so at the cost of diminished sharpness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thus, even though observation x1 is well within its generative scope, the simpler model M1 yields a higher marginal likelihood and is therefore preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In contrast, observation x0 has a higher marginal likelihood under model M2, as it is very unlikely to be generated by the simpler model M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (b) The corresponding posterior model probabilities (PMPs) given a uniform model prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' which is also tacitly defined for simulator-based models by marginalizing the joint distribution p (x, z | θj, Mj) over all possible execution paths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', random states) of the simulation program to obtain the implicit likelihood p (x | θj, Mj) = � p (x, z | θj, Mj) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (6) This integral is typically intractable for complex simula- tors (Cranmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020), which makes it impossible to evaluate the likelihood and use standard Bayesian meth- ods for parameter inference or model comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The likelihood function, be it explicit or implicit, is a key object in Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' When the parameters θ are systematically varied and the data x held constant, the likelihood quantifies the relative fit of each model instan- tiation (defined by a fixed configuration θ) to the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' When we marginalize the Bayesian joint model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 5) over its parameter space, we obtain the marginal likeli- hood or Bayesian evidence (see MacKay, 2003, Chap- ter 28): p (x | Mj) = � p (x | θj, Mj) p (θj | Mj) dθj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (7) The marginal likelihood can be interpreted as the prob- ability that we would generate data x from model Mj when we randomly sample from the model’s parame- ter prior p (θj | Mj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Moreover, the marginal likelihood is a central quantity for prior predictive hypothesis test- ing or model selection (Kass & Raftery, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' O’Hagan, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Rouder & Morey, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' It is well-known that the marginal likelihood encodes a notion of Occam’s razor arising from the basic principles of probability (Kass & Raftery, 1995, see also Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thus, the marginal like- lihood provides a foundation for the widespread use of Bayes factors (BFs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Heck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022) or posterior model probabilities (PMPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Congdon, 2006) for BMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The relative evidence for a pair of models can be com- puted through the ratio of marginal likelihoods for the two competing models Mj and Mk, BFjk = p (x | Mj) p (x | Mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (8) This ratio is called Bayes factor (BF) and is widely used for quantifying pairwise model preference in Bayesian settings (Heck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Kass & Raftery, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Ac- cordingly, a BFjk > 1 indicates preference for model j over model k given available data x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Alternatively, one can directly focus on the (marginal) posterior probability of a model Mj, p (Mj | x) = p(x | Mj) p(Mj) �J j=1 p (x | Mj) p(Mj) , (9) where p(Mj) is a categorical (typically uniform) prior distribution encoding a researcher’s prior beliefs regard- ing the plausibility of each considered model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' This prior distribution is then updated with the information con- tained in the marginal likelihood p(x | Mj) to obtain the corresponding posterior model probability (PMP), p(Mj | x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Occasionally in the text, we will refer to the vector of PMPs for all J models as π and to the individual PMPs as πj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The ratio of two PMPs, known as posterior odds, is in turn connected to the Bayes factor via the cor- responding model priors: p(Mj | x) p(Mk | x) = p (x | Mj) p (x | Mk) × p(Mj) p(Mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (10) Despite its intuitive appeal, the marginal likelihood (and thus BFs and PMPs) represents a well-known and widely appreciated source of intractability in Bayesian work- flows, since it typically involves a multi-dimensional inte- gral (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 7) over potentially unbounded parameter spaces (Gronau, Sarafoglou, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Lotfi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 3 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS Furthermore, the marginal likelihood becomes doubly in- tractable when the likelihood function is itself not avail- able (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', in simulation-based settings), thereby making the comparison of such models a challenging and some- times, up to this point, hopeless endeavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Unsurprisingly, estimating the marginal likelihood (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 7) in the context of hierarchical models becomes even more challenging, since the number of parameters over which we need to perform marginalization grows dramatically (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', parameters at all hierarchical levels enter the compu- tation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' These computational demands render probabilis- tic comparison of HMs based on BFs or PMPs analyti- cally intractable even for relatively simple models with explicit (analytical) likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Therefore, researchers need to resort to costly, approximate methods which typi- cally only work for models with explicit likelihoods (Gel- man & Meng, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Gronau, Sarafoglou, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Meng & Schilling, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3 Approximate Bayesian Model Comparison 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1 Explicit Likelihoods The most efficient approximate methods to date require all candidate models to possess explicitly available like- lihood functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For the most simple scenario in which two HMs are nested (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', through an equality constraint on a parameter), the Savage-Dickey density ratio (Dickey & Lientz, 1970) provides a convenient approximation of the BF (Wagenmakers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Typically, however, the candidate models are not nested but exhibit notable structural differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thus, a general-purpose method is needed to encompass the entire plethora of model com- parison scenarios arising in practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' A more general method, and the current state-of-the-art for comparing HMs in psychological and cognitive mod- eling (Gronau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Gronau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022), is given by bridge sampling (Bennett, 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Meng & Wong, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Bridge sampling has enabled com- parisons within families of complex process models, such as multinomial processing trees (MPTs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Gronau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2019) or evidence accumulation models (EAMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Gronau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020), and serves as a simple add-on for Markov chain Monte Carlo (MCMC) based Bayesian workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Crucially, bridge sampling relies on the posterior draws generated by an MCMC sampler (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Stan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Carpenter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017) to efficiently approximate the marginal likeli- hood of each respective model (Gronau, Sarafoglou, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Note, however, that bridge sampling requires con- siderably more random draws for stable results than stan- dard parameter estimation (usually about an order of mag- nitude more;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Gronau, Singmann, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Moreover, the approximation quality of bridge sampling is depen- dent on the convergence of the MCMC chains (Gronau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Finally, there are no strong theoretical guaran- tees that the approximations are unbiased and accurately reflect the true marginal likelihoods (Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 Implicit Likelihoods With the rise of complex, high-resolution models, in- tractable likelihood functions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', functions that do not admit a closed form or are too costly to evaluate) be- come more and more common in statistical modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Such models are not limited to psychology and cogni- tive science (Nicenboim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Van Rooij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2019), but are also common in fields such as neuro- science (Gonc¸alves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020), epidemiology (Radev, Graw, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2021), population genetics (Pudlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2016) or astrophysics (Hermans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2021) (see Cran- mer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Despite the common term likelihood- free, simulator-based models still possess an implicitly defined likelihood (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2) from which we can obtain random draws through Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' This enables model comparison through simulation-based methods, usually by means of approximate Bayesian com- putation (ABC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Marin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Mertens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Pudlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Traditional (rejection-based) ABC methods for BMC re- peatedly simulate data sets from the specified generative models, retaining only those simulations that are suffi- ciently similar to the empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' To enable the cal- culation of this (dis-)similarity even in high-dimensional cases, the information contained in the simulated data sets is reduced by computing hand-crafted summary statis- tics, such as the mean and variance (Csill´ery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Sunn˚aker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The resulting acceptance rates of the candidate models represent the approximations of their PMPs (Marin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Mertens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Even for non-hierarchical models, ABC methods are known to be notoriously inefficient and highly dependent on the concrete choice of summary statistics (Cranmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Marin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' This choice is even more challenging for HMs, as modelers now have to retain an optimal amount of information on multiple levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' More- over, the rapidly growing number of summary statistics reduces the probability that a simulated data set is similar enough to the empirical data, which vastly increases the number of required simulations (Beaumont, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Marin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Regardless of the number of summary statistics, their manual computation carries the danger of insufficiently summarizing the simulations and thereby producing bi- ased approximations (a phenomenon known as curse of insufficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Marin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' While many improve- ments of rejection-based ABC have been proposed, most 4 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS notably ABC-MCMC (Marjoram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Turner & Sederberg, 2014), ABC-SMC (Sisson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2007), as well as Gibbs ABC (Turner & Van Zandt, 2014) for Bayesian hierarchical modeling in particular (see also Clart´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Fengler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2021), these advancements are still limited by their dependence on hand-crafted summary statistics or kernel density estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Recent developments, such as ABC-RF (Pudlo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2016), combine ABC with machine learning methods to build more expressive approximators for BMC problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Accordingly, model comparison is treated as a supervised learning problem – the simulated data encompasses a training set for a machine learning algorithm that learns to recognize the true generative model from which the data set was simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The machine learning approach re- duces the inefficiency problem that haunts rejection-based ABC methods, but does not alleviate the curse of insuffi- ciency (Marin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 Bayesian Model Comparison with Neural Networks Recently, Radev, D’Alessandro, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2021) explored a method for simulation-based BMC using specialized neu- ral networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The authors proposed to jointly train two specialized neural networks using Monte Carlo simula- tions from each candidate model in M: a summary net- work and an evidential network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The goal of the summary network is to extract maximally informative (in the opti- mal case, sufficient) summary statistics from complex data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The goal of the evidential network is to approximate PMPs as accurately as possible and, optionally, to quan- tify their epistemic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Importantly, simulation-based training of neural networks enables amortized inference for both implicit and ex- plicit likelihood models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Amortization is a property that ensures rapid inference for an arbitrary amount of data sets after a potentially high computational investment for simulation and training (Mestdagh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Radev, D’Alessandro, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Radev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' As a consequence, the calibration (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Talts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2018) or the inferential adequacy (Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022) of an amortized Bayesian method are embarrassingly easy to validate in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In contrast, non-amortized methods, such as ABC- MCMC (Turner & Sederberg, 2014) or ABC-SMC (Sis- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2007) need to repeat all computations from scratch for each observed data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thereby, it is often in- feasible to assess their calibration or inferential adequacy in the pre-data phase of a Bayesian workflow (Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Unfortunately, the evidential method proposed by Radev, D’Alessandro, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2021) is not applicable to HMs due to their nested probabilistic structure which cannot be tackled via previous summary networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' This severely limits the applicability of the method in quantitative re- search, where hierarchical models have been advocated as a default choice (Lee, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' McElreath, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Rouder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In the following, we describe how to extend the original method to enable amortized BMC for HMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 3 Method At its core, our method involves a multilevel permutation invariant neural network which is aligned to the proba- bilistic symmetry of the underlying HMs (see Figure 2 for a visualization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We hold that any method which does not rely on ad hoc summary statistics should take this prob- abilistic symmetry (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', exchangeability) into account in order to ensure the structural faithfulness of its approx- imations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Moreover, respecting the probabilistic sym- metry implied by a generative model cannot only make simulation-based training easier but also suggests a par- ticular architecture for building neural Bayesian approxi- mators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1 Permutation Invariance Permutation invariance is the functional equivalent of the probabilistic notion of exchangeability (Bloem-Reddy & Teh, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Gelman, 2006), which roughly states that the order of random variables should not influence their joint probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' To illustrate this point, consider the model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4, which has two exchangeable levels by design, indexed by m ∈ {1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , M} and n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , Nm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In a setting fa- miliar to social scientists, we might have M individuals, each of whom provides Nm (multivariate) responses on some scale or in repeated trials of an experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Now, suppose that we want to compare a set of HMs M = {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , MJ} of the form given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4 that might differ in various ways (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', different prior/hyperprior as- sumptions or disparate likelihoods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Due to the structure of the models, the PMPs p(M | {xmn}) depend on nei- ther the ordering of the individuals nor the ordering of their responses (which also holds true for the correspond- ing BFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' More precisely, if S(·) is an arbitrary permutation of an index set, then p(M | {xmn}) = p(M | S({xmn})) (11) for any S(·) acting on {1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , M} × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , N1} × · · · × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , NM} where × denotes the Cartesian product of 5 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS Figure 2: Our proposed hierarchical neural network architecture for encoding permutation invariance in the trans- formation of nested, two-level data into posterior model probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' A first invariant module Σ(1) I reduces all Nm observations within each of the M groups to a single intermediary embedding vector �xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For readability of the fig- ure, we display Nm = N as constant across each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' A second invariant module Σ(2) I reduces all intermediary embedding vectors to a hierarchical embedding vector z, which gets passed through an inference network to arrive at the final vector �π of approximated posterior model probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' two (index) sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Note that this notation implies that only permuting each m and permuting each n within, but not across each group m is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The property of permu- tation invariance is immediately obvious from the right- hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4 that involves two nested products (prod- ucts being permutation invariant transformations when seen as functions operating on sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Naturally, learning permutation invariance directly from data or simulations is hardly feasible with standard neural networks, even for non-nested data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Indeed, for non-hierarchical generative models, Radev, D’Alessandro, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2021) propose to use composite permutation invariant networks as employed by Zaheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In the following section, we gen- eralize this architectural concept to the hierarchical set- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 Hierarchical Invariant Neural Network Architecture Permutation invariant networks differ from standard feed- forward networks in that they can process inputs of dif- ferent sizes and encode the probabilistic symmetry of the data directly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', remove the need to learn the symmetry implicitly during training by supervised learning alone).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For the purpose of BMC with HMs, we realize a hi- erarchical permutation invariant function via a stack of invariant modules Σ(l) I for each hierarchical level l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , L of the Bayesian model (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Each in- variant module performs an equivariant non-linear trans- formation h(l) 1 acting on the individual data points, fol- lowed by a pooling operator (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', sum or max) and a fur- ther non-linear transformation h(l) 2 acting on the pooled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In order to preserve hierarchical symmetry, we apply each Σ(l) I independently to each nested sequence of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' To make this point concrete, consider the two-level model given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4 and let data point xmn denote the multi- variate response of person m in trial n of some data col- lection experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Accordingly, the first invariant mod- ule Σ(1) I operates by reducing the trial data {xn}m of each person m to a single person-vector �xm of fixed size: �xm = Σ(1) I ({xn}m) = h(1) 2 � Nm � n=1 h(1) 1 (xmn) � , (12) where h1 and h2 are implemented as simple feedforward neural networks with trainable parameters suppressed for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The second invariant module Σ(2) I then com- 6 X11 Invariant Module Z(1) X1N Invariant Module Z(1) Invariant Inference Module Z(2) Network ZN Hierarchical M1 M2 Embedding Posterior Model Probabilities π Invariant Module Z(1) XMN Intermediary Embeddings Nested DataA DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS presses all person vectors to a final vector z of fixed size: z = Σ(2) I ({xm}) = h(2) 2 � M � m=1 h(2) 1 (�xm) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (13) In this way, the architecture becomes completely indepen- dent of the number of persons M or number of trials per person Nm, which could vary arbitrarily across persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The vector z, whose dimensionality represents a tunable hyperparameter, can be interpreted as encoding learned summary statistics for the BMC task at hand (to be dis- cussed shortly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Moreover, it is easy to see that z is inde- pendent of the ordering of persons or the ordering of trials within persons, as necessitated by the model formulation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thus, the composition Σ(2) I Σ(1) I ({xmn}) re- duces a hierarchical data set with two levels to a single vector z which respects the probabilistic symmetry im- plied by the particular hierarchical model formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3 Increasing the Capacity of Invariant Networks Encoding an entire hierarchical data set {xmn} into a sin- gle vector z forces the composite neural network to per- form massive data compression, creating a potential in- formation bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For complex generative models, this task can become rather challenging and will depend highly on the representational capacity of the neural net- work (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', its ability to extract informative data set em- beddings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Fortunately, we can enhance the simple archi- tecture described in the preceding paragraph by using in- sights from Zaheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2017) and Bloem-Reddy and Teh (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In order to increase the capacity of the previously introduced invariant transformation, we can stack to- gether multiple equivariant modules Σ(l) E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Each equivari- ant module implements a combination of equivariant and invariant transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For instance, focusing on our two-level model example (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4), the transformations at level 1 for each person m are now given by: �xm = h(1) 2 � Nm � n=1 h(1) 1 (xmn) � (14) �xmn = h(1) 3 ([xmn, �xm]) for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , Nm, (15) where h3 is also implemented as a simple feedforward neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In this way, each intermediary output �xmn of the equivariant module now contains information from all data points, so the network can learn considerably more flexible transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Moreover, we can stack K equivariant modules followed by an invariant module, in order to obtain a deep invariant module, which for the first hierarchical level (l = 1) takes the following form: �xm = (Σ(1) I Σ(K,1) E · · · ◦ Σ(1,1) E )({xn}m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (16) Compared to the simple invariant module from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 12, the deep invariant module involves a larger number of com- putations but allows the network to learn more expressive representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Accordingly, the transformation for the second hierarchical level (l = 2), which yields the final summary representation z, is given by: z = (Σ(2) I Σ(K′,2) E · · · ◦ Σ(1,2) E )({�xm}), (17) where the number of equivariant modules K′ for level 2 can differ from the number of equivariant modules K for level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In our experiments, reported in Section 4, we ob- serve a clear advantage of using deep invariant networks over their simple counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Furthermore, for two-level models, we find that the performance of the networks is largely insensitive to the choice of K or K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 Learning the Model Comparison Problem In order to get from the learned summary representation z to an approximation of the analytic PMPs �π, we ap- ply a final neural classifier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', the inference network) I(z) = �π, as visualized in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We deviate from the Dirichlet-based setting in Radev, D’Alessandro, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2021), since we found that implementing the inference network as a standard softmax classifier (Grathwohl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2019) provides slightly better calibration and leads to more stable training in the specific context of HMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Denoting the entire hierarchical neural network as fφ({x}) = �π and an arbitrary hierarchical data set as {x}, we aim to minimize the expected logarithmic loss min φ Ep(M,{x}) � �− J � j=1 IMj · log fφ({x})j � � , (18) where φ represents the vector of trainable neural network parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', weights and biases), IMj is the indicator function for the “true” model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The expectation runs over the joint generative (mixture) distribution of all models p(M, {x}), which we access through Monte Carlo sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Since the logarithmic loss is a strictly proper loss (Gneiting & Raftery, 2007), it drives the outputs of fφ({x}) to estimate the actual PMPs p(M | {x}) as best as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In practice, we approximate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 18 over a training set of B simulations from the competing HMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Each entry b for b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', B in this training set represents a hierar- chical data set {x(b)} itself along with a corresponding one-hot encoded vector for the “true” model index M(b) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The latter denotes the model from which the data set was generated and serves as the “ground truth” for supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 7 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS Similarly to Radev, D’Alessandro, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2021), our neural method encodes an implicit preference for sim- pler HMs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Occam’s razor) inherent in all marginal likelihood-based methods (see MacKay, 2003, Chap- ter 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Since our simulation-based training approximates an expectation over the marginal likelihoods of all HMs p(M) p(x | M), data sets generated by a simpler HM will tend to be more similar compared to those generated by a more complex one (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thus, data sets that are plausible under both HMs will be generated more often by the simpler model than by the more complex model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' A sufficiently expressive neural network will capture this be- havior by assigning a higher PMP for the simpler model1, thereby capturing complexity differences arising directly from the generative behavior of the HMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Finally, to increase training efficiency when working un- der a limited simulation budget, we also explore a novel pre-training method inspired by transfer learning (Ben- gio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Torrey & Shavlik, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' First, we train the networks on data sets with a reduced number of ex- changeable units (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', reducing the number of observa- tions at level l = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' This procedure accelerates training since it uses fewer simulator calls and the forward pass through the networks becomes cheaper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In a second step, we generate data with a realistic number of exchangeable units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Crucially, since we can use the pre-trained network from step one as a better-than-random initialization, we need a much smaller amount of simulations than if we trained the network from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Indeed, Experiment 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3 demonstrates the utility of this training method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4 Experiments In this section, we first conduct two simulation studies in which we extensively test the approximation perfor- mance of our hierarchical neural method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We start with a comparison of two nested toy HMs in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1, fol- lowed by a comparison of two complex non-nested HMs of cognition in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For both validation studies, we test our method internally by examining the calibra- tion of the approximated PMPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Additionally, we vali- date our method externally by benchmarking its perfor- mance against the current state-of-the-art for comparing HMs, namely, bridge sampling (Gelman & Meng, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Gronau, Singmann, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' To enable this chal- lenging benchmark, we limit our validation studies to the comparison of models with explicit likelihoods to which bridge sampling is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Finally, in a real-data application, we use our deep learn- ing method to compare four hierarchical evidence accu- mulation models (EAMs) of response times data in Sec- 1Assuming equal prior model probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Two of these models have no analytic likeli- hood, which makes the entire BMC setup intractable with current state-of-the-art methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', bridge sampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Moreover, with this example, we also address the utility of a novel EAM, the L´evy flight model (Voss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2019), that has previously been impossible to investigate directly using Bayesian HMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For all experiments, we assume uniform model pri- ors p(Mj) = 1/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' All computations are con- ducted on a single-GPU machine with an NVIDIA RTX 3070 graphics card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The reported computation times are measured as wall-clock times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Details on the implementation of our neural networks and the em- ployed training procedures are provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Code for reproducing all results from this paper is freely available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='com/elseml/ DeepHierarchicalModelComparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1 Validation study 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Hierarchical Normal Models In this first experiment, we examine a simple and control- lable model comparison setup to examine the behavior of our method under various conditions, before moving on to more complex scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Inspired by Gronau (2021), we compare two hierarchical normal models M1 and M2 that share the same hierarchical structure τ 2 ∼ Normal+(0, 1) (19) σ2 ∼ Normal+(0, 1) (20) θm ∼ Normal(µ, √ τ 2) for m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , M (21) xmn ∼ Normal(θm, √ σ2) for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , Nm, (22) with Normal+(·) denoting a zero-truncated normal distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The models differ with respect to the parameter µ that describes the location of the individual-level param- eters θm: Whereas M1 assumes the location of θm to be fixed at 0, the more flexible M2 allows for µ to vary M1: µ = 0 (23) M2: µ ∼ Normal(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (24) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1 Calibration The most important properties of an approximate infer- ence method are the trustworthiness of its results and, more pragmatically, whether we can diagnose the lack of trustworthiness in a given application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' A useful proxy for trustworthiness is the calibration of a probabilistic classi- fier, which measures how closely the predicted probabili- ties of outcomes match their true underlying probabilities (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' However, computing the calibration of a BMC procedure is hardly feasible in a non-amortized setting, since it in- 8 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS volves applying the method to a large number of simu- lated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For bridge sampling, for example, that would imply re-fitting the models via MCMC and running bridge sampling on at least hundreds, if not thousands of simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The calibration of our networks, on the other hand, can be determined almost immediately after training due to their amortization property (Radev, D’Alessandro, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In the following experiments, we assess the calibration of our networks visually (via calibration curves) and nu- merically (via a measure of calibration error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For gen- erating a calibration curve (DeGroot & Fienberg, 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Niculescu-Mizil & Caruana, 2005), we first sort the pre- dicted PMPs �π(s) j on S simulated data sets s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , S, which we then partition into I equally spaced probability bins i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , I (we use I = 15 bins for all validation experiments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For each model j and each bin i contain- ing a set Bij of predicted model indices, we compute the mean prediction for the model (predicted probability, PP) and the actual fraction of this model being true (true prob- ability, TP) as follows: PP(Bij) := 1 |Bij| � b∈Bij �π(b) j , (25) TP(Bij) := 1 |Bij| � b∈Bij IM(b) j , (26) where I again denotes the indicator function for the “true model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' These two quantities varying over the bins form the X- and Y -axis of a calibration curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' A well- calibrated model comparison method with an agreement in each bin (as indicated by a diagonal line) thus yields ap- proximations that reflect the true probabilities of the com- pared models (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We further summarize this information via the Expected Calibration Error (ECE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Naeini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2015) as a single number bounded between 0 and 1, which we estimate by averaging the individual deviations between predicted and true probability in each bin: � ECEj := I � i=1 |Bij| S ����PP(Bij) − TP(Bij) ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (27) If follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 27 that a perfect ECE can be achieved by always predicting indifferent probabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', �π1 = �π2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5 when comparing two models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We therefore complement our calibration assessment by measuring the accuracy of recovery, for which we dichotomize the pre- dicted PMPs �π(s) j on S simulated data sets into one-vs-rest model predictions � M(s) j : Accj := 1 S I � M(s) j =M(s) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (28) Thus, in our BMC context, accuracy roughly is to ECE what sharpness is to posterior calibration in Bayesian pa- rameter estimation (B¨urkner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 Fixed data set sizes In the first calibration experiment, we examine the performance of our method for the most simple application case of learning a model comparison problem on a specific (fixed) data set size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Here, all data sets simulated for training and validating the network con- sist of M = 50 groups and Nm = 50 observations for each group m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We train the network for 10, 000 backpropagation steps, taking 12 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Subsequently, we calculate its calibra- tion on 5, 000 held-out validation data sets and repeat this process 25 times to obtain stable results with uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 3a depicts the resulting median calibration curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Its close alignment to the dashed di- agonal line representing perfect calibration indicates that the PMP approximations are very well-calibrated (median ECE over all repetitions of � ECE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The curve’s coverage of the full range of predicted probabilities and the median accuracy of � Acc = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='88 confirm that the ex- cellent calibration does not stem from indifferent predic- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The subsequent comparison of our method to bridge sampling suggests that this accuracy is indeed close to the upper bound imposed by the aleatoric uncertainty in the model-implied data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Data sets with varying numbers of observations We now train our hierarchical network to approximate BMC over a range of hierarchical data sets with varying num- bers of observations within groups Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' This amortiza- tion over observation sizes would provide a substantial efficiency gain if a researcher desires to compare HMs on multiple data sets with differing Nm, as only a sin- gle network would have to be trained for all data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3 In our validation setup, each simulated data set still consist of M = 50 groups, but now the number of observations within those groups varies in Nm = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We train the network for 20, 000 training steps, taking 25 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' At each training step, we draw the number of observations for the current batch of simulations from a discrete uniform distribution Nm ∼ UniformD(1, 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For each Nm used during training, we evaluate the cali- bration 25 times on 5, 000 held-out simulated validation 2We focus on the accuracy, since we use a uniform model prior p(M), but other metrics of predictive performance, such as the logarithmic scoring rule, would have been expedient as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 3Note that we refer to variability between data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We de- scribe an approach for handling within data set variability of nested trials in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 9 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 Predicted probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 True probability Median 50% CI 95% CI (a) Median calibration curve and confidence intervals (CIs) for data sets of M = 50 groups with Nm = 50 observations within each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 20 40 60 80 100 Number of observations (Nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='10 d ECE Median 50% CI 95% CI (b) Median expected calibration errors (ECEs) and confidence intervals for data sets of M = 50 groups with differing numbers of observations Nm within each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 3: Validation study 1: Calibration results for (a) the neural network trained on fixed data set sizes and (b) the neural network trained on data sets with varying numbers of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Medians and confidence intervals (CIs) are computed over 25 repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' This repetition procedure allows us to quantify the uncertainty of our ECE estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 3b plots the median ECE values for each observa- tion size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The neural network achieves high calibration with a median ECE over all observation sizes (and rep- etitions) of � ECE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Moreover, the unsystematic pattern of the median curve and the homoskedastic vari- ation between the observation sizes indicate that the net- work has learned the model comparison task equally well for all settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Together, the low calibration error and the accurate model predictions (median accuracy � Acc = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='88) indicate that our method incurs no trade-off between cali- bration and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Data sets with varying numbers of groups and obser- vations In the third calibration experiment, we test the ability of the network to learn a model comparison prob- lem over a range of data sets with varying numbers of groups M and varying observations per group Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' This training scheme allows for amortized model comparison on multiple data sets with different sizes, which can be especially useful for a priori sample size determination on simulated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Additionally, the trained network can be stored and reused on future data sets with yet-unknown sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For this experiment, training and valida- tion data sets are simulated with M = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , 100 groups and Nm = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , 100 observations, resulting in a vast variability of data set sizes between 1 up to 10, 000 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Given the complexity of the learning task, we now train the network for 40, 000 training steps, taking 49 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' At each training step, we draw the number of groups and observations from discrete uniform distributions M ∼ UniformD(1, 100) and Nm ∼ UniformD(1, 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We es- timate calibration on 5, 000 held-out simulations for each combination of M and Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' As this implies simulating 50, 000, 000 data sets, we forego the repetition procedure employed in the previous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 4 depicts the calibration and accuracy results for all combinations of M and Nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We observe low ECEs for the vast majority of settings in Figure 4a (median ECE over all settings of � ECE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In other words, the trained network is capable of generating highly calibrated PMPs over a broad range of data set sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Moreover, the BMC results are sensitive to the number of nested obser- vations Nm, but not to the number of groups M, in our experimental setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The only systematic drop in cali- bration occurs for data sets containing just a few nested observations (Nm ≤ 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Considering that we observed high calibration even for this low number of observations in a network trained on data sets with varying Nm (see Figure 3b), we surmise that the drop in the edge areas in Figure 4a arises from the challenging learning task over vastly different data set sizes (a phenomenon known as 10 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS Number of groups (M) 20 40 60 80 100 Number of observations (Nm) 20 40 60 80 100 d ECE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='10 (a) Expected Calibration Error (ECE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Number of groups (M) 20 40 60 80 100 Number of observations (Nm) 20 40 60 80 100 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 (b) Accuracy of recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 4: Validation study 1: Results for the neural network trained and tested over variable data set sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' amortization gap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Cremer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The overall low (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', good) ECEs for all cases but the poorly identifiable Nm = 1 setting suggest that the networks’ approxima- tions are generally trustworthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' This is further confirmed by Figure 4b, where the observable accuracy pattern as- sures that this high calibration does not arise from a trade- off with predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Despite the demanding amortization setting, the network achieves an excellent median accuracy of � Acc = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='88, similar to the earlier ex- periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 Bridge Sampling Comparison After validating the general trustworthiness of our method, we now benchmark it against the current gold standard for comparing HMs, namely, bridge sampling, as implemented by Gronau, Singmann, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' As the non-amortized nature of bridge sampling restricts the feasible number of test sets, we conduct the benchmarking on 100 test sets which are simulated equally from M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' All simulated data sets consist of M = 50 groups and Nm = 50 observations per group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The fixed sample sizes of the test sets allow us to compare the two most dis- tinct networks from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1 to bridge sampling: The fixed network that is trained for this specific sample size and the more complex variable network that is trained for amortized model comparison over variable sample sizes between M = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , 100 groups and Nm = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , 100 observations per group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For bridge sampling, we first run four parallel MCMC chains with a warm-up period of 1, 000 draws and 49, 000 post-warmup posterior draws per chain in Stan (Carpenter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Stan Development Team, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We assess convergence through a visual inspection of the MCMC chains and an assessment of the �R, bulk ESS and tail ESS metrics (Vehtari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Afterwards, we use the posterior draws to approximate PMPs and BFs with the bridgesampling R package (Gronau, Singmann, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We confirm the sufficiency of the total of 196, 000 posterior draws by assessing the variability between mul- tiple runs as in Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2022), which yields highly similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Further insights via our calibration di- agnostics are precluded by bridge sampling being a non- amortized method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Approximation performance As we compare approxi- mate PMPs, we can use a number of complementary met- rics commonly employed to evaluate the quality of prob- abilistic predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' First, we quantify the fraction of times the correct model M(s) j underlying a simulated data set s was detected, that is, the accuracy of recovery (see Equation 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Second, we assess the Mean Absolute Error (MAE) to investigate the average deviation of the approx- imated model probabilities �π(s) j from a perfect classifica- tion: MAEj := 1 S S � s=1 ����π(s) j − I(s) Mj ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (29) Third, we measure the Root Mean Squared Error (RMSE), which places particular emphasis on large prediction er- rors, to detect whether one method produces highly incor- 11 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS Table 1: Validation study 1: Performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Accuracy MAE RMSE Log-Score SBC Bridge sampling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='86 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='19 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='32 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='32 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='06) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='02 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='04) Fixed network 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='85 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='04) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='33 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='33 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='06) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='02 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='04) Variable network 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='86 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='19 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='32 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='32 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='06) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='04) Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Bootstrapped mean values and standard errors (in parentheses) are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We use 1000 bootstrap versions of the test data sets and estimate the standard errors from the bootstrap standard deviations of the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' rect approximations more frequently than the other: RMSEj := � � � � 1 S S � s=1 � �π(s) j − I(s) Mj �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (30) Fourth, we calculate the Log-Score following the logarith- mic scoring rule: LogScorej := − 1 S S � s=1 � I(s) Mj · log�π(s) j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (31) Its property as a strictly proper scoring rule implies that it is asymptotically minimized if and only if the approxi- mate probabilities equal the true probabilities (Gneiting & Raftery, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Lastly, we measure simulation-based cal- ibration (SBC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Talts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2018) as adapted by Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2022) for model inference by the difference between the prior probability for a model and its average posterior probability in the test sets: SBCj := p(Mj) − 1 S S � s=1 �π(s) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (32) We evaluate all metrics for M2, so that a bias towards M1 is indicated by positive SBC values and a bias towards M2 by negative SBC values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Table 1 depicts the comparison results for our experi- mental setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' All metrics show equal performances for bridge sampling and the two neural network variants, with any differences being well within the range of the standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Approximation convergence In the following, we ana- lyze the degree of convergence between the two methods at the level of individual data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We explore this vi- sually by contrasting the PMP and (natural logarithmic) BF approximations of bridge sampling with the two neu- ral network variants in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We observe that the two methods’ PMP approximations agree for the easy cases where the true underlying model is clearly clas- sifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thus, discrepancies between the two methods arise mainly for data sets with predicted PMPs close to �π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Even for the data sets with the largest discrepan- cies, the two methods do not map to qualitatively different decisions: �π(bridge) 2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='67 and �π(neural) 2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='83 for the fixed network, �π(bridge) 2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='32 and �π(neural) 2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='25 for the vari- able network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Most importantly, we detect no systematic pattern in these deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' As BFs represent the ratio of marginal likelihoods, they al- low for a closer inspection of the degree of agreement be- tween the methods in those edge cases with PMPs close to 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We observe a close convergence for data sets clas- sified as stemming from M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Considering the predictions favoring M2, there are discrepancies for data sets with log BFs > 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4 Since this corresponds to BFs > 13, 000 and PMPs > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='9999, it is not visible in the PMP approx- imation plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We obtain such extreme results only for M2, as this model allows for deviations of the group level parameters’ location from 0 and enables the occurrence of extreme evidence in its favor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The divergence in this area of extreme evidence emerges most likely from the loss function employed for training the neural networks: The logarithmic loss obtained from a minuscule deviation of the PMP from 1 is near 0, which results in a negligible in- centive for further optimization of the network’s weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We could reject a competing explanation based on limited floating-point precision, since training with an increased floating-point precision from 32-bit to 64-bit resulted in identical patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The divergence we encountered provides insights into the technical nature of our method, but only arises in cases of extreme evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thus, it is far from altering the substan- tive conclusions derived from the simulated BMC setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Considering the convergence between the two methods in the realm of practical relevance, we can conclude that our method produces highly similar approximations to bridge sampling in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4For visibility purposes, we exclude the 27 data sets for which bridge sampling approximated a BF > 1, 000, 000 for the BF plots, all continuing the observed plateau pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Plots with all 100 data sets are provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 12 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 p(M2 |x) - bridge sampling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': 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+page_content='0 p(M2 |x) - bridge sampling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 p(M2 |x) - variable network Simulated from M1 Simulated from M2 Posterior Model Probabilities 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 log(BF21) - fixed network Simulated from M1 Simulated from M2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 log(BF21) - bridge sampling 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 log(BF21) - variable network Simulated from M1 Simulated from M2 Log Bayes Factors Figure 5: Validation study 1: Comparison of approximation results obtained via bridge sampling vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' the neural network trained on fixed data set sizes (left) and the neural network trained on variable data set sizes (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Note that the Bayes factor plots contain only those 73 data sets for which bridge sampling approximated a BF21 < 1, 000, 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 13 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS Approximation time Both bridge sampling and our deep learning method can be divided into two computa- tional phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For bridge sampling, the first phase con- sists of drawing from the posterior parameter distributions (taking 52 seconds per data set on average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Bridge sam- pling itself takes place in the second phase (taking 38 sec- onds on average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Notably, in contrast to amortized in- ference with neural networks, both phases need to be re- peated for each (simulated or observed) data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Taking the initial compilation time of 42 seconds into account, bridge sampling consequently took 152 minutes for BMC on our 100 test data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For the neural networks, the first phase (training) is resource-intensive (taking 12 minutes for the fixed net- work and 49 minutes for the variable network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The sec- ond phase (inference) is then performed in near real-time (taking 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='003 seconds for both networks on all 100 test data sets) and thus amortizes the training cost over mul- tiple applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For the simple HMs compared here, the amortization gains of our networks over bridge sam- pling come into effect after performing BMC on 8 (fixed network) or 39 (variable network) data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We acknowledge our likely suboptimal choices of com- putational steps for the bridge sampling workflow or the neural networks and hence wish to stress the general pat- terns of non-amortized vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' amortized methods demon- strated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In general, we expect an advantage of bridge sampling in terms of efficiency in situations where only one or a few data sets are available and obtaining a large number of posterior draws is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The demonstrated amortization property of our method might not be so rel- evant for inference on a single hierarchical data set, but it becomes crucial for performing calibration or recov- ery studies, which necessitate multiple re-fits of the same model (Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 Validation study 2: Hierarchical SDT vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' MPT Models We now extend our validation experiments from the sim- ple setup with nested HMs to the comparison of non- nested HMs of cognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In this simulation study, we examine the ability of our method to distinguish between data sets generated either from an HM based on signal de- tection theory (SDT model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Green, Swets, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 1966) or a hierarchical multinomial processing tree model (MPT model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Riefer & Batchelder, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For illustrative pur- poses, we embed our simulation study within an old-new recognition scenario, where participants indicate whether or not a stimulus was previously presented to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We ensure a challenging model comparison setting via three design aspects: First, we specify both models to possess a similar generative behavior, that is, hardly dis- tinguishable prior predictive distributions of hit rates and false alarm rates (prior predictive plots are provided in Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Second, data sets of old-new recognition typically contain low information as they only consist of binary variables indicating the stimulus type and re- sponse, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Third, we further amplify the infor- mation sparsity of the data sets by choosing a particularly small size for all data sets of M = 25 simulated partici- pants and Nm = 50 observations per participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' A major difference between the compared cognitive model classes lies in the assumption of a continuous la- tent process by the SDT model and discrete processes (or states) by the MPT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Our specification of the SDT model follows the hierarchical formulation of the standard equal-variance model by Rouder and Lu (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' As the competing MPT model, we specify a hierarchical latent- trait two-high-threshold model (Klauer, 2010), which, in contrast to the SDT model, explicitly models correlations between its parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We follow the convention of re- stricting the parameters that describe the probability of recognizing a previously presented stimulus as old and a distractor stimulus as new to be equal, DO = DN, to render the MPT model identifiable (Erdfelder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Singmann & Kellen, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Our prior choices for the pa- rameters of both models are described in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We train the neural network for 30, 000 training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' As in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1, we first leverage the amortization prop- erty of our method to inspect its calibration for the cur- rent model comparison task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 6a shows that the trained neural network generates well-calibrated PMP ap- proximations (median ECE over 25 repetitions of � ECE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Next, we assess whether the observed calibration of the network translates into a competitive performance relative to bridge sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The benchmarking setup (50 simu- lated data sets from each model) and the implementation of the bridge sampling workflow follow the procedure de- scribed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The classification metrics depicted in Table 2 reveal the excellent performance of both methods, despite the chal- lenging BMC scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We further observe a high degree of convergence between approximate PMPs derived by the two methods (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' As depicted in Figure 6c, obtaining PMP approximations for the 100 test data sets took more than 6 hours for bridge sampling and 36 min- utes for the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For this comparison of more complex cognitive models than in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2, the amor- tization advantage of our method emerges when analyz- ing 10 or more data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Note that this advantage would quickly show up in validation studies involving multiple 14 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 Predicted probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 True probability Median 50% CI 95% CI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 p(MPT|x) - bridge sampling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 p(MPT|x) - neural network Simulated from SDT Simulated from MPT 20 40 60 80 100 Number of data sets 0 50 100 150 200 250 300 350 Computation time in minutes Bridge sampling Neural network (a) Calibration of the neural network over 25 repetitions with 5, 000 data sets each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (b) Convergence of approximate PMPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (c) Computation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 6: Validation study 2: Results for the comparison between hierarchical SDT and MPT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Table 2: Validation study 2: Performance metrics for the comparison between hierarchical SDT and MPT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Accuracy MAE RMSE Log Score SBC Bridge sampling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='95 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='02) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='02) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='22 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='16 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='04) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='01 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='04) Neural network 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='95 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='02) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='02) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='21 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='03) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='16 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='04) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='00 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='04) Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Bootstrapped mean values and standard errors (in parentheses) are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We use 1000 bootstrap versions of the test data sets and estimate the standard errors from the bootstrap standard deviations of the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' model re-fits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', bootstrap, sensitivity analysis or cross- validation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The converging results from the two validation stud- ies demonstrate that our neural method generates well- calibrated and accurate PMP approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Despite our method only accessing the likelihood function indirectly via simulations, it can successfully compete with bridge sampling, which has direct access to the likelihood func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3 Application: Hierarchical Evidence Accumulation Models In the following, we showcase the utility of our method by comparing complex hierarchical evidence accumulation models (EAMs) in a real-data situation where likelihood- based methods such as bridge sampling would not be ap- plicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' More precisely, we seek to test the explana- tory power of different stochastic diffusion model formu- lations proposed by Voss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2019) for experimental response times data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The so-called L´evy flight model increases the flexibility of the standard Wiener diffusion model (Ratcliff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2016) but renders its likelihood function intractable with standard numerical approximations (Voss & Voss, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The complete incorporation of all information through hi- erarchical modeling and the realization of BMC has con- sequently been infeasible so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thus, in a recent study, Wieschen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2020) had to resort to a separate com- putation of the Bayesian Information Criterion (BIC) for each participant with subsequent aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We aim to extend the study of Wieschen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2020) by comparing fully hierarchical EAMs through PMPs and BFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' More- over, we intend to answer the question formulated by Wi- eschen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020 as to whether the superior performance of the more complex models in their study stems from an insufficient punishment of model flexibility by the BIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In addition to addressing a substantive research question in this application, we also demonstrate multiple advantages of our deep learning method on empirical data: Compare HMs with intractable likelihoods: As our method is simulation-based, including models with intractable likelihood functions in the comparison set does not alter its feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Adequately model nested data: Our method allevi- ates computational challenges that prevent modelers from adequately capturing the information contained in nested data structures through HMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 15 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS Re-use trained networks via fine-tuning: We acceler- ate the training of our neural network by pre-training it on less complex simulated data and subsequently fine-tuning it on simulated data resembling the ac- tual experimental setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Handle missing data: We train a neural network that can handle varying amounts of missing data by ran- domly masking simulated data during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Validate a trained network on simulated data: The amortized nature of our method allows for extensive validation of a trained network prior to its application to empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1 Model specification For this application, we consider a L´evy flight model with non-Gaussian noise (Voss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The L´evy flight process is driven by the following stochastic ordinary dif- ferential equation (SDE): dx = v dt + σdξ (33) ξ ∼ AlphaStable(α, µ = 0, σ, β = 0), (34) which represents a L´evy walk characterized by a fat-tailed stable noise distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In the above equation, x de- notes the accumulated (perceptual) evidence, v denotes the rate of accumulation and α controls the tail exponent of the noise variate ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Voss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2019) and Wieschen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2020) argue that the more abrupt changes in the information accumulation process that this model allows for could provide a better description of human decision- making than a Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The addition of L´evy noise renders the standard numerical approximation of the dif- fusion model likelihood intractable (Voss & Voss, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Consequently, neither standard MCMC nor bridge sam- pling are applicable for Bayesian parameter estimation and BMC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' There is an ongoing debate about the inclusion of addi- tional parameters that account for inter-trial variability in the diffusion model parameters: While they can provide a better model fit, the estimation of inter-trial variabil- ity parameters is often difficult and can result in unsta- ble results (Boehm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Lerche & Voss, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thus, Wieschen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2020) also compared basic (with- out inter-trial variability parameters) to full (with inter- trial variability parameters) versions of the drift-diffusion and L´evy flight model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Consequently, the set of candidate models considered here consists of four EAMs with increasing flexibility (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', the scope of possible data patterns that they can generate): M1, the most parsimonious basic diffusion model with the parameter v describing the mean rate of information uptake, the parameter a describing the threshold at which a decision is made, the parameter zr describing a bias of the starting point towards one decision alternative and the parameter t0 describing the non-decision time, that is, the time spent encod- ing the stimulus and executing the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' M2, the basic L´evy flight model in which the as- sumption of a Wiener diffusion process with Gaus- sian noise is replaced by the above introduced L´evy flight process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The additional free parameter α de- notes the heaviness of the noise distribution’s tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Note that α = 2 equals a Gaussian distribution, while α = 1 describes a Cauchy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' M3, the full diffusion model, which extends M1 with the parameters sv, sz and st that denote the vari- ability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', standard deviations) of drift rate, starting point bias and non-decision time, respectively, be- tween trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' M4, the full L´evy flight model that possesses the largest flexibility by including inter-trial variability parameters as well as the flexible L´evy noise distri- bution controlled by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Parameter priors and prior predictive checks are provided in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 Data The reanalyzed data set by Wieschen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2020) con- tains 40 participants who completed a total of 900 trials of binary decision tasks (color discrimination and lexical decision) each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' On average, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='17% of trials per partici- pant were excluded due to extremely short or long reac- tion times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3 Simulation-based training Since simulating data from EAMs can be challenging, es- pecially when they include non-Gaussian noise, we lever- age the advantage that neural networks are capable of transfer learning as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Transfer learning describes the utilization of representations that had been previously learned by a neural network in a par- ticular task for a new, related task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In this way, neural networks can be applied in small data settings by re-using the training knowledge encoded from similar (possibly big data) settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For the purpose of model comparison, we first pre-train the network for 20 epochs (passes over the whole training data) on 10, 000 simulated data sets per model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' These data 16 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 True probability d ECE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='003 M1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 d ECE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='003 M2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 Predicted probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 True probability d ECE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='010 M3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 Predicted probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 d ECE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='009 M4 M1 M2 M3 M4 Predicted model M1 M2 M3 M4 True model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 (a) Calibration curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (b) Confusion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 7: Application experiment: Validation results for the evidence accumulation models on 2, 000 simulated data sets per model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' sets resemble the empirical data in that they consist of 40 simulated participants, but differ in that the number of tri- als is reduced by a factor of 9 (100 instead of 900 trials per participant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Afterwards, we fine-tune the network for ad- ditional 30 epochs on 2, 000 simulated data sets per model that match the empirical data set with 40 simulated partic- ipants and 900 trials per participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thereby, we consid- erably reduce the computational demand of the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We further speed up the training phase by simu- lating all data prior to the training of the network in the high-performance programming language Julia (Bezan- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Pre-training took 60 minutes for the simulations and 39 minutes for training of the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Fine-tuning took 110 minutes for the simulations and 74 minutes for training of the networks, resulting in a total of 4 hours and 43 minutes for the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' To fully adapt the network to the characteristics of the em- pirical data, we also simulate missing data during fine- tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In each training epoch, we generate a random bi- nary mask f coding the simulated missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We sample the number of masked trials from a (discretized) normal distribution truncated between 1 and the number of trials, 900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The distributions’ mean and standard de- viation match the amount and variability of missing trials in the empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We then perform an element-wise multiplication ˜x = x ⊗ f and feed the “contaminated” data ˜x to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' This procedure results in a robust network that can process various proportions of missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We show the stability of our results even in the pres- ence of up to 25% missing data in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 Results Before applying our trained network to the empirical data, we validate it on 2, 000 simulated data sets per model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' First, the individual calibration curves in Figure 7a show excellent calibration for all models with � ECEs close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The calibration curves now consist of 10 instead of 15 in- tervals to obtain stable results despite the smaller amount of validation data sets per model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Second, we evaluate the accuracy of recovery and patterns of misclassification through the confusion matrix depicted in Figure 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The confusion matrix confirms that the excellent calibration of the network does not stem from chance performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' It also reveals that the selection of the “true” model be- comes more difficult with increasing model complexity, which is a direct consequence of the Occam’s razor prop- erty inherent in BMC (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Table 3 presents the model comparison results on the em- pirical data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Additionally, Figure 8 displays the model posteriors under different data perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We find lit- tle evidence for both the basic diffusion model M1 and the basic L´evy flight model M2, meaning that the addi- tional complexity of allowing parameters to vary between trials in M3 and M4 is outweighed by better model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Consistent with the results of Wieschen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2020), the 17 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS M1 M2 M3 M4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 Posterior model probability Bootstrapped trials M1 M2 M3 M4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 Bootstrapped participants M1 M2 M3 M4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 Leave-one-participant-out Figure 8: Application experiment: Model posteriors on the empirical data set with uncertainty under different data perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We use 100 bootstrap samples for the bootstrapped results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Table 3: Application experiment: Bayes factors (BFs) and posterior model probabilities (PMPs) on the empirical data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The preferred model is indicated by an asterisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' M1 M2 M3 M4 BFj4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='46e-07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='76e-04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='03 BF4j 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='85e+06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='62e+03 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='96 PMP 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='42e-07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='68e-04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='97 full L´evy flight model M4 explains the experimental data best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' While the magnitude and robustness of this advan- tage were unclear in Wieschen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2020) due to the indirect approach of calculating separate BIC values, we now obtain clear evidence of BF43 = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='96 for the full L´evy flight model M4 over the second best performing full diffusion model M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Remarkably, the strong evi- dence for the most complex model M4 occurs despite the strict penalization of prior-predictive flexibility in BMC and proves to be robust to multiple data perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 5 Discussion Nested data are ubiquitous in the quantitative sciences, including psychological and cognitive research (Farrell & Lewandowsky, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Yet, to avoid dealing with the complex dependencies resulting from these data, re- searchers often resort to simpler analyses, ignoring poten- tially important structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Hierarchical mod- els (HMs) provide a flexible way to represent the multi- level structure of nested data, but this flexibility can make Bayesian model comparison a daunting undertaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In this work, we proposed a powerful remedy to this prob- lem: Building on the BayesFlow framework (Radev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020), we developed a neural network architecture that enables approximate BMC for arbitrarily complex HMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' In two simulation studies, we showed that our deep learn- ing method is well-calibrated and performs as accurately as bridge sampling, which is the current state-of-the-art for comparing HMs with simple likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Moreover, in a subsequent real-data application, we compared the relatively new L´evy flight model with existing evidence accumulation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thus, we argue that our method is well-suited to enhance the applicability of (complex) HMs in psychological research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Below, we summarize the key properties and limitations of our method while also out- lining future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1 Amortized Inference Our method offloads the computational demands for com- paring HMs onto the training phase of a custom neural network, allowing for near real-time model comparison using the trained network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The resulting amortization of- fers several advantages over non-amortized methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' First, it enables thorough validation of a trained network on thousands of simulated data sets, allowing large-scale simulation-based diagnostics to become an integral part of the BMC workflow (Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Schad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Second, the trained and validated network can be used not only for point estimates of BFs or PMPs on em- pirical data but also for exploring the robustness of the results against multiple data perturbations, as showcased in our real-data application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Third, we demonstrated the feasibility of amortizing over variable data set sizes in our first validation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' This is particularly advantageous in the context of HMs since nested data sets often contain multiple exchangeable lev- els with variable sizes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', different numbers of clusters, participants and observations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Analyzing multiple hier- 18 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS archical data sets with variable sizes only requires a sin- gle network that has seen different data set sizes during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The same network could also be used for var- ious simulation studies, such as the challenging task of designing maximally informative experiments in a hierar- chical BMC setting (Heck & Erdfelder, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Myung & Pitt, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Lastly, we showed that researchers do not even need to consider all possible shapes of future data sets when train- ing such a network, as they can use transfer learning to efficiently adapt a trained network to a related setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Be- yond allowing more flexibility in reusing networks across experiments, researchers or even fields, transfer learning can also considerably reduce the computational demands associated with comparing complex HMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' As demon- strated in our real-data application, a network can be pre- trained on simulated data sets with reduced size and fine- tuned afterwards on sizes matching the empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 Independence From Explicit Likelihoods Unlike other popular methods for performing BMC on HMs, such as the Savage-Dickey density ratio or bridge sampling, our method is not constrained by the availabil- ity of an explicit likelihood function for all competing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' As long as the models in question can be imple- mented as simulators, the neural network can be trained to perform BMC on these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The value of such a method is evident, as it decouples the substantive task of model specification from concerns about the feasibility of estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Statistical models are instantiations of substantive knowl- edge or hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' As such, we argue that model speci- fication should not be unduly restricted by considerations of computational tractability – a sentiment that is closely related to what Haaf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2021) call the “specification- first-principle”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Our proposed deep learning method sat- isfies this principle, as model specification may be guided exclusively by substantive arguments with few concerns about tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thus, we believe that our method makes a contribution to the recent upsurge of innova- tive psychological models (Collins & Shenhav, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Ghaderi-Kangavari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Heathcote & Matzke, 2022) by allowing for an efficient assessment of their in- cremental value in a hierarchical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3 Limitations and Outlook One of the main challenges of approximate methods and, more broadly, statistical inference is ensuring the faithful- ness of the obtained results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The outlined possibilities for validating the network and examining the robustness of the results are important contributions of our method but come with open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Concerning the validation of the network, framing model comparison as a supervised learning problem allows us to draw from the rich litera- ture on classification performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Nevertheless, determining a “good-enough” score for an approximate BMC method remains challenging, as the optimally pos- sible performance is application-specific and usually un- known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Concerning the application of the network to empirical data, our robustness checks are a practical proxy for the stability of BMC results in a closed-world setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' How- ever, these checks cannot possibly capture the (lack of) absolute evidence for an HM: As a relative method, BMC may indicate that one model fits the data better than a set of competing models, but it does not provide any measure of how well (or poorly) the model itself approximates the underlying data-generating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' A promising direc- tion to address this limitation could be the combination of our method with the recently proposed meta-uncertainty framework for BMC (Schmitt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022), which can be greatly accelerated with amortized methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' This combi- nation could provide a principled delineation of different uncertainty sources, enabling the detection of model mis- specification cases where none of the competing HMs can explain the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Since BMC is a marginal likelihood (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', prior predictive) approach, the priors should be informed by scientific the- ory and will thus have a decisive influence on the results (Vanpaemel, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We do not intend to re-iterate the ongoing discussion about this property of BMC (Gronau & Wagenmakers, 2019a, 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Haaf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Ve- htari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2019), but want to highlight a specific dif- ficulty that arises for HMs: Parameter priors of an HM are connected via multilevel dependencies, increasing the risk that poor prior choices may dominate the final re- sults (for a recent discussion of this problem in cognitive modeling, see Sarafoglou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Therefore, prior predictive checks and prior sensitivity analyses become especially important when conducting BMC on compet- ing HMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' While transfer learning reduces the computa- tional demands of retraining a neural network for sensi- tivity analyses, another avenue for future research would be the amortization over different prior choices, enabling immediate prior sensitivity assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Finally, it should be noted that the version of our method explored here can only compare HMs assum- ing exchangeable data at each hierarchical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Al- though the majority of HMs in social science research follow this probabilistic symmetry, some researchers may want to compare non-exchangeable HMs, for example, to study within-person dynamics (Driver & Voelkle, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Lodewyckx et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Schumacher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For- 19 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS tunately, the modularity of our method allows easy adap- tation of the neural network architecture to handle non- exchangeable HMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' To compare hierarchical time se- ries models with temporal dependencies at the lowest level, for instance, the first invariant module could be ex- changed for a recurrent network, as proposed in Radev, D’Alessandro, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Thus, future research could extend and validate our method in BMC settings involving non-exchangeable HMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Acknowledgments The authors thank L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Schumacher for reading this article and providing helpful feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' LE and MS were supported by a grant from the Deutsche Forschungsgemeinschaft (DFG, German Research Foun- dation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' GRK 2277) to the research training group Statisti- cal Modeling in Psychology (SMiP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' LE was additionally supported by the Google Cloud Research Credits program with the award GCP19980904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' PCB was supported by the Deutsche Forschungsgemeinschaft under Germany’s Ex- cellence Strategy – EXC-2075 - 390740016 (the Stuttgart Cluster of Excellence SimTech).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' STR was supported by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy – EXC-2181 - 390900948 (the Hei- delberg Cluster of Excellence STRUCTURES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' References Abadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Agarwal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Barham, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Brevdo, E.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Gelman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Simpson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Carpenter, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', & B¨urkner, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Rank-normalization, fold- ing, and localization: An improved r for as- sessing convergence of mcmc (with discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Bayesian analysis, 16(2), 667–718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Vehtari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Simpson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Yao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', & Gelman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Limitations of “limitations of bayesian leave- one-out cross-validation for model selection”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Computational Brain & Behavior, 2(1), 22–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 23 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS Voss, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Lerche, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Mertens, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', & Voss, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Se- quential sampling models with variable bound- aries and non-normal noise: A comparison of six models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Psychonomic bulletin & review, 26(3), 813–832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Voss, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', & Voss, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Fast-dm: A free program for efficient diffusion model analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Behavior re- search methods, 39(4), 767–775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Wagenmakers, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Lodewyckx, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Kuriyal, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', & Gras- man, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Bayesian hypothesis testing for psychologists: A tutorial on the savage–dickey method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Cognitive psychology, 60(3), 158–189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Wiecki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Sofer, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', & Frank, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Hddm: Hierarchical bayesian estimation of the drift- diffusion model in python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Frontiers in Neuroin- formatics, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Wieschen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Voss, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', & Radev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Jump- ing to conclusion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' a l´evy flight model of decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The Quantitative Methods for Psychol- ogy, 16(2), 120–132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Zaheer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Kottur, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Ravanbakhsh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Poczos, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', Salakhutdinov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', & Smola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Deep sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Advances in neural information pro- cessing systems, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 24 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS Appendix A Neural network implementation and training The neural networks were implemented in the Python li- brary TensorFlow (Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2015) and jointly opti- mized via backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' During training, we use mini- batch gradient descent with batches of size B = 32 per backpropagation update (training step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We employ the Adam optimizer (Kingma & Ba, 2015) with an initial learning rate of 5e-4 and a cosine decay schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For all validation studies, we use online training, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' simulate new training data sets flexibly right before each training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For the application study, we simulate all data sets efficiently a priori in the Julia programming language and therefore use offline training, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' training with a predeter- mined amount of data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We use the following neural network architectures for all experiments: The hierarchical summary network is com- posed of two deep invariant modules, each consisting of K = 2 equivariant modules followed by an invariant module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The first deep invariant module reduces the infor- mation within each group to vectors �xm of size 32, while the second deep invariant module reduces the information between the groups to a single vector z of size 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The inference network is realized through a standard feedfor- ward network with two hidden layers and the number of output units equaling the number of competing HMs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We did not conduct a thorough search for optimal hyper- parameter settings of the neural networks and the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' B Validation study 1 details Figure 9 displays the log BFs approximated by bridge sampling and the neural network variants for all 100 test data sets, including those 27 data sets for which bridge sampling approximated a BF > 1, 000, 000 and that were therefore excluded in Figure 5 for visibility purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' C Validation study 2 details Here, we provide details on our model specifications and prior choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We reformulate the observation-level struc- ture of the MPT model as a binomial instead of a multino- mial process to obtain identical response generation im- plementations for both models xh mn ∼ Bernoulli(hm) for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , Nm (35) xf mn ∼ Bernoulli(fm) for n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' , Nm, (36) where hm denotes the probability of detecting an old item as old (”hit”) and fm denotes the probability of detecting a new item as old (”false alarm”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The generating processes of these probabilities with our distributional choices are described in Tables 4 and 5 for the SDT model and Tables 6 and 7 for the MPT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 10 shows the prior predictive patterns of hit rates and false alarm rates arising from 5, 000 simulated data sets for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' D Application study details D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1 Application experiment: Parameter priors and prior predictive checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We base our priors upon the comprehensive collection of diffusion model parameter estimates by Tran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For the L´evy flight models, M2 and M4, we inform the prior on the additional α parameter by the estimates for comparable tasks (those completed under speed instruc- tions) in Voss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' For the inter-trial variability parameters included in M3 and M4, we follow the non- hierarchical priors that Wiecki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=', 2013 suggest to use in hierarchical drift-diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Table 8 contains the hyperprior choices and table 9 the group-level priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' To ensure that the informed priors for our HMs accurately reflect prior knowledge at both levels, we conduct prior predictive checks based on 10, 000 simulations (displayed in Figures 11, 12 and 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 Robustness against artificial noise Here, we inspect the stability of our neural network against additional noise injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 14 displays the model comparison results as increasing percentages of tri- als per participant are artificially masked as missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' We repeat the random masking of trials 100 times per per- centage step to assess the sensitivity of the results to spe- cific parts of the empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The low variability indi- cated by the thin-shaded areas means that our model com- parison results do not depend on a specific subset of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Further, consistent with our model comparison re- sults, the PMPs of M1 and M2 are indistinguishable in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Despite our network being trained on the em- pirical amount of missing data, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='17% over both tasks, we observe robust model comparison results for up to 25% missing data per participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 25 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS 0 20 40 60 80 log(BF21) - bridge sampling 0 20 40 60 80 log(BF21) - fixed network Simulated from M1 Simulated from M2 0 20 40 60 80 log(BF21) - bridge sampling 0 20 40 60 80 log(BF21) - variable network Simulated from M1 Simulated from M2 Figure 9: Validation study 1: Full comparison results for the log Bayes factors (all 100 test data sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Table 4: Validation study 2: Hyperprior distributions of the SDT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Parameter Symbol Prior distribution Probit-transformed hit probability µh′ Normal(1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5) σh′ Gamma(1, 1) Probit-transformed false alarm probability µf ′ Normal(−1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5) σf ′ Gamma(1, 1) Table 5: Validation study 2: Group-level prior distributions and transformations of the SDT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Parameter Symbol Prior distribution / transformation Probit-transformed hit probability h′ m Normal(µh′, σh′) Probit-transformed false alarm probability f ′ m Normal(µf ′, σf ′) Hit probability hm Φ(h′ m) False alarm probability fm Φ(f ′ m) Table 6: Validation study 2: Hyperprior distributions and transformations of the MPT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Parameter Symbol Prior distribution / transformation Probit-transformed recognition probability hd′ Normal(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='25) Probit-transformed guessing probability hg′ Normal(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='25) Covariance matrix λd′ Uniform(0, 2) λg′ Uniform(0, 2) Q InvWishart(3, I) Σ Diag(λd′, λg′) Q Diag(λd′, λg′) 26 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS Table 7: Validation study 2: Group-level prior distributions and transformations of the MPT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Parameter Symbol Prior distribution / transformation Probit-transformed recognition probability d′ m Normal ��µd′ µg′ � , Σ � Probit-transformed guessing probability g′ m Recognition probability dm Φ(d′ m) Guessing probability gm Φ(g′ m) Hit probability hm dm + (1 − dm) ∗ gm False alarm probability fm (1 − dm) ∗ gm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0 10000 20000 30000 40000 50000 Hit rates 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0 10000 20000 30000 40000 50000 False alarm rates Hierarchical SDT Model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0 5000 10000 15000 20000 25000 30000 35000 40000 Hit rates 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0 5000 10000 15000 20000 25000 30000 35000 40000 False alarm rates Hierarchical MPT Model Figure 10: Validation study 2: Prior predictive checks for the SDT and the MPT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The green vertical lines indicate the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 27 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS Table 8: Application experiment: Hyperprior distributions of the evidence accumulation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Parameter Symbol Prior distribution Threshold separation µa Normal(5, 1) σa Normal+(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='15) Relative starting point µzr Normal(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='25) σzr Normal+(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='05) Drift rate for blue/non-word stimuli µv0 Normal(5, 1) σv0 Normal+(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='25) Drift rate for orange/word stimuli µv1 Normal(5, 1) σv1 Normal+(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='25) Non-decision time µt0 Normal(5, 1) σt0 Normal+(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='05) Stability parameter of the noise distribution µα Normal(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='65, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='15) σα Normal+(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='1) Table 9: Application experiment: Group-level prior distributions of the evidence accumulation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Parameter Symbol Prior distribution Threshold separation am Gamma(µa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' σa) Relative starting point zrm invlogit(Normal(µzr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' σzr)) Drift rate for blue/non-word stimuli v0m Gamma(µv0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' σv0) Drift rate for orange/word stimuli v1m Gamma(µv1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' σv1) Non-decision time t0m Gamma(µt0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' σt0) Stability parameter of the noise distribution αm TruncatedNormal(µα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' σα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 2) Inter-trial variability of starting point sz Beta(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 3) Inter-trial variability of drift sv Normal+(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 2) Inter-trial variability of non-decision time st Normal+(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='3) 28 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS Figure 11: Application experiment: Prior predictive checks for the hyperpriors in the comparison of evidence accu- mulation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The green vertical lines indicate the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 12: Application experiment: Prior predictive checks for the hierarchical group-level priors in the comparison of evidence accumulation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The green vertical lines indicate the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' Figure 13: Application experiment: Prior predictive checks for the non-hierarchical group-level priors in the compar- ison of evidence accumulation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The green vertical lines indicate the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 29 A DEEP LEARNING METHOD FOR COMPARING BAYESIAN HIERARCHICAL MODELS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='40 Percentage of missing data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content='0 Posterior model probability M1 M2 M3 M4 Figure 14: Application experiment: Robustness of the model comparison results against artificially injected random noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' The lines represent the average probabilities of 100 repetitions per percentage step (in each repetition masking a random subset of the data), while the shaded areas indicate the standard deviation between these repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} +page_content=' 30' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FKT4oBgHgl3EQfqS4B/content/2301.11873v1.pdf'} diff --git a/d9AzT4oBgHgl3EQfn_2x/content/tmp_files/2301.01590v1.pdf.txt b/d9AzT4oBgHgl3EQfn_2x/content/tmp_files/2301.01590v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ee094fd29e16003161bcfe5f45420252a0bb19e --- /dev/null +++ b/d9AzT4oBgHgl3EQfn_2x/content/tmp_files/2301.01590v1.pdf.txt @@ -0,0 +1,1114 @@ +FATE in AI: Towards Algorithmic Inclusivity and Accessibility +Isa Inuwa-Dutse +University of Huddersfield, UK +i.inuwa-dutse@hud.ac.uk +Abstract +One of the defining phenomena in this age is the widespread deployment of systems powered +by artificial intelligence (AI) technology. With AI taking the center stage, many sections of +society are being affected directly or indirectly by algorithmic decisions. Algorithmic decisions +carry both economical and personal implications which have brought about the issues of fairness, +accountability, transparency and ethics (FATE) in AI geared towards addressing algorithmic +disparities. Ethical AI deals with incorporating moral behaviour to avoid encoding bias in AI’s +decisions. However, the present discourse on such critical issues is being shaped by the more +economically developed countries (MEDC), which raises concerns regarding neglecting local +knowledge, cultural pluralism and global fairness. This study builds upon existing research on +responsible AI, with a focus on areas in the Global South considered to be under-served vis-a-vis +AI. Our goal is two-fold (1) to assess FATE-related issues and the effectiveness of transparency +methods and (2) to proffer useful insights and stimulate action towards bridging the accessibility +and inclusivity gap in AI. Using ads data from online social networks, we designed a user study +(n = 43) to achieve the above goals. Among the findings from the study include: explanations +about decisions reached by the AI systems tend to be vague and less informative. To bridge the +accessibility and inclusivity gap, there is a need to engage with the community for the best way +to integrate fairness, accountability, transparency and ethics in AI. This will help in empowering +the affected community or individual to effectively probe and police the growing application of +AI-powered systems. +Keywords: AI fairness, explainable AI, FATE in AI, AI and Society, under-served communities +1. Introduction +Technological developments could inadvertently lead to individual and societal harm. With +the growing applications of systems powered by artificial intelligence (AI) technology, reliance +on the algorithmic decision could amplify discrimination and stereotype [1, 2]. For instance, +some of the past instances of bias and discrimination regarding the use of AI include applications +in domains such as in court decision [1], job hiring1, online ads2, and credit worthiness rating. +Algorithmic decisions carry both economical and personal implications for the individual, +which have brought about the issues of fairness, accountability, transparency and ethics (FATE) +in AI [3, 4], especially in high-stake domains [1, 5, 6, 7, 8, 9, 10]. Ethics is about making +1https://reut.rs/2UghQQS +2https://wapo.st/3FkU3IN +arXiv:2301.01590v1 [cs.CY] 3 Jan 2023 + +choices based on concepts of right and wrong, duty and obligation, and FATE in AI is geared +towards addressing societal challenges brought about by digital systems. However, the present +discourse on FATE-related issues is being shaped by the more economically developed countries +(MEDC), which raises concerns regarding neglecting local knowledge, cultural pluralism and +global fairness [11]. As AI systems continue to be weaved into multiple types of products +[7, 8, 12, 10, 13, 14], AI technology is a major driver for the Fourth Industrial Revolution and +transformation. With AI taking the center stage, it is crucial to have an understanding of the +FATE-related concerns and needs of various types of communities. Because of the diverse +audience affected by AI technology, ensuring effective transparency cannot be monolithic [15] +or dominated by certain viewpoints [11]; such viewpoint can disproportionately affect different +communities [16]. Therefore, there is a need for more contextualised and interdisciplinary +research to underscore best practices that can inform algorithmic fairness and transparency +[17, 18, 19]. Thus high regard for diversity and sociodemographics should be taken into account +in the design and governance of algorithms that affect the public [20]. One of the most effective +approaches is to involve the affected public, and the AI developers to incorporate community- +specific FATE needs. Relevant stakeholders within the community will ensure better policing of +AI’s operations. Adhering to social values is a core requirement for AI practitioners to ensure +algorithmic fairness for public good [21]. Through cooperative, inclusive, and community-led +design of AI applications, algorithmic disparities could be addressed effectively. +As the most populous country in Africa, we take a community of online users in Nigeria +from the Global South as a case study to examine aspects of FATE in AI as viewed by the public. +Nigeria is chosen due to its population and the deployment of AI-powered products and services +is on the rise. Moreover, the country is ranked 8th for the global Internet users [22], thus, setting +the pace for a vibrant AI workforce in Africa. Focusing on areas considered to be under-served +vis-a-vis AI, this study builds upon existing research on responsible AI (1) to assess FATE-related +issues and the effectiveness of transparency methods and (2) to offer some insights that will +stimulate action towards bridging the accessibility and inclusivity gap in AI. Using ads data from +online social networks, we designed a user study (n = 43) to achieve the above goals. To bridge +the accessibility and inclusivity gap, the study3 contributes the following: +- the study examines the prevailing issues in AI applications and how FATE in AI might +better serve in places not traditionally served by AI systems. +- we offer some recommendations on how to promote inclusivity and wider public access +towards addressing FATE-related challenges. +Leveraging the aforementioned contributions will bring within the purview of mainstream AI +discourse and research (in both academia and industry) to ensure an accessible and inclusive +AI ecosystem that is fairer to all and sundry. The endeavour will help in improving awareness, +privacy, democtratisation of AI systems and better distribution of the economic benefits from the +AI technology leading to positive pro-societal changes [24]. +3part of the result in this study was presented as a poster [23] +2 + +2. Background +Our approach in this study borrows from socio-technical disciplines to help in examining +FATE in AI issues. Thus, we review relevant literature in responsible AI, FATE in AI, and +human-computer interaction (HCI) disciplines. The novelty of our approach is involving the +affected communities towards the development of responsible and inclusive AI systems. +2.1. Algorithmic Fairness +Algorithms could also promote a form of discrimination and stereotype as seen in the past, +such as the compas system4 for assessing the likelihood of becoming a recidivist, Amazon’s hiring +process favouring male applicants over female5, and the Google’s jobs ad algorithm showing +high-paying jobs to men compared to women6. Algorithmic decisions are capable of reproducing +or amplifying disparities for many reasons. For instance, discrimination is often inherent because +the data used to train the AI model relied on past decisions which may have themselves been +biased and discriminatory [2]. As such, fairness, accountability, transparency and ethics in +AI are geared towards developing and ensuring responsible AI that will incorporate moral +behaviour and avoid encoding bias to AI’s decisions [25]. Ethics is about making choices based +on concepts of right and wrong, duty and obligation. Thus, it is possible to formulate a hierarchy +of goals that embody ethical concepts in digital systems [26]. Among the measures to tackle +algorithmic disparities include legislation, relevant policies and positive action play a crucial +role in improving access to opportunity [27]. In the Equality Act 2010 UK7, positive action +is rooted in the anti-discrimination legislative process to curtail the imbalance of opportunity +affecting individuals from under-represented communities. In the same vein, algorithmic fairness +is viewed through the lens of positive action to improve equal representation [28]. +Ethical frameworks such as the UNESCO’s recommendation on the Ethics of AI8, World +Economic Forum (WEF) and Global Future Council on Human Rights9 have been developed to +address rising concerns over human rights resulting from the use of AI systems. Also, oversight +and regulatory bodies such as the steering group of the European AI Alliance [29] are in place to +police AI’s operation and address concerns over human rights in the digital age [30]. Ensuring +fairness requires a critical look at how inclusive is the approach in factoring demographics and +local context in the development process. Through data generated leverage [31], the public +can exert a certain degree of power to tackle algorithmic unfairness by demanding changes or +neutralising societal power imbalances [9, 12, 25, 24, 32]. +2.2. Improving Algorithmic Experience +Earlier studies have pointed to the need for contextualised and interdisciplinary research to +underscore best practices that can inform algorithmic fairness and transparency [17, 18, 19]. The +principle of transparency is at the centre of ensuring ethical AI, and constitutes about 90% of the +discourse. However, there exist significant variations in terms of interpretation, justification, the +4https://bit.ly/3VJxEu3 +5https://reut.rs/2UghQQS +6https://wapo.st/3FkU3IN +7shorturl.at/cqr19 +8https://bit.ly/3XJoj7o +9https://bit.ly/3gQEwHx +3 + +domain of application, and mode of achievement [11]. In explainable AI (XAI), explanations +are crucial for humans to better understand AI systems [5, 33], and they offer an effective +interface for human-in-the-loop to identify and address algorithmic fairness issues [2, 34, 35]. +Explanations should be able to satisfy specific goals, expectations, needs, and demands regarding +AI systems. It is not always the case that the end-user must understand the decision process, +and explanations should be offered based on the person requesting it [36]. A user-centric +approach to providing explanations will facilitate the shift from the majorly developer-driven +explanations [37]. Moreover, an appropriate explanation about why an algorithm arrived at a +given recommendation should be able to satisfy the user’s curiosity and improve knowledge +about the technology [38]. Because of the diverse audience affected by AI technology, an +explanation cannot be monolithic and many factors need to be taken into consideration [15]. +Discourse on AI issues is heavily influenced by Western viewpoint due to the prevalence of data, +measurement scales, and legal and philosophical dimensions [16]. This led to most of the AI’s +path being shaped by its originating contexts in Western nations [39]. +Creating transparent, trustworthy and human-centric AI that is in line with ethical needs +requires input from diverse stakeholders. Thus high regard for diversity and sociodemographics +should be taken into account in the design and governance of algorithms that affect the public +[20]. Failure to incorporate local context or sociodemographic factors could lead to (in)advertent +or (un)intended discrimination. It is crucial to incorporate interdisciplinary approaches that +will be beneficial to various stakeholders’ desiderata and for evaluation, [40, 41, 19]. Relevant +approaches have been put forward within the human-computer interaction (HCI) research to find +better ways of improving the transparency of digital systems [42, 19, 43, 44]. For algorithmic +intuitiveness and improved transparency, the work of [42] focuses on design principles for +explanation interfaces that inform users about algorithmic decisions. Information about the inner +workings of the algorithm and interactivity are effective [42]. To better improve inclusivity and +leverage AI’s capability, the approach in [45] involves various stakeholders to sketch out plans +for AI applications that recognise local context or needs. +2.3. AI and Policy Response in Africa +AI technologies are increasingly intersecting with new user groups, applications, datasets, +and regulations [39]. AI in Africa is making inroads into the areas of governance, commercial +activities, education, public and private engagements and other societal activities. With the +increasing population size, many countries in the Global South are creating a huge amount of data +that is generating value and competitive advantage for numerous technology companies. With +the increasing application of AI techniques, there are growing calls for relevant laws, policies +and guidelines to regulate the use and application of AI [46, 47, 48, 49, 30, 50, 51]. Access to +representative and quality data is critical for ensuring responsible and ethical AI requirements. +The term data colonialism is associated with the commercialisation and weaponisation of +data facilitated by local and foreign AI models [52, 53, 54, 54]. Relying on foreign data +generation and processing tools has been criticised over privacy and data protection concerns +10. Many countries in Africa, such as Tunisia [55], Mauritius [56], and Botswana [57], strive to +incorporate strategies that meet national interest and facilitate AI development. Moreover, many +countries in the continent have enacted comprehensive data protection and privacy legislation +10see shorturl.at/dpNO1 +4 + +Figure 1: An overview of the user study (n = 43) to understand the degree of AI awareness and effectiveness +of explanation. The approach requires the participants to examine some recommendations and corresponding +explanations about the decision. This is followed by some broader questions about FATE in AI. +[58]. Organisations such as the African Development Bank offer AI-supported services across +some countries in Africa [59]. For a successful national AI strategy, there is a need to ensure +algorithmic accountability, data protection, and explainability of decision-making by AI models. +3. Study Design +To gather diverse perceptions and insights, our approach involves both qualitative and +quantitative methods. +- Online User Survey (n = 43) is conducted to collect relevant data and assess FATE- +related issues. Because online social networks are widely used and diverse users encounter +various ads controlled by algorithmic decision (AI), our case study involves online ad- +serving data and associated explanations from relevant sources. +- Post-survey Session follows the online user survey to gather feedback from some of the +participants. Some participants11 volunteered to take part in the further discussion. +Figure 1 shows an overview of the research process. Stemming from the aforementioned +methods, our focus will be on the following: +- exposure to AI: to examine the level of AI awareness and how the public view and interact +with AI-powered systems. +- explanation basis: central to explainable AI is for models to be able to explain how +a decision is reached [60]. For instance, the EU General Data Protection Regulation +(GDPR) requires organisations deploying AI systems to offer meaningful information to +the affected individuals about the systems’ output [29]. In the context of this work, we +will be examining the suitability of the explanations being offered to the users. +11a rule of thumb suggests that stable psychometric estimate requires 5-10 respondents [38] +5 + +Recruitment of +the Research +3.2 +Participants +Quantitative +Common use case +analysis +of Al deployment: +Online ad +3 User Study +recommendation +3.1 +- engagement with Al's +recommendation +- participants' asseement +Post-user study +sessionTable 1: Demographics of the research participants. Sec. Edu means the highest qualification is a secondary school +leaving certificate and Higher Inst. refers to other institutions of higher education. +Gender +Age +Digital Skill +Education +Employment +Female 27.9% +min. 18yrs +Satisfactory 12% +Sec. Edu 11.6% +Student 44.2% +Male 72.1% +max. 48yrs +Good 42% +Higher Inst. 11.6% +Self-employed 20.9% +— +– +Excellent 46% +BSc 62.8% +Full-time 25.6% +— +– +– +MSc 14% +Unspecified 9.3% +- concerns and needs: to understand public concerns over AI and their needs or expectation +of AI. +3.1. Online User Study +With the help of AI models, online platforms enable ad customisation making it possible +for two online users to be presented with different ads based on certain information about them. +Because the research participants come from diverse backgrounds and varying digital literacy, +we chose a common use case that will be easier to engage with. Noting how the online platforms +enable targeted marketing with the aid of AI, we utilise ads data and corresponding explanations +collected from Facebook12,Twitter13 and Instagram14 online social networks. For transparency +and privacy requirements, online social networks are required to present an explanation about +each recommendation or ad shown to their online users. Figure 4 shows some of the explanations +provided by the three online social platforms. The explanations used in the survey were left +unaltered in order to assess how relevant the explanations are and to understand the participants’ +perceptions about the transparency of the recommendation system. +3.1.1. Survey Data Collection +The survey was designed in Qualtrics15 questionnaire and participants recruitment using +Jinga, a local platform that enables recruitment of the research participants. A total of n = 43 +participants have been recruited for the study. The ethical data collection process has been +followed, and the respondents were compensated for their time. The scenario involves an AI- +generated ad recommendation and a corresponding explanation to describe the rationale behind +the recommendation. The participants were presented with a brief introduction about the task +and some examples to get started. The activity took about 10-15 minutes to complete. Table 1 +shows relevant demographic information about the research participants. +Post-survey Session. On completion of the online study, we engage some volunteers out of the +participants for further discussion around the following issues: +- awareness of algorithmic deployments and FATE-related issues: will enable us to access +how inclusive and accessible AI models are to the public. The inclusivity aspect addresses +or describes whether the explanations take into account the local context (demographics, +language, educational qualification, norm, etc). Accessibility deals with whether (it’s clear +users know the role of AI in the decision process). +12https://facebook.com/ +13https://twitter.com/ +14https://www.instagram.com/ +15https://www.qualtrics.com/uk/ +6 + +- context and sociodemographics: how best to harness perceptions from specific communi- +ties, often under-served by AI systems, to shape the field for positive societal impact? +- hindrance: what are the barriers to algorithmic awareness and what they considered to be +required or missing? +4. Participants’ Responses +After each recommended ad, the participants respond to some questions which are based on +the metrics provided in [38] to assess FATE-related issues. Table 2 shows the constructs and +sample questions answered by the participants. Using a Likert scale (5-1, with 5 denoting strong +agreement and 1 strong disagreement), the research participants report their agreements with +each of the statements in Table 2. +Figure 2: The aggregated ratings of the participants in response to knowledge about the AI, relevance of the AI’s +recommendation, satisfaction with the explanation and trust in the AI system based on educational qualification and +self-reported digital skill. +In Figure 2, the participants proclaiming ’excellent’ and ’good’ digital skills rated knowledge +about AI very high across all levels of educational qualification. Trust in the AI and satisfaction +with the explanation are relatively high. Essentially, satisfaction with the explanation is less under +the ’MSc’ category compared to the remaining educational qualification. Similarly, Figure 3 +shows that participants under the ’satisfactory’ digital skill are more willing to share data for an +improved and personalised recommendation. The need for transparency and privacy concerns +are higher across all groups. +4.1. Reliability Analysis +We use the following constructs based on participants’ responses to answer the following +research questions: +7 + +012345 +012345 +Excelent +Good +Satisfactory +Excelent +Good +Satisfactory +Sec. Edu. +口 +Sec. Edu. +MSc +自 +MSc +Higher Inst. +Higher Inst. +BSc +BSc +012345 +345 +0 +45 +KnowledgeaboutAl +RelevanceoftheAl'srecommendation +012345 +012345 +Excelent +Good +Satisfactory +Excelent +Good +Satisfactory +Sec. Edu. +Sec. Edu. +MSc +MSc +Higher Inst. +Higher Inst. +BSc +BSc +012345 +012345 +012345 +012345 +Satisfactionabouttherecommendation +Trust inthe AlFigure 3: The aggregated ratings of the participants in response to perception of the AI’s robustness, willingness to +share data for personalised service, need for transparency and privacy concerns based on educational qualification +and self-reported digital skill. +- awareness of AI’s role construct includes questions about general knowledge about AI and +its role recommendation services +- relevance of the recommendation construct relates to the relevance of the recommendation +to the user +- transparency of the explanation construct includes the transparency or how relatable the +explanation is to the user +- privacy concern construct includes privacy and willingness to share data for better recom- +mendation services +For reliability analysis, the response from the participants regarding the set of questions under +each construct should be correlated in some ways. The degree of such correlation is captured +using Cronbach’s alpha to measure the internal consistencies among the responses under each +construct [61]. Cronbach’s alpha is a measure of the internal consistency of a scale given by: +α = +N¯c +¯v + (N − 1)¯c +where N is the number of items, ¯v is the average variance and ¯c is the average inter-item +covariance between items. A value of α > 0.7 suggests that each experimental construct +is reliable and consistent. Table 2 reports the reliability analysis for each of the applicable +constructs. With the exception of the privacy concern construct, all the constructs are reliable. +The questions for the privacy concern constructs seem not reliable or consistent, which could be +due to the using two seemingly different questions (willingness to share data and trusting online +8 + +012345 +012345 +Excelent +Good +Satisfactory +Excellent +Good +Satisfactory +Sec. Edu. +Sec. Edu. +口二 +MSc +MSc +Higher Inst. +Higher Inst. +BSc +BSc +01 +0 +345 +2345 +2345 +Alrobustness +Willingnesstosharedata +012345 +2 +3 +4 +5 +Excelent +Good +Satisfactory +Excelent +Good +Satisfactory +Sec. Edu. +Sec. Edu. +MSc +MSc +Higher Inst. +Higher Inst. +BSco +口 +BSc +012345 +012345 +45 +Needforrecommendationtransparency +ConcernonprivacyTable 2: Survey questions and reliability analysis for all constructs in the study using Cronbach’s alpha (α). The +alpha value for all the constructs is 0.87 +Construct: Recommendation Relevance +Measure: +1. The above ad is relevant to me +2. The explanation is convincing +3. I want to know that I understand this AI system correctly +α = 0.70 +Construct: Awareness of AI’s role +Measure: +1. I want to know what AI is +2. I want to know what the AI would have done if something had +been different +3. I now have better understanding of AI +α = 0.66 +Construct: Need for Transparency +Measure: +1. I want to understand how the AI works +2. I want to know why the AI did not make some other decision +α = 0.78 +Construct: Privacy Concern +Measure: +1. I want to know that my information is safe with the social media +platforms +2. I want to share more information in order to get personalised +recommendations +α = 0.17 +social networks with data). In addition to the questions in Table 2, the participants chose from +the following three options: satisfactory, good and excellent to report their perceived digital skill +or literacy. +4.2. Variation in Responses +We are interested in determining whether there is any difference in the perception or rating +of algorithmic transparency and other FATE-related needs among the research participants based +on the following: +9 + +- using the self-reported digital skill to determine if there is any difference in the ratings +provided by the participants +- using the educational qualification to determine whether there is any difference in the +ratings of FATE-related issues +- using the source of the recommendation, i.e. the online social networks, to determine if +there is any difference in the ratings provided by the participants. +To explore any variation according to the above categorisations, we leverage the Mann- +Whitney-U test to determine how the participants’ responses vary across self-reported digital +skills, educational qualification, and the online social media platforms offering the ads and corre- +sponding explanations. Essentially, we put forward the following hypotheses for investigation: +- H0: there is no difference in the ranking of the variables by the participants across digital +skill, education and the source of the recommendations and corresponding explanations. +- H1: there is a difference in the ranking of the variables by the participants across digital +skill, education and the source of the recommendations and corresponding explanations. +Before proceeding with determining any difference between the relevant groups, we check +for normality in the data using the Shapiro-Wilk normality test. In Table 3, the p−value < 0.001 +for the individual variable indicates that the Mann-Whitney-U test can be applied since the data is +not normally distributed. For the self-reported digital skill, we focus on the differences between +’excellent’ and ’good’ skills because the number of samples under the satisfactory self-reported +digital skill is quite low (9 out of 77 instances). +Table 3: Reliability analysis for an individual item (n = 77 per variable) using Cronbach’s (α) without and with +item dropping. +Variable +Mean Value +α Without Item Dropping +α With Item Dropping +Normality Test +Relevance +3.3 +0.61 +0.87 +W = 0.84, p − value = 0.001 +Satisfaction +3.8 +0.72 +0.85 +W = 0.83, p − value = 0.001 +AI Knowledge +3.9 +0.77 +0.84 +W = 0.82, p − value = 0.001 +Transparency Need +4.0 +0.80 +0.84 +W = 0.78, p − value = 0.001 +Alternate Decision Need +3.8 +0.70 +0.85 +W = 0.83, p − value = 0.001 +Recommendation Correctness +4.0 +0.75 +0.85 +W = 0.82, p − value = 0.001 +Privacy Concern +4.3 +0.65 +0.85 +W = 0.70, p − value = 0.001 +To Share Data +3.5 +0.49 +0.86 +W = 0.85, p − value = 0.001 +AI Knowledge - Post +3.5 +0.60 +0.86 +W = 0.90, p − value = 0.001 +AI Robustness +3.7 +0.67 +0.85 +W = 0.86, p − value = 0.001 +Trust in AI +3.8 +0.54 +0.87 +W = 0.80, p − value = 0.001 +See Tables 5, 6 and 7 for details about the respective statistical results. Table 4 shows the +samples with significant differences according to the participants’ ratings. Digital skill plays +a role towards the participants’ rating of FATE in AI questions. High digital literacy means +the participants are accustomed to engaging with various AI-powered recommendations and +decisions compared to those with less exposure to digital systems. The difference is more +pronounced in the need for further transparency. Thus, we can conclude that there is a difference +in the ranking of the transparency variables by the participants based on digital skills. However, +there is no significant difference based on educational qualification and source of the ads and +corresponding explanations. There exists a significant difference in the ratings of satisfaction +10 + +Table 4: Mann-Whitney-U Test for samples with some differences according to self-reported digital skill, educational +qualification and online social network platforms. +Pair +Measure +Statistic +p − value +Remark +good/excellent +need for transparency +269.5 +0.001 +digital skill +good/excellent +need for alternate explanation +291 +0.003 +digital skill +MSc/BSc +satisfaction about the recommendation +130 +0.052 +education +MSc/Higher Inst. +satisfaction about the recommendation +0 +0.013 +education +BSc/Higher Inst. +need for alternate explanation +24 +0.048 +education +Fabcebook/Instagram +recommendation relevance +176 +0.017 +online platform +Twitter/Instagram +recommendation relevance +194 +0.045 +online platform +about the recommendation and the need for alternate decisions based on educational qualification. +Similarly, there is a difference in the rating of the recommendation relevance with respect to the +source of the recommendations and corresponding explanations. The recommendation offered +by Facebook and Twitter tends to be more relevant if compared with Instagram. +5. Towards Accessible and Inclusive AI +We offer our discussion according to the participants’ perceptions about FATE-related issues +and how to overcome or improve community-specific AI challenges. +5.1. Explanation Style +It is widely acknowledged that algorithms are meant to offer personalised content and +services that are relevant. Explanations are then provided to convince the affected people about +the recommendation process. In the same spirit, the explanation should be personalised to +serve the respective needs of the affected individuals or communities. The style of explanation +varies across online social networks (Figure 4). While the explanations associated with ads from +Facebook and Twitter offer similar explanation styles, Instagram offers a lengthy and generic +explanation. For instance, one of the participants (#P3) commented that the explanation tends to +contain too much information, it could have been simplified[sic]. Focusing on contextualised and +value-driven explanations will be useful in facilitating wider awareness creation and engagement +with AI technologies. Moreover, explainable AI should factor in the local context and take +into account the various social strata the AI deployment is reaching. As noted earlier, if +most users rely on local language for online engagements, then good explanations should +incorporate contextualised needs (demographics) that will promote awareness and enable users +better understand the technology they interact with. Regarding the self-reported digital skill, +participants with ’excellent’ digital skills show no significant changes in their assessment across +the evaluation metrics. From the other extreme, participants with ’satisfactory’ digital skills tend +to incline towards a strong agreement with the statements (see Figures 2 and 3). +5.2. Improving Awareness and AI Engagement +Because most of the discourse on ethical AI and FATE-related issues are being shaped by +the more economically advanced countries (MEDC) [11, 16], this kind of monolithic approach +could ignore local knowledge, cultural pluralism and global fairness [11]. Ensuring accessible +and inclusive AI technology will require (1) sound and inclusive policies from governments to +foster useful AI development and (2) technology companies to implement AI systems that will +11 + +operate within the remit of regulation, sociodemographics and other contextual knowledge. This +will require inputs from various stakeholders, which is in line with the need of involving public +actors to shape the technologies that affect them [62, 45]. Adhering to social values is a core +requirement for AI practitioners to ensure algorithmic fairness for public good [21]. Through +cooperative, inclusive, and community-led design of AI applications, algorithmic disparities +could be addressed effectively. +5.2.1. Community Involvement +Using biased and discriminatory training data to develop AI systems would reproduce or +amplify disparities [1, 2]. +There are regulations, legislation and policies at various levels geared towards fostering +useful and responsible AI development. A case in point include the UNESCO’s recommendation +on the Ethics of AI16, World Economic Forum (WEF) and Global Future Council on Human +Rights17, which have been developed to address rising concerns over human rights resulting from +the use of AI systems. One of the most effective approaches is to involve the community, and the +AI developers to incorporate community-specific FATE needs. Communities can be empowered +to shape the development of accessible and transparent AI systems [63]. Relevant stakeholders +within the community will ensure better policing of AI’s operations, and dictate norms, values +and other ethical requirements to be reflected in AI and its operation. Through positive action, +equal representation can be improved [27, 28]. In line with the theme of positive action and +data leverage, the public or community can dictate or help integrate local context and norms in +the development of AI systems. This is crucial since ethical AI deals with incorporating moral +behaviour to avoid encoding bias in AI’s decisions. At this critical juncture, it is essential to +explore ways by which community voice and power can inform how AI systems that affect them +should be developed. Accordingly, the following community-led initiative was proposed as a +way of engaging the public to help improve responsible AI and mitigate FATE-related concerns: +- document of concerns: is a publicly available document to share concerns or unethical +issues +- document of values: is a publicly available document to share norms and values that will +inform and shape FATE in AI. +Based on the alignment or discrepancies between the perceived AI’s operations and community +values, the public or community can demand change through various means. Within the more +economically developed countries, for instance, wider awareness about the value placed on data +often results in data strikes or limited engagement with online platforms resulting in disruption +of access to relevant online services [32]. This sort of online activism is due to wider awareness +about how such platforms value public data for their success [64]. Although unequal access to +data leverage is putting a certain section of society at a disadvantaged point [24], empowering +the community to exercise data leverage will be useful. The public can exert a certain degree of +influence via data leverage to demand better service [31]. The data can be leveraged to neutralise +societal power imbalances [9, 12, 25, 24, 32]. +16https://bit.ly/3XJoj7o +17https://bit.ly/3gQEwHx +12 + +6. Conclusion +As AI systems continue to be deployed across various domains [7, 8, 10, 13, 14], algorith- +mic decisions carry both economic and personal implications for the affected individuals or +communities. Failure to incorporate sociodemographic factors and neglecting viewpoints from +the affected communities will result in promoting what is being set to be avoided – bias and +unfair algorithmic decisions. Noting that one of the tenets of explainable AI is that models +should be able to explain how a decision is reached, this study presented a useful approach to +examine FATE in AI issues with emphasis on areas not traditionally served by AI systems. A +central motivation for this was the need to offer complementary/different perspectives from +the West-centric viewpoints on AI. Among the findings from the study include: explanations +about decisions reached by the AI systems tend to be vague and less informative. Creating +awareness and understanding of the best way to communicate with specific communities will +improve algorithmic accessibility and inclusivity. This will help in empowering the affected +community or individual to effectively probe and police the growing application of AI-powered +systems. Among the contributions from the study include ways of incorporating insights from +under-served communities and to open doors for further exploration of FATE-related issues. +Future work will involve engagement with stakeholders across various disciplines involving +rights activists, data economists, technologies/developers, researchers and policymakers to +improve diversity and equity in AI utilisation. +Further Information +Figure 4 shows some explanation types from three online social networks. +Table 5: Mann-Whitney-U Test to compare samples based on the participants’ self-reported digital skills. +Pair +Measure +Statistic +p − value +Remark +good/excellent +need for transparency +269.5 +0.001 +Some difference +good/excellent +need for alternate explanation +291 +0.003 +Some difference +good/excellent +satisfaction about the recommendation +458.5 +0.603 +No difference +References +[1] A. Julia, L. Jeff, M. Surya, K. Lauren, Machine bias: There’s software used across the +country to predict future criminals. and it’s biased against blacks, Online (2016). +[2] J. Dodge, Q. V. Liao, Y. Zhang, R. K. Bellamy, C. Dugan, +Explaining models: an +empirical study of how explanations impact fairness judgment, in: Proceedings of the 24th +International Conference on Intelligent User Interfaces, 2019, pp. 275–285. +[3] D. Shin, B. Zhong, F. A. Biocca, Beyond user experience: What constitutes algorithmic +experiences?, International Journal of Information Management 52 (2020) 102061. +[4] A. Ferrario, M. Loi, E. Viganò, In ai we trust incrementally: A multi-layer model of trust +to analyze human-artificial intelligence interactions, Philosophy & Technology 33 (2020) +523–539. +13 + +Figure 4: Explanation types from (A) Twitter (B) Facebook and (C) Instagram social networks. These examples are +used to describe the ads and corresponding explanations. +Table 6: Mann-Whitney-U Test to compare samples based on the participants’ educational qualifications +Pair +Variable +Statistic +p − value +Remark +MSc/BSc +satisfaction about the recommendation +130 +0.052 +Some difference +MSc/Higher Inst. +satisfaction about the recommendation +0 +0.013 +Some difference +BSc/Higher Inst. +satisfaction about the recommendation +31.5 +0.089 +Some difference +BSc/Sec. Edu. +satisfaction about the recommendation +281 +0.910 +Some difference +Higher Inst./Sec. Edu. +satisfaction about the recommendation +30 +0.069 +Some difference +MSc/BSc +need for transparency +196.5 +0.657 +Some difference +MSc/Higher Inst. +need for transparency +4.5 +0.088 +Some difference +MSc/Sec. Edu. +need for transparency +46.5 +0.597 +Some difference +BSc/Higher Inst. +need for transparency +36 +0.121 +Some difference +BSc/Sec. Edu. +need for transparency +285 +0.961 +Some difference +Higher Inst./Sec. Edu. +need for transparency +28.5 +0.110 +Some difference +MSc/BSc +need for alternate explanation +252 +0.420 +Some difference +MSc/Higher Inst. +need for alternate explanation +4.5 +0.086 +Some difference +MSc/Sec. Edu. +need for alternate explanation +49.5 +0.760 +Some difference +BSc/Higher Inst. +need for alternate explanation +24 +0.048 +Some difference +BSc/Sec. Edu. +need for alternate explanation +225 +0.230 +Some difference +Higher Inst./Sec. Edu. +need for alternate explanation +28.5 +0.110 +Some difference +[5] A. Adadi, M. Berrada, Peeking inside the black-box: a survey on explainable artificial +intelligence (xai), IEEE access 6 (2018) 52138–52160. +14 + +A +B +c +X +Why this ad +x + Only you can see this +AboutInstagramads +[ +Howads work on Instagram +HowdoesInstagramdecidewhich +adsto show me? +We want to show you ads from businesses +that are interesting and relevant to you, and to +One reason you might be seeing this ad is that +Why you're seeing this ad +do that we may use information about what +you do on Instagram and Facebook (our +you are part of or similar to an audience from +Harvard Business School Online wants to reach +parent company) as well as your activity on +Binance. There might be other reasons you're +people like you who may have: +third-party sites and apps you use. We also +seeing this ad, including that Binance wants to +may use information provided by businesses +reachpeopleabovetheageof18and located +Shown interest in Retail, Property and +outside of Instagram or Meta Company +here: Nigeria. +more +Products to decide which ads to show you. +Communicated in English (UK) or +For example, you might see ads based on the +You canviewandmanage information +English(Us) +people you follow and posts you like on +connected toyouraccountthatTwittermight +Instagram, your information and interests on +useforadspurposes.View your Twitter data. +Set their age to 21 and older +Facebook (if you have a Facebook account) +the websites and apps you visit, or information +Twitter also personalizes ads using +advertisers, their partners, and our marketing +information received from partners andyour +A primary location in Nigeria +partners share with us that they already have +app and website visits.You can control these +like your email address. +interest-based ads using the"Personalize +ads" setting. +What else influences your ads +Yourpersonalised ads maybebased on other +advertiser choices, your profile and activities +Learn more about: +such as websites that you visit and ads that you +interact with - as well as other information not +. Your ad topic preferences and how to +listed here. Learn more about how ads work +adjust ther +Was this explanation useful? +Yes +NoTable 7: Mann-Whitney test to compare samples based on the recommendation source and corresponding explana- +tions +Pair +Measure +Statistic +p − value +Remark +Facebook/Twitter +satisfaction about the recommendation +313 +0.609 +– +Facebook/Instagram +satisfaction about the recommendation +269.5 +0.706 +– +Twitter/Instagram +satisfaction about the recommendation +335.5 +0.312 +– +Facebook/Twitter +need for transparency +319 +0.503 +– +Facebook/Instagram +need for transparency +272 +0.727 +– +Twitter/Instagram +need for transparency +239 +0.284 +– +Facebook/Twitter +need for alternate explanation +278.5 +0.847 +– +Facebook/Instagram +need for alternate explanation +267 +0.658 +– +Twitter/Instagram +need for alternate explanation +273.5 +0.762 +– +Facebook/Twitter +recommendation relevance +288 +1.000 +– +Facebook/Instagram +recommendation relevance +176 +0.017 +some difference +Twitter/Instagram +recommendation relevance +194 +0.045 +some difference +[6] K. 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Welles, # HashtagActivism: Networks of race and gender +justice, Mit Press, 2020. +19 + diff --git a/d9AzT4oBgHgl3EQfn_2x/content/tmp_files/load_file.txt b/d9AzT4oBgHgl3EQfn_2x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c569544e71fd96b93132d4ed3764e5266354b7d4 --- /dev/null +++ b/d9AzT4oBgHgl3EQfn_2x/content/tmp_files/load_file.txt @@ -0,0 +1,840 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf,len=839 +page_content='FATE in AI: Towards Algorithmic Inclusivity and Accessibility Isa Inuwa-Dutse University of Huddersfield, UK i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='inuwa-dutse@hud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='uk Abstract One of the defining phenomena in this age is the widespread deployment of systems powered by artificial intelligence (AI) technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' With AI taking the center stage, many sections of society are being affected directly or indirectly by algorithmic decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Algorithmic decisions carry both economical and personal implications which have brought about the issues of fairness, accountability, transparency and ethics (FATE) in AI geared towards addressing algorithmic disparities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Ethical AI deals with incorporating moral behaviour to avoid encoding bias in AI’s decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' However, the present discourse on such critical issues is being shaped by the more economically developed countries (MEDC), which raises concerns regarding neglecting local knowledge, cultural pluralism and global fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' This study builds upon existing research on responsible AI, with a focus on areas in the Global South considered to be under-served vis-a-vis AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Our goal is two-fold (1) to assess FATE-related issues and the effectiveness of transparency methods and (2) to proffer useful insights and stimulate action towards bridging the accessibility and inclusivity gap in AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Using ads data from online social networks, we designed a user study (n = 43) to achieve the above goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Among the findings from the study include: explanations about decisions reached by the AI systems tend to be vague and less informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' To bridge the accessibility and inclusivity gap, there is a need to engage with the community for the best way to integrate fairness, accountability, transparency and ethics in AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' This will help in empowering the affected community or individual to effectively probe and police the growing application of AI-powered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Keywords: AI fairness, explainable AI, FATE in AI, AI and Society, under-served communities 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Introduction Technological developments could inadvertently lead to individual and societal harm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' With the growing applications of systems powered by artificial intelligence (AI) technology, reliance on the algorithmic decision could amplify discrimination and stereotype [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' For instance, some of the past instances of bias and discrimination regarding the use of AI include applications in domains such as in court decision [1], job hiring1, online ads2, and credit worthiness rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Algorithmic decisions carry both economical and personal implications for the individual, which have brought about the issues of fairness, accountability, transparency and ethics (FATE) in AI [3, 4], especially in high-stake domains [1, 5, 6, 7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Ethics is about making 1https://reut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='rs/2UghQQS 2https://wapo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='st/3FkU3IN arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='01590v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='CY] 3 Jan 2023 choices based on concepts of right and wrong, duty and obligation, and FATE in AI is geared towards addressing societal challenges brought about by digital systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' However, the present discourse on FATE-related issues is being shaped by the more economically developed countries (MEDC), which raises concerns regarding neglecting local knowledge, cultural pluralism and global fairness [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' As AI systems continue to be weaved into multiple types of products [7, 8, 12, 10, 13, 14], AI technology is a major driver for the Fourth Industrial Revolution and transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' With AI taking the center stage, it is crucial to have an understanding of the FATE-related concerns and needs of various types of communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Because of the diverse audience affected by AI technology, ensuring effective transparency cannot be monolithic [15] or dominated by certain viewpoints [11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' such viewpoint can disproportionately affect different communities [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Therefore, there is a need for more contextualised and interdisciplinary research to underscore best practices that can inform algorithmic fairness and transparency [17, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Thus high regard for diversity and sociodemographics should be taken into account in the design and governance of algorithms that affect the public [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' One of the most effective approaches is to involve the affected public, and the AI developers to incorporate community- specific FATE needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Relevant stakeholders within the community will ensure better policing of AI’s operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Adhering to social values is a core requirement for AI practitioners to ensure algorithmic fairness for public good [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Through cooperative, inclusive, and community-led design of AI applications, algorithmic disparities could be addressed effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' As the most populous country in Africa, we take a community of online users in Nigeria from the Global South as a case study to examine aspects of FATE in AI as viewed by the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Nigeria is chosen due to its population and the deployment of AI-powered products and services is on the rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Moreover, the country is ranked 8th for the global Internet users [22], thus, setting the pace for a vibrant AI workforce in Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Focusing on areas considered to be under-served vis-a-vis AI, this study builds upon existing research on responsible AI (1) to assess FATE-related issues and the effectiveness of transparency methods and (2) to offer some insights that will stimulate action towards bridging the accessibility and inclusivity gap in AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Using ads data from online social networks, we designed a user study (n = 43) to achieve the above goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' To bridge the accessibility and inclusivity gap, the study3 contributes the following: the study examines the prevailing issues in AI applications and how FATE in AI might better serve in places not traditionally served by AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' we offer some recommendations on how to promote inclusivity and wider public access towards addressing FATE-related challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Leveraging the aforementioned contributions will bring within the purview of mainstream AI discourse and research (in both academia and industry) to ensure an accessible and inclusive AI ecosystem that is fairer to all and sundry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The endeavour will help in improving awareness, privacy, democtratisation of AI systems and better distribution of the economic benefits from the AI technology leading to positive pro-societal changes [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 3part of the result in this study was presented as a poster [23] 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Background Our approach in this study borrows from socio-technical disciplines to help in examining FATE in AI issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Thus, we review relevant literature in responsible AI, FATE in AI, and human-computer interaction (HCI) disciplines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The novelty of our approach is involving the affected communities towards the development of responsible and inclusive AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Algorithmic Fairness Algorithms could also promote a form of discrimination and stereotype as seen in the past, such as the compas system4 for assessing the likelihood of becoming a recidivist, Amazon’s hiring process favouring male applicants over female5, and the Google’s jobs ad algorithm showing high-paying jobs to men compared to women6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Algorithmic decisions are capable of reproducing or amplifying disparities for many reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' For instance, discrimination is often inherent because the data used to train the AI model relied on past decisions which may have themselves been biased and discriminatory [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' As such, fairness, accountability, transparency and ethics in AI are geared towards developing and ensuring responsible AI that will incorporate moral behaviour and avoid encoding bias to AI’s decisions [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Ethics is about making choices based on concepts of right and wrong, duty and obligation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Thus, it is possible to formulate a hierarchy of goals that embody ethical concepts in digital systems [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Among the measures to tackle algorithmic disparities include legislation, relevant policies and positive action play a crucial role in improving access to opportunity [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' In the Equality Act 2010 UK7, positive action is rooted in the anti-discrimination legislative process to curtail the imbalance of opportunity affecting individuals from under-represented communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' In the same vein, algorithmic fairness is viewed through the lens of positive action to improve equal representation [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Ethical frameworks such as the UNESCO’s recommendation on the Ethics of AI8, World Economic Forum (WEF) and Global Future Council on Human Rights9 have been developed to address rising concerns over human rights resulting from the use of AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Also, oversight and regulatory bodies such as the steering group of the European AI Alliance [29] are in place to police AI’s operation and address concerns over human rights in the digital age [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Ensuring fairness requires a critical look at how inclusive is the approach in factoring demographics and local context in the development process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Through data generated leverage [31], the public can exert a certain degree of power to tackle algorithmic unfairness by demanding changes or neutralising societal power imbalances [9, 12, 25, 24, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Improving Algorithmic Experience Earlier studies have pointed to the need for contextualised and interdisciplinary research to underscore best practices that can inform algorithmic fairness and transparency [17, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The principle of transparency is at the centre of ensuring ethical AI, and constitutes about 90% of the discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' However, there exist significant variations in terms of interpretation, justification, the 4https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='ly/3VJxEu3 5https://reut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='rs/2UghQQS 6https://wapo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='st/3FkU3IN 7shorturl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='at/cqr19 8https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='ly/3XJoj7o 9https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='ly/3gQEwHx 3 domain of application, and mode of achievement [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' In explainable AI (XAI), explanations are crucial for humans to better understand AI systems [5, 33], and they offer an effective interface for human-in-the-loop to identify and address algorithmic fairness issues [2, 34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Explanations should be able to satisfy specific goals, expectations, needs, and demands regarding AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' It is not always the case that the end-user must understand the decision process, and explanations should be offered based on the person requesting it [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' A user-centric approach to providing explanations will facilitate the shift from the majorly developer-driven explanations [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Moreover, an appropriate explanation about why an algorithm arrived at a given recommendation should be able to satisfy the user’s curiosity and improve knowledge about the technology [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Because of the diverse audience affected by AI technology, an explanation cannot be monolithic and many factors need to be taken into consideration [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Discourse on AI issues is heavily influenced by Western viewpoint due to the prevalence of data, measurement scales, and legal and philosophical dimensions [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' This led to most of the AI’s path being shaped by its originating contexts in Western nations [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Creating transparent, trustworthy and human-centric AI that is in line with ethical needs requires input from diverse stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Thus high regard for diversity and sociodemographics should be taken into account in the design and governance of algorithms that affect the public [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Failure to incorporate local context or sociodemographic factors could lead to (in)advertent or (un)intended discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' It is crucial to incorporate interdisciplinary approaches that will be beneficial to various stakeholders’ desiderata and for evaluation, [40, 41, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Relevant approaches have been put forward within the human-computer interaction (HCI) research to find better ways of improving the transparency of digital systems [42, 19, 43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' For algorithmic intuitiveness and improved transparency, the work of [42] focuses on design principles for explanation interfaces that inform users about algorithmic decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Information about the inner workings of the algorithm and interactivity are effective [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' To better improve inclusivity and leverage AI’s capability, the approach in [45] involves various stakeholders to sketch out plans for AI applications that recognise local context or needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' AI and Policy Response in Africa AI technologies are increasingly intersecting with new user groups, applications, datasets, and regulations [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' AI in Africa is making inroads into the areas of governance, commercial activities, education, public and private engagements and other societal activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' With the increasing population size, many countries in the Global South are creating a huge amount of data that is generating value and competitive advantage for numerous technology companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' With the increasing application of AI techniques, there are growing calls for relevant laws, policies and guidelines to regulate the use and application of AI [46, 47, 48, 49, 30, 50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Access to representative and quality data is critical for ensuring responsible and ethical AI requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The term data colonialism is associated with the commercialisation and weaponisation of data facilitated by local and foreign AI models [52, 53, 54, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Relying on foreign data generation and processing tools has been criticised over privacy and data protection concerns 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Many countries in Africa, such as Tunisia [55], Mauritius [56], and Botswana [57], strive to incorporate strategies that meet national interest and facilitate AI development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Moreover, many countries in the continent have enacted comprehensive data protection and privacy legislation 10see shorturl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='at/dpNO1 4 Figure 1: An overview of the user study (n = 43) to understand the degree of AI awareness and effectiveness of explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The approach requires the participants to examine some recommendations and corresponding explanations about the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' This is followed by some broader questions about FATE in AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Organisations such as the African Development Bank offer AI-supported services across some countries in Africa [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' For a successful national AI strategy, there is a need to ensure algorithmic accountability, data protection, and explainability of decision-making by AI models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Study Design To gather diverse perceptions and insights, our approach involves both qualitative and quantitative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Online User Survey (n = 43) is conducted to collect relevant data and assess FATE- related issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Because online social networks are widely used and diverse users encounter various ads controlled by algorithmic decision (AI), our case study involves online ad- serving data and associated explanations from relevant sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Post-survey Session follows the online user survey to gather feedback from some of the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Some participants11 volunteered to take part in the further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Figure 1 shows an overview of the research process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Stemming from the aforementioned methods, our focus will be on the following: exposure to AI: to examine the level of AI awareness and how the public view and interact with AI-powered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' explanation basis: central to explainable AI is for models to be able to explain how a decision is reached [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' For instance, the EU General Data Protection Regulation (GDPR) requires organisations deploying AI systems to offer meaningful information to the affected individuals about the systems’ output [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' In the context of this work, we will be examining the suitability of the explanations being offered to the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 11a rule of thumb suggests that stable psychometric estimate requires 5-10 respondents [38] 5 Recruitment of the Research 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='2 Participants Quantitative Common use case analysis of Al deployment: Online ad 3 User Study recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content="1 engagement with Al's recommendation participants' asseement Post-user study sessionTable 1: Demographics of the research participants." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu means the highest qualification is a secondary school leaving certificate and Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' refers to other institutions of higher education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Gender Age Digital Skill Education Employment Female 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='9% min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 18yrs Satisfactory 12% Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='6% Student 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='2% Male 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='1% max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 48yrs Good 42% Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='6% Self-employed 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='9% — – Excellent 46% BSc 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='8% Full-time 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='6% — – – MSc 14% Unspecified 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='3% concerns and needs: to understand public concerns over AI and their needs or expectation of AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Online User Study With the help of AI models, online platforms enable ad customisation making it possible for two online users to be presented with different ads based on certain information about them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Because the research participants come from diverse backgrounds and varying digital literacy, we chose a common use case that will be easier to engage with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Noting how the online platforms enable targeted marketing with the aid of AI, we utilise ads data and corresponding explanations collected from Facebook12,Twitter13 and Instagram14 online social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' For transparency and privacy requirements, online social networks are required to present an explanation about each recommendation or ad shown to their online users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Figure 4 shows some of the explanations provided by the three online social platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The explanations used in the survey were left unaltered in order to assess how relevant the explanations are and to understand the participants’ perceptions about the transparency of the recommendation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Survey Data Collection The survey was designed in Qualtrics15 questionnaire and participants recruitment using Jinga, a local platform that enables recruitment of the research participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' A total of n = 43 participants have been recruited for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The ethical data collection process has been followed, and the respondents were compensated for their time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The scenario involves an AI- generated ad recommendation and a corresponding explanation to describe the rationale behind the recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The participants were presented with a brief introduction about the task and some examples to get started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The activity took about 10-15 minutes to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Table 1 shows relevant demographic information about the research participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Post-survey Session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' On completion of the online study, we engage some volunteers out of the participants for further discussion around the following issues: awareness of algorithmic deployments and FATE-related issues: will enable us to access how inclusive and accessible AI models are to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The inclusivity aspect addresses or describes whether the explanations take into account the local context (demographics, language, educational qualification, norm, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Accessibility deals with whether (it’s clear users know the role of AI in the decision process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 12https://facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='com/ 13https://twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='com/ 14https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='instagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='com/ 15https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='qualtrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='com/uk/ 6 context and sociodemographics: how best to harness perceptions from specific communi- ties, often under-served by AI systems, to shape the field for positive societal impact?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' hindrance: what are the barriers to algorithmic awareness and what they considered to be required or missing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Participants’ Responses After each recommended ad, the participants respond to some questions which are based on the metrics provided in [38] to assess FATE-related issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Table 2 shows the constructs and sample questions answered by the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Using a Likert scale (5-1, with 5 denoting strong agreement and 1 strong disagreement), the research participants report their agreements with each of the statements in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Figure 2: The aggregated ratings of the participants in response to knowledge about the AI, relevance of the AI’s recommendation, satisfaction with the explanation and trust in the AI system based on educational qualification and self-reported digital skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' In Figure 2, the participants proclaiming ’excellent’ and ’good’ digital skills rated knowledge about AI very high across all levels of educational qualification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Trust in the AI and satisfaction with the explanation are relatively high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Essentially, satisfaction with the explanation is less under the ’MSc’ category compared to the remaining educational qualification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Similarly, Figure 3 shows that participants under the ’satisfactory’ digital skill are more willing to share data for an improved and personalised recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The need for transparency and privacy concerns are higher across all groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Reliability Analysis We use the following constructs based on participants’ responses to answer the following research questions: 7 012345 012345 Excelent Good Satisfactory Excelent Good Satisfactory Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 口 Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' MSc 自 MSc Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=" BSc BSc 012345 345 0 45 KnowledgeaboutAl RelevanceoftheAl'srecommendation 012345 012345 Excelent Good Satisfactory Excelent Good Satisfactory Sec." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' MSc MSc Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' BSc BSc 012345 012345 012345 012345 Satisfactionabouttherecommendation Trust inthe AlFigure 3: The aggregated ratings of the participants in response to perception of the AI’s robustness, willingness to share data for personalised service, need for transparency and privacy concerns based on educational qualification and self-reported digital skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' awareness of AI’s role construct includes questions about general knowledge about AI and its role recommendation services relevance of the recommendation construct relates to the relevance of the recommendation to the user transparency of the explanation construct includes the transparency or how relatable the explanation is to the user privacy concern construct includes privacy and willingness to share data for better recom- mendation services For reliability analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' the response from the participants regarding the set of questions under each construct should be correlated in some ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The degree of such correlation is captured using Cronbach’s alpha to measure the internal consistencies among the responses under each construct [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Cronbach’s alpha is a measure of the internal consistency of a scale given by: α = N¯c ¯v + (N − 1)¯c where N is the number of items, ¯v is the average variance and ¯c is the average inter-item covariance between items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' A value of α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='7 suggests that each experimental construct is reliable and consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Table 2 reports the reliability analysis for each of the applicable constructs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' With the exception of the privacy concern construct, all the constructs are reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The questions for the privacy concern constructs seem not reliable or consistent, which could be due to the using two seemingly different questions (willingness to share data and trusting online 8 012345 012345 Excelent Good Satisfactory Excellent Good Satisfactory Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 口二 MSc MSc Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' BSc BSc 01 0 345 2345 2345 Alrobustness Willingnesstosharedata 012345 2 3 4 5 Excelent Good Satisfactory Excelent Good Satisfactory Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' MSc MSc Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' BSco 口 BSc 012345 012345 45 Needforrecommendationtransparency ConcernonprivacyTable 2: Survey questions and reliability analysis for all constructs in the study using Cronbach’s alpha (α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The alpha value for all the constructs is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='87 Construct: Recommendation Relevance Measure: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The above ad is relevant to me 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The explanation is convincing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' I want to know that I understand this AI system correctly α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='70 Construct: Awareness of AI’s role Measure: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' I want to know what AI is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' I want to know what the AI would have done if something had been different 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' I now have better understanding of AI α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='66 Construct: Need for Transparency Measure: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' I want to understand how the AI works 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' I want to know why the AI did not make some other decision α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='78 Construct: Privacy Concern Measure: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' I want to know that my information is safe with the social media platforms 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' I want to share more information in order to get personalised recommendations α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='17 social networks with data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' In addition to the questions in Table 2, the participants chose from the following three options: satisfactory, good and excellent to report their perceived digital skill or literacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Variation in Responses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='We are interested in determining whether there is any difference in the perception or rating ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='of algorithmic transparency and other FATE-related needs among the research participants based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='on the following: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='using the self-reported digital skill to determine if there is any difference in the ratings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='provided by the participants ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='using the educational qualification to determine whether there is any difference in the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='ratings of FATE-related issues ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='using the source of the recommendation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' the online social networks, to determine if there is any difference in the ratings provided by the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' To explore any variation according to the above categorisations, we leverage the Mann- Whitney-U test to determine how the participants’ responses vary across self-reported digital skills, educational qualification, and the online social media platforms offering the ads and corre- sponding explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Essentially, we put forward the following hypotheses for investigation: H0: there is no difference in the ranking of the variables by the participants across digital skill, education and the source of the recommendations and corresponding explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' H1: there is a difference in the ranking of the variables by the participants across digital skill, education and the source of the recommendations and corresponding explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Before proceeding with determining any difference between the relevant groups, we check for normality in the data using the Shapiro-Wilk normality test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' In Table 3, the p−value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='001 for the individual variable indicates that the Mann-Whitney-U test can be applied since the data is not normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' For the self-reported digital skill, we focus on the differences between ’excellent’ and ’good’ skills because the number of samples under the satisfactory self-reported digital skill is quite low (9 out of 77 instances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Table 3: Reliability analysis for an individual item (n = 77 per variable) using Cronbach’s (α) without and with item dropping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Variable Mean Value α Without Item Dropping α With Item Dropping Normality Test Relevance 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='87 W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='84, p − value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='001 Satisfaction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='85 W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='83, p − value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='001 AI Knowledge 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='84 W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='82, p − value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='001 Transparency Need 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='84 W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='78, p − value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='001 Alternate Decision Need 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='85 W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='83, p − value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='001 Recommendation Correctness 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='85 W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='82, p − value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='001 Privacy Concern 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='85 W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='70, p − value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='001 To Share Data 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='86 W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='85, p − value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='001 AI Knowledge - Post 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='86 W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='90, p − value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='001 AI Robustness 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='85 W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='86, p − value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='001 Trust in AI 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='87 W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='80, p − value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='001 See Tables 5, 6 and 7 for details about the respective statistical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Table 4 shows the samples with significant differences according to the participants’ ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Digital skill plays a role towards the participants’ rating of FATE in AI questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' High digital literacy means the participants are accustomed to engaging with various AI-powered recommendations and decisions compared to those with less exposure to digital systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The difference is more pronounced in the need for further transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Thus, we can conclude that there is a difference in the ranking of the transparency variables by the participants based on digital skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' However, there is no significant difference based on educational qualification and source of the ads and corresponding explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' There exists a significant difference in the ratings of satisfaction 10 Table 4: Mann-Whitney-U Test for samples with some differences according to self-reported digital skill, educational qualification and online social network platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Pair Measure Statistic p − value Remark good/excellent need for transparency 269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='001 digital skill good/excellent need for alternate explanation 291 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='003 digital skill MSc/BSc satisfaction about the recommendation 130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='052 education MSc/Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' satisfaction about the recommendation 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='013 education BSc/Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' need for alternate explanation 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='048 education Fabcebook/Instagram recommendation relevance 176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='017 online platform Twitter/Instagram recommendation relevance 194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='045 online platform about the recommendation and the need for alternate decisions based on educational qualification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Similarly, there is a difference in the rating of the recommendation relevance with respect to the source of the recommendations and corresponding explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The recommendation offered by Facebook and Twitter tends to be more relevant if compared with Instagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Towards Accessible and Inclusive AI We offer our discussion according to the participants’ perceptions about FATE-related issues and how to overcome or improve community-specific AI challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Explanation Style It is widely acknowledged that algorithms are meant to offer personalised content and services that are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Explanations are then provided to convince the affected people about the recommendation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' In the same spirit, the explanation should be personalised to serve the respective needs of the affected individuals or communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The style of explanation varies across online social networks (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' While the explanations associated with ads from Facebook and Twitter offer similar explanation styles, Instagram offers a lengthy and generic explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' For instance, one of the participants (#P3) commented that the explanation tends to contain too much information, it could have been simplified[sic].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Focusing on contextualised and value-driven explanations will be useful in facilitating wider awareness creation and engagement with AI technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Moreover, explainable AI should factor in the local context and take into account the various social strata the AI deployment is reaching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' As noted earlier, if most users rely on local language for online engagements, then good explanations should incorporate contextualised needs (demographics) that will promote awareness and enable users better understand the technology they interact with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Regarding the self-reported digital skill, participants with ’excellent’ digital skills show no significant changes in their assessment across the evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' From the other extreme, participants with ’satisfactory’ digital skills tend to incline towards a strong agreement with the statements (see Figures 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Improving Awareness and AI Engagement Because most of the discourse on ethical AI and FATE-related issues are being shaped by the more economically advanced countries (MEDC) [11, 16], this kind of monolithic approach could ignore local knowledge, cultural pluralism and global fairness [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Ensuring accessible and inclusive AI technology will require (1) sound and inclusive policies from governments to foster useful AI development and (2) technology companies to implement AI systems that will 11 operate within the remit of regulation, sociodemographics and other contextual knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' This will require inputs from various stakeholders, which is in line with the need of involving public actors to shape the technologies that affect them [62, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Adhering to social values is a core requirement for AI practitioners to ensure algorithmic fairness for public good [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Through cooperative, inclusive, and community-led design of AI applications, algorithmic disparities could be addressed effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Community Involvement Using biased and discriminatory training data to develop AI systems would reproduce or amplify disparities [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' There are regulations, legislation and policies at various levels geared towards fostering useful and responsible AI development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' A case in point include the UNESCO’s recommendation on the Ethics of AI16, World Economic Forum (WEF) and Global Future Council on Human Rights17, which have been developed to address rising concerns over human rights resulting from the use of AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' One of the most effective approaches is to involve the community, and the AI developers to incorporate community-specific FATE needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Communities can be empowered to shape the development of accessible and transparent AI systems [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Relevant stakeholders within the community will ensure better policing of AI’s operations, and dictate norms, values and other ethical requirements to be reflected in AI and its operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Through positive action, equal representation can be improved [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' In line with the theme of positive action and data leverage, the public or community can dictate or help integrate local context and norms in the development of AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' This is crucial since ethical AI deals with incorporating moral behaviour to avoid encoding bias in AI’s decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' At this critical juncture, it is essential to explore ways by which community voice and power can inform how AI systems that affect them should be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Accordingly, the following community-led initiative was proposed as a way of engaging the public to help improve responsible AI and mitigate FATE-related concerns: document of concerns: is a publicly available document to share concerns or unethical issues document of values: is a publicly available document to share norms and values that will inform and shape FATE in AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Based on the alignment or discrepancies between the perceived AI’s operations and community values, the public or community can demand change through various means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Within the more economically developed countries, for instance, wider awareness about the value placed on data often results in data strikes or limited engagement with online platforms resulting in disruption of access to relevant online services [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' This sort of online activism is due to wider awareness about how such platforms value public data for their success [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Although unequal access to data leverage is putting a certain section of society at a disadvantaged point [24], empowering the community to exercise data leverage will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The public can exert a certain degree of influence via data leverage to demand better service [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' The data can be leveraged to neutralise societal power imbalances [9, 12, 25, 24, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 16https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='ly/3XJoj7o 17https://bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='ly/3gQEwHx 12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Conclusion As AI systems continue to be deployed across various domains [7, 8, 10, 13, 14], algorith- mic decisions carry both economic and personal implications for the affected individuals or communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Failure to incorporate sociodemographic factors and neglecting viewpoints from the affected communities will result in promoting what is being set to be avoided – bias and unfair algorithmic decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Noting that one of the tenets of explainable AI is that models should be able to explain how a decision is reached, this study presented a useful approach to examine FATE in AI issues with emphasis on areas not traditionally served by AI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' A central motivation for this was the need to offer complementary/different perspectives from the West-centric viewpoints on AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Among the findings from the study include: explanations about decisions reached by the AI systems tend to be vague and less informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Creating awareness and understanding of the best way to communicate with specific communities will improve algorithmic accessibility and inclusivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' This will help in empowering the affected community or individual to effectively probe and police the growing application of AI-powered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Among the contributions from the study include ways of incorporating insights from under-served communities and to open doors for further exploration of FATE-related issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Future work will involve engagement with stakeholders across various disciplines involving rights activists, data economists, technologies/developers, researchers and policymakers to improve diversity and equity in AI utilisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Further Information Figure 4 shows some explanation types from three online social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Table 5: Mann-Whitney-U Test to compare samples based on the participants’ self-reported digital skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Pair Measure Statistic p − value Remark good/excellent need for transparency 269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='001 Some difference good/excellent need for alternate explanation 291 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='003 Some difference good/excellent satisfaction about the recommendation 458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Dugan, Explaining models: an empirical study of how explanations impact fairness judgment, in: Proceedings of the 24th International Conference on Intelligent User Interfaces, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 275–285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Shin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Zhong, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Biocca, Beyond user experience: What constitutes algorithmic experiences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=', International Journal of Information Management 52 (2020) 102061.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Ferrario, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Loi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Viganò, In ai we trust incrementally: A multi-layer model of trust to analyze human-artificial intelligence interactions, Philosophy & Technology 33 (2020) 523–539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 13 Figure 4: Explanation types from (A) Twitter (B) Facebook and (C) Instagram social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' These examples are used to describe the ads and corresponding explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Table 6: Mann-Whitney-U Test to compare samples based on the participants’ educational qualifications Pair Variable Statistic p − value Remark MSc/BSc satisfaction about the recommendation 130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='052 Some difference MSc/Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' satisfaction about the recommendation 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='013 Some difference BSc/Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' satisfaction about the recommendation 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='089 Some difference BSc/Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' satisfaction about the recommendation 281 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='910 Some difference Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='/Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' satisfaction about the recommendation 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='069 Some difference MSc/BSc need for transparency 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='657 Some difference MSc/Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' need for transparency 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='088 Some difference MSc/Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' need for transparency 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='597 Some difference BSc/Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' need for transparency 36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='121 Some difference BSc/Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' need for transparency 285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='961 Some difference Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='/Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' need for transparency 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='110 Some difference MSc/BSc need for alternate explanation 252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='420 Some difference MSc/Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' need for alternate explanation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='086 Some difference MSc/Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' need for alternate explanation 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='760 Some difference BSc/Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' need for alternate explanation 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='048 Some difference BSc/Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' need for alternate explanation 225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='230 Some difference Higher Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='/Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' need for alternate explanation 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='110 Some difference [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Adadi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Berrada, Peeking inside the black-box: a survey on explainable artificial intelligence (xai), IEEE access 6 (2018) 52138–52160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 14 A B c X Why this ad x Only you can see this AboutInstagramads [ Howads work on Instagram HowdoesInstagramdecidewhich adsto show me?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=" We want to show you ads from businesses that are interesting and relevant to you, and to One reason you might be seeing this ad is that Why you're seeing this ad do that we may use information about what you do on Instagram and Facebook (our you are part of or similar to an audience from Harvard Business School Online wants to reach parent company) as well as your activity on Binance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=" There might be other reasons you're people like you who may have: third-party sites and apps you use." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' We also seeing this ad, including that Binance wants to may use information provided by businesses reachpeopleabovetheageof18and located Shown interest in Retail, Property and outside of Instagram or Meta Company here: Nigeria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' more Products to decide which ads to show you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Communicated in English (UK) or For example, you might see ads based on the You canviewandmanage information English(Us) people you follow and posts you like on connected toyouraccountthatTwittermight Instagram, your information and interests on useforadspurposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='View your Twitter data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Set their age to 21 and older Facebook (if you have a Facebook account) the websites and apps you visit, or information Twitter also personalizes ads using advertisers, their partners, and our marketing information received from partners andyour A primary location in Nigeria partners share with us that they already have app and website visits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='You can control these like your email address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' interest-based ads using the"Personalize ads" setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' What else influences your ads Yourpersonalised ads maybebased on other advertiser choices, your profile and activities Learn more about: such as websites that you visit and ads that you interact with - as well as other information not .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Your ad topic preferences and how to listed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Learn more about how ads work adjust ther Was this explanation useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Yes NoTable 7: Mann-Whitney test to compare samples based on the recommendation source and corresponding explana- tions Pair Measure Statistic p − value Remark Facebook/Twitter satisfaction about the recommendation 313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='609 – Facebook/Instagram satisfaction about the recommendation 269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='706 – Twitter/Instagram satisfaction about the recommendation 335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='312 – Facebook/Twitter need for transparency 319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='503 – Facebook/Instagram need for transparency 272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='727 – Twitter/Instagram need for transparency 239 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='284 – Facebook/Twitter need for alternate explanation 278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='847 – Facebook/Instagram need for alternate explanation 267 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='658 – Twitter/Instagram need for alternate explanation 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='762 – Facebook/Twitter recommendation relevance 288 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='000 – Facebook/Instagram recommendation relevance 176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='017 some difference Twitter/Instagram recommendation relevance 194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content='045 some difference [6] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Ava, How facial recognition can ruin your life – the intercept, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Niklas, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Sztandar-Sztanderska, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' Szymielewicz, Profiling the unemployed in poland: Social and political implications (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} 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of race and gender justice, Mit Press, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9AzT4oBgHgl3EQfn_2x/content/2301.01590v1.pdf'} diff --git a/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf b/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..670d29149d2c21a6db723e23a8f3a5ed0f2f85a6 --- /dev/null +++ b/d9FRT4oBgHgl3EQfUTfv/content/2301.13536v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:076a4735a107d4868f857aec5adef6e8c8f7a22b7eab19cda9871f36938c3698 +size 2216551 diff --git a/d9FRT4oBgHgl3EQfUTfv/vector_store/index.faiss b/d9FRT4oBgHgl3EQfUTfv/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..533f5e18b8e82f515f38a20572c28d07ee65f750 --- /dev/null +++ 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China +Abstract +We investigate the 2D combustion model with Dirichlet boundary conditions and slip +boundary conditions in bounded domains. The global existence of weak and strong solutions +and the uniqueness of strong solutions are obtained provided the initial density is small in some +precise sense. Using the energy method and the estimates of boundary integrals, we obtain +the a priori bounds of the density and velocity field. In addition, we prove the local existence +of the strong solutions via iterative method and the contraction mapping theorem. Finally, we +extend the well-known Serrin’s blowup criterion to the 2D combustion model. Under the suit- +able boundary conditions, the Serrin’s condition on the velocity can be removed in this criteria. +Keywords: combustion model; Dirichlet boundary conditions; slip boundary conditions; +strong solutions; weak solutions; Serrin’s condition +1 +Intrduction and main results +In this paper, we will study the following combustion model: + + + + + +ρt + div(ρu) = 0, ρ ≥ 0, +(ρu)t + div(ρu ⊗ u) − div(2µD) + ∇π = 0, +div u = c0∆ψ(ρ), ψ(ρ) := ρ−1, +(1.1) +for t > 0 and x ∈ Ω, where Ω ⊂ R2 is a bounded simply connected domain with smooth boundary. +Here u = (u1, u2), ρ and π stand for the unknown velocity field, density and pressure respectively, +c0 > 0 is a fixed constant and +0 < µ = µ(s) ∈ C∞[0, ∞). +We denote +D = D(u) = 1 +2(∇u + (∇u)t) = 1 +2(∂iuj + ∂jui), +for 1 ≤ i, j ≤ 2. +From the physical viewpoint, combustion model is the low Mach number limit of the fully +compressible Navier-Stokes equations + + + + + +ρt + div(ρu) = 0, +(ρu)t + div(ρu ⊗ u) − div S + ∇p = 0, +(ρe)t + div(ρue) − div(k∇θ) + p div u = S : D(u), +(FCNS) +where e, θ, p stands for the internal enery, temperature and pressure respectively and A : B strands +for the inner product of matrices +A : B := tr(ABt) = +2 +� +i,j=1 +aijbji. +∗Email address: zhangjiawen@amss.ac.cn +1 + +S is the viscous strain tensor given by +S = 2µD(u) + λ div uIn×n, +where In×n is the n×n indentity matrix. The thermal conductivity k and the viscosity coefficients +µ, λ are functions of ρ and θ. From Lions’s book [27], if we define the Mach number ǫ as |u|/ +� +p′(ρ) +and let (ρ, u, θ) be smooth solution of system (FCNS) corresponding to the small ǫ, after rescaling +the time variable by +ρǫ(x, t) = ρ +� +x, t +ǫ +� +, +uǫ(x, t) = 1 +ǫ ρ +� +x, t +ǫ +� +, +θǫ(x, t) = θ +� +x, t +ǫ +� +, +then, (ρǫ, uǫ, θǫ) satisfies + + + + + +ρǫ +t + div(ρǫuǫ) = 0, +(ρǫuǫ)t + div(ρuǫ ⊗ uǫ) − div Sǫ + ǫ−2∇pǫ = 0, +(ρǫeǫ)t + div(ρǫuǫeǫ) − div(kǫ∇θǫ) + pǫ div uǫ = ǫ2Sǫ · D(uǫ), +(FCNS’) +where +Sǫ = 2µǫD(uǫ) + λǫ div uǫIn×n, +and +pǫ = p +� +x, t +ǫ +� +, +µǫ = 1 +ǫ µ +� +x, t +ǫ +� +, +λǫ = 1 +ǫ λ +� +x, t +ǫ +� +, +kǫ = 1 +ǫ k +� +x, t +ǫ +� +. +Considerig the ideal gas laws: +p = Rρθ, +e = CV θ, +(1.2) +where R, CV denote the ideal gas constant and the specific heat constant, respectively. Then, +letting the Mach number ǫ go to 0, the momentum equation (FCNS’)2 implies that +pǫ = P(t) + π(t, x)ǫ2 + o +� +ǫ2� +. +Plugging this formula into the energy equation (FCNS’)3 entails that P(t) is independent of t, pro- +vided uǫ and ∇θǫ vanish at infinity. From now on, we shall denote this constant by P0. Therefore, +denoting CP = γCV = γR/(γ − 1), the low Mach number limit system reads + + + + + +ρCP (∂tθ + u · ∇θ) − div(k∇θ) = 0, +ρut + ρu · ∇u − div S + ∇π = 0, +γP0 div u = (γ − 1) div(k∇θ). +(1.3) +Plugging (1.2) into (1.3) with constant heat conductivity coefficient k implies the following system + + + + + +∂tρ + div(ρu) = 0, +ρut + ρu · ∇u − div S + ∇π = 0, +div u = k(γ − 1)(Rγ)−1∆ρ−1, +(1.4) +which is exactly the equations (1.1). +If we particularly take the diffusion coefficient c0 = 0, (1.1) will become the classical non- +homogeneous incompressible Navier-Stokes equations + + + + + +ρt + div(ρu) = 0, ρ ≥ 0, +(ρu)t + div(ρu ⊗ u) − div(2µD) + ∇π = 0, +div u = 0. +(N) +The study of the combustion model may date back to the 1980s. It has been introduced by +A. Majda [29] and studied in particular by P. Embid [14] who has proved the local-in-time well- +posedness of the system (1.1). For the system (1.1) replacing (1.1)3 by Fick’s law with ψ(ρ) = log ρ, +the local well-posedness was considered by H. B. da Veiga [12]. +Danchin-Liao [13] established +2 + +the local existence and uniqueness of a solution in critical homogeneous Besov spaces provided +the density is closed to a positive constant and they proved the local well-posedness in non- +homogeneous Besov space arbitrarily large data. +On the other hand, there are also a large number of works investigating the global-in-time +existence of weak and strong solutions for the combustion model. P. Secchi [32] proved that there +exists a unique global strong solution in the two-dimensional domain under Fick’s law providing the +diffusion coefficient c0 is small enough. They also considered the limiting behavior of the solutions +when c0 → 0 for dimensions 2 and 3 and the convergence towards the corresponding solutions of +(N). Under the small initial data assumption, P. Lions [28] showed, in R2 or periodic boundary +condition, that a small perturbation of a constant density gives a global existence of weak solutions +without any restriction on the initial velocity. Danchin-Liao [13] proved the existence of solutions +in critical homogeneous Besov spaces by assuming the initial density is close to a constant and the +initial velocity is small enough. Recently, W. Tan [37] proved the global existence of the weak and +strong solutions of the system (1.1) with general viscosity coefficient µ(ρ) in (1.1)2 and ψ(ρ) in +(1.1)3 provided the density is closed to a positive constant in some precise sense. For large data, +Bresch-Essoufi-Sy [5] showed the global existence of the weak solutions for the combustion model +in dimensions 2 and 3 by taking µ(ρ) = c0 +2 log ρ. In [6], Bresch-Giovangigli-Zatorska relaxed the +restriction on µ(ρ) by using the idea of the renormalized solution. +If one takes the decomposition u = v + c0∇ρ−1 with div v = 0 and converts the system (1.1) +to the equations for (ρ, v), then (1.1) will be reduced to the Kazhikhov-Smagulov type model, +see (1.18). In [9, 10], Cai-Liao-Sun established the global-in-time existence of strong solutions to +the initial-boundary value problem of a 2D Kazhikhov-Smagulov type model for incompressible +non-homogeneous fluids with mass diffusion for the arbitrary size of initial data. For works on the +classical Kazhikhov-Smagulov’s model, we refer the reader to [2, 4]. +For the general non-homogeneous incompressible Navier-Stokes equations (N) with the viscosity +coefficient µ(ρ) depending on ρ, global weak solutions were derived by P. Lions [27]. +Abidi- +Zhang [1] obtained the global strong solutions strictly away from vacuum whenever ∥u0∥L2 ∥∇u0∥L2 +and ∥µ(ρ0) − 1∥L∞ are small enough. For the initial density containing vacuum, Cho-Kim [11] +established the existence of the local strong solutions under compatibility conditions similar to +[23]. In addition, Huang-Wang [22], J. Zhang [40] established the global strong solutions with +small ∥∇u0∥L2 in 3D bounded domains. For the Cauchy problem, He-Li-L¨u [18] obtained the +global strong ones to with small ∥u0∥ ˙Hβ for some β ∈ (1/2, 1] and some extra restrictions on µ(ρ) +via the exponential decay-in-time estimates. More recently, Cai-L¨u-Peng [8] studied the global +existence of strong solutions in 3D exterior domains with nonslip or slip boundary conditions +provided that the gradient of the initial velocity is suitably small. +Finally, for the study of the mechanism of blowup and structure of possible singularities of +strong (or smooth) solutions to the Navier-Stokes system can be traced to Serrin’s criterion [33] +on the Leray-Hopf weak solutions to the 3D incompressible homogeneous Navier-Stokes equations, +which can showed that if a weak solution u satisfies +u ∈ Ls(0, T ; Lr), +2 +s + 3 +r ≤ 1, +3 < r ≤ ∞, +(1.5) +then it is regular. Later, He-Xin [19] showed that the Serrin’s criterion (1.5) still holds even for +the strong solution to the incompressible MHD equations. For non-homogeneous incompressible +Navier–Stokes equations (N), H. Kim [24] established the Serrin-type blowup criterion. +They +showed that if (ρ, u) blows up at T ∗, then +lim +t→T ∗ ∥u∥Ls(0,T ;Lrw) = ∞ +for all +2 +s + 3 +r ≤ 1, +3 < r ≤ ∞. +(1.6) +Recently, X. Zhong [41] obtained a blowup criterion (1.5) to the non-homogeneous incompressible +heat conducting Navier–Stokes flows with non-negative density in bounded domain of R3. For +the compressible fluids, Huang-Li-Xin [21] first extend Serrin’s blow-up criterion to the barotropic +compressible Navier-Stokes equations. Later, Xu-Zhang [39] extended the results of [21] to the +isentropic compressible MHD system and Huang-Li-Wang [20] improve the all previous blowup +criterion results to the full compressible Navier–Stokes system. +However, for the general viscosity coefficient, the theory of the combustion model in the bounded +domain is still blank. Therefore, our goal is obtaining the global existence of solutions with small +3 + +initial data and the local existence for (1.1) in the general domain under different initial-boundary +conditions and trying to extend Serrin’s blow-up criterion to (1.1). +More precisely, we impose the initial data +u0(x) := u(x, 0), +0 < α ≤ ρ0(x) := ρ(x, 0) ≤ β < ∞, +x ∈ Ω +(1.7) +and one of the following boundary conditions: +(1) ρ satisfies the Neumann condition and u satisfies the slip boundary condition, that is, +n · ∇ρ = 0, +u · n = 0 and curl u = −n⊥ · B · u +on ∂Ω × (0, T ), +(A) +where n = (n1, n2) denotes the unit outer normal vector of the boundary ∂Ω, n⊥ = (n2, −n1) +is the unit tangential vector on the boundary and B = B(x) is a bounded smooth symmetric +matrix which is positive semi-definite; +(2) (ρ, u) satisfies the non-homogeneous Dirichlet condition, that is, +ρ = ˜ρ, +u = c0∇ρ−1 +on ∂Ω × (0, T ), +(B) +where ˜ρ is a positive constant such that α ≤ ˜ρ ≤ β; +(3) ρ satisfies the Neumann condition and u satisfies the non-slip condition, that is, +n · ∇ρ = 0, +u = 0 +on ∂Ω × (0, T ). +(C) +Before giving the main results, we explain some notations and conventions used throughout the +paper. For simplicity, we set +� +f := +� +Ω +f dx, +� +∂ +f := +� +∂Ω +f dS, +�� +f := +�� +QT +f dxdt, +where QT := Ω × (0, T ), and +fΩ := 1 +|Ω| +� +f, +where |E| stands for the Lebesgue measure of the measurable set E. +Also, for all integer k and 1 ≤ p < ∞, W k,p is the standard Sobolev spaces as defined as follows: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +Lp := Lp(Ω), W k,p = W k,p(Ω), +Hk := W k,2, H∞ := � +k≥1 Hk, +W k,p +0 += C∞ +0 +closure in the norm of W k,p, +∥·∥B1∩B2 := ∥·∥B1 + ∥·∥B2 for two Banach spaces B1 and B2, +H1 +ω := {u ∈ H1 : u · n = 0, curl u = −n⊥ · B · u on ∂Ω}, +H1 +nd := {u ∈ H1 : u = c0∇ρ−1 on ∂Ω}, +V 0,2 := {u ∈ L2 : div u = 0, u · n = 0 on ∂Ω}, +V 1,2 +0 +:= {u ∈ H1 +0 : div u = 0}, +V −1,2 +0 +:= [V 1,2 +0 +]∗. +For 0 < γ < 1, we denote by Cγ(Ω) the standard H¨older space and ρ ∈ Cγ, γ +2 (QT ) the parabolic +one, that is, +Cγ, γ +2 (QT ) := + + + + + +f ∈ C(QT ) : +sup +(x,t),(x′,t′)∈QT +(x,t)̸=(x′,t′) +|f(x, t) − f(x′, t′)| +|x − x′|γ + |t − t′| +γ +2 < ∞ + + + + + +. +The weak, weak* and strong convergence of a sequence {f n} are respectively denoted by +f n +w +−−⇀ f, +f n +w∗ +−−⇀ f, +f n +s +−−→ f. +4 + +Finally, the transpose gradient is given by +∇⊥ := (∂2, −∂1). +With this notion, one can write +� +curl u = ∇⊥ · u, +∆u = ∇ div u + ∇⊥ curl u. +Now, we give the definitions of weak solutions and strong ones. +Definition 1.1 (Weak Solutions). (ρ, u) is called a global weak solution, if the following regularity +properties hold: + + + + + + + + + +α ≤ ρ ≤ β, +ρ ∈ C([0, T ]; H1) ∩ L2(0, T ; H2), +ρt ∈ L2(0, T ; L2), +� +u ∈ L∞(0, T ; L2) ∩ L2(0, T ; H1 +ω), +case (A), +u ∈ L∞(0, T ; L2) ∩ L2(0, T ; H1 +nd), +case (B), +(1.8) +and (ρ, u) statisfies (1.1) in the sense of distributions for all T ∈ (0, ∞). More precisely, (1.1)1, +(1.1)3 hold almost everywhere in Ω × (0, T ) and (1.1)2 is satisfied in the following sense: +�� +ρu · φt + ρu ⊗ u : ∇φ − 2µD(u) : D(φ) = − +� +ρ0u0 · φ(x, 0), +(1.9) +for φ ∈ C∞(QT ) with div φ = 0, φ(x, T ) = 0, x ∈ Ω and φ = 0 on ∂Ω × (0, T ). +Definition 1.2 (Strong Solutions). If (ρ, u, π) is a solution such that (1.1) holds almost everywhere +in Ω × (0, T ), T ∈ (0, ∞), such that + + + + + + + + + + + + + + + + + + + +α ≤ ρ ≤ β, +ρ ∈ C([0, T ]; H2) ∩ L2(0, T ; H3), +ρt ∈ C([0, T ]; L2) ∩ L2(0, T ; H1), +u ∈ C([0, T ]; H1) ∩ L2(0, T ; H2), +ut ∈ L2(0, T ; L2), +π ∈ L2(0, T ; H1), +(1.10) +we call (ρ, u, π) the strong solution on Ω × (0, T ). In particular, if (ρ, u, π) satisfies (1.10) for all +T ∈ (0, ∞), we say that (ρ, u, π) is a global strong solution of the system (1.1). +Our main results sate as following. The first two theorems concern with the existence results +for (ρ, u) satisfying (A) or (B). +Theorem 1.3. Suppose that u0 ∈ L2 and (ρ0, u0) satisfies the following compatibility condition +� +div u0 = c0∆ρ−1 +0 , +x ∈ Ω +u0 · n = n · ∇ρ−1 +0 , +x ∈ ∂Ω +(1.11) +Assume that (ρ, u) satisfies the condition (A) or (B), then there exist a positive constant δ which +only depends on Ω, α, β, c0 and ∥v0∥L2 such that, if ∥∇ρ0∥L2 ≤ δ, problem (1.1) and (1.7) admits +at least one gobal weak solution. +Theorem 1.4. Suppose that (ρ0, u0) satisfies (1.11) and (ρ, u) satisfies the condition (A) or (B). +Let u0 ∈ H1 +ω provided u satisfying the condition (A); u0 ∈ H1 +nd provided u satisfying the condition +(B). In addition, let π satisfy the normalized condition +� +π = 0. +(1.12) +Then, if ∥∇ρ0∥L2 ≤ δ with the same δ obtained in Theorem 1.3, the problem (1.1) and (1.7) admits +a unique global strong solution (ρ, u, π). +5 + +Next, for the case when (ρ, u) satisfying the condition (C), we have +Theorem 1.5. Suppose that u0 ∈ H1 +0 and (ρ0, u0) satisfies (1.11). Suppose that (ρ, u) satisfies the +condition (C) and π satisfies (1.12). There exists a positive constant δ which only depends on Ω, +α, β, c0 such that, if ∥∇u0∥L2 ≤ δ, then the problem (1.1) and (1.7) admits a unique global strong +solution (ρ, u, π). +At last, we give the local existence result and the corresponding Serrin-type blowup criterion. +Theorem 1.6. Assume that (ρ0, u0) satisfies (1.11) and u0 ∈ H1 +ω, H1 +nd provided u satisfying the +condition (A) and (B), respectively. +Let π saitisfies the condition (1.12). +Then there exists a +positive time T1 < ∞ depending on Ω, c0, α, β and ∥u0∥H1 so that the problem (1.1) and (1.7) +admits an unique strong solution (ρ, u, π) on Ω × (0, T1). +Moreover, if µ(ρ) = µ is a positive constant and u0 ∈ H1 +0, then, the same result holds for (ρ, u) +satisfying the condition (C) +Theorem 1.7. If (ρ, u, π) is a strong solution of (1.1) on Ω × (0, T ∗) and T ∗ < ∞ is the maximal +time of existence, then, one has +(1) +lim +T →T ∗ ∥∇ρ∥Ls(0,T ;Lr) = ∞ +(1.13) +provided (ρ, u) satisfying the condition (A) or (B); +(2) +lim +T →T ∗ ∥u∥Ls(0,T ;Lr) = ∞ +(1.14) +provided (ρ, u) satisfying the condition (C). +Here, r and s satisfy the relation +2 +s + 2 +r ≤ 1, +2 < r ≤ ∞. +(1.15) +Remark 1.8. The definition of v0 in Theorem 1.3–1.4 will be given at the end of this section. +Remark 1.9. Theorem 1.3–1.6 are first results concerning with the weak and strong solutions +for the combustion model in bounded domain. Theorem 1.5 can be seen as a kind of extension of +the global existence results in [22, 40] with div u = c0∆ρ−1. Theorem 1.7 can be regarded as an +extension to the classical Serrin’s condition. +Remark 1.10. For some technical reasons, in the proof of Theorem 1.4, we need the following +consistency condition +ρ0|∂Ω = ˜ρ +(1.16) +to ensure the continuity of ρ, which is crucial to the higher order estimates of v, see subsection 3.3 +for details. On the other hand, one should notice that the restriction α ≤ ˜ρ ≤ β and the condition +(1.16) are not necessary for the proof of Theorem 1.3. However, for simiplicity, we may always +impose these requirements. +Remark 1.11. Noticing that, in Theorem 1.3–1.6, we only impose the regularity restrictions on +u0 for given initial data (ρ0, u0). This is due to the compatiability condition (1.11) from which one +can find that the regularity of ρ0 can be totally determinded by that of u0. Indeed, for example, +if u0 ∈ H1 +0 as we assumed in Theorem 1.5, it follows from the following epllitic problem +� +c0∆ρ−1 +0 += div u0, +x ∈ Ω, +n · ∇ρ−1 +0 += 0, +x ∈ ∂Ω. +that, for all 1 < p < ∞, +� +∥∇ρ0∥Lp ≤ C(p) ∥u0∥Lp , +∥∇ρ0∥H1 ≤ C ∥∇u0∥L2 . +Thus, alonging with the fact that ρ0 ∈ L∞, ρ0 ∈ H2. We will come to this point again many times +in later sections. +6 + +At the end of this section, we make a short comment on the analysis of this paper. Formally +speaking, we treat Theorem 1.3–1.5 via two different types of decomposition and the proofs of +Theorem 1.6–1.7 are based on those of Theorem 1.4–1.5. +The proofs of Theorem 1.3–1.4 are based on the decomposition u = v + c0∇ρ−1, which may +convert system (1.1) into the Kazhikhov-Smagulov type model. In this case, one can find that v +satisfies either the Dirichlet boundary condition or the slip one. More precisely, we may first write +in view of (1.1)3 +v = u − c0∇ρ−1. +(1.17) +Of course, such v can be found for given (ρ, u) with the boundary condition (A) or (B). Next, +using (1.17), we write +ρu = ρv + c0ρ∇ρ−1 = ρv − c0∇ log ρ. +Therefore, combining this equality and (1.17), the original system (1.1) can be changed into the +following equivalent formulations: + + + + + + + + + + + + + + + + + + + +ρt + v · ∇ρ + c0ρ−2 |∇ρ|2 − c0ρ−1∆ρ = 0, +� +(ρv)t + div(ρv ⊗ v) − div [2µD(v)] + ∇π1 = c0 div +� +2µ∇2ρ−1� +−c0 div +� +ρv ⊗ ∇ρ−1� +− div +� +c0ρ∇ρ−1 ⊗ v +� +− c2 +0 div +� +ρ∇ρ−1 ⊗ ∇ρ−1� +, +div v = 0, +(1.18) +where π1 = π − c0(log ρ)t is a modified pressure. +Next, we give a precise defintion for the initial-boundary value of v. For given initial data +(ρ0, u0) satisfying the initial conditions (1.11), one can deduce that there exists a unique v0 satis- +fying +v(x, 0) := v0 = u0 − c0∇ρ−1 +0 , +div v0 = 0, +v0 · n = 0 on ∂Ω, +(1.19) +sharing with the similar compatibility conditions of u0, that is, v0|∂Ω = 0 provided u0 ∈ H1 +nd and +curl v0 = −n⊥ · B · (v0 + c0∇ρ−1 +0 ) +provided u0 ∈ H1 +ω. Furthermore, from the relation (1.17), we can define the boundary conditions +of v as follows: +(1) v · n = 0 and curl v = curl u = −n⊥ · B · (v + c0∇ρ−1) on ∂Ω × (0, T ), if (ρ, u) satisfies the +condition (A). In this case, from Remark 2.6 in Section 2, one has +∥v∥H2 ≤ C(∥∆v∥L2 + +��∆ρ−1�� +L2). +(2) v = 0 on ∂Ω × (0, T ), if (ρ, u) satisfies the condition (B). In this case, we have +∥v∥H2 ≤ C ∥∆v∥L2 . +An interesting observation is that, once the solution (ρ, v) of (1.18), which is defined as in +Definition 1.1, incorporating with the initial-boundary conditions given above, is established, one +can expect to obtain u from (1.17) and, consequently, (ρ, u) becomes the solution of the original +system (1.1). Therefore, in Section 3, we mainly establish the a priori estimates of (ρ, v). The +details for proving the existence of (ρ, u) will be shown in Section 6. +To sum up, we may impose (ρ, v) satisfying one of the following two boundary conditions +(1) if (ρ, u) satisfies (A), we impose +n · ∇ρ = 0, +v · n = 0 and curl v = −n⊥ · B · (v + c0∇ρ−1) on ∂Ω × (0, T ); +(A’) +(2) if (ρ, u) satisfies (B), we impose +ρ = ˜ρ, +v = 0 on ∂Ω × (0, T ) +(B’) +7 + +and our strategy of the proof can be concluded as follows: +given (ρ0, u0) =⇒ (ρ0, v0) =⇒ ∃ (ρ, v) =⇒ ∃ (ρ, u). +Unfortunately, for Theorem 1.5, such decomposition may cause some serious problems when it +comes to the boundary estimates, that is, if we extract v as we did above, v|∂Ω = −c0∇ρ−1, which +will hinder us to integrate by parts. As a consequence, we consider another type of decomposition +u = w + Q coming from Lemma 2.4. In every case that follows, w is divergence-free and enjoys +vanished boundary condition and Q can be dominanted by ∇ρ, which allows us to overcome the +bounardy integrals, see Section 4 for details. +The scond part we are interested in is the local well-posedness for system (1.1). +To prove +Theorem 1.6, we mainly follow the proof from Kim-Cho [11] by using the iterative appoarch. This +method will be based on the linearized model associated with (1.1) , we refer to Section 5 for +details. +To the proof of Theorem 1.7, at least for the case when (ρ, v) satisfies condition (A’) or (B’), +the key obeservation is that, if ρ is a weak solution of system (1.1) satisfying ∇ρ ∈ Ls(0, T ; Lr), +then (ρ, v) is regular, since we can close the lower bounds for (ρ, v) merely under the condition +(1.13). Here is an interpretation for (1.13) and (1.14): for (ρ, u) satisfying the condition (A) or +(B), the Ls(0, T ; Lr)-norm of v does not blowup during the finite time [0, T ∗), which is parallel +to the classical Serrin’s condition for 2D non-homogeneous Navier-Stokes equations (N) (since, in +such case, problem (N), at least for ρ away from the vacuum, automatically satisies the Serrin’s +condition and admits a unique global strong solution without any smallness assumption, here v +can be seen as the velocity field u in (N)). However, for the case when (ρ, u) satisfies the condition +(C), we can not get rid off the the blowup behavior of v, since, in this case, v|∂Ω = −c0∇ρ−1, +which leads to some issue on the boundary estimates, we will come to this point again in Section +7. +At last, we explain some techniques used in Section 3 and 7. Since our main difficulty arises +from the boundary integrals, in order to overcome it, we adapt the ideas from Cai-Li [7]: observing +that the condition v · n|∂Ω = 0 leads to +v = (v · n⊥)n⊥, +which implies that +� +∂ +v · ∇f = +� +∂ +(v · n⊥)n⊥ · ∇f = +� +∇f · ∇⊥(v · n⊥). +This observation can allow us to avoid some higher derivatives of f, which has advantages over +directly using the trace inequality, since the latter needs the second order derivative of f. +The rest of this paper is organized as follows. In Section 2, we give some elementary results +which will be used in later. Section 3 is devoted to the lower order estimates, compactness results +for weak solutions and the higher order estimates for Theorem 1.3–1.4, while Section 4 is devoted +to the a priori estimets for Theorem 1.5. In Section 5, we will use the contraction mapping theorem +to prove Theorem 1.6 and, then, in Section 6, use this result to establish the global existence for +Theorem 1.3–1.5. At last, in Section 7, we will give the proof of Theorem 1.7. +2 +Preliminaries +First, we give the following Gagliardo-Nirenberg’s inequalities which will be frequently used through- +out the whole paper. +Lemma 2.1 (Gagliardo-Nirenberg’s inequality [26, 30]). For all ui ∈ H1, i = 1, 2, q1 ∈ (2, ∞) +and q2 ∈ (4, ∞), there exist positive constants Ci, ˜Ci depending on qi, Ω, i = 1, 2, such that +∥u1∥Lq1 ≤ C1 ∥u1∥2/q1 +L2 +∥∇u1∥1−2/q1 +L2 ++ ˜C1 ∥u1∥L2 , +∥u2∥Lq2 ≤ C2 ∥u2∥4/q2 +L4 +∥∇u2∥1−4/q2 +L2 ++ ˜C2 ∥u2∥L2 . +In particular, if ui satisifies ui · n = 0 on ∂Ω or (ui)Ω = 0, then one can take ˜C1 = ˜C2 = 0. +8 + +Lemma 2.2 ([3, 38]). Let Ω be a simply connected bounded domain in R2 with smooth boundary. +Assume that 1 < p < ∞. There exists a positive constant C = C(p, Ω) such that +∥∇u∥Lp ≤ C (∥div u∥Lp + ∥curl u∥Lp) , +for all u ∈ W 1,p with u · n = 0 on ∂Ω. Furthermore, for u ∈ W 2,p with u · n = 0 on ∂Ω, there +exists a constant C = C(p, Ω) such that +∥u∥W 2,p ≤ C (∥div u∥W 1,p + ∥curl u∥W 1,p + ∥u∥Lp) . +Remark 2.3. For case of use, we list the following equivalent norms for ρ satisfying the Neumann +or the non-homogeneous Dirichlet condition. Let 1 < p < ∞, using Lemma 2.1–2.2, if ρ satsifies +the Neumann condition, one has, for all t ≥ 0, +ρΩ = (ρ0)Ω, +∥∇ρ∥Lp ≤ C∥∇2ρ∥Lp ≤ C ∥∆ρ∥Lp ≤ C ∥∇∆ρ∥Lp , +and +C−1(∥∇ρ∥Lp + (ρ0)Ω) ≤ ∥ρ∥W 1,p ≤ C(∥∇ρ∥Lp + (ρ0)Ω), +C−1(∥∆ρ∥Lp + (ρ0)Ω) ≤ ∥ρ∥W 2,p ≤ C(∥∆ρ∥Lp + (ρ0)Ω), +C−1(∥∇∆ρ∥Lp + (ρ0)Ω) ≤ ∥ρ∥W 3,p ≤ C(∥∇∆ρ∥Lp + (ρ0)Ω), +for some positive constant C = C(p, Ω). +If ρ satisfies the the non-homogeneous Dirichlet condition, then there exists a positive constant +C = C(p, Ω) such that +C−1 ∥∇ρ∥Lp ≤ ∥ρ − ˜ρ∥W 1,p ≤ C ∥∇ρ∥Lp , +C−1 ∥∆ρ∥Lp ≤ ∥ρ − ˜ρ∥W 2,p ≤ C ∥∆ρ∥Lp , +C−1 ∥∇∆ρ∥Lp ≤ ∥ρ − ˜ρ∥W 3,p ≤ C ∥∇∆ρ∥Lp . +In both cases, the following Gagliardo-Nirenberg’s inequalities are established +∥∇ρ∥Lq1 ≤ C ∥∇ρ∥2/q1 +L2 +∥∆ρ∥1−2/q1 +L2 +, +∥∇ρ∥Lq2 ≤ C ∥∇ρ∥4/q2 +L4 +∥∆ρ∥1−4/q2 +L2 +, +∥∆ρ∥Lq1 ≤ C ∥∆ρ∥2/q1 +L2 +∥∇∆ρ∥1−2/q1 +L2 +. +where q1, q2 as in Lemma 2.1. +Next, for the problem +� +div v = f, +x ∈ Ω, +v = Φ, +x ∈ ∂Ω +(2.1) +one has the following conclusion which will be frequently used to eliminate the non-homogeneity +of equations in Section 4. +Lemma 2.4 ([16], Theorem III.3.3). Suppose that Φ · n = 0 on ∂Ω and fΩ = 0. Then, +1) If Φ = 0, there exists a bounded linear operator B = [B1, B2], +B : {f ∈ Lp : fΩ = 0} �→ +� +W 1,p +0 +�2 +such that +∥B[f]∥W 1,p ≤ C(p)∥f∥Lp, +for all p ∈ (1, ∞), and the function Q = B[f] solves the problem (2.1). Moreover, if f = div g +with a certain g ∈ Lr, g · n|∂Ω = 0, then for any r ∈ (1, ∞) +∥B[f]∥Lr ≤ C(r)∥g∥Lr. +B is so-called the Bogovskiˇi operator. +9 + +2) If f = 0, there exists a bounded linear operator C = [C1, C2], +C : {Φ : Φ · n|∂Ω = 0, div Φ ∈ Lp} �→ +� +W 1,p�2 +such that +∥C[Φ]∥W 1,p ≤ C(p) ∥div Φ∥Lp , +for all p ∈ (1, ∞) and the function R = C[Φ] sovles the problem (2.1). +Proof. We only give a brief proof for (2). By a simply change +˜v = v − Φ, +it follows from (2.1) that ˜v satisfies +� +div ˜v = − div Φ, +x ∈ Ω, +˜v = 0, +x ∈ ∂Ω. +(2.2) +Thus, applying (1) for (2.2), we finish the proof. +Lemma 2.5–2.9 are a series of results relating to the Stokes system which are vital to the higher +order estimates of v and the construction of smooth initial data. These lemmas will be frequently +used in Section 3–7. +Lemma 2.5. Let Ω be a simply connected bounded domain in R2 with smooth boundary. Let (u, p) +satisfy the following equations +� +−∆u + ∇p = F, +x ∈ Ω, +div u = 0, +x ∈ Ω, +(2.3) +where F ∈ L2, +� +p = 0. There exists a positive constant C depending only on Ω such that +(1) if u|∂Ω = Φ, where Φ ∈ H2 is a function defined on Ω, then +∥u∥H2 + ∥p∥H1 ≤ C(∥F∥L2 + ∥Φ∥H2); +(2.4) +(2) if u · n = 0, curl u = ϕ on ∂Ω, where ϕ ∈ H1 is a function defined on Ω, then +∥u∥H2 + ∥p∥H1 ≤ C(∥F∥L2 + ∥ϕ∥H1). +(2.5) +Proof. Since (2.4) can be finded from [16], Theorem IV.6.1, we only prove the case of slip condition. +Using the identity ∆u = ∇ div u + ∇⊥ curl u and integrating by parts, one has +� +| curl u|2 − +� +∂ +ϕ(u · n⊥) = +� +F · u, +which implies that, using Lemma 2.1–2.2 and the trace inequality, +∥u∥H1 ≤ C(∥F∥L2 + ∥ϕ∥H1). +(2.6) +Since ∇p is bounded in H−1, it follows form the condition +� +p = 0 that p is bounded in L2. Next, +taking curl on the both side of (2.3)1 leads to +−∆(curl u − ϕ) = curl F − ∆ϕ, +with boundary condition curl u − ϕ = 0. +Then, using the regularity result of elliptic partial +differential equations, we have +∥curl u∥H1 ≤ C∥ curl F − ∆ϕ∥H−1 + C ∥ϕ∥H1 ≤ C(∥F∥L2 + ∥ϕ∥H1). +Then, using again Lemma 2.2 and (2.6) gives +∥u∥H2 ≤ C(∥F∥L2 + ∥ϕ∥H1 + ∥u∥L2). +(2.7) +and, consequently, the estimate of p is followed easily. It remains to omit the terms ∥u∥L2 on the +right-hand side of (2.7). Indeed, this is a simple consequence of the uniqueness of (2.3) and we +leave the proof to the reader. +10 + +Remark 2.6. Slimilar results for the Laplace equations −∆u = F instead of (2.3) with the same +boundary conditions can be found in [17]. +Next, we give a lemma which indicates that ρ ∈ Cγ, γ +2 (QT ) for some γ ∈ (0, 1) provided v +satisfying the Serrin’s condition. This result is critial to the estimate of ∆v which will be used in +Section 3 and 7. The observation is based on the fact that div v = 0. +Lemma 2.7 ([10, 36]). Let v ∈ Ls(0, T ; Lr) for some r, s satisfying (1.15), div v = 0, v · n = 0 +and ρ ∈ C([0, T ]; L2) ∩ L2(0, T ; H1) be the weak solution of equation (1.18)1 (in the sense of +distributions), α ≤ ρ ≤ β. Let ρ satisfy either the Neumann condition +n · ∇ρ = 0 on ∂Ω × (0, T ) +or the non-homogeneous Dirichlet condition +ρ = ˜ρ on ∂Ω × (0, T ). +Suppose that ρ0 ∈ Cγ0(Ω) for some γ0 ∈ (0, 1), then ρ is H¨older continuous. +More precisely, +ρ ∈ Cγ, γ +2 (QT ), for some γ depending only on γ0, α and β. +Proof. We only give the proof for ρ|∂Ω = ˜ρ, since the case for ρ satisfying the Neumann boundary +condition has been proved in [10, 36]. Let ζ be a cut-off function, supp ζ ⊂ Br × [t0, t0 + τ], where +Br is an arbitrary ball contained in Ω and [t0, t0 + τ] ⊂ (0, T ), 0 < τ < 1. Multiplying ζ2(ρ − k)+ +on (1.18)1 and integrating by parts leads to +1 +2 +sup +t∈[t0,t0+τ] +∥ζ(ρ − k)+∥2 +L2 + ν ∥ζ∇(ρ − k)+∥2 +L2 +≤ 1 +2 ∥ζ(ρ − k)+∥2 +L2 (t0) + C +� t0+τ +t0 +� +Ω +� +|∇ζ|2 + ζ |ζt| +� +(ρ − k)2 ++ dxdt +− +� t0+τ +t0 +� +Ω +(v · ∇ρ)ζ2(ρ − k)+ dxdt. +(2.8) +For the last term on the right-hand side of (2.8), using Lemma 2.1, we have +���� +� t0+τ +t0 +� +Ω +(v · ∇ρ)ζ2(ρ − k)+ dxdt +���� += +���� +� t0+τ +t0 +� +Ω +(v · ∇ζ)ζ(ρ − k)2 ++ dxdt +���� +≤ ∥v∥ +L +2r +r−2 +t +Lrx +∥ζ(ρ − k)+∥ +Lr +t L +2r +r−2 +x +∥|∇ζ| (ρ − k)+∥L2 +t,x +≤ Cετ +rs−2s−2r +2rs +∥|∇ζ| (ρ − k)+∥2 +L2 +t,x + ε +� +sup +t∈[t0,t0+τ] +∥ζ(ρ − k)+∥2 +L2 + ∥ζ∇(ρ − k)+∥2 +L2 +t,x +� +, +which, alonging with (2.8), implies that +sup +t∈[t0,t0+τ] +∥ζ(ρ − k)+∥2 +L2 + ν ∥ζ∇(ρ − k)+∥2 +L2 +≤ ∥ζ(ρ − k)+∥2 +L2 (t0) + C +� t0+τ +t0 +� +Ω +� +|∇ζ|2 + ζ |ζt| +� +(ρ − k)2 ++ dxdt. +(2.9) +The inequality above is valid for all k ∈ R. Then, It follows from [25] Theorem 10.1 that ρ ∈ +Cγ, γ +2 (QT ), for some γ ∈ (0, 1). +For the boundary estimates, if ρ = ˜ρ on ∂Ω, we still use ζ and choose arbitrary Br×[t0, t0+τ] ⊂ +R2 × [0, T ], where Br may intersect Ω. Then, (2.9) holds for k sufficiently large, since (ρ − k)+ has +vanished boundary, which implies that ρ ∈ Cγ, γ +2 (QT ). +Once ρ is H¨older continuous, µ(ρ(x, t)) is continuous on QT and, thus, we have the following +estiamtes for the non-divergence type Stokes system. +11 + +Lemma 2.8. Let (v, p) be a strong solution of the following Stokes system, +� +−µ(x)∆v + ∇p = F, +x ∈ Ω +div v = 0, +x ∈ Ω +(2.10) +where µ(x) ∈ C(Ω), µ ∈ [µ, µ], +� +p = 0 and F ∈ L2. Then there exists a positive constant C +depending only on µ, µ, continuity module of µ and Ω such that +(1) if u|∂Ω = Φ, where Φ ∈ H2 is a function defined on Ω, then +∥u∥H2 + ∥p∥H1 ≤ C(∥F∥L2 + ∥Φ∥H2); +(2.11) +(2) if u · n = 0, curl u = ϕ on ∂Ω, where ϕ ∈ H1 is a function defined on Ω, then +∥u∥H2 + ∥p∥H1 ≤ C(∥F∥L2 + ∥ϕ∥H1). +(2.12) +Proof. The proof of Lemma 2.8 can be simply derived by using the freezing point argument, since +we already have the conclusion when µ ≡ constant from Lemma 2.5. +Furthermore, in order to prove Lemma 4.3 (see Section 4), we need the following auxiliary +lemma. The purpose for using such result will be explained in the proof of Lemma 4.3. +Lemma 2.9. Let (v, p) be a strong solution of the following Stokes system, +� +− div[2µD(v)] + ∇p = F, +x ∈ Ω +div v = 0, +x ∈ Ω +(2.13) +where ∇µ(ρ) ∈ L4, µ is smooth and 0 < µ ≤ µ ≤ µ < ∞, � p = 0 and F ∈ L2. Then there exists +a positive constant C depending only on µ, µ and Ω such that +(1) if v|∂Ω = Φ, where Φ ∈ H2 is a function defined on Ω, then +∥v∥H2 + ∥p∥H1 ≤ C +�� +∥∇µ∥2 +L4 + 1 +� +(∥F∥L2 + ∥∇Φ∥H1) + ∥∇µ∥2 +L4 ∥∇v∥L2 +� +; +(2.14) +(2) if v · n = 0, curl v = ϕ on ∂Ω, where ϕ ∈ H1 is a function defined on Ω, then +∥v∥H2 + ∥p∥H1 ≤ C +�� +∥∇µ∥2 +L4 + 1 +� +(∥F∥L2 + ∥ϕ∥H1) + ∥∇µ∥2 +L4 ∥∇v∥L2 +� +. +(2.15) +Proof. We only give the proof for (1). +First of all, we can use Lemma 2.4 to find a function +R = C[Φ] such that div R = 0 and R|∂Ω = Φ, then (2.13)1 becomes +− div[2µ(ρ)D(v − R)] + ∇p = F + div[2µ(ρ)D(R)]. +Using standard energy approach and the fact ∥R∥H1 ≤ C ∥∇Φ∥L2, one has +∥∇v∥L2 + ∥p∥L2 ≤ C(∥F∥H−1 + ∥∇Φ∥L2) ≤ C(∥F∥L2 + ∥∇Φ∥H1). +(2.16) +Next, rewritting (2.13)1 into the form +−∆v + ∇ +� +p +µ(ρ) +� += +F +µ(ρ) + 2µ′∇ρ · D(v) +µ(ρ) +− +pµ′ +µ(ρ)2 ∇ρ, +using Lemma 2.5, we have +∥v∥H2 + ∥p∥H1 ≤ C [∥F∥L2 + ∥∇Φ∥H1 + ∥∇µ(ρ)∥L4 (∥∇v∥L4 + ∥p∥L4)] , +which, using Lemma 2.1 and (2.16), leads to +∥v∥H2 + ∥p∥H1 ≤ C +�� +∥∇µ(ρ)∥2 +L4 + 1 +� +(∥F∥L2 + ∥∇Φ∥H1) + ∥∇µ(ρ)∥2 +L4 ∥∇v∥L2 +� +. +Thus, we complete the proof. +12 + +At last, in subsection 3.2 and Section 6, we need the following lemma. +Lemma 2.10 (Simon [31, 34]). Let X ֒→ B ֒→ Y be three Banach spaces with compact imbedding +X ֒→֒→ Y . Further, let there eixst 0 < θ < 1 and M > 0 such that +∥v∥B ≤ M ∥v∥1−θ +X +∥v∥θ +Y , for all v ∈ X ∩ Y. +Denote for T > 0, +W(0, T ) := W s0,r0(0, T ; X) ∩ W s1,r1(0, T ; Y ) +with s0, s1 ∈ R, r1, r0 ∈ [1, ∞], and +sθ := (1 − θ)s0 + θs1, +1 +rθ +:= 1 − θ +r0 ++ θ +r1 +, s∗ := sθ − 1 +rθ +. +Assume that sθ > 0 and F is a bounded set in W(0, T ). +(1) If s∗ ≤ 0, then F is precompact in Lp(0, T ; B) for all 1 ≤ p < − 1 +s∗ . +(2) If s∗ > 0, then F is precompact in C([0, T ]; B). +3 +A Priori Estimates (I): Case (A) and (B) +In this section, we are going to establish the a priori bounds for (ρ, v) which will be used in the +proof of Theorem 1.3. Throughout this section, let T ∈ (0, ∞) and (ρ, v) be a smooth solution to +(1.18) with smooth data (ρ0, v0). Moreover, in order to simplify the notation, we always denote +by ε, εi, i ∈ N+, the arbitrarily small number belongs to (0, 1/2], and we use the subscripts Cε, +Cεi to emphasize the dependency of the constant C on ε, εi +3.1 +Lower Order Estimates +The first lemma is a consequence of the standard maximal principle. +Lemma 3.1. Let α ≤ ρ0 ≤ β and (ρ, v) satisfy the condition (A’) or (B’), one has α ≤ ρ(x, t) ≤ β +for x ∈ Ω and all t ∈ [0, T ]. +Proof. We only prove the upper bound, since the lower one can be derived in a similar way. Using +(1.17), we convert the equation (1.18) into the form +ρt + v · ∇ρ − c0∆ log ρ = 0. +(3.1) +If (ρ, v) satisfies the condition (A’), set k = β and multiply (3.1) by (ρ − k)+ := max{ρ − k, 0}. +After integrating by parts, we obtain +d +dt +� 1 +2(ρ − k)2 ++ + +� +c0ρ−1 |∇(ρ − k)+|2 = 0, +(3.2) +where we have used the identity +� +v · ∇ρ(ρ − k)+ = +� +v · ∇(ρ − k)+(ρ − k)+ = 0, +since v is divergence-free and v ·n = 0 on ∂Ω. Hence, integrating (3.2) from 0 to T and then, using +(1.7) implies that +sup +t∈[0,T ] +� +(ρ − k)2 ++ ≤ +� +(ρ0 − k)2 ++ = 0, +which implies that ρ ≤ k = β for all x ∈ Ω and t ∈ [0, T ]. The case when (ρ, v) satisfies the +condition (B’) can be proved analogously. Thus, we complete the proof of Lemma 3.1. +Our main purpose in this subsection is establishing the lower order bounds. We aim to prove +the following proposition. +13 + +Proposition 3.2. Let (ρ, v) satisfy the condition (A’) or (B’). Suppose that +sup +t∈[0,T ] +∥∇ρ∥2 +L2 + +� T +0 +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� +dt ≤ 2. +(3.3) +There exists a positive constant δ depending on Ω, c0, α, β and ∥v0∥L2 such that, if ∥∇ρ0∥L2 ≤ δ, +sup +t∈[0,T ] +∥∇ρ∥2 +L2 + +� T +0 +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� +dt ≤ 1. +(3.4) +We give the proof of Proposition 3.2 in several steps. First, we estimate the first order derivative +of ρ, which is given by the following lemma. +Lemma 3.3. There exists a positive constant C depending only on c0 and β such that, if (ρ, v) +satisfies the condition (A’), +sup +t∈[0,T ] +∥ρ∥L2 + ∥∇ρ∥L2(0,T ;L2) ≤ C ∥ρ0∥L2 ; +(3.5) +if (ρ, v) satisfies the condition (B’), +sup +t∈[0,T ] +∥ρ − ˜ρ∥L2 + ∥∇ρ∥L2(0,T ;L2) ≤ C ∥ρ0 − ˜ρ∥L2 . +(3.6) +Proof. Multiplying (3.1) by ρ (if ρ satisfies the condition (B’), multiply ρ − ˜ρ), integrating over Ω +and computing in the same way of Lemma 3.1, one has +d +dt ∥ρ∥2 +L2 + 2c0β−1 ∥∇ρ∥2 +L2 ≤ 0. +(3.7) +Using Gr¨onwall’s inequality leads to +sup +t∈[0,T ] +∥ρ∥L2 + +� +2c0β−1 ∥∇ρ∥L2(0,T ;L2) ≤ ∥ρ0∥L2 , +where we use the fact that ρ ≤ β from Lemma 3.1. This completes the proof. +Next, the following lemma shows that the second order derivative of ρ can be dominated by +the norm of v provided ∥∇ρ∥L2 (t) is small enough. +Lemma 3.4. Let (ρ, v) satisfy the condition (A’) or (B’). Then there exist a positive constant +δ1 depending on Ω, c0, α, β and a positive constant C depending on Ω, c0, α and β such that, if +∥∇ρ∥L2 (t) ≤ δ1 for all t ∈ [0, T ], +sup +t∈[0,T ] +∥∇ρ∥2 +L2 + +� T +0 +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� +dt ≤ exp +� +C +� T +0 +∥v∥4 +L4 dt +� +∥∇ρ0∥2 +L2 . +(3.8) +Proof. Multiplying (1.18) by (−∆ρ) and integrating over Ω, we obtain +d +dt +� 1 +2 |∇ρ|2 + +� +c0ρ−1 |∆ρ|2 = +� +(v · ∇ρ)∆ρ − +� +c0ρ−2 |∇ρ|2 ∆ρ, +which implies that, using Lemma 3.1, +d +dt +� 1 +2 |∇ρ|2 + c0β−1 +� +|∆ρ|2 ≤ C +� � +|∇ρ|2 + |v| |∇ρ| +� +|∆ρ| +≤ C +� � +|∇ρ|4 + |v|2 |∇ρ|2� ++ c0(2β)−1 +� +|∆ρ|2 . +Hence, by Lemma 2.1 and 3.1, we have +d +dt ∥∇ρ∥2 +L2 + ν ∥∆ρ∥2 +L2 ≤ C ∥∇ρ∥2 +L2 ∥∆ρ∥2 +L2 + C ∥v∥4 +L4 ∥∇ρ∥2 +L2 , +(3.9) +14 + +for some positive constant ν = ν(c0, β) and C = C(Ω, c0, α, β). +Thus, if we choose δ1 = +ν1/2(2C)−1/2 and set ∥∇ρ∥L2 (t) ≤ δ1 for all t ∈ [0, T ], using the Gr¨onwall’s inequality, we can +deduce from (3.9) and Lemma 2.1 that +sup +t∈[0,T ] +∥∇ρ∥2 +L2 + ν +� T +0 +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� +dt ≤ exp +� +C +� T +0 +∥v∥4 +L4 dt +� +∥∇ρ0∥2 +L2 , +which concludes the proof of (3.8). The case when (ρ, v) satisfies the condition (B’) can be com- +puted in the same way, since ρt has vanished boundary. +From the observation of Lemma 3.4, in order to derive the bounds for ρ, we need to control the +L4(0, T ; L4) norm of v, which is given by the following lemma. +Lemma 3.5. Let (ρ, v) satisfy the condition (A’) or (B’). Suppose that condition (3.3) holds. +Then there exists a positive constant C depending on Ω, c0, α and β such that +sup +t∈[0,T ] +∥v∥2 +L2 + +� T +0 +� +∥v∥4 +L4 + ∥∇v∥2 +L2 +� +dt ≤ C(1 + ∥v0∥2 +L2). +(3.10) +Proof. In order to simplify our proof, we only consider the case when ρ satisfies (B’) and v satisfies +(A’), since other cases can be established in the same way and are much easier. We first deal with +a special case for curl v = −n⊥ · B · v on the boundary. +We write (1.18)2, using (1.17), into the form +ρvt + ρu · ∇v − div [2µD(v)] + ∇π1 += c0 div +� +2µ∇2ρ−1� +− c0 div +� +ρv ⊗ ∇ρ−1� +− c2 +0 div +� +ρ∇ρ−1 ⊗ ∇ρ−1� +, +(3.11) +Multiplying (3.11) by v and integrating over Ω, one has +d +dt +� 1 +2ρ |v|2 − +� +div [2µD(v)] · v += +� +c0 div +� +2µ∇2ρ−1� +· v − +� +c0 div +� +ρv ⊗ ∇ρ−1� +· v +− +� +c2 +0 div +� +ρ∇ρ−1 ⊗ ∇ρ−1� +· v := +3 +� +i=1 +Ii. +(3.12) +Next, for the last term on the left-hand side of (3.12), we use again Lemma 2.1 and 3.1 to get +− +� +div [2µD(v)] · v = − +� +2µ∆v · v − +� +2µ′∇ρ · D(v) · v += +� +2µ| curl v|2 + +� +∂ +2µv · B · v ++ +� +2µ′∇⊥ρ · v(curl v) − +� +2µ′∇ρ · D(v) · v, +≥ µ +� +|curl v|2 − +� +Cε ∥∇ρ∥4 +L4 ∥√ρv∥2 +L2 + ε ∥∇v∥2 +L2 +� +, +(3.13) +where µ := mins∈[α,β] µ(s) and the last inequality follows from the fact that B is positive semi- +definite. +For I1, we use the similar approach and write it in the component form, +I1 = +� +∂ +−2c0µ∂ijρ−1vinj + +� +2c0µ∂ijρ−1∂jvi += +� +∂ +−4c0µρ−3∂iρ∂jρvinj + +� +∂ +2c0µρ−2∂ijρvinj + +� +2c0µ∂ijρ−1∂jvi := +3 +� +i=1 +Ji. +15 + +The estimate of J3 can be simply derived by using Lemma 2.1 and 3.1, that is, +|J3| ≤ C +���� +� +µ +� +2ρ−3∂iρ∂jρ − ρ−2∂ijρ +� +∂jvi +���� +≤ Cε(∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2) + ε ∥∇v∥2 +L2 . +(3.14) +To the boundary parts J1 and J2, it suffices to estimate +J′ +1 = +� +∂ +φ(ρ)∂iρ∂jρvinj, +J′ +2 = +� +∂ +φ(ρ)∂ijρvinj = − +� +∂ +φ(ρ)vi∂inj∂jρ, +where φ(·) is a positive smooth function defined on (0, ∞). Using Lemma 2.1 and 3.1, one has +|J′ +1| = +���� +� +∂ +φ(ρ)(n · ∇ρ)(v · n⊥)n⊥ · ∇ρ +���� += +���� +� +∇⊥[φ(ρ)(n · ∇ρ)] · ∇ρ(v · n⊥) + +� +φ(ρ)(n · ∇ρ)∇ρ · ∇⊥(v · n⊥) +���� +≤ C +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� ++ Cε1 ∥∇ρ∥4 +L4 ∥√ρv∥2 +L2 + ε1 ∥∇v∥2 +L2 +(3.15) +and +|J′ +2| = +���� +� +∂ +φ(ρ)(v · n⊥)n⊥ · ∇n · ∇ρ +���� += +���� +� +∇⊥φ(ρ) · (∇n · ∇ρ)(v · n⊥) − +� +Ω +φ(ρ)∇⊥ · (∇n · ∇ρ)(v · n⊥) dx +− +� +φ(ρ)∇⊥(v · n⊥) · (∇n · ∇ρ) +���� +≤ C +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� ++ Cε2 ∥∇ρ∥4 +L4 ∥√ρv∥2 +L2 + ε2 ∥∇v∥2 +L2 . +(3.16) +Combining (3.14)–(3.16), we deduce that +|I1| ≤ C +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� ++ Cε ∥∇ρ∥4 +L4 ∥√ρv∥2 +L2 + ε ∥∇v∥2 +L2 . +(3.17) +Similar computation can be applied for I2 and I3, that is, +|I2| ≤ Cε3 ∥∇ρ∥4 +L4 ∥√ρv∥2 +L2 + ε3 ∥∇v∥2 +L2 , +(3.18) +|I3| ≤ C +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� ++ Cε4 ∥∇ρ∥4 +L4 ∥√ρv∥2 +L2 + ε4 ∥∇v∥2 +L2 . +(3.19) +Therefore, we go back to the estimate of v, combining (3.12)–(3.13) and (3.17)–(3.19) and then, +using Lemma 2.2 implies that +d +dt ∥√ρv∥2 +L2 + ν ∥∇v∥2 +L2 ≤ C +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� � +1 + ∥√ρv∥2 +L2 +� +, +(3.20) +for some constant C depending on Ω, c0, α and β. +In view of the condition (3.3), we obtain +the bound (3.10) via Gr¨onwall’s inequality and Lemma 2.1. For the general case when curl v = +−n⊥ · B · (v + c0∇ρ−1) on ∂Ω × (0, T ), we can also obtain the desire bounds (3.10) by calculating +the extra boundary term +� +∂ +2c0µv · B · ∇ρ−1. +However, this term is nothing but a special case of J′ +2 with ∇n replaced by B. Therefore, following +the same computation of J′ +2, we complete the proof for the general case. +Now, we can turn back to prove Proposition 3.2. +16 + +Proof of Proposition 3.2. Using Lemma 3.5, we obtain the bound of v, (3.10), under the condition +(3.3). Next, using Lemma 3.4 and (3.10) leads to (3.8), that is, if ∥∇ρ∥L2 (t) ≤ δ1 for all t ∈ [0, T ], +sup +t∈[0,T ] +∥∇ρ∥2 +L2 + +� T +0 +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� +dt ≤ C ∥∇ρ0∥2 +L2 , +(3.21) +where C is a positive constant depending on Ω, c0, α, β and ∥v0∥L2. +Hence, if we set the constant δ > 0 such that +δ = min +� +C− 1 +2 , δ1C− 1 +2 +� +, +(3.22) +and let ∥∇ρ0∥L2 ≤ δ, then it is easy to check that (3.4) is established. Consequently, we complete +the proof. +Remark 3.6. It follows from the equation (1.18)1 and (∇ρ, v) ∈ L4(0, T ; L4) that ρt is bounded +in L2(0, T ; L2). +3.2 +Compactness Results +Before establishing higher order estimates, we tend to prove the compactness results for (ρ, v), +which plays a crucial role in the proof of Theorem 1.3, see Section 6. Concerning a sequence of +weak solutions (ρn, vn) with π1 replaced by πn +1 satisfying the condition (A’) or (B’) and the initial +conditions +ρn|t=0 = ρn +0, vn|t=0 = vn +0 . +(3.23) +We assume that (ρn, vn) satisfy, uniformly in n ≥ 1, the a priori bounds that derived in the pre- +ceeding section and ∇ρn +0, vn +0 +s +−−→ ∇ρ0, v0 in L2. Without loss of generality, extracting subsequences +if necessary, we assume + + + + + + + + + + + +ρn +w∗ +−−⇀ ρ +in L∞(0, T ; H1), +ρn +w +−−⇀ ρ +in L2(0, T ; H2), +vn +w∗ +−−⇀ v +in L∞(0, T ; L2), +vn +w +−−⇀ v +in L2(0, T ; H1). +(3.24) +We may now state our compactness results whose proof is followed by [27]. +Lemma 3.7. Under the hypothesis above, we have, for all p ∈ [1, ∞), +ρn +s +−−→ ρ +in C([0, T ]; Lp), +(3.25) +vn +s +−−→ v +in L2(0, T ; L2). +(3.26) +Proof. Since we have (3.24)2 and ρn +t is bounded in L2(0, T ; L2) from Remark 3.6, (3.25) can be +directly derived by using Lemma 2.10. To prove (3.26), observing that (1.18)2 leads to +|⟨(ρnvn)t, φ⟩| ≤ C ∥φ∥L2(0,T ;H1) , +for all φ ∈ C∞ +c (Ω × [0, T ]) such that div φ = 0, which implies that (ρnvn)t is bounded in +L2(0, T ; V −1,2). On the other hand, since ρnvn is bounded in L2(0, T ; H1), it follows from Lemma +2.10 that ρnvn is precompact in L2(0, T ; L2), that is, +ρnvn +s +−−→ ρv +in L2(0, T ; L2). +(3.27) +Thanks to (3.24)4 and (3.25), we have +ρv = ρv, +vn +s +−−→ v +in L2(0, T ; L2), +which gives (3.26). +17 + +3.3 +Higher Order Estimates +In this subsection, we will show the a priori estimates for strong solutions of (1.18). We still use +the assumption at the begining of Section 3. Furthermore, throughout this subsection, we always +keep +∥∇ρ0∥L2 ≤ δ +small enough so that Proposition 3.2 is valid. In a word, we have all the estimates of (ρ, v) from +Lemma 3.1–3.5. +For convenience, we set +F(t) := ∥∇v∥2 +L2 + ∥∆ρ∥2 +L2 + ∥ρt∥2 +L2 , +G(t) := ∥∆v∥2 +L2 + ∥vt∥2 +L2 + ∥∇∆ρ∥2 +L2 + ∥∇ρt∥2 +L2 , +M1(t) := +� +∂ +µv · B · v, +M2(t) := +� +c0µ∇⊥(v · n⊥) · B · ∇ρ−1. +We now state the proposition we are aimming for in this subsection. +Proposition 3.8. Let (ρ, v) satisfy the condition (A’) or (B’). Then +sup +t∈[0,T ] +F(t) + +� T +0 +� +G(t) + ∥π∥2 +H1 +� +dt ≤ C, +(3.28) +where C is a positive constant depending on Ω, α, β, c0, ∥ρ0∥H2 and ∥v0∥H1. +In order to prove Proposition 3.8, we need several auxiliary lemmas. +Lemma 3.9. Under the assumptions at the begining of this section, +(1) if (ρ, v) satisfies the condition (A’), then, for all ε1 ∈ (0, 1/2], +d +dt +� +∥∆ρ∥2 +L2 + ∥ρt∥2 +L2 +� ++ ∥∇∆ρ∥2 +L2 + ∥∇ρt∥2 +L2 +≤ Cε1A1(t)F(t) + ε1 +� +∥v∥2 +H2 + ∥vt∥2 +L2 +� +; +(3.29) +(2) if (ρ, v) satisfies the condition (B’), then +∥∆ log ρ∥2 +L2 ≤ C +� +∥(log ρ)t∥2 +L2 + ∥∇v∥2 +L2 +� +(3.30) +and for all ε2, ε3 ∈ (0, 1], +d +dt ∥(log ρ)t∥2 +L2 + ∥∇(log ρ)t∥2 +L2 ≤ Cε2A2(t) ∥(log ρ)t∥2 +L2 + ε2 ∥vt∥2 +L2 . +(3.31) +∥∇∆ log ρ∥2 +L2 ≤ Cε3A3(t) +� +∥∆ log ρ∥2 +L2 + ∥∇v∥2 +L2 +� ++ Cε3 ∥∇(log ρ)t∥2 +L2 + ε3 ∥∆v∥2 +L2 . +(3.32) +Here, C, Cε1 − Cε3 are positive constants depending on Ω, α, β, c0 with Cε1 − Cε3 extra depending +on ε1–ε3 respectively, A1–A3 are all nonnegative integrable functions defined on [0, ∞). +Proof. We first consider the case when (ρ, v) satisfies the condition (A’). Taking −(∇∆ρ)∇ on the +both sides of (1.18)1 and integrating by parts, we have +d +dt +� 1 +2 |∆ρ|2 + +� +c0ρ−1 |∇∆ρ|2 = +� +∇∆ρ · ∇v · ∇ρ + +� +v · ∇2ρ · ∇∆ρ +− +� +2c0ρ−3 |∇ρ|2 ∇ρ · ∇∆ρ + +� +c0ρ−2∇(|∇ρ|2) · ∇∆ρ ++ +� +c0ρ−2∆ρ∇ρ · ∇∆ρ +:= +5 +� +i=1 +Ki. +(3.33) +18 + +For K1–K5, we use Lemma 2.1 and 3.1 to find that + + + + + + + + + + + + + + + + + + + +|K1| ≤ Cε1,ε2 ∥∇ρ∥4 +L4 ∥∇v∥2 +L2 + ε1 ∥∇∆ρ∥2 +L2 + ε2 ∥∆v∥2 +L2 +|K2| ≤ Cε3 ∥v∥4 +L4 ∥∆ρ∥2 +L2 + ε3 ∥∇∆ρ∥2 +L2 +|K3| ≤ Cε4 ∥∇ρ∥6 +L6 + ε4 ∥∇∆ρ∥2 +L2 +≤ Cε4 ∥∇ρ∥4 +L4 ∥∆ρ∥2 +L2 + ε4 ∥∇∆ρ∥2 +L2 +|K4| ≤ Cε5 ∥∇ρ∥4 +L4 ∥∆ρ∥2 +L2 + ε5 ∥∇∆ρ∥2 +L2 +|K5| ≤ Cε6 ∥∇ρ∥4 +L4 ∥∆ρ∥2 +L2 + ε6 ∥∇∆ρ∥2 +L2 . +(3.34) +Combining (3.33) and (3.34), we have, for some ν > 0, +d +dt ∥∆ρ∥2 +L2+ν ∥∇∆ρ∥2 +L2 ≤ Cε +� +∥∇ρ∥4 +L4 + ∥v∥4 +L4 +� +∥∆ρ∥2 +L2+Cε ∥∇ρ∥4 +L4 ∥∇v∥2 +L2+ε ∥∆v∥2 +L2 , (3.35) +Next, we estimate the bound of ρt. Differentiating (1.18)1 with respect to t, one has +ρtt − c0ρ−1∆ρt = −vt · ∇ρ − v · ∇ρt + 2c0ρ−3ρt |∇ρ|2 − c0ρ−1 � +|∇ρ|2� +t − c0ρ−2ρt∆ρ. +(3.36) +Multiplying ρt on both sides of (3.36) and integrating over Ω, we have +d +dt +� 1 +2 |ρt|2 + ν +� +|∇ρt|2 ≤ +� +|vt| |∇ρ| |ρt| + C +� +|ρt|2 |∇ρ|2 ++ +� +|ρt| |∇ρt| |∇ρ| + C +� +|ρt|2 |∆ρ| +:= +4 +� +i=1 +Li. +(3.37) +Similarly, we use Lemma 2.1 and 3.1 to obtain + + + + + + + + + +|L1| ≤ Cε1,ε2 ∥∇ρ∥4 +L4 ∥ρt∥2 +L2 + ε1 ∥vt∥2 +L2 + ε2 ∥∇ρt∥2 +L2 , +|L2| ≤ Cε3 ∥∇ρ∥4 +L4 ∥ρt∥2 +L2 + ε3 ∥∇ρt∥2 +L2 , +|L3| ≤ Cε4 ∥∇ρ∥4 +L4 ∥ρt∥2 +L2 + ε4 ∥∇ρt∥2 +L2 , +|L4| ≤ Cε5 ∥∆ρ∥2 +L2 ∥ρt∥2 +L2 + ε5 ∥∇ρt∥2 +L2 . +(3.38) +Thus, from (3.37) and (3.38), one has, for some ν > 0, +d +dt ∥ρt∥2 +L2 + ν ∥∇ρt∥2 +L2 ≤ Cε +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� +∥ρt∥2 +L2 + ε ∥vt∥2 +L2 , +(3.39) +Combining (3.35) and (3.39) leads to the estimate (3.29). +Next, we trun to the case when (ρ, v) satisfies the condition (B’). The main difficulty in this case +is that, although we still have the estimate (3.39), we can not use the energy method by integrating +by parts to derive the bound of ∇∆ρ. To overcome it, we estimate directly from (1.18)1. More +precisely, we first renormalize (1.18)1 by writting +(log ρ)t + v · ∇ log ρ − c0ρ−1∆ log ρ = 0. +(3.40) +Next, differentiating in x on both sides of (3.40), one has +∇(log ρ)t + ∇v · ∇ log ρ + v · ∇2 log ρ + c0ρ−2∇ρ∆ log ρ − c0ρ−1∇∆ log ρ = 0. +(3.41) +Then, applying L2-norm for ∇∆ log ρ, then, using Lemma 2.1 and 3.1 leads to +∥∇∆ log ρ∥L2 ≤ C +� +∥∇(log ρ)t∥L2 + ∥|∇v|·|∇ log ρ|∥L2 + ∥ |v|·|∇2 log ρ|∥L2 + ∥ |∇ρ|·|∆ log ρ|∥L2� +. +Thus, using Lemma 2.1, for all ε1, ε2 ∈ (0, 1/2], there exists a constant Cε1,ε2 depending on Ω, c0, +α, β, ε1 and ε2 such that +∥∇∆ log ρ∥2 +L2 ≤ Cε1,ε2 +� +∥∇(log ρ)t∥2 +L2 + ∥v∥4 +L4 ∥∆ log ρ∥2 +L2 + ∥∇ρ∥4 +L4 ∥∆ log ρ∥2 +L2 +� ++ Cε2 ∥∇ρ∥4 +L4 ∥∇v∥2 +L2 + ε1 ∥∇∆ log ρ∥2 +L2 + ε2 ∥∆v∥2 +L2 , +19 + +Consequently, +∥∇∆ log ρ∥2 +L2 ≤ Cε ∥∇(log ρ)t∥2 +L2 + Cε +� +∥∇ρ∥4 +L4 + ∥v∥4 +L4 +� +∥∆ log ρ∥2 +L2 ++ Cε ∥∇ρ∥4 +L4 ∥∇v∥2 +L2 + ε ∥∆v∥2 +L2 , +(3.42) +which gives (3.32). +It remains to show (3.30) and (3.31). From (3.40), we use Lemma 2.1 and 3.1 to obtain, +∥∆ log ρ∥2 +L2 ≤ C ∥(log ρ)t∥2 +L2 + C ∥∇ log ρ∥2 +L4 ∥v∥2 +L4 +≤ C ∥(log ρ)t∥2 +L2 + Cε ∥∇ρ∥2 +L2 ∥v∥2 +L2 ∥∇v∥2 +L2 + ε ∥∆ log ρ∥2 +L2 , +that is, +∥∆ log ρ∥2 +L2 ≤ C +� +∥(log ρ)t∥2 +L2 + ∥∇v∥2 +L2 +� +. +For (3.31), we can follow the proof from (3.36) to (3.39) by applying (log ρ)t∂t on both sided +(3.40) and integrating over Ω, that is, +d +dt +� 1 +2|(log ρ)t|2 + +� +c0ρ−1|∇(log ρ)t|2 += − +� +c0ρ−1∇(log ρ)t · ∇ log ρ(log ρ)t − +� +vt · ∇ log ρ(log ρ)t ++ +� +c0ρ−1|(log ρ)t|2|∇ log ρ|2 +:= +3 +� +i=1 +Pi, +(3.43) +where, applying Lemma 2.1 and 3.1, + + + + + + + + + + + + + + + + + + + +|P1| ≤ C ∥∇ log ρ∥L4 ∥(log ρ)t∥L4 ∥∇(log ρ)t∥L2 +≤ Cε1 ∥∇ρ∥4 +L4 ∥(log ρ)t∥2 +L2 + ε1 ∥∇(log ρ)t∥2 +L2 +|P2| ≤ ∥∇ log ρ∥L4 ∥(log ρ)t∥L4 ∥vt∥L2 +≤ Cε2 ∥∇ρ∥4 +L4 ∥(log ρ)t∥2 +L2 + ε2 ∥vt∥2 +L2 +|P3| ≤ ∥∇ log ρ∥2 +L4 ∥(log ρ)t∥2 +L4 +≤ Cε3 ∥∇ρ∥4 +L4 ∥(log ρ)t∥2 +L2 + ε3 ∥∇(log ρ)t∥2 +L2 . +(3.44) +Combining (3.43) and (3.44) leads to +d +dt ∥(log ρ)t∥2 +L2 + ν ∥∇(log ρ)t∥2 +L2 ≤ Cε ∥∇ρ∥4 +L4 ∥(log ρ)t∥2 +L2 + ε ∥vt∥2 +L2 . +This completes the proof of the lemma. +Remark 3.10. One may find that we estimate log ρ instead of ρ in the proof of (ρ, v) satisfying +(B’). This is based on the observation that (1.18)1 has the dissipative term ∆ log ρ, that is, +ρt + v · ∇ρ − c0∆ log ρ = 0. +Thus, such conversion can avoid the occurrence of the nonlinear term |∇ρ|2, otherwise, if we +estimate ∆ρ in the proof of (3.30), we need additional smallness assumption on ∇ρ0 to handle +��|∇ρ|2�� +L2, which is not what we expect (this point will also be seen in Section 7). +The next lemma shows that v can be bounded by the norm of ρ provided ∥∇ρ0∥L2 is small. +Lemma 3.11. Under the assumptions at the begining of this section, +20 + +(1) if (ρ, v) satisfy the condition (A’), then for all ε ∈ (0, 1/2], +d +dt +� +M1(t) + ∥√µ curl v∥2 +L2 +� ++ ∥vt∥2 +L2 + d +dtM2(t) +≤ CεA4(t)F(t) + ε +� +∥∇ρt∥2 +L2 + ∥∇∆ρ∥2 +L2 +� ++ A5(t), +(3.45) +and +∥v∥2 +H2 + ∥π∥2 +H1 ≤ A6(t)F(t) + C +� +∥∇∆ρ∥2 +L2 + ∥vt∥2 +L2 + ∥∇ρt∥2 +L2 +� ++ A7(t), +(3.46) +where C and Cε are positive constants depending on Ω, c0, α, β with Cε extra depending on +ε; A4–A7 are nonnegative integrable functions defined on [0, ∞); +(2) if (ρ, v) satisfies the conditin (B’), one still has the estimates (3.45) and (3.46) with M1(t) = +M2(t) = 0 in (3.45). More precisely, +d +dt ∥√µ|D(v)|∥2 +L2 + ∥vt∥2 +L2 +≤ CεA8(t) +� +∥∇v∥2 +L2 + ∥∆ log ρ∥2 +L2 +� ++ ε ∥∇∆ log ρ∥2 +L2 , +(3.47) +and +∥v∥2 +H2 + ∥π∥2 +H1 ≤ CεA9(t) +� +∥∇v∥2 +L2 + ∥∆ log ρ∥2 +L2 +� ++ ε ∥∇∆ log ρ∥2 +L2 ++ C +� +∥vt∥2 +L2 + ∥∇(log ρ)t∥2 +L2 +� +. +(3.48) +where C and Cε as in (1); A8 and A9 are nonnegative integrable functions defined on [0, ∞). +Proof. Rewrite (1.18)2 as +ρvt − div [2µD(v)] + ∇π1 += −ρu · ∇v + c0 div +� +2µ∇2ρ−1� +− c0 div +� +ρv ⊗ ∇ρ−1� +− c2 +0 div +� +ρ∇ρ−1 ⊗ ∇ρ−1� +. +(3.49) +We first come to the case when (ρ, v) satisfies the condition (A’) and consider the special case when +curl v = −n⊥ · B · v on ∂Ω × (0, T ). Multiplying vt on both sides of (3.49) and integrating over Ω, +one gets +� +ρ |vt|2 − +� +div[2µD(v)] · vt += − +� +ρu · ∇v · vt + +� +c0 div +� +2µ∇2ρ−1� +· vt − +� +c0 div +� +ρv ⊗ ∇ρ−1� +· vt +− +� +c2 +0 div +� +ρ∇ρ−1 ⊗ ∇ρ−1� +· vt := +4 +� +i=1 +Mi. +(3.50) +For the second term on the left-hand side of (3.50), we have +− +� +div[2µD(v)] · vt = − +� +2µ∆v · vt − +� +2µ′∇ρ · D(v) · vt := +2 +� +i=1 +Qi. +First, to estimate Q1, we have +Q1 = − +� +2µ∇⊥(curl v) · vt += − +� +∂ +2µ curl v(vt · n⊥) + +� +µ d +dt| curl v|2 + +� +µ′∇⊥ρ · vt(curl v) += d +dt +� +M1(t) + ∥√µ curl v∥2 +L2 +� +− +� +∂ +µt(v · B · v) − +� +µt| curl v|2 + +� +µ′∇⊥ρ · vt(curl v), +21 + +where, for the last three terms, we use Lemma 2.1 and 3.1 to obtain +���� +� +∂ +µt(v · B · v) +���� = +���� +� +∂ +µt(v · n⊥)n⊥ · B · v +���� += +���� +� +µt∇⊥(v · n⊥) · B · v + +� +v · n⊥∇⊥ · [µtB · v] +���� +≤ Cε1 +� +∥∇ρ∥4 +L4 + ∥v∥4 +L4 +� +∥ρt∥2 +L2 + Cε1 ∥v∥4 +L4 + ε1 +� +∥∇ρt∥2 +L2 + ∥v∥2 +H2 +� +, +���� +� +µt |curl v|2 +���� ≤ Cε2 ∥ρt∥2 +L2 ∥∇v∥2 +L2 + ε2 ∥v∥2 +H2 , +���� +� +µ′∇⊥ρ · vt(curl v) +���� ≤ Cε3 ∥∇ρ∥4 +L4 ∥∇v∥2 +L2 + ε3 +� +∥vt∥2 +L2 + ∥v∥2 +H2 +� +, +which gives +Q1 ≥ d +dt +� +M1(t) + ∥√µ curl v∥2 +L2 +� +− Cε4 +� +∥ρt∥2 +L2 + ∥v∥4 +L4 + ∥∇ρ∥4 +L4 +� � +∥∇v∥2 +L2 + ∥ρt∥2 +L2 +� ++ Cε4 ∥v∥4 +L4 ++ ε4 +� +∥∇ρt∥2 +L2 + ∥vt∥2 +L2 + ∥v∥2 +H2 +� +. +(3.51) +On the other hand, +|Q2| ≤ Cε5 ∥∇ρ∥4 +L4 ∥∇v∥2 +L2 + ε5 +� +∥vt∥2 +L2 + ∥v∥2 +H2 +� +. +(3.52) +Combining (3.51) and (3.52) leads to +− +� +div[2µD(v)] · vt ≥ d +dt +� +M1(t) + ∥√µ curl v∥2 +L2 +� +− Cε6 +� +∥ρt∥2 +L2 + ∥v∥4 +L4 + ∥∇ρ∥4 +L4 +� � +∥∇v∥2 +L2 + ∥ρt∥2 +L2 +� ++ Cε6 ∥v∥4 +L4 + ε6 +� +∥∇ρt∥2 +L2 + ∥vt∥2 +L2 + ∥v∥2 +H2 +� +. +(3.53) +Next, using Lemma 2.1 and 3.1 again, we estimate M1 − M4, that is, +|M1| ≤ Cε7 +� +∥v∥4 +L4 + ∥∇ρ∥4 +L4 +� +∥∇v∥2 +L2 + ε7 +� +∥vt∥2 +L2 + ∥v∥2 +H2 +� +, +(3.54) +|M2| = 2c0 +���� +� +µ′∂jρ∂ijρ−1(vt)i + +� +µ(ρ)∂ijjρ−1(vt)i +���� += 2c0 +���� +� +µ′∂jρ∂ijρ−1(vt)i − +� +µ′∂iρ∂jjρ−1(vt)i +���� +≤ C +� +(|∇ρ|3 + |∇ρ| |∇2ρ|) |vt| +≤ Cε8 ∥∇ρ∥4 +L4 ∥∆ρ∥2 +L2 + ε8 +� +∥vt∥2 +L2 + ∥∇∆ρ∥2 +L2 +� +, +(3.55) +|M3 + M4| = +����c0 +� +∂j (uj∂i log ρ) (vt)i +���� += +����c0 +� +∂j (log ρ∂ivj) (vt)i + c2 +0 +� +∂j +� +log ρ∂ijρ−1� +(vt)i +���� +≤ C +� � +|∇v| |∇ρ| + |∇ρ|3 + |∇ρ| +��∇2ρ +�� +� +|vt| +≤ Cε9 ∥∇ρ∥4 +L4 +� +∥∇v∥2 +L2 + ∥∆ρ∥2 +L2 +� ++ ε9 +� +∥vt∥2 +L2 + ∥v∥2 +H2 + ∥∇∆ρ∥2 +L2 +� +. +(3.56) +22 + +Combining (3.53)–(3.56), we have, for all ε ∈ (0, 1/2], there exists a positive constant Cε depending +on Ω, c0, α, β and ε such that +d +dt +� +M1(t) + ∥√µ curl v∥2 +L2 +� ++ ∥vt∥2 +L2 +≤ Cε +� +∥ρt∥2 +L2 + ∥v∥4 +L4 + ∥∇ρ∥4 +L4 +� +F(t) + Cε ∥v∥4 +L4 ++ ε +� +∥∇ρt∥2 +L2 + ∥v∥2 +H2 + ∥∇∆ρ∥2 +L2 +� +. +(3.57) +For general boundary case, that is, curl v = −n⊥ · B · (v + c0∇ρ−1), it suffices to calculate the +following extra term +� +∂ +φ(ρ)vt · B · ∇ρ = +� +∂ +φ(ρ)(vt · n⊥)n⊥ · B · ∇ρ += +� +(vt · n⊥)∇⊥φ(ρ) · B · ∇ρ + +� +φ(ρ)(vt · n⊥)∇⊥ · (B · ∇ρ) ++ +� +φ(ρ)∇⊥(vt · n⊥) · B · ∇ρ := +3 +� +i=1 +Gi, +(3.58) +where φ(s) := c0µ(s)s−2. For the first two terms of (3.58), using Lemma 2.1 and 3.1, we have +|G1 + G2| ≤ Cε1 +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� ++ ε1 ∥vt∥2 +L2 . +(3.59) +For G3, since we can not handle the term ∇⊥(vt · n⊥), it shall be converted into +G3 = d +dtM2(t) − +� +φ(ρ)t∇⊥(v · n⊥) · B · ∇ρ − +� +φ(ρ)∇⊥(v · n⊥) · B · ∇ρt +≥ d +dtM2(t) − +� +Cε2 +� +∥ρt∥2 +L2 + ∥∇ρ∥4 +L4 ∥∇v∥2 +L2 + ∥∇v∥2 +L2 +� ++ ε2 +� +∥v∥2 +H2 + ∥∇ρt∥2 +L2 +�� +. +Thus, combining (3.60)–(3.57), we deduce the estimate which is simliar with (3.57), that is +d +dt +� +M1(t) + ∥√µ curl v∥2 +L2 +� ++ ∥vt∥2 +L2 + d +dtM2(t) +≤ CεA4(t)F(t) + ε +� +∥∇ρt∥2 +L2 + ∥v∥2 +H2 + ∥∇∆ρ∥2 +L2 +� ++ A5(t), +(3.60) +for some integrable functions A4 and A5 defined on [0, ∞). +We still need estimate ∥v∥H2. Let us rewrite (3.49) as +− µ∆v + ∇π = F, +(3.61) +where +F := −ρvt + ∇(log ρ)t − ρu · ∇v + 2µ′∇ρ · D(v) + 2c0 div +� +µ∇2ρ−1� +− c0 div +� +ρv ⊗ ∇ρ−1� +− c2 +0 div +� +ρ∇ρ−1 ⊗ ∇ρ−1� +Since µ(ρ(x, t)) is bounded contiuous on QT from Lemma 2.7, it follows from Lemma 2.8 with +ϕ = −n⊥ · B · (v + c0∇ρ−1) that +∥v∥H2 + ∥π∥H1 ≤ C (∥F∥L2 + ∥ϕ∥H1) , +(3.62) +where +∥ϕ∥H1 ≤ C +� +∥∇v∥L2 + ∥∆ρ∥L2 + ∥∇ρ∥2 +L4 +� +, +∥F∥L2 ≤ C +� +∥vt∥L2 + ∥∇ρt∥L2 + ∥∇ρ∥2 +L4 ∥ρt∥L2 + ∥|v|·|∇v|∥L2 + ∥|∇ρ|·|∇v|∥L2 + ∥∇ρ∥3 +L6 ++ ∥|∇ρ|·|∇2ρ|∥L2 + ∥|v|·|∇2ρ|∥L2 + ∥|v|·|∇ρ|2∥L2� +≤ C (∥vt∥L2 + ∥∇ρt∥L2) + Cε +� +∥v∥2 +L4 + ∥∇ρ∥2 +L4 +� +(∥∇v∥L2 + ∥∆ρ∥L2 + ∥ρt∥L2) ++ ε (∥v∥H2 + ∥∇∆ρ∥L2) . +23 + +Hence, using the Poincaré’s inequality, we deduce from (3.62) that +∥v∥2 +H2 + ∥π∥2 +H1 ≤ C +� +∥v∥4 +L4 + ∥∇ρ∥4 +L4 +� � +∥∇v∥2 +L2 + ∥∆ρ∥2 +L2 + ∥ρt∥2 +L2 +� ++ C +� +∥vt∥2 +L2 + ∥∇ρt∥2 +L2 +� ++ C +� +∥∇∆ρ∥2 +L2 + ∥∇v∥2 +L2 +� +, +(3.63) +which gives (3.46). Finally, substituting (3.46) into (3.60), we obtain (3.45) and complete the proof +of the lemma. +For (ρ, v) satisfying condition (B’), we first convert (3.49) into +ρvt − div [2µD(v)] + ∇π1 += −ρv · ∇v + c0∇ log ρ · ∇v + c0 div +� +2µρ−1 � +∇2 log ρ − ∇ log ρ ⊗ ∇ log ρ +�� ++ c0 div (v ⊗ ∇ log ρ) − c2 +0 div +� +ρ−1∇ log ρ ⊗ ∇ log ρ +� +. +(3.64) +Then, following the calculations from (3.49) to (3.57) and from (3.61) to (3.63), we can derive the +similar estimates (even much easier, since vt is vanished on the boundary), that is, +d +dt ∥µD(v)∥2 +L2 + ν ∥vt∥2 +L2 ≤ Cε +� +∥∇ρ∥4 +L4 + ∥v∥4 +L4 + ∥ρt∥2 +L2 +� � +∥∇v∥2 +L2 + ∥∆ log ρ∥2 +L2 +� ++ ε +� +∥∇∆ log ρ∥2 +L2 + ∥v∥2 +H2 +� +, +(3.65) +∥F∥L2 ≤ C (∥vt∥L2 + ∥∇ log ρt∥L2) + Cε +� +∥v∥2 +L4 + ∥∇ρ∥2 +L4 +� +(∥∇v∥L2 + ∥∆ log ρ∥L2) ++ ε (∥v∥H2 + ∥∇∆ log ρ∥L2) +(3.66) +and, using Lemma 2.8 with Φ = 0, together with (3.66), +∥v∥2 +H2 + ∥π∥2 +H1 ≤ C ∥F∥2 +L2 +≤ Cε +� +∥v∥4 +L4 + ∥∇ρ∥4 +L4 +� � +∥∇v∥2 +L2 + ∥∆ log ρ∥2 +L2 +� ++ ε ∥∇∆ log ρ∥2 +L2 ++ C +� +∥vt∥2 +L2 + ∥∇(log ρ)t∥2 +L2 +� +. +(3.67) +Thus, we complete the proof by plugging (3.67) into (3.65). +Now, combining Lemma 3.9–3.11, we can complete the proof of Proposition 3.8. +Proof of Proposition 3.8. We first prove the case when (ρ, v) satisfies the condition (A’). Using +Lemma 2.2, (3.29) in Lemma 3.9 and (3.45), (3.46) in Lemma 3.11 leads to +d +dt +� +∥√µ curl v∥2 +L2 + ∥∆ρ∥2 +L2 + ∥ρt∥2 +L2 + M1(t) +� ++ ∥vt∥2 +L2 + ∥∇∆ρ∥2 +L2 + ∥∇ρt∥2 +L2 +≤ − d +dtM2(t) + ˜ +A1(t)F(t) + ˜ +A2(t), +≤ − d +dtM2(t) + ˜ +A1(t) +� +∥√µ curl v∥2 +L2 + ∥∆ρ∥2 +L2 + ∥ρt∥2 +L2 + M1(t) +� ++ ˜ +A2(t), +(3.68) +where ˜ +A1 and ˜ +A2 are positive integrable functions defined on [0, ∞). Using Gr¨onwall’s inequality +and Lemma 2.2 once again, we deduce the bound +sup +t∈[0,T ] +F(t) + +� T +0 +� +∥∇∆ρ∥2 +L2 + ∥∇ρt∥2 +L2 + ∥vt∥2 +L2 +� +dt +≤ C(∥v0∥2 +H1 + ∥ρ0∥2 +H2 + ∥v0∥2 +H1 ∥ρ0∥2 +H2 + 1) ≤ C, +(3.69) +where we have used, denote by ρt,0 = ρt(x, 0), +∥ρt,0∥L2 ≤ ∥ρ0∥H2 + ∥v0∥H1 ∥ρ0∥H2 , +24 + +M1 ≥ 0 for B positively semi-definited and +�����e +� T +0 h(t) dt +� T +0 +d +dtM2(t)e−� t +0 h(s) ds dt +����� ≤ ε sup +t∈[0,T ] +∥∇v∥2 +L2 + Cε sup +t∈[0,T ] +∥∇ρ∥2 +L2 , +where h(t) is an integrable function on [0, ∞). +Next, integrating (3.46) over [0, T ] and using the bound (3.69) gives +� T +0 +� +∥v∥2 +H2 + ∥π∥2 +H1 +� +dt ≤ C, +(3.70) +which shows (3.28). +To prove the case when (ρ, v) satisfies the condition (B’), we use (3.31) in Lemma 3.9 and (3.47) +in Lemma 3.11. It follows from Lemma 3.1 and +2 +� +|D(v)|2 = +� +|∇v|2 +that +d +dt +� +∥∇v∥2 +L2 + ∥(log ρ)t∥2 +L2 +� ++ ∥vt∥2 +L2 + ∥∇(log ρ)t∥2 +L2 +≤ ˜ +A3(t) +� +∥∇v∥2 +L2 + ∥(log ρ)t∥2 +L2 +� ++ ε ∥∇∆ log ρ∥2 +L2 , +(3.71) +for some nonegative integrable functions ˜ +A3. Using the Gr¨onwall’s inequality, one has the bound +sup +t∈[0,T ] +∥∇v∥2 +L2 + sup +t∈[0,T ] +∥(log ρ)t∥2 +L2 + +� T +0 +� +∥vt∥2 +L2 + ∥∇(log ρ)t∥2 +L2 +� +dt +≤ C(∥v0∥2 +H1 + ∥ρ0∥2 +H2 + ∥v0∥2 +H1 ∥ρ0∥2 +H2 + 1) ≤ C, +(3.72) +With the aid of the estimate (3.30), (3.32), (3.48) and (3.72), one has +sup +t∈[0,T ] +∥∆ log ρ∥2 +L2 + +� T +0 +� +∥∇∆ log ρ∥2 +L2 + ∥v∥2 +H2 + ∥π∥2 +H1 +� +dt ≤ C. +(3.73) +At last, noticing that +∆ρ = ρ∆ log ρ + ρ−1|∇ρ|2, +and +∇∆ρ = ∇ρ∆ log ρ + ρ∇∆ log ρ − ρ−2∇ρ|∇ρ|2 + 2ρ−1∇ρ · ∇2ρ, +we complete the proof of (3.28). +4 +A Priori Estimates (II): Case (C) +In this section, we will prove Theorem 1.5 via a different approach. The main difficulty lines here +is that, in this situation, v · n = 0 and v = −c0∇ρ−1 on ∂Ω, which makes one impossible to handle +the high order derivatives of ρ appeared in the boundary integrals when we deal with the energy +estimates of v. +To over come it, we may take a different decomposition on u. This idea mainly comes from +Lemma 2.4, which pushes us to introduce a new function Q = B[c0∆ρ−1] to eliminate the non- +divergence-free condition on u. More precisely, we split u into two parts, u = w + Q, and one can +find that w possesses the nice properties, that is, w is divergence-free and w|∂Ω = 0. Therefore, +we can use w to get the energy estimates for system (1.1)2. +Fortunately, in spite of this difficulty, we still has the estimates on ρ, which has been derived in +Section 3 and 3.3 such as (3.9), (3.39), etc. This is because those estimates only require v · n = 0 +on Ω. Then, using the relation v = u − c0∇ρ−1, one can easily change the norm of v into that of +u and ρ. +25 + +Before giving the proof of Theorem 1.5, we make some comments on the analysis in this section. +In this section, we devote to establish the higher order estimates for (ρ, u). One should notice that, +if Q = B[c0∆ρ−1], then Qt = B[c0∆ρ−1 +t ]. With this fact, using Remark 2.3 and Lemma 2.4, one +has the following estimates, + + + + + + + +∥Q∥Lp ≤ C ∥∇ρ∥Lp , +∥Q∥H1 ≤ C +� +∥∆ρ∥L2 + ∥∇ρ∥2 +L4 +� +, +∥Qt∥Lp ≤ C (∥∇ρt∥Lp + ∥|ρt||∇ρ|∥Lp) ; +(4.1) +for 1 < p < ∞ and some constants C depending only on Ω, c0 and p. In addition, the following +equivalence will be used frequently, that is, +∥u∥Lp + ∥∇ρ∥Lp ∼ ∥v∥Lp + ∥∇ρ∥Lp , +∥∇u∥Lp + ∥∆ρ∥Lp + ∥∇ρ∥2 +L2p ∼ ∥∇v∥Lp + ∥∆ρ∥Lp + ∥∇ρ∥2 +L2p . +(4.2) +Now, we turn to the proof. The key of the proof is deriving the following proposition. Using +the idea from [22], we first assume the bounds (4.3) and obtain the a priori estimates of (ρ, u), see +Lemma 4.2–4.3. Then, these bounds lead to smaller ones (4.4) provided ∥∇u0∥L2 suitably small, +which means that we can close the energy estimates of (ρ, u) and, consequently, we complete the +proof of the theorem. +Proposition 4.1. Suppose that (ρ, u, π) is a smooth solution of (1.1). There exists a positive +constant δ depending only on Ω, α, β and c0 such that, if ∥∇u0∥L2 ≤ δ and +sup +t∈[0,T ] +∥∇ρ∥L4 ≤ 2, +� T +0 +� +∥∆ρ∥4 +L2 + ∥∇u∥4 +L2 +� +dt ≤ 2 ∥∇u0∥2 +L2 , +(4.3) +then the following estimates hold +sup +t∈[0,T ] +∥∇ρ∥L4 ≤ 1, +� T +0 +� +∥∆ρ∥4 +L2 + ∥∇u∥4 +L2 +� +dt ≤ ∥∇u0∥2 +L2 . +(4.4) +In order to prove Proposition 4.1, we need the following estimates. +Lemma 4.2. Suppose that (ρ, u, π) is a smooth solution of (1.1). +There exists some positive +constant C depending on Ω, α, β and c0 such that, for all T ∈ (0, ∞), α ≤ ρ ≤ β and +sup +t∈[0,T ] +∥ρ − (ρ0)Ω∥2 +L2 + +� T +0 +∥∇ρ∥2 +L2 dt ≤ ∥ρ0 − (ρ0)Ω∥2 +L2 . +(4.5) +Furthermore, if ∥∇u0∥L2 ≤ 1 and (4.3) holds, one has +sup +t∈[0,T ] +∥∇ρ∥2 +L2 + +� T +0 +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� +dt ≤ C ∥∇ρ0∥2 +L2 . +(4.6) +Lemma 4.3. Suppose that ∥∇u0∥L2 ≤ 1 and (4.3) is established, then one has +sup +t∈[0,T ] +F(t) + +� T +0 +� +G(t) + ∥π∥2 +H1 +� +dt ≤ C ∥∇u0∥2 +L2 , +(4.7) +where λ and C are positive constants depending on Ω, α, β and c0, +F(t) := ∥∇u∥2 +L2 + ∥ρt∥2 +L2 + ∥∆ρ∥2 +L2 , +G(t) := ∥ut∥2 +L2 + ∥∆u∥2 +L2 + ∥∇∆ρ∥2 +L2 + ∥∇ρt∥2 +L2 . +Remark 4.4. One should keep in mind that we always have +∥ρ0 − (ρ0)Ω∥L2 ≤ C ∥∇ρ0∥L2 ≤ C ∥u0∥L2 ≤ C ∥∇u0∥L2 . +(4.8) +26 + +We temporarily assume that Lemma 4.2–4.3 are established and prove Proposition 4.1. +Proof of Proposition 4.1. Since Lemma 4.3 is established, we have, for some C1 > 0 depending +only on Ω, +sup +t∈[0,T ] +∥∇ρ∥L4 ≤ C1 sup +t∈[0,T ] +∥∆ρ∥L2 ≤ C1C ∥∇u0∥L2 ≤ 1 +provided +∥∇u0∥L2 ≤ δ1 := (C1C)−1, +(4.9) +and, by Lemma 2.1, +� T +0 +� +∥∆ρ∥4 +L2 + ∥∇u∥4 +L2 +� +dt ≤ +� +sup +t∈[0,T ] +∥∆ρ∥2 +L2 + sup +t∈[0,T ] +∥∇u∥2 +L2 +� � T +0 +� +∥∆ρ∥2 +L2 + ∥∇u∥2 +L2 +� +dt +≤ C2 ∥∇u0∥4 +L2 ≤ ∥∇u0∥2 +L2 +provided +∥∇u0∥L2 ≤ δ2 := C−1, +(4.10) +where C is the constant in Lemma 4.3. +Thus, if we choose +∥∇u0∥L2 ≤ δ := min {1, δ1, δ2} , +then the proof of Proposition 4.1 is completed. +Now, we trun back to prove Lemma 4.2–4.3. Since Lemma 4.2 has already been proved in +Section 3, we only give the proof for Lemma 4.3. +Proof of Lemma 4.3. We start with the lower order estimate of u. Multiplying w on the both sides +of (1.1)2 and integrating over Ω, one has +d +dt +� 1 +2ρ|u|2 + +� +2µ|D(u)|2 = +� +ρut · Q + +� +ρu · ∇u · Q − +� +div[2µD(u)] · Q += +� +ρut · Q + +� +ρu · ∇u · Q + +� +2µD(u) · ∇Q +:= +3 +� +i=1 +Si, +(4.11) +where, using estimates (4.1), Lemma 2.1 and 3.1, + + + + + + + +|S1| ≤ C ∥Q∥L2 ∥ut∥L2 ≤ Cε1 ∥∇ρ∥2 +L2 + ε1 ∥ut∥2 +L2 , +|S2| ≤ C ∥u∥L4 ∥∇u∥L2 ∥Q∥L4 ≤ Cε2 ∥∆ρ∥4 +L2 ∥u∥2 +L2 + ε2 ∥∇u∥2 +L2 , +|S3| ≤ C ∥∇Q∥L2 ∥∇u∥L2 ≤ Cε3 +� +∥∆ρ∥2 +L2 + ∥∇ρ∥4 +L4 +� ++ ε3 ∥∇u∥2 +L2 . +(4.12) +Here, we still use the notation εi ∈ (0, 1/2] and the constant Cεi as before. Combining (4.11) and +(4.12) leads to, ∃ ν > 0, +d +dt ∥u∥2 +L2 + ν ∥∇u∥2 +L2 ≤ Cε +� +∥∆ρ∥4 +L2 ∥u∥2 +L2 + ∥∆ρ∥2 +L2 + ∥∇ρ∥4 +L4 +� ++ Cε ∥∇ρ∥2 +L2 + ε ∥ut∥2 +L2 , +(4.13) +For the estimate of ut, multiplying wt on the both sides of (1.1)2, one has +� +ρ|ut|2 + d +dt +� +µ|D(u)|2 = − +� +ρut · Qt − +� +ρu · ∇u · wt ++ +� +µt|D(u)|2 − +� +div[2µD(u)] · Qt +:= +4 +� +i=1 +Ui. +(4.14) +27 + +Using Lemma 2.1 and 3.1, (4.1)–(4.3) and Poincaré’s inequality, we have +∥Qt∥2 +L2 ≤ C +� +∥∇ρt∥2 +L2 + ∥ρt∥2 +L4 ∥∇ρ∥2 +L4 +� +≤ C ∥∇ρt∥2 +L2 +(4.15) +and, thus, + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +|U1| ≤ C ∥Qt∥L2 ∥ut∥L2 ≤ Cε1 ∥∇ρt∥2 +L2 + ε1 ∥ut∥2 +L2 , +|U2| ≤ C ∥u∥L4 ∥∇u∥L4 ∥wt∥L2 +≤ Cε2 ∥u∥2 +L4 ∥∇u∥2 +L4 + ε2 ∥ut∥2 +L2 + C ∥∇ρt∥2 +L2 +≤ Cε2,ε3 ∥∇u∥4 +L2 ∥∇u∥2 +L2 + C ∥∇ρt∥2 +L2 + ε2 ∥ut∥2 +L2 + ε3 ∥∆u∥2 +L2 , +|U3| ≤ C ∥ρt∥L2 ∥∇u∥2 +L4 ≤ Cε4 ∥∇u∥2 +L2 ∥ρt∥2 +L2 + ε4 ∥∆u∥2 +L2 +≤ Cε4 ∥∇u∥4 +L2 ∥ρt∥2 +L2 + ε4 +� +∥∇ρt∥2 +L2 + ∥∆u∥2 +L2 +� +, +|U4| ≤ C +� +∥∇ρ∥L4 ∥∇u∥L4 + ∥∇2u∥L2� +∥∇ρt∥L2 +≤ Cε5 ∥∆ρ∥4 +L2 ∥∇u∥2 +L2 + Cε5 ∥∇ρt∥2 +L2 + ε5 ∥∆u∥2 +L2 . +(4.16) +Thus, combining (4.14) and (4.16), one has +d +dt∥√µD(u)∥2 +L2 + ν ∥ut∥2 +L2 ≤ Cε +� +∥∇u∥4 +L2 + ∥∆ρ∥4 +L2 +� +∥∇u∥2 +L2 + Cε ∥∇u∥4 +L2 ∥ρt∥2 +L2 ++ Cε ∥∇ρt∥2 +L2 + ε ∥∆u∥2 +L2 . +(4.17) +From the observation of (4.17), one have to derive the estimate of ∆u, or that of ∆v. Unfortu- +nately, we can not directly use, for example, (3.63) in the Section 3.3. The main obstacle here is +that (3.63) strongly depends on the conitnuity of ρ and we have not closed the lower bounds of v +yet, so that we can not apply Lemma 2.7–2.8 (notice that Lemma 2.7 requiring v ∈ Ls(0, T ; Lr)). +Consequently, we apply Lemma 2.9 with Φ = −c0∇ρ−1 on +− div[2µD(v)] + ∇π = F, +(4.18) +where +F = −ρut − ρu · ∇u + c0 div +� +2µ∇2ρ−1� +and, using condition (4.3), Lemma 2.1 and 3.1 and Poincaré’s inequality, +∥F∥L2 ≤ C ∥ut∥L2 + Cε2 ∥u∥2 +L4 ∥∇u∥L2 + C ∥∇ρ∥2 +L4 ∥∆ρ∥L2 ++ ε2 ∥∆u∥L2 + C ∥∇∆ρ∥L2 +≤ C ∥ut∥L2 + Cε2 +� +∥u∥2 +L4 + ∥∇ρ∥2 +L4 +� +(∥∇u∥L2 + ∥∆ρ∥L2) ++ ε2 ∥v∥H2 + C ∥∇∆ρ∥L2 , +(4.19) +where we have applied the estimate (4.2) and +∥∆u∥L2 ≤ C +� +∥∆v∥L2 + +��∇∆ρ−1�� +L2 +� +≤ C +� +∥∆v∥L2 + ∥∇ρ∥2 +L4 ∥∆ρ∥L2 + ∥∇∆ρ∥L2 +� +use (4.3) +≤ C (∥∆v∥L2 + ∥∇∆ρ∥L2) . +(4.20) +Then, using Lemma 2.1, Lemma 2.9 and Poincaré’s inequality, combining the condition (4.3) and +the estimates (4.19), we can derive a similar estimate of (3.63), that is, +∥v∥2 +H2 + ∥∇π∥2 +L2 ≤ C ∥ut∥2 +L2 + C +� +∥u∥4 +L4 + ∥∇ρ∥4 +L4 +� � +∥∇u∥2 +L2 + ∥∆ρ∥2 +L2 +� ++ C ∥∇∆ρ∥2 +L2 . +Using again (4.20) we derive that +∥∆u∥2 +L2 + ∥∇π∥2 +L2 ≤ C ∥ut∥2 +L2 + C +� +∥u∥4 +L4 + ∥∇ρ∥4 +L4 +� � +∥∇u∥2 +L2 + ∥∆ρ∥2 +L2 +� ++ C ∥∇∆ρ∥2 +L2 . +(4.21) +28 + +Then, substituting (4.21) into (4.17), ∃ ν > 0, +d +dt ∥∇u∥2 +L2 + ν ∥∆u∥2 +L2 + ν ∥ut∥2 +L2 ≤ Cε +� +∥∇u∥4 +L2 + ∥∆ρ∥4 +L2 +� +F(t) ++ Cε ∥∇ρt∥2 +L2 + ε ∥∇∆ρ∥2 +L2 . +(4.22) +Next, in order to close the estimate (4.22), we turn to get the bounds of ∇ρt and ∇∆ρ. From +the estimate (3.39) and +∥vt∥L2 ≤ C(∥ut∥L2 + +��∇ρ−1 +t +�� +L2) +≤ C +� +∥ut∥L2 + ∥∇ρt∥L2 + ∥∇ρ∥2 +L4 ∥ρt∥L2 +� +use (4.3) +≤ C(∥ut∥L2 + ∥∇ρt∥L2) +we have +d +dt ∥ρt∥2 +L2 + ν ∥∇ρt∥2 +L2 ≤ Cε +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� +∥ρt∥2 +L2 + ε ∥ut∥2 +L2 . +(4.23) +On the other hand, for ∇∆ρ, since the estimate (3.35) we derived in the Section 3.3 is still +valid, replacing v by u and ∇ρ, one has +d +dt ∥∆ρ∥2 +L2 + ν ∥∇∆ρ∥2 +L2 ≤ Cε +� +∥∆ρ∥4 +L2 + ∥∇u∥4 +L2 +� � +∥∆ρ∥2 +L2 + ∥∇u∥2 +L2 +� ++ ε ∥∆u∥2 +L2 . +(4.24) +Using this inequality together with (4.21), we eliminate term ∆u and, then, we subsititute it, +alonging with(4.23), into (4.22) to deduce that +d +dtF(t) + νG(t) ≤ C +� +∥∇u∥4 +L2 + ∥∆ρ∥4 +L2 + ∥∆ρ∥2 +L2 +� +F(t). +(4.25) +Since we have ∥∇ρ0∥H1 ≤ ∥∇u0∥L2 from Remark 1.11, applying Gr¨onwall’s inequality for (4.25) +and using Lemma 4.2, we have +sup +t∈[0,T ] +F(t) + +� T +0 +G(t) dt ≤ C +� +∥∇u0∥2 +L2 + ∥∇u0∥4 +L2 +� +≤ C ∥∇u0∥2 +L2 , +(4.26) +which, turning back to (4.21) to get the bound for π, implies the estimate (4.7). Therefore, we +complete the proof of Lemma 4.3. +5 +Proof of Theorem 1.6 +In this section, we devote to accomplish the proofs of Theorem 1.6 in several steps. Our proofs are +basically relying on the approach in [11]. In subsection 5.1, we are going to solve the linearized +system and give some basic uniform estimates, which is critical for the existence proofs in next +few subsections. Next, in subsection 5.2–5.3, we will construct an approximate system and use +the contraction mapping theorem to show that it admits a unique smooth solution. Finally, in +subsection 5.4, we will prove Theorem 1.6. +5.1 +Linearized Problem +Consider the following linearized problem + + + + + + + + + + + + + + + + + + + +ρt + Φ · ∇ρ − div(ϕ−1∇ρ) = 0, +ρut + ρ(Φ + ∇ϕ−1) · ∇u − div(2µD) + ∇p = 0, +div u = c0∆ρ−1, +ρ|t=0 = ρ0, +u|t=0 = u0, +α ≤ ρ0 ≤ β, (ρ0, u0) ∈ [C∞(Ω)]4 satisfying (1.11), +(ρ, u) satisfies one of the bundary conditions (A) − (C), +(L.P.) +where +� +p = 0, µ = µ(x, t) ∈ H1(0, T ; C∞(Ω)) is a positive function. +29 + +Lemma 5.1 (Linearized problem). Assume that the hypotheses of Theorem 1.6 are satisfied by the +data (ρ0, u0). If Φ and ϕ satisifes the following conditions + + + + + +Φ ∈ C([0, T ]; H1) ∩ L2(0, T ; H2), +Φt ∈ L2(0, T ; L2), +ϕ ∈ C([0, T ]; H2) ∩ L2(0, T ; H3), +ϕt ∈ C([0, T ]; L2) ∩ L2(0, T ; H1), +c−1 ≤ ϕ ≤ c, +div Φ = 0 in Ω, +(5.1) +then there exists a unique global strong solution (ρ, u, p) to the problem (L.P.) satisfying (1.10). +Proof. We only give the a priori estimates. The unique sovablity is obvious, since we can first solve +(L.P.)1 by the theories of linear parabolic equations, see [25] and, then, derive u from (L.P.)2, see +[11, 35]. +Firstly, the sup-bound and lower order estimates of ρ has been proved in Section 3. See Lemma +3.1 and 3.3, that is, +α ≤ ρ ≤ β, +∥ρ∥2 +L2 + +� t +0 +∥∇ρ∥2 +L2 ds ≤ C(Ω, c) (or C(Ω, c, ˜ρ)). +(5.2) +Next, we multiply −∆ρ on (L.P.)1 and integrate over Ω, then, using Lemma 2.1, we have +d +dt ∥∇ρ∥2 +L2 + ν ∥∆ρ∥2 +L2 ≤ C +� +(|Φ| + |∇ϕ|) |∇ρ||∆ρ| +≤ C +� +∥Φ∥4 +L4 + ∥∇ϕ∥4 +L4 +� +∥∇ρ∥2 +L2 + ν +2 ∥∆ρ∥2 +L2 . +Thus, +d +dt ∥∇ρ∥2 +L2 + ν ∥∆ρ∥2 +L2 ≤ C +� +∥Φ∥4 +L4 + ∥∇ϕ∥4 +L4 +� +∥∇ρ∥2 +L2 . +(5.3) +By virtue of Gr¨onwall’s inequality, we derive +∥∇ρ∥2 +L2 (t) + ν +� t +0 +∥∆ρ∥2 +L2 ds ≤ ∥∇ρ0∥2 +L2 exp +� +C +� t +0 +� +∥Φ∥4 +L4 + ∥∇ϕ∥4 +L4 +� +ds +� +. +(5.4) +To get the higher order bounds, if n · ∇ρ = 0 on ∂Ω, we apply −∇∆ρ∇ on both sides of (L.P.)1 +and integrate over Ω, then +d +dt +� 1 +2 |∆ρ|2 + +� +ϕ−1 |∇∆ρ|2 = +� +∇∆ρ · ∇Φ · ∇ρ + +� +Φ · ∇2ρ · ∇∆ρ +− +� +2ϕ−3|∇ϕ|2∇ρ · ∇∆ρ + +� +ϕ−2∇ϕ · ∇2ρ · ∇∆ρ ++ +� +ϕ−2∇ρ · ∇2ϕ · ∇∆ρ − +� +ϕ−2∆ρ∇ϕ · ∇∆ρ. +Then, applying Lemma 2.1 and Poincar e’s inequality, we have +d +dt ∥∆ρ∥2 +L2 + ν ∥∇∆ρ∥2 +L2 +≤ (∥∇Φ∥L4 ∥∇ρ∥L4 + ∥Φ∥L4 ∥∆ρ∥L4) ∥∇∆ρ∥L2 + C ∥∇ϕ∥2 +L8 ∥∇ρ∥L4 ∥∇∆ρ∥L2 ++ C +� +∥∇ϕ∥L4 ∥∆ρ∥L4 + +��∇2ϕ +�� +L4 ∥∇ρ∥L4 +� +∥∇∆ρ∥L2 +≤ C +� +∥Φ∥4 +W 1,4 + ∥ϕ∥4 +W 2,4 + ∥∇ϕ∥8 +L8 +� +∥∆ρ∥2 +L2 + ν +2 ∥∇∆ρ∥2 +L2 , +that is +d +dt ∥∆ρ∥2 +L2 + ν ∥∇∆ρ∥2 +L2 ≤ C +� +∥Φ∥4 +W 1,4 + ∥ϕ∥4 +W 2,4 + ∥∇ϕ∥8 +L8 +� +∥∆ρ∥2 +L2 , +(5.5) +which, using Gr¨onwall’s inequality, leads to +∥∆ρ∥2 +L2 (t) + ν +� t +0 +∥∇∆ρ∥2 +L2 ds +≤ C ∥∆ρ0∥2 +L2 exp +�� t +0 +� +∥Φ∥4 +W 1,4 + ∥ϕ∥4 +W 2,4 + ∥∇ϕ∥8 +L8 +� +ds +� +. +(5.6) +30 + +For ρ satisfying the non-homogeneous Dirichlet condition, that is, ρ|∂Ω = ˜ρ, taking ρt∂t on +(L.P.)1, one has +d +dt +� 1 +2 |ρt|2 + ν +� +|∇ρt|2 ≤ +� +|Φt| |∇ρ| |ρt| + C +� +|ϕt| |ρt||∇ϕ||∇ρ| ++ C +� +|ρt| |∇ρt| |∇ϕ| + C +� +|ρt| |∇ϕt| |∇ρ| ++ C +� +|ϕt||ρt| |∆ρ| . +Simiarly, it follows from Lemma 2.1 that +d +dt ∥ρt∥2 +L2 + ν ∥∇ρt∥2 +L2 +≤ C (∥Φt∥L2 + ∥∇ϕ∥L4 ∥ϕt∥L4) ∥∇ρ∥L4 ∥ρt∥L4 ++ C (∥∇ϕ∥L4 ∥∇ρt∥L2 + ∥∇ρ∥L4 ∥∇ϕt∥L2) ∥ρt∥L4 + C ∥∆ρ∥L2 ∥ϕt∥L4 ∥ρt∥L4 +≤ +� +∥Φt∥2 +L2 + ∥ϕt∥4 +L4 + ∥∇ϕ∥4 +L4 + ∥∇ϕt∥2 +L2 +� +∥ρt∥2 +L2 + ν +2 ∥∇ρt∥2 +L2 ++ C +� +∥Φt∥2 +L2 + ∥ϕt∥4 +L4 + ∥∇ϕ∥4 +L4 + ∥∇ϕt∥2 +L2 +� +∥∇ρ∥2 +L2 + C ∥∆ρ∥2 +L2 . +that is, +d +dt ∥ρt∥2 +L2 + ν ∥∇ρt∥2 +L2 +≤ C +� +∥Φt∥2 +L2 + ∥ϕt∥4 +L4 + ∥∇ϕ∥4 +L4 + ∥∇ϕt∥2 +L2 +� � +∥ρt∥2 +L2 + ∥∇ρ∥2 +L2 +� ++ C ∥∆ρ∥2 +L2 . +(5.7) +Then, using Gr¨onwall’s inequality and (5.4), one has +∥ρt∥2 +L2 (t) + ν +� t +0 +∥∇ρt∥2 +L2 ds ≤ C(Ω, c, α, β, Φ, ϕ, u0). +(5.8) +Noticing that (5.8) also holds for the Neumann case. +Next, we take L2-norm on (L.P.)1 and use Lemma 2.1 to get +∥∆ρ∥2 +L2 ≤ C ∥ρt∥2 +L2 + C +� +∥Φ∥4 +L4 + ∥∇ϕ∥4 +L4 +� +∥∇ρ∥2 +L2 +(5.9) +and take ∇ on both sides of (L.P.)1 to get +∥∇∆ρ∥2 +L2 ≤ C ∥∇ρt∥2 +L2 + C +� +∥Φ∥4 +W 1,4 + ∥∇ϕ∥8 +L8 + ∥∇ϕ∥4 +L4 +� +∥∆ρ∥2 +L2 . +(5.10) +Thus, we use (5.9) and (5.10), alonging with (5.4) and (5.7), to deduce that +∥∆ρ∥2 +L2 (t) + +� t +0 +∥∇∆ρ∥2 +L2 ds ≤ C(Ω, c, α, β, Φ, ϕ, u0). +(5.11) +In conclusion, for both cases, it follows from (5.2), (5.4), (5.6), (5.8) and (5.11) that +∥ρt∥2 +L2 (t) + ∥∇ρ∥2 +H1 (t) + +� t +0 +� +∥ρt∥2 +H1 + ∥∇ρ∥2 +H2 +� +ds ≤ C(Ω, c, α, β, Φ, ϕ, u0). +(5.12) +The next part is estimating v (for case (A) or (B)) or u (for case (C)). We first treat the case +for u satisfying (C). Note that (L.P.)1 is equivalent to +ρt + div[ρ(Φ + ∇ϕ−1)] = ∆(ρϕ−1). +Thus, if we multiply (L.P.)2 by w := u − Q with Q = B[c0∆ρ−1] and integrate over Ω, we have +d +dt +� 1 +2ρ|u|2 + +� +2µ|D(u)|2 += +� +∆ +� +ρϕ−1� |u|2 +2 ++ +� +ρut · Q + +� +ρ(Φ + ∇ϕ−1) · ∇u · Q + +� +2µD(u) · ∇Q. +31 + +Then, using Lemma 2.1 and 2.2, we have +d +dt ∥√ρu∥2 +L2 + ν ∥∇u∥2 +L2 ≤ C +� +∥∆ρ∥L2 + ∥∇ρ∥L4 ∥∇ϕ∥L4 + ∥∇ϕ∥2 +L4 + ∥∆ϕ∥L2 +� +∥u∥2 +L4 ++ C +��Φ + ∇ϕ−1�� +L4 ∥Q∥L4 ∥∇u∥L2 + C ∥∇Q∥L2 ∥∇u∥L2 ++ C ∥Q∥L2 ∥ut∥L2 +≤ C +� +∥∆ρ∥2 +L2 + ∥∇ρ∥4 +L4 + ∥∇ϕ∥4 +L4 + ∥∆ϕ∥2 +L2 +� +∥u∥2 +L2 ++ C +��Φ + ∇ϕ−1��4 +L4 ∥Q∥2 +L2 + C ∥∇Q∥2 +L2 + ν +2 ∥∇u∥2 +L2 ++ Cε ∥Q∥2 +L2 + ε ∥ut∥2 +L2 , +that is, +d +dt ∥√ρu∥2 +L2 + ν ∥∇u∥2 +L2 ≤ C +� +∥∆ρ∥2 +L2 + ∥∇ρ∥4 +L4 + ∥∇ϕ∥4 +L4 + ∥∆ϕ∥2 +L2 +� +∥u∥2 +L2 ++ C +��Φ + ∇ϕ−1��4 +L4 ∥∇ρ∥2 +L2 + Cε ∥Q∥2 +H1 + ε ∥ut∥2 +L2 , +≤ C +� +∥∆ρ∥2 +L2 + ∥∇ρ∥4 +L4 + ∥∇ϕ∥4 +L4 + ∥∆ϕ∥2 +L2 +� +∥u∥2 +L2 ++ C +��Φ + ∇ϕ−1��4 +L4 ∥∇ρ∥2 +L2 + Cε +� +∥∇ρ∥4 +L4 + ∥∇ρ∥2 +H1 +� ++ ε ∥ut∥2 +L2 , +(5.13) +Next, for the estimate of ut, multiplying wt = ut − Qt on the both sides of (L.P.)2, one has +� +ρ|ut|2 + d +dt +� +µ|D(u)|2 = − +� +ρut · Qt − +� +ρ(Φ + ∇ϕ−1) · ∇u · wt ++ +� +µt|D(u)|2 − +� +div[2µD(u)] · Qt. +Using again Lemma 2.1, applying Poincaré’s inequality and the fact that +∥Qt∥2 +L2 ≤ C +� +∥∇ρt∥2 +L2 + ∥ρt∥2 +L4 ∥∇ρ∥2 +L4 +� +, +we obtain +∥ut∥2 +L2 + d +dt ∥√µD(u)∥2 +L2 +≤ C +���Φ + ∇ϕ−1��2 +L4 + ∥µt∥L2 + ∥∇µ∥2 +L4 +� +∥∇u∥2 +L4 + C ∥Qt∥2 +L2 + C ∥Qt∥L2 ∥∆u∥L2 +≤ Cε +���Φ + ∇ϕ−1��4 +L4 + ∥µt∥2 +L2 + ∥∇µ∥4 +L4 + ∥∇ρ∥4 +L4 +� +∥∇u∥2 +L2 + ε ∥∆u∥2 +L2 ++ Cε +� +∥∇ρ∥4 +L4 ∥ρt∥2 +L2 + ∥∇ρt∥2 +L2 +� +(5.14) +To estimate ∆u, we change (L.P.)2 into the form +−µ∆v + ∇p = 2∇µ · D(u) − ρut − ρ(Φ + ∇ϕ−1) · ∇u + 2c0µ∇∆ρ−1, +which, using Lemma 2.8, leads to +∥v∥2 +H2 + ∥p∥2 +H1 ≤ C ∥ut∥2 +L2 + C +��∇∆ρ−1��2 +L2 ++ C +� +∥∇µ∥2 +L4 + +��Φ + ∇ϕ−1��2 +L4 +� +∥∇u∥2 +L4 , +(5.15) +that is, using Lemma 2.1, +∥∆u∥2 +L2 + ∥p∥2 +H1 ≤ C ∥ut∥2 +L2 + C +��∇∆ρ−1��2 +L2 ++ C +� +∥∇µ∥4 +L4 + +��Φ + ∇ϕ−1��4 +L4 +� +∥∇u∥2 +L2 . +(5.16) +32 + +Then, using this bound together with (5.14), we have +∥ut∥2 +L2 + d +dt ∥√µD(u)∥2 +L2 +≤ Cε +���Φ + ∇ϕ−1��4 +L4 + ∥µt∥2 +L2 + ∥∇µ∥4 +L4 + ∥∇ρ∥4 +L4 +� +∥∇u∥2 +L2 + ε +��∇∆ρ−1��2 +L2 ++ C ∥∇ρ∥4 +L4 ∥ρt∥2 +L2 + C ∥∇ρt∥2 +L2 +(5.17) +Finally, combining (5.13) and (5.17), then, using Gr¨onwall’s inequality and the bound (5.12), we +obtain the a priori estimates for u. +For case (A) or (B), as we have said at the end of Section 1, we convert (L.P.) into + + + + + + + + + + + + + + + + + + + +ρt + Φ · ∇ρ − div(ϕ−1∇ρ) = 0, +� +ρvt + ρ(Φ + ∇ϕ−1) · ∇v − div(2µD(v)) + ∇p += c0∇(log ρ)t − c0ρ(Φ + ∇ϕ−1) · ∇2ρ−1 + c0 div(2µ∇2ρ−1), +div v = 0. +(5.18) +Then, we can apply the energy arguements analogous to the case (C). More precisely, multiplying +v on both sides of (5.18)2 and integrating over Ω, we have, for all ε ∈ (0, 1/2], +d +dt +� 1 +2ρ |v|2 − +� +div [2µD(v)] · v += +� +∆(ρϕ−1)|v|2 +2 ++ +� +c0 div +� +2µ∇2ρ−1� +· v − +� +c0ρ(Φ + ∇ϕ−1) · ∇2ρ−1 · v +≤ C +� +∥∆ρ∥L2 + ∥∇ρ∥L4 ∥∇ϕ∥L4 + ∥∇ϕ∥2 +L4 + ∥∆ϕ∥L2 +� +∥v∥2 +L4 ++ C ∥∇µ∥L4 +��∇2ρ−1�� +L2 ∥v∥L4 + C +��∇∆ρ−1�� +L2 ∥v∥L2 ++ C (∥Φ∥L4 + ∥∇ϕ∥L4) +��∇2ρ−1�� +L2 ∥v∥L4 +≤ Cε +� +∥∆ρ∥2 +L2 + ∥∇ρ∥4 +L4 + ∥∇µ∥4 +L4 + ∥Φ∥4 +L4 + ∥∇ϕ∥4 +L4 + ∥∆ϕ∥2 +L2 +� +∥v∥2 +L2 ++ ε +� +∥∇v∥2 +L2 + +��∇∆ρ−1��2 +L2 +� +(5.19) +where we have used Poincaré’s inequality for the last inequality. For the term − +� +div [2µ(ρ)D(v)]·v, +we directly use the results in Lemma 3.5, that is, for case (B’), +− +� +div [2µD(v)] · v = +� +2µ|D(v)|2 ≥ ν ∥∇v∥2 +L2 , (use Lemma 2.2), +(5.20) +while, for case (A’), +− +� +div [2µD(v)] · v ≥ ν ∥∇v∥2 +L2 − +� +Cε ∥∇µ∥4 +L4 ∥√ρv∥2 +L2 + Cε +��∆ρ−1��2 +L2 + ε ∥∇v∥2 +L2 +� +. +(5.21) +Thus, combining (5.19)–(5.21), in both cases, +d +dt ∥√ρv∥2 +L2 + ν ∥∇v∥2 +L2 +≤ C +� +∥∆ρ∥2 +L2 + ∥∇ρ∥4 +L4 + ∥∇µ∥4 +L4 + ∥Φ∥4 +L4 + ∥∇ϕ∥4 +L4 + ∥∆ϕ∥2 +L2 +� +∥v∥2 +L2 ++ C +��∇∆ρ−1��2 +L2 +(5.22) +which, using Gr¨onwall’s inequality and (5.12), gives +∥v∥2 +L2 (t) + +� t +0 +∥∇v∥2 +L2 ds ≤ C(Ω, c, α, β, Φ, ϕ, ρ0, v0). +(5.23) +33 + +Next, multiplying (5.18)2 by vt and integrating over Ω, one has, using Lemma 2.1, +� +ρ |vt|2 − +� +div[2µD(v)] · vt += − +� +ρ(Φ + ∇ϕ−1) · ∇v · vt + +� +c0 div +� +2µ∇2ρ−1� +· vt +− +� +c0ρ(Φ + ∇ϕ−1) · ∇2ρ−1 · vt +≤ C (∥Φ∥L4 + ∥∇ϕ∥L4) ∥∇v∥L4 ∥vt∥L2 ++ C +� +∥∇µ∥L4 +��∇2ρ−1�� +L4 + +��∇∆ρ−1�� +L2 +� +∥vt∥L2 ++ C (∥Φ∥L4 + ∥∇ϕ∥L4) +��∇2ρ−1�� +L4 ∥vt∥L2 +≤ Cε +� +∥Φ∥4 +L4 + ∥∇ϕ∥4 +L4 +� +∥∇v∥2 +L2 + Cε +��∇∆ρ−1��2 +L2 ++ Cε +� +∥Φ∥4 +L4 + ∥∇ϕ∥4 +L4 + ∥∇µ∥4 +L4 +� ��∆ρ−1��2 +L2 + ε +� +∥vt∥2 +L2 + ∥v∥2 +H2 +� +. +(5.24) +For the term − � div[2µD(v)] · vt, if (ρ, v) satisfies the condition (A’), we use the proof from (3.53) +to (3.60), +− +� +div[2µD(v)] · vt ≥ d +dt +� +M1(t) + ∥√µ curl v∥2 +L2 +� ++ d +dtM2(t) +− Cε +� +∥µt∥2 +H1 + ∥∇µ∥4 +L4 + 1 +� +∥v∥2 +H1 +− Cε ∥∇µ∥4 +L4 +��∇ρ−1��2 +L2 − Cε +��∆ρ−1��2 +L2 +− ε +� +∥vt∥2 +L2 + ∥v∥2 +H2 + +��∇ρ−1 +t +��2 +L2 +� +. +(5.25) +Recalling that +M1(t) = +� +∂ +µv · B · v, +M2(t) = +� +c0µ∇⊥(v · n⊥) · B · ∇ρ−1. +For case (B’), it is much easier, +− +� +div[2µD(v)] · vt = +� +µ d +dt|D(v)|2 += d +dt +� +µ|D(v)|2 − +� +µt|D(v)|2 +≥ d +dt +� +µ|D(v)|2 − +� +Cε ∥µt∥2 +L2 ∥∇v∥2 +L2 + ε ∥v∥2 +H2 +� +. +(5.26) +Furthermore, from (5.15), we have +∥v∥2 +H2 + ∥p∥2 +H1 ≤ C ∥vt∥2 +L2 + C ∥∇ log ρt∥2 +L2 + C +��∆ρ−1��2 +L2 ++ C +� +∥∇µ∥4 +L4 + ∥Φ∥4 +L4 + ∥∇ϕ∥4 +L4 +� � +∥∇v∥2 +L2 + +��∆ρ−1��2 +L2 +� +. +(5.27) +Therefore, combining (5.24)–(5.27), without loss of generality, one has +∥vt∥2 +L2 + d +dt +� +M1(t) + ∥√µ curl v∥2 +L2 +� ++ d +dtM2(t) +≤ Cε +� +∥Φ∥4 +L4 + ∥∇ϕ∥4 +L4 + ∥µt∥2 +H1 + ∥∇µ∥4 +L4 + 1 +� +∥v∥2 +H1 + Cε +��∇∆ρ−1��2 +L2 ++ Cε +� +∥Φ∥4 +L4 + ∥∇ϕ∥4 +L4 + ∥∇µ∥4 +L4 +� ��∆ρ−1��2 +L2 + ε +� +∥∇ log ρt∥2 +L2 + +��∇ρ−1 +t +��2 +L2 +� +. +(5.28) +Finally, using Gr¨onwall’s inequality, (5.12) and (5.23), we deduce that +∥∇v∥2 +L2 (t) + +� t +0 +∥vt∥2 +L2 ds ≤ C(Ω, c, α, β, Φ, ϕ, ρ0, v0). +(5.29) +Therefore, we complete the proof of Lemma 5.1. +34 + +5.2 +Preliminary Reductions +We claim that it is enough to prove the existence results for smooth initial data (ρ0, u0) satisfying +the compatiblity conditions (1.11). Once this is established, for general data (ρ0, u0), we can build +a sequence of smooth initial data (ρn +0, un +0) such that it converges to (ρ0, u0) in some appropriate +functional spaces. +Then, we can obtain a corresponding sequence of solutions (ρn, vn, πn) (or +(ρn, un, πn)), which is uniformly bounded with respect of n, satisfying the initial data (ρn +0, vn +0 ) +(or (ρn +0, un +0)). We may use the weak convergence method and compactness reults to deduce that +(ρn, vn, πn) (or (ρn, un, πn)) converges to (ρ, v, π) (or (ρ, u, π)) in some functional spaces. As a +result, (ρ, v, π) (or (ρ, u, π)) will be the solution we expect, which proves our claim. +Now, we explain how we obtain such smooth data. We begin with α ≤ ρ0 ≤ β, u0 ∈ H1 +ω (the +case u0 ∈ H1 +nd or H1 +0 can be done analogously). First, as we have said in Remark 1.11, we can +derive that ρ0 ∈ H2 from the compatiability condition (1.11) +� +∆ρ−1 +0 += c−1 +0 +div u0, +x ∈ Ω, +n · ∇ρ−1 +0 += 0, +x ∈ ∂Ω. +(5.30) +Consequently, we get v0 ∈ H1 by setting v0 = u0 − c0∇ρ−1 +0 . Then, we can construct a smooth +sequence (ˆρn +0, ˆvn +0 ) ∈ [C∞(Ω)]4 via flatten method and partition of unity such that +ˆρn +0 +s +−−→ ρ0 +in H2, +ˆvn +0 +s +−−→ v0 +in H1. +(5.31) +For details, see [15] Chapter 5. +However, the sequence (ˆρn +0 , ˆvn +0 ) may be failed to satisfy the boundary conditions and divergence- +free condition, which means that we need further construction. First of all, we solve the following +ellptic problem + + + + + +∆ρn +0 = ∆ˆρn +0 , +x ∈ Ω, +n · ∇ρn +0 = 0, +x ∈ ∂Ω, +(ρn +0 )Ω = (ρ0)Ω. +Of course, for each n ≥ 1, ρn +0 ∈ C∞(Ω) is unique and +∥∇(ρn +0 − ρm +0 )∥H1 ≤ C ∥∇(ˆρn +0 − ˆρm +0 )∥H1 → 0, +as n, m → ∞. +(5.32) +It follows from (5.31) that {ρn +0} is a Cauchy sequence, and, thus, ρn +0 +s +−−→ ρ0 in H2. Using Sobolev +embedding theorem, H2 ֒→ C(Ω), we deduce that ρn +0 converges uniformly to ρ0 and thus, without +loss of generality, we may assume that ρn +0 ∈ [α, β]. +Next, to construct vn +0 , we borrow from the construction method in [27], Appendix A. More +precisely, consider following Stokes problem of vn +0 + + + + + +−∆vn +0 + ∇pn = −∆ˆvn +0 , +x ∈ Ω, +div vn +0 = 0, +x ∈ Ω, +vn +0 · n = 0, curl vn +0 = −n⊥ ·B ·[vn +0 + c0∇(ρn +0 )−1], +x ∈ ∂Ω, +where +� +pn = 0 and {ρn +0} is the smooth sequence we just obtain. In view of Lemma 2.5, there +exists a unique smooth solution (vn +0 , pn) ∈ [C∞(Ω)]4 such that +∥vn +0 ∥H1 + ∥pn∥L2 ≤ C(∥ˆvn +0 ∥H1 + ∥ρn +0∥H2). +(5.33) +Thus, we obtain a Cauchy sequence +∥vn +0 − vm +0 ∥H1 + ∥pn − pm∥L2 ≤ C(∥ˆvn +0 − ˆvm +0 ∥H1 + ∥ρn +0 − ρm +0 ∥H2) −→ 0, as n, m → ∞, +because of (5.31) and the strong covergence of {ρn +0}. Without loss of generality, let +vn +0 +s +−−→ v0 in H1 and pn +s +−−→ p in L2. +35 + +Then, V0 := v0 − v0 solves + + + + + +−∆V0 + ∇p = 0, +x ∈ Ω, +div V0 = 0, +x ∈ Ω, +V0 · n = 0, curl V0 = −n⊥ · B · V0, +x ∈ ∂Ω. +It follows form the uniqueness of Stokes equations that V0 ≡ 0, that is, v0 = v0. Thus, we have +found a smooth divergence-free sequence vn +0 , which satisfies the condtion (A’), that converges +strongly to v0 in H1. +If we treat the case (C), we just turn back to un +0 by setting +un +0 := vn +0 + c0∇(ρn +0 )−1. +Then, it is easy to check that un +0 ∈ C∞(Ω), un +0|∂Ω = 0 and (ρn +0 , un +0) satisfies the compatiablity +condition (1.11). +5.3 +Approximate System +In order to get the existence for (1.1), we first try to establish the smooth solutions for the following +system: + + + + + + + + + + + + + + + + + + + +ρt + vη · ∇ρ − c0 div +� +ρ−1 +η ∇ρ +� += 0, +ρut + ρuη · ∇u − div[2µǫD(u)] + ∇π = 0, +div u = c0∆ρ−1, +ǫ, η ∈ (0, 1], +ρ|t=0 = ρ0, +u|t=0 = u0, +α ≤ ρ0 ≤ β, (ρ0, u0) ∈ [C∞(Ω)]4 satisfying (1.11), +u0, (ρ, u) satisfies one of the bundary conditions (A) − (C). +(A.P.) +Let us give an explaination about the new elements in (A.P.). We define +uη := vη + ρη, +µǫ := µ(ρǫ), +and ρǫ, ρη, vη are constructed as we did in preceeding subsection, that is, ρǫ, ρη, vη ∈ C∞(Ω), +div vη = 0, ρǫ, ρη ∈ [α, β] and ρǫ, ρη, vη satisfying corresponding boundary conditions. +For convenience, we collect some bounds here which will be used later. Obviously, we always +have µǫ ∈ C∞(Ω) for every fixed t, ǫ and, for all 1 ≤ r ≤ ∞, k ∈ N, +∥µǫ∥Lr ≤ C(r, Ω) ∥ρ∥L∞ , +∥∇µǫ∥Lr ≤ C(r, ǫ, Ω) ∥ρ∥L∞ , +��∇kρη +�� +Lr ≤ C(k, r, η, Ω) ∥ρ∥H1 , +��∇kρǫ +�� +Lr ≤ C(k, r, ǫ, Ω) ∥ρ∥H1 , +��∇kvη +�� +Lr ≤ C(k, r, η, Ω) ∥v∥L2 . +(5.34) +Also, we have the following uniform controls, for all 1 ≤ q < ∞, +∥vη∥W ℓ,q ≤ C ∥v∥W ℓ,q , +ℓ = 0, 1, +∥ρη∥W ℓ,q ≤ C ∥ρ∥W ℓ,q , +ℓ = 0, 1, 2. +(5.35) +Our aim is proving the following theorem. +Theorem 5.2. For every fixed ǫ, η ∈ (0, 1], the problem (A.P.) admits an unique smooth solution +on QT1 for some positive time T1. +Our proof is organized as follows. In the first part, we use iteration arguements and contraction +mapping theorem to establish the unique smooth solution of (A.P.) for every fixed η and ǫ. Then, +we recover the original system (1.1) by letting η, ǫ tend to 0 in turn with help of the uniform +estimates. +36 + +5.3.1 +Uniform Bounds + + + + + +ρn +t + vn−1 +η +· ∇ρn − c0 div +� +(ρn−1 +η +)−1∇ρn� += 0, +ρnun +t + ρnun−1 +η +· ∇un − div[2µn +ǫ D(un)] + ∇πn = 0, +div un = c0∆(ρn)−1, +(5.36) +with the initial-boundary conditions +(ρn, un)(x, 0) = (ρ0, u0), +in Ω, +(5.37) +n · ∇ρn = 0, +un = 0 +on ∂Ω × (0, T ). +(5.38) +where we use the following notations +µn = µ(ρn), +Qn = B[c0∆(ρn)−1], +vn = un + c0∇(ρn)−1, +wn = un − Qn +To prove the existence for (5.36), we construct approximate solutions as follows. We first define +(ρ0, u0) = (C, 0) and, then, assume that (ρn−1, un−1) was defined for n ≥ 1, let (ρn, un, πn) be the +unique global strong solution to the problem (5.36). +To prove the uniform bounds for the approximate solutions, we introduce the function HN(t) +defined by +HN(t) := + + + +max1≤n≤N +� +1 + ∥ρn∥2 +H2 + ∥vn∥2 +H1 + ∥ρn +t ∥2 +L2 +� +, +case (A) or (B) +max1≤n≤N +� +1 + ∥ρn∥2 +H2 + ∥un∥2 +H1 + ∥ρn +t ∥2 +L2 +� +, +case (C) +Observe that, in all cases, it follows from the maximal principle and energy estimates that +α ≤ ρn ≤ β, +sup +t∈[0,T ] +∥ρn∥2 +L2 + +� T +0 +∥∇ρn∥2 +L2 ≤ C, for all T ∈ (0, ∞). +(5.39) +Moreover, let N be a fixed large number, we have +Lemma 5.3. There exists a positive constant C depending on Ω, c0, α, β and ρ0 such that +∥∇ρn∥2 +L2 (t) + +� t +0 +∥∆ρn∥2 +L2 ds ≤ C + C +� t +0 +HN(s)3 ds, +(5.40) +for all n, 1 ≤ n ≤ N. +Proof. Let n ≥ 2. From (5.3), +d +dt ∥∇ρn∥2 +L2 + ν ∥∆ρn∥2 +L2 ≤ C +���vn−1 +η +��4 +L4 + +��∇ρn−1 +η +��4 +L4 +� +∥∇ρn∥2 +L2 +≤ C +���un−1��4 +L4 + +��∇ρn−1��4 +L4 +� +∥∇ρn∥2 +L2 +≤ CHN(t)3. +Then, we integrate from 0 to t with respect of time and finish the proof of lemma. +The next Lemma concerns with the uniform bounds for case (C). +Lemma 5.4. Let (ρ, u) satisfy the condition (C). There exists a positive constant C depending on +Ω, c0, α, β and u0 such that +� +∥un∥2 +H1 (t) + ∥∇ρn∥2 +H1 (t) + ∥ρn +t ∥2 +L2 (t) +� ++ +� t +0 +� +∥un∥2 +H2 + ∥∆ρn∥2 +H1 + ∥ρn +t ∥2 +H1 +� +ds +≤ C + C +� t +0 +HN(s)4 ds, +(5.41) +for all n, 1 ≤ n ≤ N. +37 + +Proof. For the higher regularity, we apply −∇∆ρn∇ on both sides of (5.36) and, then, integrate +over Ω to derive the analogue of (3.33) +d +dt +� 1 +2 |∆ρn|2 + +� +c0 +ρn−1 +η +|∇∆ρn|2 += +� +∇∆ρn · ∇vn−1 +η +· ∇ρn + +� +vn−1 +η +· ∇2ρn · ∇∆ρn +− +� +2c0 +(ρn−1 +η +)3 +��∇ρn−1 +η +��2 ∇ρn · ∇∆ρn + +� +c0 +(ρn−1 +η +)2 ∇(∇ρn · ∇ρn−1 +η +) · ∇∆ρn ++ +� +c0 +ρn−1 +η +∆ρn∇ρn−1 +η +· ∇∆ρn. +Then, applying Lemma 2.1, we can obtain the following inequality which is similar with (3.35), +d +dt ∥∆ρn∥2 +L2 + ν ∥∇∆ρn∥2 +L2 +≤ Cε +���∇ρn−1��4 +L4 + ∥∇ρn∥4 +L4 + +��vn−1��4 +L4 +� +× +���∆ρn−1��2 +L2 + ∥∆ρn∥2 +L2 + +��∇vn−1��2 +L2 +� ++ ε +��∇vn−1��2 +H1 +≤ Cε +���∇ρn−1��4 +L4 + ∥∇ρn∥4 +L4 + +��un−1��4 +L4 +� +× +���∆ρn−1��2 +L2 + +��∇ρn−1��4 +L4 + ∥∆ρn∥2 +L2 + +��∇un−1��2 +L2 +� ++ ε +��∇vn−1��2 +H1 +≤ CεHN(t)4 + ε +���∆un−1��2 +L2 + +��∇∆ρn−1��2 +L2 + +��∇ρn−1��4 +L4 +��∆ρn−1��2 +L2 +� +, +which gives +d +dt ∥∆ρn∥2 +L2 + ν ∥∇∆ρn∥2 +L2 ≤ Cε1HN(t)4 + ε1 +��∆un−1��2 +L2 +(5.42) +Moreover, from (3.39),we also have +d +dt ∥ρn +t ∥2 +L2 + ∥∇ρn +t ∥2 +L2 ≤ Cε2HN(t)3 + ε2 +��un−1 +t +��2 +L2 , +(5.43) +To get the bounds for un, noticing that the mass equation can be written as +ρn +t + div(ρnun−1 +η +) = c0∆(ρn/ρn−1 +η +). +Then, we follow the proof from (5.13) to get, for all n ≥ 2 +d +dt +��√ρnun��2 +L2 + ν ∥∇un∥2 +L2 +≤ Cε3 ∥∇ρn∥2 +L2 + C +���un−1��4 +L4 + +��∇ρn−1��4 +L4 + ∥∇ρn∥4 +L4 + ∥∆ρn∥2 +L2 +� ++ C +� +∥∇ρn∥4 +L4 + +��∇ρn−1��4 +L4 + ∥∆ρn∥2 +L2 + +��∆ρn−1��2 +L2 +� +∥un∥2 +L2 + ε3 ∥un +t ∥2 +L2 +≤ Cε3HN(t)3 + ε3 ∥un +t ∥2 +L2 . +(5.44) +Similarly, for un +t , it follows from (5.14) that +∥un +t ∥2 +L2 + d +dt +��� +µnǫ D(un) +��2 +L2 +≤ C +���un−1��2 +L4 + ∥ρn +t ∥L2 + ∥∇ρn∥2 +L4 +� +∥∇un∥2 +L4 + C ∥Qn +t ∥2 +L2 + C ∥Qn +t ∥L2 ∥∆un∥L2 +≤ Cε4 +� +HN(t)3 + ∥∇ρn +t ∥2 +L2 +� ++ ε4 ∥∆un∥2 +L2 . +(5.45) +Combining (5.44)–(5.45), we obtain +� +∥un +t ∥2 +L2 + ∥∇un∥2 +L2 +� ++ d +dt +���� +µnǫ D(un) +��2 +L2 + +��√ρnun��2 +L2 +� +≤ Cε5 +� +HN(t)3 + ∥∇ρn +t ∥2 +L2 +� ++ ε5 ∥∆un∥2 +L2 . +(5.46) +38 + +Moreover, it follows from (5.16) that +∥∆un∥2 +L2 + ∥πn∥2 +H1 ≤ C +� +HN(t)4 + ∥∇∆ρn∥2 +L2 + ∥un +t ∥2 +L2 +� +. +(5.47) +Plugging this into (5.42) and (5.46) and, then, combining two of them, we have +d +dt +���� +µnǫ D(un) +��2 +L2 + ∥∆ρn∥2 +L2 +� ++ ν +� +∥∇∆ρn∥2 +L2 + ∥un +t ∥2 +L2 +� +≤ C ∥∇ρn +t ∥2 +L2 + Cε4HN(t)4 + ε4 +���∇∆ρn−1��2 +L2 + +��un−1 +t +��2 +L2 +� +. +Alonging with (5.43) and choosing ε2 small enough and ε4 = 1/2, one has +d +dt +���� +µnǫ D(un) +��2 +L2 + ∥∆ρn∥2 +L2 + 2C ∥ρn +t ∥2 +L2 +� ++ +� +∥∇∆ρn∥2 +L2 + ∥un +t ∥2 +L2 + ∥∇ρn +t ∥2 +L2 +� +≤ CHN(t)4 + 1 +2 +���∇∆ρn−1��2 +L2 + +��un−1 +t +��2 +L2 +� +. +(5.48) +For simplicity, we denote by the above +d +dtPn(t) + +� +∥∇∆ρn∥2 +L2 + ∥un +t ∥2 +L2 +� +≤ CHN(t)4 + 1 +2 +���∇∆ρn−1��2 +L2 + +��un−1 +t +��2 +L2 +� +. +Then, integrating over [0, t], one has +Pn(t) + +� t +0 +� +∥∇∆ρn∥2 +L2 + ∥un +t ∥2 +L2 +� +ds +≤ C +� +1 + +� t +0 +HN(s)4 ds +� ++ 1 +2 +� t +0 +���∇∆ρn−1��2 +L2 + +��un−1 +t +��2 +L2 +� +ds. +Using this recursive inequality for +� t +0 +� +∥∇∆ρn∥2 +L2 + ∥un +t ∥2 +L2 +� +ds, we obtain +� t +0 +� +∥∇∆ρn∥2 +L2 + ∥un +t ∥2 +L2 +� +ds ≤ +� +1 + 1 +2 + · · · + 1 +2n +� +C +� +1 + +� t +0 +HN(s)4 ds +� +≤ 2C +� +1 + +� t +0 +HN(s)4 ds +� +and hence, turning back to (5.48), we get +Pn(t) + +� t +0 +� +∥∇∆ρn∥2 +L2 + ∥un +t ∥2 +L2 + ∥∇ρn +t ∥2 +L2 +� +ds ≤ C +� +1 + +� t +0 +HN(s)4 ds +� +, +for all 2 ≤ n ≤ N and, thus, for all 1 ≤ n ≤ N. Finally, using (5.47), we get the bounds for ∆un +and πn which concludes the lemma. +Next, we give the uniform estimates for condition (A) or (B). +Lemma 5.5. Let (ρ, v) satisfy the condition (A’) or (B’). There exists a positive constant C +depending on Ω, c0, α, β, ρ0 and v0 such that +� +∥vn∥2 +H1 (t) + ∥∇ρn∥2 +H1 (t) + ∥ρn +t ∥2 +L2 (t) +� ++ +� t +0 +� +∥vn∥2 +H2 + ∥∆ρn∥2 +H1 + ∥ρn +t ∥2 +H1 +� +ds +≤ C + C +� t +0 +HN(s)8 ds, +(5.49) +for all n, 1 ≤ n ≤ N. +39 + +Proof. We still only give the proof for case (A’). From (5.22) and the proof of Lemma 3.11, we get +d +dt +��√ρnvn��2 +L2 + ν ∥∇vn∥2 +L2 +≤ C +� +∥∆ρn∥2 +L2 + ∥∇ρn∥4 +L4 + ∥∇µn +ǫ ∥4 +L4 + +��vn−1 +η +��4 +L4 + +��∇ρn−1 +η +��4 +L4 + +��∆ρn−1 +η +��2 +L2 +� +∥vn∥2 +L2 ++ C +��∇∆(ρn)−1��2 +L2 +≤ CHN(t)3 + C ∥∇∆ρn∥2 +L2 +(5.50) +and +∥vn +t ∥2 +L2 + d +dt +� +Mn +1(t) + ∥√µ curl v∥2 +L2 +� ++ d +dtMn +2(t) +≤ CHN(t)3 + CHN(t)2 ∥∇vn∥2 +L4 +≤ CHN(t)3 + CHN(t)2 ∥∇vn∥L2 ∥v∥H2 , +(5.51) +while, for ∥v∥H2, we apply Lemma 2.9 for +− div[2µn +ǫ D(vn)] + ∇πn += −ρnvn +t + c0∇ log ρn +t − ρn[vn−1 +η ++ c0∇(ρn−1)−1] · ∇[vn +η + c0∇(ρn)−1] ++ div[2µn +ǫ ∇2(ρn)−1] := F n +(5.52) +to obtain +∥v∥H2 + ∥π∥H1 ≤ C +� +∥∇µn +ǫ ∥2 +L4 + 1 +� � +∥F n∥L2 + +��∆(ρn)−1�� +L2 +� ++ ∥∇µn +ǫ ∥2 +L4 ∥∇vn∥L2 +≤ CHN(t) +� +∥vn +t ∥L2 + ∥∇ log ρn +t ∥L2 + +��∇∆(ρn)−1�� +L2 +� ++ CHN(t) +3 +2 (∥∇vn∥L4 + +��∆(ρn)−1�� +L4) + CHN(t) +3 +2 +≤ CHN(t) +� +∥vn +t ∥L2 + ∥∇ρn +t ∥L2 + ∥∇∆ρn∥L2 + HN(t) +3 +2 +� ++ C +� +HN(t)4 + HN(t) +3 +2 +� ++ 1 +2 ∥vn∥H2 , +which leads to +∥v∥H2 + ∥π∥H1 ≤ CHN(t) (∥vn +t ∥L2 + ∥∇ρn +t ∥L2 + ∥∇∆ρn∥L2) + CHN(t)4. +(5.53) +Substituting this into (5.51), we have +∥vn +t ∥2 +L2 + d +dt +� +Mn +1(t) + ∥√µ curl v∥2 +L2 +� ++ d +dtMn +2(t) +≤ CHN(t)3 + CHN(t)4 (∥vn +t ∥L2 + ∥∇ρn +t ∥L2 + ∥∇∆ρn∥L2) + CHN(t)6 +≤ Cε1H8 +N(t) + ε1 +� +∥vn +t ∥2 +L2 + ∥∇ρn +t ∥2 +L2 + ∥∇∆ρn∥2 +L2 +� +. +(5.54) +On the other hand, following the proofs of (5.42)–(5.43), one has +d +dt ∥∆ρn∥2 +L2 + ν ∥∇∆ρn∥2 +L2 +≤ CHN(t)4 + C ∥∇ρn∥2 +L4 +��∇vn−1��2 +L4 +≤ CHN(t)4 + C ∥∇ρn∥2 +L4 +��∇vn−1�� +L2 +��vn−1�� +H2 +≤ Cε2HN(t)6 + ε2 +���vn−1 +t +��2 +L2 + +��∇ρn−1 +t +��2 +L2 + +��∇∆ρn−1��2 +L2 +� +(5.55) +and +d +dt ∥ρn +t ∥2 +L2 + ∥∇ρn +t ∥2 +L2 ≤ Cε3HN(t)3 + ε3 +���vn−1 +t +��2 +L2 + +��∇ρn−1 +t +��2 +L2 +� +, +(5.56) +Therefore, combining Lemma 5.3, (5.50) and (5.54)–(5.56) and, then, using the same recursive +arguements at the end of the proof of Lemma 5.4, we can obtain the desire bound (5.49). +40 + +Remark 5.6. It follows from (5.47) in Lemma 5.4 that the constant C in (5.41) depends on +ǫ ∈ (0, 1], which indicates that we can only obtain the local existence for the case (C) with µ = µǫ, +in particular, µ being a positive constant. However, from the proof of Lemma 5.5, since we used +Lemma 2.9 to get the estimate (5.53), the constant C in (5.49) is independent with ǫ and that is +why we could extend the local existence for cases (A) and (B) to general viscosity coefficient µ(ρ). +In conclusion, we have the bounds +HN(t) ≤ C +� +1 + +� t +0 +HN(s)q ds +� +, for some q > 1. +(5.57) +Thanks to this integral inequality, we can easily show that there exists a time T1 ∈ (0, T ) depending +only on Ω, c0, α, β and u0 such that +sup +t∈[0,T1] +HN(t) ≤ C0, +(5.58) +for some C0 independing with N. Therefore, we obtain the bounds, for all n ≥ 1, +sup +t∈[0,T 1] +� +∥un∥2 +H1 + ∥ρn∥2 +H2 + ∥ρn +t ∥2 +L2 +� ++ +� T1 +0 +� +∥un∥2 +H2 + ∥ρn∥2 +H3 + ∥ρn +t ∥2 +H1 +� +ds ≤ C0. +(5.59) +5.3.2 +Convergence +We next show that the whole sequence (ρn, un) converges to a solution to (A.P.) in a sufficiently +strong sense. Let +σn+1 := ρn+1 − ρn, +an+1 := un+1 − un, +bn+1 := vn+1 − vn, +cn+1 := Qn+1 − Qn +and +Yn(t) := +� +∥an∥2 +H1 + ∥σn∥2 +H2 + ∥σn +t ∥2 +L2 , +case (C) +∥bn∥2 +H1 + ∥σn∥2 +H2 + ∥σn +t ∥2 +L2 , +case (A) or (B) +Zn(t) := +� +∥an +t ∥2 +L2 + ∥σn∥2 +H3 + ∥σn +t ∥2 +H1 , +case (C) +∥bn +t ∥2 +L2 + ∥σn∥2 +H3 + ∥σn +t ∥2 +H1 , +case (A) or (B) +In addition, we always let In(t) and Bn(t) be generic functions associated with the bounds +(5.59) such that +� T1 +0 +In(t) dt + +sup +t∈[0,T1] +Bn(t) ≤ C0, +where C0 is the constant as in (5.59). +Case (C): +It follows from the linearized mass equation that +σn+1 +t ++ vn +η · ∇σn+1 − c0 div +� 1 +ρnη +∇σn+1 +� += −bn +η · ∇ρn − c0 div +� +σn +η +ρn−1 +η +ρnη +∇ρn +� +:= Gn +(5.60) +where +∥Gn∥H−1 ≤ C ∥∇ρn∥L4 +���bn +η +�� +L4 + +��σn +η +�� +L4 +� +≤ C ∥ρn∥H2 (∥an∥H1 + ∥σn∥H1) +∥Gn∥L2 ≤ ∥∇ρn∥L4 +��bn +η +�� +L4 + C ∥∆ρn∥L2 +��σn +η +�� +L∞ + C +��σn +η +�� +W 1,4 ∥∇ρn∥L4 +≤ C ∥ρn∥H2 (∥an∥H1 + ∥σn∥H2) +41 + +∥∇Gn∥L2 ≤ ∥∇ρn∥L4 +��∇bn +η +�� +L4 + ∥∆ρn∥L2 +��bn +η +�� +L∞ + C ∥∇ρn∥L4 +��∆σn +η +�� +L4 ++ C ∥∆ρn∥L2 +��∇σn +η +�� +L∞ + C +��(|∇ρn +η| + |∇ρn−1 +η +|)|∇ρn| +�� +L4 +��∇σn +η +�� +L4 ++ C +��� +|∇ρn +η|2 + |∇ρn−1 +η +|2 + |∇2ρn−1 +η +| + |∇2ρn +η| +� +|∇ρn| +�� +L2 +��σn +η +�� +L∞ ++ C +��(|∇ρn +η| + |∇ρn−1 +η +|)|∇2ρn| +�� +L2 +��σn +η +�� +L∞ + C ∥∇∆ρn∥L2 +��σn +η +�� +L∞ +≤ Cη +� +∥ρn∥H2 + ∥ρn∥2 +W 1,4 +� +(∥an∥L2 + ∥σn∥H2) ++ Cη ∥∇∆ρn∥L2 ∥σn∥H1 . +use (5.34) +Then, using the simplified notations, the above bounds can be written as follows +∥Gn∥H−1 + ∥Gn∥L2 ≤ CBn(t)Yn(t), +∥∇Gn∥L2 ≤ CηBn(t)Yn(t) + Cη +� +In(t) ∥σn∥H1 . +(5.61) +Next, we are going to establish the bounds for σn+1 and an+1. Multiplying (5.60) by σn+1 and +integrating over Ω, we obtain +d +dt +��σn+1��2 +L2 + ν +��∇σn+1��2 +L2 ≤ C ∥Gn∥H−1 +��σn+1�� +H1 , +then, using (5.61), we deduce that +d +dt +��σn+1��2 +L2 + ν +��∇σn+1��2 +L2 ≤ C ∥Gn∥2 +H−1 ≤ CBn(t)Yn(t). +(5.62) +Similar with (5.3), multiplying −∆σn+1 on both sides of (5.60) and integrating over Ω, one has +d +dt +��∇σn+1��2 +L2 + ν +��∆σn+1��2 +L2 ≤ C +���vn +η +��4 +L4 + +��∇ρn +η +��4 +L4 +� ��∇σn+1��2 +L2 + C ∥Gn∥2 +L2 +≤ CIn(t) +��∇σn+1��2 +L2 + CBn(t)Yn(t), +(5.63) +where we have used (5.61) for the last inequality. If we integrate (5.63) between [0, t], t < T1, we +have +��∇σn+1��2 +L2 (t) + +� t +0 +��∆σn+1��2 +L2 ds ≤ C +� t +0 +Yn(s) ds. +(5.64) +For ρn satisfying the Neumann condition, we copy the proof from (5.5) by applying −∇∆σn+1∇ +on (5.60), integrating over Ω and using (5.61), that is +d +dt +��∆σn+1��2 +L2 + ν +��∇∆σn+1��2 +L2 +≤ C +���vn +η +��4 +W 1,4 + +��ρn +η +��4 +W 2,4 + +��∇ρn +η +��8 +L8 +� ��∆σn+1��2 +L2 + C ∥∇Gn∥2 +L2 +≤ CIn(t) +��∆σn+1��2 +L2 + CBn(t)Yn(t) + CIn(t) ∥σn∥2 +H1 . +This, alonging with (5.64), implies that +d +dt +��∆σn+1��2 +L2 + ν +��∇∆σn+1��2 +L2 +≤ CIn(t) +��∆σn+1��2 +L2 + CBn(t)Yn(t) + CIn(t) +� t +0 +Yn(s) ds. +(5.65) +For σn+1 +t +, we multiply σn+1 +t +on the both sides of (5.60) and integrating over Ω, it follows analogously +from (5.7) that +d +dt +��σn+1 +t +��2 +L2 + ν +��∇σn+1 +t +��2 +L2 +≤ Cε +� +∥un +t ∥2 +L2 + ∥ρn +t ∥4 +L4 + ∥∇ρn∥4 +L4 + ∥∇ρn +t ∥2 +L2 +� ���σn+1 +t +��2 +L2 + +��∇σn+1��2 +L2 +� ++ ε +���∆σn+1��2 +L2 + ∥an +t ∥2 +L2 +� ++ C ∥∆ρn∥2 +L2 ∥σn∥2 +H1 +≤ CεIn(t) +���σn+1 +t +��2 +L2 + +��σn+1��2 +H1 +� ++ CBn(t)Yn(t) + ε +���∆σn+1��2 +L2 + ∥an +t ∥2 +L2 +� +(5.66) +42 + +On the other hand, we differeniate the equations between those of un+1 and un to get +ρn+1an+1 +t ++ ρn+1un +η · ∇an+1 − div[2µn+1 +η +D(an+1)] + ∇(πn+1 − πn) += −σn+1(un +t + un +η · ∇un) − ρnan +η · ∇un + div[2(µn+1 +ǫ +− µn +ǫ )D(un)] := Kn, +(5.67) +where +∥Kn∥H−1 ≤ +��un +t + un +η · ∇un�� +L2 +��σn+1�� +L∞ + ∥∇un∥L2 +��an +η +�� +L∞ + ∥∇un∥L2 +��σn+1 +η +�� +L∞ +∥Kn∥L2 ≤ +��un +t + un +η · ∇un�� +L2 +��σn+1�� +L∞ + ∥∇un∥L2 +��an +η +�� +L∞ + ∥∆un∥L2 +��σn+1 +η +�� +L∞ ++ C ∥∇un∥L4 +��∇ρn +η +�� +L4 +��σn+1 +η +�� +L∞ + C ∥∇un∥L4 +��∇σn+1 +η +�� +L4 , +that is, +∥Kn∥H−1 + ∥Kn∥L2 ≤ C +� +In(t) +��σn+1�� +H2 + CηBn(t) ∥an∥L2 , +(5.68) +Next, following the proof of (5.13), we multilpy an+1 − cn+1 on both sides of (5.67), integrate +over Ω and use (5.68) to obtain +d +dt +��� +� +ρn+1an+1��� +2 +L2 + ν +��∇an+1��2 +L2 +≤ C +���∆ρn+1��2 +L2 + ∥∆ρn∥2 +L2 + +��∇ρn+1��4 +L4 + ∥∇ρn∥4 +L4 +� ��an+1��2 +L2 ++ C +��un +η +��2 +L4 +��cn+1��2 +L4 + Cε +��cn+1��2 +H1 + ε +��an+1 +t +��2 +L2 + C ∥Kn∥2 +H−1 +≤ CεIn(t) +���an+1��2 +L2 + +��σn+1��2 +H2 +� ++ CBn(t) ∥an∥2 +L2 + C +��∆σn+1��2 +L2 + ε +��an+1 +t +��2 +L2 . +(5.69) +Here, for the last inequality, we have used Lemma 2.1 and +��cn+1�� +Lp ≤ C +��σn+1�� +W 1,p , +��cn+1�� +H1 ≤ C +���∇ρn+1��2 +L4 + ∥∇ρn∥2 +L4 + ∥∇ρn∥4 +L8 + ∥∆ρn∥2 +L4 +� ��σn+1�� +H1 ++ C +��∆σn+1�� +L2 +≤ C +� +In(t) +��σn+1�� +H1 + C +��∆σn+1�� +L2 +To get the higher bound for an, multiplying (5.67) by an+1 +t +− cn+1 +t +, it follows from (5.14) that +d +dt +��∇an+1��2 +L2 + ν +��an+1 +t +��2 +L2 +≤ C +���un +η +��2 +L4 + +��ρn+1 +η,t +�� +L2 + +��∇ρn+1 +η +��2 +L4 +� ��∇an+1��2 +L4 + C +��cn+1 +t +��2 +L2 ++ C +��cn+1 +t +�� +L2 +��∆an+1�� +L2 + C ∥Kn∥2 +L2 +≤ CεIn(t) +���∇an+1��2 +L2 + +��σn+1��2 +H2 + +��σn+1 +t +��2 +L2 +� ++ Cε +��∇σn+1 +t +��2 +L2 ++ CBn(t) ∥an∥2 +L2 + ε +��∆an+1��2 +L2 +(5.70) +where we have used the fact that +��cn+1 +t +�� +L2 ≤ C +��∇σn+1 +t +�� +L2 + C (∥∇ρn +t ∥L2 + ∥ρn +t ∥L4 ∥∇ρn∥L4) +��σn+1�� +L∞ ++ C ∥ρn +t ∥L4 +��∇σn+1�� +L4 + C +��∇ρn+1�� +L4 +��σn+1 +t +�� +L4 +≤ C +��∇σn+1 +t +�� +L2 + C +� +In(t) +���σn+1�� +H2 + +��σn+1 +t +�� +L2 +� +. +At last, in order to get the estimate of ∆an+1, we use (5.16) with the additional term Kn, +��∆an+1��2 +L2 + +��πn+1 − πn��2 +H1 ≤ C +��an+1 +t +��2 +L2 + C +��∇∆[(ρn+1)−1 − (ρn)−1] +��2 +L2 ++ C +���∇ρn +η +��4 +L4 + +��un +η +��4 +L4 +� ��∇an+1��2 +L2 + C ∥Kn∥2 +L2 +≤ C +��an+1 +t +��2 +L2 + C +��∇∆σn+1��2 +L2 +use (5.68) ++ CIn(t) +���∇an+1��2 +L2 + +��σn+1��2 +H2 +� ++ CBn(t) ∥an∥2 +L2 . +(5.71) +43 + +We substitute above into (5.70) to get +d +dt +��∇an+1��2 +L2 + ν +��an+1 +t +��2 +L2 +≤ CεIn(t) +���∇an+1��2 +L2 + +��σn+1��2 +H2 + +��σn+1 +t +��2 +L2 +� ++ Cε +��∇σn+1 +t +��2 +L2 ++ CBn(t) ∥an∥2 +L2 + ε +��∇∆σn+1��2 +L2 +(5.72) +Therefore, combining (5.62)–(5.63), (5.65)–(5.66), (5.69) and (5.72), we eventually get +d +dtYn+1(t) + νZn+1(t) ≤ C +� +In(t)Yn+1(t) + Bn(t)Yn(t) + In(t) +� t +0 +Yn(s) ds +� +. +(5.73) +Then, applying the Gr¨onwall’s inequality and recalling that Yn(0) = 0 and the definitions of +In(t), Bn(t), one has, for all t ∈ (0, T1), +Yn+1(t) ≤ C0 +� t +0 +� +CBn(s)Yn(s) ds + CIn(s) +� s +0 +Yn(τ) dτ +� +ds +≤ C +� t +0 +Yn(s) ds, +which reduces to the Volterra-type integral equation. After a simple recursive argument, we can +show that +sup +t∈[0,T1] +Yn+1(t) ≤ C (CT1)n−1 +(n − 1)! +� T1 +0 +Y1(t) dt. +(5.74) +Applying the contraction mapping theorem and using this inequality together with (5.73), we show +that the sequence (ρn, un) converges strongly to an unique limit (ρ, u) and, as a consequence, πn +converges strongly to a function π. More precisely, we have +ρn +s +−−→ ρ +in C([0, T ]; H2) ∩ L2(0, T ; H3), +un +s +−−→ u +in C([0, T ]; H1) ∩ L2(0, T ; H2), +un +t +s +−−→ ut +in L2([0, T ]; L2), +πn +s +−−→ π +in L2(0, T ; H1). +Of course, (ρ, u, π) is the unique strong solution in Ω×(0, T1) for (A.P.). Furthermore, we can show +(ρ, u, π) is acually smooth. Indeed, sicne u ∈ L2(0, T ; H2) ∩ H1(0, T ; L2), vη, ρη ∈ H1(0, T ; H∞). +With this regularity on vη, ρη, using the regularity theories of parabolic equations for (A.P.)1, +we can derive that ρ ∈ H2([0, T ]; H∞). Then, applying the Lp-theory ([35]) for (A.P.)2, we get +u ∈ H2(0, T ; H∞) and, hence, we can bootstrap and gain more time regualrity on vη, ρη then ρ, +which implies that (ρ, u) ∈ C∞(QT ). Therefore, we finish the proof of Theorem 5.2. +Case (A) or (B): +We only consider the case (A) here and case (B) can be proved identically. Firstly, it follows +from (5.68) that +∥Kn∥H−1 + ∥Kn∥L2 ≤ C +� +In(t) +��σn+1�� +H2 + CηBn(t)(∥bn∥L2 + ∥σn∥H1). +(5.75) +Then, applying the estimates (5.22) and (5.28) with ϕ = ρn +η and Φ = vn +η and using (5.75), we can +44 + +obtain +d +dt +��� +� +ρn+1bn+1��� +2 +L2 + ν +��∇bn+1��2 +L2 +≤ C +���∆ρn+1��2 +L2 + +��∇ρn+1��4 +L4 + +��∇µn+1 +ǫ +��4 +L4 +� ��bn+1��2 +L2 ++ C +���vn +η +��4 +L4 + +��∇ρn +η +��4 +L4 + +��∆ρn +η +��2 +L2 +� ��bn+1��2 +L2 ++ C +��∇∆[(ρn+1)−1 − (ρn)−1] +��2 +L2 + C ∥Kn∥2 +H−1 +≤ CIn(t) +���bn+1��2 +L2 + +��σn+1��2 +H2 +� ++ C +��∇∆σn+1��2 +L2 + CBn(t) +� +∥bn∥2 +L2 + ∥σn∥2 +H1 +� +(5.76) +and +��bn+1 +t +��2 +L2 + d +dt +� +Mn +1(t) + +��∇bn+1��2 +L2 +� ++ d +dtMn +2(t) +≤ Cε +���vn +η +��4 +L4 + +��∇ρn +η +��4 +L4 + +��µn+1 +ǫ,t +��2 +H1 + +��∇µn+1 +ǫ +��4 +L4 + 1 +� ��bn+1��2 +H1 ++ Cε +���vn +η +��4 +L4 + +��∇ρn +η +��4 +L4 + +��∇µn+1 +ǫ +��4 +L4 +� ��∆[(ρn+1)−1 − (ρn)−1] +��2 +L2 ++ Cε +��∇∆[(ρn+1)−1 − (ρn)−1] +��2 +L2 ++ ε +���∇(log ρn+1 − log ρn)t +��2 +L2 + +��∇[(ρn+1)−1 − (ρn)−1]t +��2 +L2 +� ++ C ∥Kn∥2 +L2 +≤ CεIn(t) +���bn+1��2 +H1 + +��σn+1��2 +H2 +� ++ Cε +��∇∆σn+1��2 +L2 + ε +��∇σn+1 +t +��2 +L2 ++ CBn(t) +� +∥bn∥2 +L2 + ∥σn∥2 +H1 +� +, +(5.77) +where +Mn +1(t) := +� +∂ +µn+1 +ǫ +bn+1 · B · bn+1, +Mn +2(t) := +� +c0µn+1 +ǫ +∇⊥(bn+1 · n⊥) · B · ∇ +� +(ρn+1)−1 − (ρn)−1� +. +Therefore, combining (5.62)–(5.63), (5.65)–(5.66), (5.76)–(5.77), we have +d +dtYn+1(t) + d +dtMn +2(t) + νZn+1(t) +≤ C +� +In(t)Yn+1(t) + Bn(t)Yn(t) + In(t) +� t +0 +Yn(s) ds +� +. +(5.78) +Here, we have used the fact that Mn +1(t) ≥ 0. +Noticing that +|Mn +2(t)| ≤ ε +��bn+1��2 +H1 (t) + Cε +���∇σn+1��2 +L2 (t) + ∥∇ρn∥4 +L4 (t) +��σn+1��2 +L2 (t) +� +≤ ε +��bn+1��2 +H1 (t) + Cε +��σn+1��2 +H1 (t) +≤ ε +��bn+1��2 +H1 (t) + Cε +� t +0 +Yn(t) +use (5.64) +Applying Gr¨onwall’s inequality and using (5.78), we finally get the Volterra-type integral equation +Yn+1(t) ≤ C +� t +0 +Yn(s) ds, +t ∈ (0, T1), +(5.79) +and, hence, following the proof of case (C), we complete the proof for the case (A). +In conclusion, we finish the proof for Theorem 5.2. +45 + +5.4 +Proofs of Theorem 1.6: Recover ǫ and η +We temporarily fix ǫ ∈ (0, 1] to let η → 0+. We still first consider the case (C). +The recovering process is standard. Using Theorem 5.2, we get a smooth sequence +(ρǫ,η, uǫ,η, πǫ,η) ∈ C∞(QT1) +which solves the problem (A.P.) for each ǫ, η ∈ (0, 1] (for simplicity, we use the notation (ρη, uη, πη)). +Next, we can follow the proofs in Lemma 5.3–5.4 step by step and the uniform bounds (5.35) to +obtain the following control +d +dtFη(t) + νGη(t) ≤ CFη(t)3, +(5.80) +where +Fη(t) := ∥uη∥2 +L2 + ∥ρη +t ∥2 +L2 + ∥ρη∥2 +H2 , +Gη(t) := ∥uη +t ∥2 +L2 + ∥uη∥2 +H2 + ∥ρη∥2 +H3 + ∥ρη +t ∥2 +H1 . +and C is a constant which is not depend on η. Using the inequality (5.80), we can easily deduce +that there eixsts a positive time T2 such that +sup +t∈[0,T2] +Fη(t) + +� T2 +0 +Gη(t) dt ≤ C2. +(5.81) +Therefore, using the above uniform esitmate and Lemma 2.10, we can derive that (ρη, uη, πη) +converges in some proper sense to the limit (ρ, u, π) such that + + + + + +ρt + div(ρu) = 0, +ρut + ρu · ∇u − div[2µǫD(u)] + ∇π = 0, +div u = c0∆ρ−1. +(5.82) +The convergence is easy to check, we left it to the reader. Of course, as a special case, we can +let µǫ be a constant µ and, thus, we have proved the uniqueness and existence of the local strong +solutions for the case (C). +For the case (A) or (B), the proof is basically the same. However, the difference lies in this +case is that we can recover ǫ → 0+ because of the uniform estimates of (ρǫ, uǫ, πǫ), ǫ ∈ (0, 1], see +Remark 5.6. The convergence is easy to check and we omit it. Thus, we have completed the proof +of the existence results for Theorem 1.6. +It remains to check the uniqueness for the case (A) or (B). However, this can be done by +following the proof in 5.3.2. Indeed, for example, if we consider the case (A) (another case can +be proved analogously), let (ρi, ui, πi), i = 1, 2, be two strong solutions on Ω × (0, T1) with same +initial data and set +σ := ρ1 − ρ2, +a := u1 − u2, +b := v1 − v2, +c := Q1 − Q2, +Y(t) := ∥a∥2 +H1 + ∥σ∥2 +H2 + ∥σt∥2 +L2 , +Z(t) := ∥at∥2 +L2 + ∥σ∥2 +H3 + ∥σt∥2 +H1 . +Then, we can derive the similar type of equations to (5.60) and (5.67), that is, +� +σt + v2 · ∇σ − c0 div +� +ρ−1 +2 ∇σ +� += −b · ∇ρ2 − c0 div +� +σρ−1 +1 ρ−1 +2 ∇ρ2 +� +, +ρ1at + ρ1u1 · ∇a − div[2µD(a)] + ∇(π1 − π2) = −σ(u1,t + u1 · ∇u1) − ρ2a · ∇u1. +Applying the same discussions from 5.3.2, we can get the following type inequality +d +dtY(t) + Z(t) ≤ CI(t)Y(t), +where I stands for some integrable functions on time interval (0, T1). +Thus, using Gr¨onwall’s +inequality and the fact that Y(0) = 0, we can easily deduce that Y(t) ≡ 0, which yields the +uniqueness. +46 + +6 +Proof of Theorem 1.3–1.5 +6.1 +Proof of Theorem 1.4–1.5 +Since we have already show the existence and uniqueness of strong solution on Ω×(0, T1) for some +positive times T1, the proof of global ones is quite standard with the a priori estiamtes we obtained +in Section 3–4. One thing we should mention is that there is a gap between the local existence and +the global one when (ρ, u) satisfies the condition (C) in that we only established the unique local +strong solution for µ = µǫ. In every case that follows, one should first recover ǫ → 0+ to get the +global existence and, then, show their uniqueness under the smallness assumption ∥∇u0∥L2 ≤ δ. +Fourtunately, the proof of either is simpe and indentical with that in subsection 5.4. The only +thing one should notice is that, under the restriction ∥∇u0∥L2 ≤ δ, Proposition 4.1 holds and, +thus, we always have +sup +t∈[0,T ] +∥∇ρ∥L4 ≤ 1, +which allows us to use Lemma 2.9 (in such case, there is no difference between the estimates of +Lemma 2.8 and those of Lemma 2.9) and get the uniqueness. +6.2 +Proof of Theorem 1.3 +Following the construction process in subsection 5.2, one can find a smooth sequence (ρn +0 , vn +0 ) such +that +ρn +0 +s +−−→ ρ0 +in H1, +α ≤ ρn +0 ≤ β, +vn +0 +s +−−→ v0 +in L2, +div vn +0 = 0, +vn +0 · n = 0 +on ∂Ω, +(ρn +0 , vn +0 ) satisfying (A’) or (B’). +(6.1) +If we define un +0 := vn +0 + c0∇(ρn +0 )−1, it is easy to check that un +0 is smooth and (ρn +0, un +0) satisfies +all the conditions in Theorem 1.4. Thus, by using Theorem 1.4, there exists a sequence of global +strong solutions (ρn, un) of (1.1) with initial data (ρn +0 , un +0). Then, using the uniform bounds we +get from subsection 3.1, extracting subsequences if necessary, we can derive a weak convergent +subsequence satisfying + + + + + + + + + + + + + + + +ρn +w∗ +−−⇀ ρ +in L∞(0, T ; H1), +ρn +w +−−⇀ ρ +in L2(0, T ; H2), +ρn +t +w +−−⇀ ρt +in L2(0, T ; L2), +un +w∗ +−−⇀ u +in L∞(0, T ; L2), +un +w +−−⇀ u +in L2(0, T ; H1). +(6.2) +Next, we can apply Lemma 3.7 to obtain (3.25)–(3.26). With these hold in hand, one can imme- +diately get +un +s +−−→ u +in L2(0, T ; L2). +(6.3) +Indeed, since vn +s +−−→ v in L2(0, T ; L2), it suffices to show the strong convergence for ∇(ρn)−1, that +is, +∇(ρn)−1 +s +−−→ ∇ρ−1 +in L2(0, T ; L2). +(6.4) +However, +∇(ρn)−1 = −(ρn)−2∇ρn +and ρn +s +−−→ ρ in L2(0, T ; H1), since we have (6.2)2–(6.2)3 and, then, use Lemma 2.10. Therefore, +(6.4) is an easy consequence of (3.25) and Egorov theorem. +Finally, using (3.25), (6.2)–(6.3), we can recover the weak solutions (ρ, u) for system (1.1) and +complete the prove of Theorem 1.3. +47 + +7 +Proof of Theorem 1.7 +In the last section, we come to prove the blowup criterion for (ρ, u) satisfying one of three conditions +(A), (B) and (C). Let (ρ, u, π) be a local strong solution as being described in Theorem 1.6 and +suppose that (1.13) or (1.14) was false, that is, for some r and s satisfying (1.15), +lim +T →T ∗ ∥∇ρ∥Ls(0,T ;Lr) ≤ M0 < ∞. +(7.1) +or +lim +T →T ∗ ∥u∥Ls(0,T ;Lr) ≤ M0 < ∞. +(7.2) +We also let ˜C be a positive generic constant depending on Ω, c0, α, β, T ∗, M0 and ∥u0∥H1. Then, +our goal is proving the following estimate. +Proposition 7.1. Suppose that (7.1) holds for (ρ, u) satisfying the condition (A) or (B) and (7.2) +holds for (ρ, u) satisfying the condition (C). Then, one has, for all T ∈ (0, T ∗), +sup +t∈[0,T ] +� +∥ρt∥2 +L2 + ∥ρ∥2 +H2 + ∥u∥2 +H1 +� ++ +� T +0 +� +∥ρt∥2 +H1 + ∥ρ∥2 +H3 + ∥u∥2 +H2 +� +dt ≤ ˜C. +(7.3) +Before proving the proposition, let us show how to derive the blowup criterion in Theorem 1.6 +from Proposition 7.1. +Proof of Theorem 1.7. For simplicity, we give the prove for the case when (ρ, u) satisfies the con- +dition (A), since other cases can be proved identically. Note that ˜C, in (7.3), is uniformly bounded +for all T ≤ T ∗, so +(ρ, u)(x, T ∗) := lim +t→T ∗(ρ, u)(x, t) in the sense of H2 × H1 +satisfying the conditions imposed on the initial data, that is, α ≤ ρ0 ≤ β, u0 ∈ H1 +ω, at the time +t = T ∗. Furthermore, +� +div u|t=T ∗ = c0∆ρ−1|t=T ∗, +x ∈ Ω +u|t=T ∗ · n = n · ∇ρ−1|t=T ∗, +x ∈ ∂Ω +Thus, (ρ, u)(x, T ∗) satisfies (1.11) also. Therefore, we can take (ρ, u)(x, T ∗) as the initial data and +apply the existence result in Theorem 1.6 to extend the local strong solution beyond T ∗. This +contradicts the maximality of T ∗ and, hence, we finish the proof of Theorem 1.6. +7.1 +Case for (ρ, u) satisfying (A) or (B) +In this subsection, we always let (ρ, u) satisfy the condition (A) or (B). Recall that it is also +equivalent to require (ρ, v) satisfying the condition (A’) or (B’). +The proof for the first part of Proposition 7.1 will be separated into the following few steps. The +key of the proof is obtaining the lower order estimates for (ρ, v), that is, (∇ρ, v) ∈ C([0, T ]; L2) ∩ +L2(0, T ; H1), then, following the proof in Section 3.3, the weak solution is automatically a strong +one. +The first lemma is just the combination of Lemma 3.1 and 3.3, we give it here for convenience. +Lemma 7.2. The following bounds hold for all T ∈ (0, T ∗), that is, +α ≤ ρ ≤ β, +sup +t∈[0,T ] +∥ρ∥2 +L2 + ν +� T +0 +∥∇ρ∥2 +L2 dt ≤ C. +(7.4) +The next crucial lemma gives the lower bounds of (ρ, v), that is, +Lemma 7.3. Suppose that (7.1) holds and (ρ, v) satisfies the condition (A’) or (B’), then one has +sup +t∈[0,T ] +� +∥∇ρ∥2 +L2 + ∥v∥2 +L2 +� ++ +� T +0 +� +∥∆ρ∥2 +L2 + ∥∇v∥2 +L2 +� +dt ≤ ˜C. +(7.5) +48 + +Proof. We first follow the proof of Lemma 3.4, applying Lemma 2.1 and 7.2, to get +d +dt ∥∇ρ∥2 +L2 + ν ∥∆ρ∥2 +L2 ≤ C +� � +|∇ρ|2 + |v| |∇ρ| +� +|∆ρ| +≤ C ∥∇ρ∥Lr +� +∥∇ρ∥ +L +2r +r−2 + ∥v∥ +L +2r +r−2 +� +∥∆ρ∥L2 +≤ C ∥∇ρ∥ +2r +r−2 +Lr +� +∥∇ρ∥2 +L2 + ∥v∥2 +L2 +� ++ ν +2 ∥∆ρ∥L2 , +which implies that +d +dt ∥∇ρ∥2 +L2 + ν ∥∆ρ∥2 +L2 ≤ C(∥∇ρ∥s +Lr + 1) +� +∥∇ρ∥2 +L2 + ∥v∥2 +L2 +� +. +(7.6) +On the onther hand, as we did in (3.12), multiplying v on both sides of (1.18)2 and integrating +over Ω, +d +dt +� 1 +2ρ |v|2 − +� +div [2µD(v)] · v = +3 +� +i=1 +Ii, +(7.7) +where Ii, i = 1, 2, 3, as in (3.12). From (3.13), applying Lemma 2.1, 2.2 and 7.2, the second term +on the left-hand side can be controlled by +− +� +div [2µD(v)] · v ≥ µ +� +|curl v|2 − C +� +∥∇ρ∥Lr ∥√ρv∥ +L +2r +r−2 ∥∇v∥L2 +� +≥ ν ∥∇v∥2 +L2 − +� +Cε(∥∇ρ∥s +Lr + 1) ∥√ρv∥2 +L2 + ε ∥∇v∥2 +L2 +� +. +(7.8) +Following the proof from (3.14) to (3.17), since +|J′ +1| = +���� +� +∂ +φ(ρ)(n · ∇ρ)(v · n⊥)n⊥ · ∇ρ +���� += +���� +� +∇⊥[φ(ρ)(n · ∇ρ)] · ∇ρ(v · n⊥) + +� +φ(ρ)(n · ∇ρ)∇ρ · ∇⊥(v · n⊥) +���� +≤ Cε1 ∥∇ρ∥2 +Lr +� +∥√ρv∥2 +L +2r +r−2 + ∥∇ρ∥2 +L +2r +r−2 +� ++ ε1 +� +∥∇v∥2 +L2 + ∥∆ρ∥2 +L2 +� +≤ Cε1(∥∇ρ∥s +Lr + 1) +� +∥√ρv∥2 +L2 + ∥∇ρ∥2 +L2 +� ++ ε1 +� +∥∇v∥2 +L2 + ∥∆ρ∥2 +L2 +� +(7.9) +|J′ +2| = +���� +� +∂ +φ(ρ)(v · n⊥)n⊥ · ∇n · ∇ρ +���� += +���� +� +∇⊥φ(ρ) · (∇n · ∇ρ)(v · n⊥) − +� +Ω +φ(ρ)∇⊥ · (∇n · ∇ρ)(v · n⊥) dx +− +� +φ(ρ)∇⊥(v · n⊥) · (∇n · ∇ρ) +���� +≤ Cε2 +� +∥v∥2 +L2 + ∥∇ρ∥2 +L2 +� ++ Cε2 ∥∇ρ∥2 +Lr ∥√ρv∥2 +L +2r +r−2 + ε2 +� +∥∇v∥2 +L2 + ∥∆ρ∥2 +L2 +� +≤ Cε2 ∥∇ρ∥2 +L2 + Cε2(∥∇ρ∥s +Lr + 1) ∥√ρv∥2 +L2 + ε2 +� +∥∇v∥2 +L2 + ∥∆ρ∥2 +L2 +� +. +(7.10) +and +|J3| = +���� +� +2c0µ(ρ)∂ijρ−1∂jvi +���� += +���� +� +∂ +2c0µ(ρ)∇ρ−1 · ∇v · n − +� +2c0µ′∇ρ−1 · ∇v · ∇ρ +���� += +����− +� +∂ +2c0µ(ρ)∇ρ−1 · ∇n · v − +� +2c0µ′∇ρ−1 · ∇v · ∇ρ +���� +≤ Cε2(∥∇ρ∥s +Lr + 1) +� +∥√ρv∥2 +L2 + ∥∇ρ∥2 +L2 +� ++ ε2 +� +∥∇v∥2 +L2 + ∥∆ρ∥2 +L2 +� +, +(7.11) +49 + +we deduce that +|I1| ≤ Cε(∥∇ρ∥s +Lr + 1) +� +∥√ρv∥2 +L2 + ∥∇ρ∥2 +L2 +� ++ ε +� +∥∇v∥2 +L2 + ∥∆ρ∥2 +L2 +� +, +(7.12) +Similarly, for I2–I3, one has +|I2| ≤ Cε(∥∇ρ∥s +Lr + 1) ∥√ρv∥2 +L2 + ε ∥∇v∥2 +L2 , +|I3| ≤ Cε(∥∇ρ∥s +Lr + 1) ∥∇ρ∥2 +L2 + ε +� +∥∇v∥2 +L2 + ∥∆ρ∥2 +L2 +� +. +(7.13) +Substituting (7.12)–(7.14) into (7.7) and, then, alonging with (7.6) leads to +d +dt +� +∥√ρv∥2 +L2 + ∥∇ρ∥2 +L2 +� ++ ν +� +∥∇v∥2 +L2 + ∥∆ρ∥2 +L2 +� +≤ C(∥∇ρ∥s +Lr + 1) +� +∥∇ρ∥2 +L2 + ∥√ρv∥2 +L2 +� +. +(7.14) +Finally, applying the Gr¨onwall’s inequality to (7.14), we finish the proof of Lemma 7.3. +Now, we can prove the first part of Proposition 7.1. +Proof of Proposition 7.1. It follows from Lemma 7.2 and 7.3 that +sup +t∈[0,T ] +� +∥ρ∥2 +H1 + ∥v∥2 +L2 +� ++ +� T +0 +� +∥ρ∥2 +H2 + ∥v∥2 +H1 +� +dt ≤ ˜C. +(7.15) +Thus, by Lemma 2.1, we get the bounds +� T +0 +∥(∇ρ, v)∥4 +L4 dt ≤ ˜C. +This, together with (7.15), allows us to follow the proof of Proposition 3.8 step by step, since +the lower order bounds are enough to deduce the higher ones, according to Lemma 3.9 and 3.11 +(noticing that, the proofs of Lemma 3.9 and 3.11 are merely based on the smallness assumption +we derived from Proposition 3.2, that is, ∥∇ρ0∥L2 ≤ δ, without any additional restriction, see also +Remark 3.10). We omit the remaining proof here and leave it to the reader. +7.2 +Case for (ρ, u) satisfying (C) +Now, we assume that (ρ, u) satisfies the condition (C). One should notice that condition (7.2) is +also equivalent with +lim +T →T ∗ +� +∥v∥Ls(0,T ;Lr) + ∥∇ρ∥Ls(0,T ;Lr) +� +≤ ˜ +M0 < ∞, +(7.16) +since ρ is bounded from above and below and the identity (1.17). +Our aim is proving the rest of Proposition 7.1 under (7.2). First, we give the following lemma, +which concludes some results we need later. This nothing but a directly application of Lemma 2.7 +and Lemma 7.2. +Lemma 7.4. Let (ρ, u, π) be a local strong solution as being described in Theorem 1.6. Then, +Lemma 7.2 still holds. +Moreover, under the condition (7.2) (or, equivalently, (7.16)), one has +ρ ∈ Cγ, γ +2 (QT ) for some γ ∈ (0, 1) and for all T ∈ (0, T ∗). +Next, with help of the Serrin’s condition (7.2), one can get the lower bound of ρ. +Lemma 7.5. Suppose that (7.16) holds and (ρ, u) satisfies (C), then +sup +t∈[0,T ] +∥∇ρ∥2 +L2 + +� T +0 +� +∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 +� +dt ≤ ˜C. +(7.17) +50 + +Proof. As we did in Lemma (3.4), applying Lemma 2.1, +d +dt ∥∇ρ∥2 +L2 + ν ∥∆ρ∥2 +L2 ≤ Cε +� +∥∇ρ∥2 +Lr + ∥v∥2 +Lr +� +∥∇ρ∥2 +L +2r +r−2 + ε ∥∆ρ∥2 +L2 +≤ Cε (∥∇ρ∥s +Lr + ∥v∥s +Lr + 1) ∥∇ρ∥2 +L2 + ε ∥∆ρ∥2 +L2 , +that is, +d +dt ∥∇ρ∥2 +L2 + ν ∥∆ρ∥2 +L2 ≤ C (∥∇ρ∥s +Lr + ∥v∥s +Lr + 1) ∥∇ρ∥2 +L2 . +(7.18) +Thus, using Gr¨onwall’s inequality and Lemma 2.1, we conclude the proof. +Remark 7.6. With this Lemma (7.5) and condition (7.16), we deduce from (1.18)1 that +� T +0 +∥ρt∥2 +L2 dt ≤ ˜C. +(7.19) +Now, we can prove Proposition 7.1. +Proof of Proposition 7.1. We start with (4.11) +d +dt +� 1 +2ρ|u|2 + +� +2µ(ρ)|D(u)|2 := +3 +� +i=1 +Si, +(7.20) +where Si as in (4.11). Using Lemma 2.1, + + + + + + + +|S1| ≤ C ∥Q∥L2 ∥ut∥L2 ≤ Cε1 ∥∇ρ∥2 +L2 + ε1 ∥ut∥2 +L2 , +|S2| ≤ C ∥Q∥Lr ∥u∥ +L +2r +r−2 ∥∇u∥L2 ≤ Cε2 (∥∇ρ∥s +Lr + 1) ∥u∥2 +L2 + ε2 ∥∇u∥2 +L2 , +|S3| ≤ C ∥∇Q∥L2 ∥∇u∥L2 ≤ Cε3 +� +∥∆ρ∥2 +L2 + ∥∇ρ∥4 +L4 +� ++ ε3 ∥∇u∥2 +L2 . +(7.21) +Combining (7.20) and (7.21) leads to, +d +dt ∥u∥2 +L2 + ν ∥∇u∥2 +L2 ≤ Cε (∥∇ρ∥s +Lr + 1) ∥u∥2 +L2 + Cε +� +∥∆ρ∥2 +L2 + ∥∇ρ∥4 +L4 +� ++ ε ∥ut∥2 +L2 , +(7.22) +Similarly, we deduce from (4.14)–(4.17) that +d +dt∥ +� +µ(ρ)|D(u)|∥2 +L2 + ν ∥ut∥2 +L2 ≤ Cε +� +∥u∥s +Lr + ∥∇ρ∥4 +L4 + ∥ρt∥2 +L2 + 1 +� +∥∇u∥2 +L2 ++ Cε ∥∇ρt∥2 +L2 + ε ∥∆u∥2 +L2 . +(7.23) +By Lemma 7.4, µ(ρ) ∈ C(QT ), hence, we can apply Lemma 2.8 for (3.61) with Φ = −c0∇ρ−1 +and, then, use Lemma 2.1 and 7.4 to deduce that +∥v∥2 +H2 + ∥π∥2 +H1 ≤ C +� +∥F∥2 +L2 + +��∇∆ρ−1��2 +L2 +� +≤ C +� +∥v∥s +Lr + ∥∇ρ∥4 +L4 + 1 +� � +∥∇v∥2 +L2 + ∥∆ρ∥2 +L2 + ∥ρt∥2 +L2 +� ++ C +� +∥vt∥2 +L2 + ∥∇ρt∥2 +L2 +� ++ C ∥∇∆ρ∥2 +L2 , +which gives +∥∆u∥2 +L2 + ∥π∥2 +H1 ≤ C +� +∥u∥s +Lr + ∥∇ρ∥4 +L4 + 1 +� � +∥∇u∥2 +L2 + ∥∆ρ∥2 +L2 + ∥ρt∥2 +L2 +� ++ C +� +∥ut∥2 +L2 + ∥∇ρt∥2 +L2 +� ++ C ∥∇∆ρ∥2 +L2 , +(7.24) +Plugging (7.24) into (7.23) and choosing ε sufficiently small, we have, for some positive constant +ν depending on Ω, c0, α and β, +d +dt∥ +� +µ(ρ)|D(u)|∥2 +L2 + ν +� +∥∆u∥2 +L2 + ∥ut∥2 +L2 +� +≤ C +� +∥u∥s +Lr + ∥∇ρ∥4 +L4 + ∥ρt∥2 +L2 + 1 +� +∥∇u∥2 +L2 + C +� +∥∇ρt∥2 +L2 + ∥∇∆ρ∥2 +L2 +� +. +(7.25) +51 + +On the other hand, following the proof from (3.33) to (3.35), replacing ∥v∥4 +L4 by ∥v∥s +Lr, then, +replacing v by u via (1.17) and applying Lemma 2.1, one has +d +dt ∥∆ρ∥2 +L2 + ν ∥∇∆ρ∥2 +L2 ≤ Cε +� +∥∇ρ∥4 +L4 + ∥u∥s +Lr + 1 +� � +∥∆ρ∥2 +L2 + ∥∇u∥2 +L2 +� ++ ε ∥∆u∥2 +L2 , +Alonging with (4.23), we deduce that +d +dt +� +∥∆ρ∥2 +L2 + ∥ρt∥2 +L2 +� ++ ∥∇∆ρ∥2 +L2 + ∥∇ρt∥2 +L2 +≤ Cε +� +∥u∥s +Lr + ∥∇ρ∥4 +L4 + ∥∆ρ∥2 +L2 + 1 +� � +∥∆ρ∥2 +L2 + ∥ρt∥2 +L2 + ∥∇u∥2 +L2 +� ++ ε +� +∥∆u∥2 +L2 + ∥ut∥2 +L2 +� +. +(7.26) +Combining (7.22), (7.25) and (7.26), then, applying the Gr¨onwall’s inequality, condition (7.2) +and Lemma 7.5, we get, for all T ∈ (0, T ∗), +sup +t∈[0,T ] +� +∥u∥2 +H1 + ∥ρt∥2 +L2 + ∥∆ρ∥2 +L2 +� ++ +� T +0 +� +∥∇u∥2 +H1 + ∥∇ρt∥2 +L2 + ∥∇∆ρ∥2 +L2 +� +dt ≤ ˜C. +Then, we can turn back to (7.24) to get +� T +0 +∥π∥2 +H1 dt ≤ ˜C. +Therefore, we complete the proof of Proposition 7.1. +References +[1] H. 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Journal of Differential Equations, 263(8):4978–4996, 2017. +54 + diff --git a/dtE1T4oBgHgl3EQfLgMI/content/tmp_files/load_file.txt b/dtE1T4oBgHgl3EQfLgMI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7968e9aeb64f2db015660a122d07ac6d264f7240 --- /dev/null +++ b/dtE1T4oBgHgl3EQfLgMI/content/tmp_files/load_file.txt @@ -0,0 +1,2113 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf,len=2112 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='02976v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='AP] 8 Jan 2023 Well-Posedness for 2D Combustion Model in Bounded Domains and Serrin-Type Blowup Criterion Jiawen Zhang∗ School of Mathematical Sciences, University of Chinese Academy of Sciences, Bejing 100049, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' China Abstract We investigate the 2D combustion model with Dirichlet boundary conditions and slip boundary conditions in bounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The global existence of weak and strong solutions and the uniqueness of strong solutions are obtained provided the initial density is small in some precise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using the energy method and the estimates of boundary integrals, we obtain the a priori bounds of the density and velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In addition, we prove the local existence of the strong solutions via iterative method and the contraction mapping theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Finally, we extend the well-known Serrin’s blowup criterion to the 2D combustion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Under the suit- able boundary conditions, the Serrin’s condition on the velocity can be removed in this criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Keywords: combustion model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Dirichlet boundary conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' slip boundary conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' strong solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' weak solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Serrin’s condition 1 Intrduction and main results In this paper, we will study the following combustion model: \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ρt + div(ρu) = 0, ρ ≥ 0, (ρu)t + div(ρu ⊗ u) − div(2µD) + ∇π = 0, div u = c0∆ψ(ρ), ψ(ρ) := ρ−1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) for t > 0 and x ∈ Ω, where Ω ⊂ R2 is a bounded simply connected domain with smooth boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Here u = (u1, u2), ρ and π stand for the unknown velocity field, density and pressure respectively, c0 > 0 is a fixed constant and 0 < µ = µ(s) ∈ C∞[0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We denote D = D(u) = 1 2(∇u + (∇u)t) = 1 2(∂iuj + ∂jui), for 1 ≤ i, j ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' From the physical viewpoint, combustion model is the low Mach number limit of the fully compressible Navier-Stokes equations \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ρt + div(ρu) = 0, (ρu)t + div(ρu ⊗ u) − div S + ∇p = 0, (ρe)t + div(ρue) − div(k∇θ) + p div u = S : D(u), (FCNS) where e, θ, p stands for the internal enery, temperature and pressure respectively and A : B strands for the inner product of matrices A : B := tr(ABt) = 2 � i,j=1 aijbji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' ∗Email address: zhangjiawen@amss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='cn 1 S is the viscous strain tensor given by S = 2µD(u) + λ div uIn×n, where In×n is the n×n indentity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The thermal conductivity k and the viscosity coefficients µ, λ are functions of ρ and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' From Lions’s book [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' if we define the Mach number ǫ as |u|/ � p′(ρ) and let (ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' θ) be smooth solution of system (FCNS) corresponding to the small ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' after rescaling the time variable by ρǫ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' t) = ρ � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' t ǫ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' uǫ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' t) = 1 ǫ ρ � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' t ǫ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' θǫ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' t) = θ � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' t ǫ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (ρǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' uǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' θǫ) satisfies \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ρǫ t + div(ρǫuǫ) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (ρǫuǫ)t + div(ρuǫ ⊗ uǫ) − div Sǫ + ǫ−2∇pǫ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (ρǫeǫ)t + div(ρǫuǫeǫ) − div(kǫ∇θǫ) + pǫ div uǫ = ǫ2Sǫ · D(uǫ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (FCNS’) where Sǫ = 2µǫD(uǫ) + λǫ div uǫIn×n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' and pǫ = p � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' t ǫ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' µǫ = 1 ǫ µ � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' t ǫ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' λǫ = 1 ǫ λ � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' t ǫ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' kǫ = 1 ǫ k � x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' t ǫ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Considerig the ideal gas laws: p = Rρθ, e = CV θ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2) where R, CV denote the ideal gas constant and the specific heat constant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, letting the Mach number ǫ go to 0, the momentum equation (FCNS’)2 implies that pǫ = P(t) + π(t, x)ǫ2 + o � ǫ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Plugging this formula into the energy equation (FCNS’)3 entails that P(t) is independent of t, pro- vided uǫ and ∇θǫ vanish at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' From now on, we shall denote this constant by P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Therefore, denoting CP = γCV = γR/(γ − 1), the low Mach number limit system reads \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ρCP (∂tθ + u · ∇θ) − div(k∇θ) = 0, ρut + ρu · ∇u − div S + ∇π = 0, γP0 div u = (γ − 1) div(k∇θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) Plugging (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2) into (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) with constant heat conductivity coefficient k implies the following system \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ∂tρ + div(ρu) = 0, ρut + ρu · ∇u − div S + ∇π = 0, div u = k(γ − 1)(Rγ)−1∆ρ−1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4) which is exactly the equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' If we particularly take the diffusion coefficient c0 = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) will become the classical non- homogeneous incompressible Navier-Stokes equations \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ρt + div(ρu) = 0, ρ ≥ 0, (ρu)t + div(ρu ⊗ u) − div(2µD) + ∇π = 0, div u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (N) The study of the combustion model may date back to the 1980s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' It has been introduced by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Majda [29] and studied in particular by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Embid [14] who has proved the local-in-time well- posedness of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) replacing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1)3 by Fick’s law with ψ(ρ) = log ρ, the local well-posedness was considered by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' da Veiga [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Danchin-Liao [13] established 2 the local existence and uniqueness of a solution in critical homogeneous Besov spaces provided the density is closed to a positive constant and they proved the local well-posedness in non- homogeneous Besov space arbitrarily large data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' On the other hand, there are also a large number of works investigating the global-in-time existence of weak and strong solutions for the combustion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Secchi [32] proved that there exists a unique global strong solution in the two-dimensional domain under Fick’s law providing the diffusion coefficient c0 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' They also considered the limiting behavior of the solutions when c0 → 0 for dimensions 2 and 3 and the convergence towards the corresponding solutions of (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Under the small initial data assumption, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lions [28] showed, in R2 or periodic boundary condition, that a small perturbation of a constant density gives a global existence of weak solutions without any restriction on the initial velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Danchin-Liao [13] proved the existence of solutions in critical homogeneous Besov spaces by assuming the initial density is close to a constant and the initial velocity is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Recently, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Tan [37] proved the global existence of the weak and strong solutions of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) with general viscosity coefficient µ(ρ) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1)2 and ψ(ρ) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1)3 provided the density is closed to a positive constant in some precise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For large data, Bresch-Essoufi-Sy [5] showed the global existence of the weak solutions for the combustion model in dimensions 2 and 3 by taking µ(ρ) = c0 2 log ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In [6], Bresch-Giovangigli-Zatorska relaxed the restriction on µ(ρ) by using the idea of the renormalized solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' If one takes the decomposition u = v + c0∇ρ−1 with div v = 0 and converts the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) to the equations for (ρ, v), then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) will be reduced to the Kazhikhov-Smagulov type model, see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In [9, 10], Cai-Liao-Sun established the global-in-time existence of strong solutions to the initial-boundary value problem of a 2D Kazhikhov-Smagulov type model for incompressible non-homogeneous fluids with mass diffusion for the arbitrary size of initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For works on the classical Kazhikhov-Smagulov’s model, we refer the reader to [2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For the general non-homogeneous incompressible Navier-Stokes equations (N) with the viscosity coefficient µ(ρ) depending on ρ, global weak solutions were derived by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lions [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Abidi- Zhang [1] obtained the global strong solutions strictly away from vacuum whenever ∥u0∥L2 ∥∇u0∥L2 and ∥µ(ρ0) − 1∥L∞ are small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For the initial density containing vacuum, Cho-Kim [11] established the existence of the local strong solutions under compatibility conditions similar to [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In addition, Huang-Wang [22], J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Zhang [40] established the global strong solutions with small ∥∇u0∥L2 in 3D bounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For the Cauchy problem, He-Li-L¨u [18] obtained the global strong ones to with small ∥u0∥ ˙Hβ for some β ∈ (1/2, 1] and some extra restrictions on µ(ρ) via the exponential decay-in-time estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' More recently, Cai-L¨u-Peng [8] studied the global existence of strong solutions in 3D exterior domains with nonslip or slip boundary conditions provided that the gradient of the initial velocity is suitably small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Finally, for the study of the mechanism of blowup and structure of possible singularities of strong (or smooth) solutions to the Navier-Stokes system can be traced to Serrin’s criterion [33] on the Leray-Hopf weak solutions to the 3D incompressible homogeneous Navier-Stokes equations, which can showed that if a weak solution u satisfies u ∈ Ls(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lr), 2 s + 3 r ≤ 1, 3 < r ≤ ∞, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5) then it is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Later, He-Xin [19] showed that the Serrin’s criterion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5) still holds even for the strong solution to the incompressible MHD equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For non-homogeneous incompressible Navier–Stokes equations (N), H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Kim [24] established the Serrin-type blowup criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' They showed that if (ρ, u) blows up at T ∗, then lim t→T ∗ ∥u∥Ls(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='Lrw) = ∞ for all 2 s + 3 r ≤ 1, 3 < r ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6) Recently, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Zhong [41] obtained a blowup criterion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5) to the non-homogeneous incompressible heat conducting Navier–Stokes flows with non-negative density in bounded domain of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For the compressible fluids, Huang-Li-Xin [21] first extend Serrin’s blow-up criterion to the barotropic compressible Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Later, Xu-Zhang [39] extended the results of [21] to the isentropic compressible MHD system and Huang-Li-Wang [20] improve the all previous blowup criterion results to the full compressible Navier–Stokes system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' However, for the general viscosity coefficient, the theory of the combustion model in the bounded domain is still blank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Therefore, our goal is obtaining the global existence of solutions with small 3 initial data and the local existence for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) in the general domain under different initial-boundary conditions and trying to extend Serrin’s blow-up criterion to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' More precisely, we impose the initial data u0(x) := u(x, 0), 0 < α ≤ ρ0(x) := ρ(x, 0) ≤ β < ∞, x ∈ Ω (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7) and one of the following boundary conditions: (1) ρ satisfies the Neumann condition and u satisfies the slip boundary condition, that is, n · ∇ρ = 0, u · n = 0 and curl u = −n⊥ · B · u on ∂Ω × (0, T ), (A) where n = (n1, n2) denotes the unit outer normal vector of the boundary ∂Ω, n⊥ = (n2, −n1) is the unit tangential vector on the boundary and B = B(x) is a bounded smooth symmetric matrix which is positive semi-definite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2) (ρ, u) satisfies the non-homogeneous Dirichlet condition, that is, ρ = ˜ρ, u = c0∇ρ−1 on ∂Ω × (0, T ), (B) where ˜ρ is a positive constant such that α ≤ ˜ρ ≤ β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3) ρ satisfies the Neumann condition and u satisfies the non-slip condition, that is, n · ∇ρ = 0, u = 0 on ∂Ω × (0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (C) Before giving the main results, we explain some notations and conventions used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For simplicity, we set � f := � Ω f dx, � ∂ f := � ∂Ω f dS, �� f := �� QT f dxdt, where QT := Ω × (0, T ), and fΩ := 1 |Ω| � f, where |E| stands for the Lebesgue measure of the measurable set E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Also,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' for all integer k and 1 ≤ p < ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' W k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='p is the standard Sobolev spaces as defined as follows: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 Lp := Lp(Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' W k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='p = W k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='p(Ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Hk := W k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H∞ := � k≥1 Hk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' W k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='p 0 = C∞ 0 closure in the norm of W k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' ∥·∥B1∩B2 := ∥·∥B1 + ∥·∥B2 for two Banach spaces B1 and B2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1 ω := {u ∈ H1 : u · n = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' curl u = −n⊥ · B · u on ∂Ω},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1 nd := {u ∈ H1 : u = c0∇ρ−1 on ∂Ω},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' V 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 := {u ∈ L2 : div u = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' u · n = 0 on ∂Ω},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' V 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 0 := {u ∈ H1 0 : div u = 0},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' V −1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 0 := [V 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 0 ]∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For 0 < γ < 1, we denote by Cγ(Ω) the standard H¨older space and ρ ∈ Cγ, γ 2 (QT ) the parabolic one, that is, Cγ, γ 2 (QT ) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 f ∈ C(QT ) : sup (x,t),(x′,t′)∈QT (x,t)̸=(x′,t′) |f(x, t) − f(x′, t′)| |x − x′|γ + |t − t′| γ 2 < ∞ \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The weak, weak* and strong convergence of a sequence {f n} are respectively denoted by f n w −−⇀ f, f n w∗ −−⇀ f, f n s −−→ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 4 Finally, the transpose gradient is given by ∇⊥ := (∂2, −∂1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' With this notion, one can write � curl u = ∇⊥ · u, ∆u = ∇ div u + ∇⊥ curl u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Now, we give the definitions of weak solutions and strong ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 (Weak Solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (ρ, u) is called a global weak solution, if the following regularity properties hold: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 α ≤ ρ ≤ β, ρ ∈ C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H2), ρt ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2), � u ∈ L∞(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1 ω), case (A), u ∈ L∞(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1 nd), case (B), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8) and (ρ, u) statisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) in the sense of distributions for all T ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' More precisely, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1)1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1)3 hold almost everywhere in Ω × (0, T ) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1)2 is satisfied in the following sense: �� ρu · φt + ρu ⊗ u : ∇φ − 2µD(u) : D(φ) = − � ρ0u0 · φ(x, 0), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9) for φ ∈ C∞(QT ) with div φ = 0, φ(x, T ) = 0, x ∈ Ω and φ = 0 on ∂Ω × (0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 (Strong Solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' If (ρ, u, π) is a solution such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) holds almost everywhere in Ω × (0, T ), T ∈ (0, ∞), such that \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 α ≤ ρ ≤ β, ρ ∈ C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H2) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H3), ρt ∈ C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1), u ∈ C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H2), ut ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2), π ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10) we call (ρ, u, π) the strong solution on Ω × (0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In particular, if (ρ, u, π) satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10) for all T ∈ (0, ∞), we say that (ρ, u, π) is a global strong solution of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Our main results sate as following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The first two theorems concern with the existence results for (ρ, u) satisfying (A) or (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Suppose that u0 ∈ L2 and (ρ0, u0) satisfies the following compatibility condition � div u0 = c0∆ρ−1 0 , x ∈ Ω u0 · n = n · ∇ρ−1 0 , x ∈ ∂Ω (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) Assume that (ρ, u) satisfies the condition (A) or (B), then there exist a positive constant δ which only depends on Ω, α, β, c0 and ∥v0∥L2 such that, if ∥∇ρ0∥L2 ≤ δ, problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7) admits at least one gobal weak solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Suppose that (ρ0, u0) satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) and (ρ, u) satisfies the condition (A) or (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let u0 ∈ H1 ω provided u satisfying the condition (A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' u0 ∈ H1 nd provided u satisfying the condition (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In addition, let π satisfy the normalized condition � π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12) Then, if ∥∇ρ0∥L2 ≤ δ with the same δ obtained in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3, the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7) admits a unique global strong solution (ρ, u, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 5 Next, for the case when (ρ, u) satisfying the condition (C), we have Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Suppose that u0 ∈ H1 0 and (ρ0, u0) satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Suppose that (ρ, u) satisfies the condition (C) and π satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' There exists a positive constant δ which only depends on Ω, α, β, c0 such that, if ∥∇u0∥L2 ≤ δ, then the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7) admits a unique global strong solution (ρ, u, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' At last, we give the local existence result and the corresponding Serrin-type blowup criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Assume that (ρ0, u0) satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) and u0 ∈ H1 ω, H1 nd provided u satisfying the condition (A) and (B), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let π saitisfies the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then there exists a positive time T1 < ∞ depending on Ω, c0, α, β and ∥u0∥H1 so that the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7) admits an unique strong solution (ρ, u, π) on Ω × (0, T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Moreover, if µ(ρ) = µ is a positive constant and u0 ∈ H1 0, then, the same result holds for (ρ, u) satisfying the condition (C) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' If (ρ, u, π) is a strong solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) on Ω × (0, T ∗) and T ∗ < ∞ is the maximal time of existence, then, one has (1) lim T →T ∗ ∥∇ρ∥Ls(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='Lr) = ∞ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13) provided (ρ, u) satisfying the condition (A) or (B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2) lim T →T ∗ ∥u∥Ls(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='Lr) = ∞ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14) provided (ρ, u) satisfying the condition (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Here, r and s satisfy the relation 2 s + 2 r ≤ 1, 2 < r ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='15) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The definition of v0 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4 will be given at the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6 are first results concerning with the weak and strong solutions for the combustion model in bounded domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5 can be seen as a kind of extension of the global existence results in [22, 40] with div u = c0∆ρ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7 can be regarded as an extension to the classical Serrin’s condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For some technical reasons, in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4, we need the following consistency condition ρ0|∂Ω = ˜ρ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16) to ensure the continuity of ρ, which is crucial to the higher order estimates of v, see subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' On the other hand, one should notice that the restriction α ≤ ˜ρ ≤ β and the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16) are not necessary for the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' However, for simiplicity, we may always impose these requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Noticing that, in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6, we only impose the regularity restrictions on u0 for given initial data (ρ0, u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' This is due to the compatiability condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) from which one can find that the regularity of ρ0 can be totally determinded by that of u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Indeed, for example, if u0 ∈ H1 0 as we assumed in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5, it follows from the following epllitic problem � c0∆ρ−1 0 = div u0, x ∈ Ω, n · ∇ρ−1 0 = 0, x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' that, for all 1 < p < ∞, � ∥∇ρ0∥Lp ≤ C(p) ∥u0∥Lp , ∥∇ρ0∥H1 ≤ C ∥∇u0∥L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Thus, alonging with the fact that ρ0 ∈ L∞, ρ0 ∈ H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We will come to this point again many times in later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 6 At the end of this section, we make a short comment on the analysis of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Formally speaking, we treat Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5 via two different types of decomposition and the proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7 are based on those of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4 are based on the decomposition u = v + c0∇ρ−1, which may convert system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) into the Kazhikhov-Smagulov type model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In this case, one can find that v satisfies either the Dirichlet boundary condition or the slip one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' More precisely, we may first write in view of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1)3 v = u − c0∇ρ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17) Of course, such v can be found for given (ρ, u) with the boundary condition (A) or (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17), we write ρu = ρv + c0ρ∇ρ−1 = ρv − c0∇ log ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Therefore, combining this equality and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17), the original system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) can be changed into the following equivalent formulations: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ρt + v · ∇ρ + c0ρ−2 |∇ρ|2 − c0ρ−1∆ρ = 0, � (ρv)t + div(ρv ⊗ v) − div [2µD(v)] + ∇π1 = c0 div � 2µ∇2ρ−1� −c0 div � ρv ⊗ ∇ρ−1� − div � c0ρ∇ρ−1 ⊗ v � − c2 0 div � ρ∇ρ−1 ⊗ ∇ρ−1� , div v = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18) where π1 = π − c0(log ρ)t is a modified pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, we give a precise defintion for the initial-boundary value of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For given initial data (ρ0, u0) satisfying the initial conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11), one can deduce that there exists a unique v0 satis- fying v(x, 0) := v0 = u0 − c0∇ρ−1 0 , div v0 = 0, v0 · n = 0 on ∂Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='19) sharing with the similar compatibility conditions of u0, that is, v0|∂Ω = 0 provided u0 ∈ H1 nd and curl v0 = −n⊥ · B · (v0 + c0∇ρ−1 0 ) provided u0 ∈ H1 ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Furthermore, from the relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17), we can define the boundary conditions of v as follows: (1) v · n = 0 and curl v = curl u = −n⊥ · B · (v + c0∇ρ−1) on ∂Ω × (0, T ), if (ρ, u) satisfies the condition (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In this case, from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6 in Section 2, one has ∥v∥H2 ≤ C(∥∆v∥L2 + ��∆ρ−1�� L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2) v = 0 on ∂Ω × (0, T ), if (ρ, u) satisfies the condition (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In this case, we have ∥v∥H2 ≤ C ∥∆v∥L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' An interesting observation is that, once the solution (ρ, v) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18), which is defined as in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, incorporating with the initial-boundary conditions given above, is established, one can expect to obtain u from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17) and, consequently, (ρ, u) becomes the solution of the original system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Therefore, in Section 3, we mainly establish the a priori estimates of (ρ, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The details for proving the existence of (ρ, u) will be shown in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' To sum up, we may impose (ρ, v) satisfying one of the following two boundary conditions (1) if (ρ, u) satisfies (A), we impose n · ∇ρ = 0, v · n = 0 and curl v = −n⊥ · B · (v + c0∇ρ−1) on ∂Ω × (0, T );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (A’) (2) if (ρ, u) satisfies (B), we impose ρ = ˜ρ, v = 0 on ∂Ω × (0, T ) (B’) 7 and our strategy of the proof can be concluded as follows: given (ρ0, u0) =⇒ (ρ0, v0) =⇒ ∃ (ρ, v) =⇒ ∃ (ρ, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Unfortunately, for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5, such decomposition may cause some serious problems when it comes to the boundary estimates, that is, if we extract v as we did above, v|∂Ω = −c0∇ρ−1, which will hinder us to integrate by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' As a consequence, we consider another type of decomposition u = w + Q coming from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In every case that follows, w is divergence-free and enjoys vanished boundary condition and Q can be dominanted by ∇ρ, which allows us to overcome the bounardy integrals, see Section 4 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The scond part we are interested in is the local well-posedness for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6, we mainly follow the proof from Kim-Cho [11] by using the iterative appoarch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' This method will be based on the linearized model associated with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) , we refer to Section 5 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' To the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7, at least for the case when (ρ, v) satisfies condition (A’) or (B’), the key obeservation is that, if ρ is a weak solution of system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) satisfying ∇ρ ∈ Ls(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lr), then (ρ, v) is regular, since we can close the lower bounds for (ρ, v) merely under the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Here is an interpretation for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14): for (ρ, u) satisfying the condition (A) or (B), the Ls(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lr)-norm of v does not blowup during the finite time [0, T ∗), which is parallel to the classical Serrin’s condition for 2D non-homogeneous Navier-Stokes equations (N) (since, in such case, problem (N), at least for ρ away from the vacuum, automatically satisies the Serrin’s condition and admits a unique global strong solution without any smallness assumption, here v can be seen as the velocity field u in (N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' However, for the case when (ρ, u) satisfies the condition (C), we can not get rid off the the blowup behavior of v, since, in this case, v|∂Ω = −c0∇ρ−1, which leads to some issue on the boundary estimates, we will come to this point again in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' At last, we explain some techniques used in Section 3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Since our main difficulty arises from the boundary integrals, in order to overcome it, we adapt the ideas from Cai-Li [7]: observing that the condition v · n|∂Ω = 0 leads to v = (v · n⊥)n⊥, which implies that � ∂ v · ∇f = � ∂ (v · n⊥)n⊥ · ∇f = � ∇f · ∇⊥(v · n⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' This observation can allow us to avoid some higher derivatives of f, which has advantages over directly using the trace inequality, since the latter needs the second order derivative of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In Section 2, we give some elementary results which will be used in later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Section 3 is devoted to the lower order estimates, compactness results for weak solutions and the higher order estimates for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4, while Section 4 is devoted to the a priori estimets for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In Section 5, we will use the contraction mapping theorem to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6 and, then, in Section 6, use this result to establish the global existence for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' At last, in Section 7, we will give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 2 Preliminaries First, we give the following Gagliardo-Nirenberg’s inequalities which will be frequently used through- out the whole paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 (Gagliardo-Nirenberg’s inequality [26, 30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For all ui ∈ H1, i = 1, 2, q1 ∈ (2, ∞) and q2 ∈ (4, ∞), there exist positive constants Ci, ˜Ci depending on qi, Ω, i = 1, 2, such that ∥u1∥Lq1 ≤ C1 ∥u1∥2/q1 L2 ∥∇u1∥1−2/q1 L2 + ˜C1 ∥u1∥L2 , ∥u2∥Lq2 ≤ C2 ∥u2∥4/q2 L4 ∥∇u2∥1−4/q2 L2 + ˜C2 ∥u2∥L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In particular, if ui satisifies ui · n = 0 on ∂Ω or (ui)Ω = 0, then one can take ˜C1 = ˜C2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 8 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 ([3, 38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let Ω be a simply connected bounded domain in R2 with smooth boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Assume that 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' There exists a positive constant C = C(p, Ω) such that ∥∇u∥Lp ≤ C (∥div u∥Lp + ∥curl u∥Lp) , for all u ∈ W 1,p with u · n = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Furthermore, for u ∈ W 2,p with u · n = 0 on ∂Ω, there exists a constant C = C(p, Ω) such that ∥u∥W 2,p ≤ C (∥div u∥W 1,p + ∥curl u∥W 1,p + ∥u∥Lp) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For case of use, we list the following equivalent norms for ρ satisfying the Neumann or the non-homogeneous Dirichlet condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let 1 < p < ∞, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2, if ρ satsifies the Neumann condition, one has, for all t ≥ 0, ρΩ = (ρ0)Ω, ∥∇ρ∥Lp ≤ C∥∇2ρ∥Lp ≤ C ∥∆ρ∥Lp ≤ C ∥∇∆ρ∥Lp , and C−1(∥∇ρ∥Lp + (ρ0)Ω) ≤ ∥ρ∥W 1,p ≤ C(∥∇ρ∥Lp + (ρ0)Ω), C−1(∥∆ρ∥Lp + (ρ0)Ω) ≤ ∥ρ∥W 2,p ≤ C(∥∆ρ∥Lp + (ρ0)Ω), C−1(∥∇∆ρ∥Lp + (ρ0)Ω) ≤ ∥ρ∥W 3,p ≤ C(∥∇∆ρ∥Lp + (ρ0)Ω), for some positive constant C = C(p, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' If ρ satisfies the the non-homogeneous Dirichlet condition, then there exists a positive constant C = C(p, Ω) such that C−1 ∥∇ρ∥Lp ≤ ∥ρ − ˜ρ∥W 1,p ≤ C ∥∇ρ∥Lp , C−1 ∥∆ρ∥Lp ≤ ∥ρ − ˜ρ∥W 2,p ≤ C ∥∆ρ∥Lp , C−1 ∥∇∆ρ∥Lp ≤ ∥ρ − ˜ρ∥W 3,p ≤ C ∥∇∆ρ∥Lp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In both cases, the following Gagliardo-Nirenberg’s inequalities are established ∥∇ρ∥Lq1 ≤ C ∥∇ρ∥2/q1 L2 ∥∆ρ∥1−2/q1 L2 , ∥∇ρ∥Lq2 ≤ C ∥∇ρ∥4/q2 L4 ∥∆ρ∥1−4/q2 L2 , ∥∆ρ∥Lq1 ≤ C ∥∆ρ∥2/q1 L2 ∥∇∆ρ∥1−2/q1 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' where q1, q2 as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, for the problem � div v = f, x ∈ Ω, v = Φ, x ∈ ∂Ω (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) one has the following conclusion which will be frequently used to eliminate the non-homogeneity of equations in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4 ([16], Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Suppose that Φ · n = 0 on ∂Ω and fΩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, 1) If Φ = 0, there exists a bounded linear operator B = [B1, B2], B : {f ∈ Lp : fΩ = 0} �→ � W 1,p 0 �2 such that ∥B[f]∥W 1,p ≤ C(p)∥f∥Lp, for all p ∈ (1, ∞), and the function Q = B[f] solves the problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Moreover, if f = div g with a certain g ∈ Lr, g · n|∂Ω = 0, then for any r ∈ (1, ∞) ∥B[f]∥Lr ≤ C(r)∥g∥Lr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' B is so-called the Bogovskiˇi operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 9 2) If f = 0, there exists a bounded linear operator C = [C1, C2], C : {Φ : Φ · n|∂Ω = 0, div Φ ∈ Lp} �→ � W 1,p�2 such that ∥C[Φ]∥W 1,p ≤ C(p) ∥div Φ∥Lp , for all p ∈ (1, ∞) and the function R = C[Φ] sovles the problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We only give a brief proof for (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' By a simply change ˜v = v − Φ, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) that ˜v satisfies � div ˜v = − div Φ, x ∈ Ω, ˜v = 0, x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2) Thus, applying (1) for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2), we finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9 are a series of results relating to the Stokes system which are vital to the higher order estimates of v and the construction of smooth initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' These lemmas will be frequently used in Section 3–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let Ω be a simply connected bounded domain in R2 with smooth boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let (u, p) satisfy the following equations � −∆u + ∇p = F, x ∈ Ω, div u = 0, x ∈ Ω, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) where F ∈ L2, � p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' There exists a positive constant C depending only on Ω such that (1) if u|∂Ω = Φ, where Φ ∈ H2 is a function defined on Ω, then ∥u∥H2 + ∥p∥H1 ≤ C(∥F∥L2 + ∥Φ∥H2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4) (2) if u · n = 0, curl u = ϕ on ∂Ω, where ϕ ∈ H1 is a function defined on Ω, then ∥u∥H2 + ∥p∥H1 ≤ C(∥F∥L2 + ∥ϕ∥H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Since (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4) can be finded from [16], Theorem IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, we only prove the case of slip condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using the identity ∆u = ∇ div u + ∇⊥ curl u and integrating by parts, one has � | curl u|2 − � ∂ ϕ(u · n⊥) = � F · u, which implies that, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 and the trace inequality, ∥u∥H1 ≤ C(∥F∥L2 + ∥ϕ∥H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6) Since ∇p is bounded in H−1, it follows form the condition � p = 0 that p is bounded in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, taking curl on the both side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3)1 leads to −∆(curl u − ϕ) = curl F − ∆ϕ, with boundary condition curl u − ϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, using the regularity result of elliptic partial differential equations, we have ∥curl u∥H1 ≤ C∥ curl F − ∆ϕ∥H−1 + C ∥ϕ∥H1 ≤ C(∥F∥L2 + ∥ϕ∥H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, using again Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6) gives ∥u∥H2 ≤ C(∥F∥L2 + ∥ϕ∥H1 + ∥u∥L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7) and, consequently, the estimate of p is followed easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' It remains to omit the terms ∥u∥L2 on the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Indeed, this is a simple consequence of the uniqueness of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) and we leave the proof to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 10 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Slimilar results for the Laplace equations −∆u = F instead of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) with the same boundary conditions can be found in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, we give a lemma which indicates that ρ ∈ Cγ, γ 2 (QT ) for some γ ∈ (0, 1) provided v satisfying the Serrin’s condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' This result is critial to the estimate of ∆v which will be used in Section 3 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The observation is based on the fact that div v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7 ([10, 36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let v ∈ Ls(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lr) for some r, s satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='15), div v = 0, v · n = 0 and ρ ∈ C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1) be the weak solution of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)1 (in the sense of distributions), α ≤ ρ ≤ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let ρ satisfy either the Neumann condition n · ∇ρ = 0 on ∂Ω × (0, T ) or the non-homogeneous Dirichlet condition ρ = ˜ρ on ∂Ω × (0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Suppose that ρ0 ∈ Cγ0(Ω) for some γ0 ∈ (0, 1), then ρ is H¨older continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' More precisely, ρ ∈ Cγ, γ 2 (QT ), for some γ depending only on γ0, α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We only give the proof for ρ|∂Ω = ˜ρ, since the case for ρ satisfying the Neumann boundary condition has been proved in [10, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let ζ be a cut-off function, supp ζ ⊂ Br × [t0, t0 + τ], where Br is an arbitrary ball contained in Ω and [t0, t0 + τ] ⊂ (0, T ), 0 < τ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Multiplying ζ2(ρ − k)+ on (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)1 and integrating by parts leads to 1 2 sup t∈[t0,t0+τ] ∥ζ(ρ − k)+∥2 L2 + ν ∥ζ∇(ρ − k)+∥2 L2 ≤ 1 2 ∥ζ(ρ − k)+∥2 L2 (t0) + C � t0+τ t0 � Ω � |∇ζ|2 + ζ |ζt| � (ρ − k)2 + dxdt − � t0+τ t0 � Ω (v · ∇ρ)ζ2(ρ − k)+ dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8) For the last term on the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8), using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, we have ���� � t0+τ t0 � Ω (v · ∇ρ)ζ2(ρ − k)+ dxdt ���� = ���� � t0+τ t0 � Ω (v · ∇ζ)ζ(ρ − k)2 + dxdt ���� ≤ ∥v∥ L 2r r−2 t Lrx ∥ζ(ρ − k)+∥ Lr t L 2r r−2 x ∥|∇ζ| (ρ − k)+∥L2 t,x ≤ Cετ rs−2s−2r 2rs ∥|∇ζ| (ρ − k)+∥2 L2 t,x + ε � sup t∈[t0,t0+τ] ∥ζ(ρ − k)+∥2 L2 + ∥ζ∇(ρ − k)+∥2 L2 t,x � , which, alonging with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8), implies that sup t∈[t0,t0+τ] ∥ζ(ρ − k)+∥2 L2 + ν ∥ζ∇(ρ − k)+∥2 L2 ≤ ∥ζ(ρ − k)+∥2 L2 (t0) + C � t0+τ t0 � Ω � |∇ζ|2 + ζ |ζt| � (ρ − k)2 + dxdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9) The inequality above is valid for all k ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, It follows from [25] Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 that ρ ∈ Cγ, γ 2 (QT ), for some γ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For the boundary estimates, if ρ = ˜ρ on ∂Ω, we still use ζ and choose arbitrary Br×[t0, t0+τ] ⊂ R2 × [0, T ], where Br may intersect Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9) holds for k sufficiently large, since (ρ − k)+ has vanished boundary, which implies that ρ ∈ Cγ, γ 2 (QT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Once ρ is H¨older continuous, µ(ρ(x, t)) is continuous on QT and, thus, we have the following estiamtes for the non-divergence type Stokes system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 11 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let (v, p) be a strong solution of the following Stokes system, � −µ(x)∆v + ∇p = F, x ∈ Ω div v = 0, x ∈ Ω (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10) where µ(x) ∈ C(Ω), µ ∈ [µ, µ], � p = 0 and F ∈ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then there exists a positive constant C depending only on µ, µ, continuity module of µ and Ω such that (1) if u|∂Ω = Φ, where Φ ∈ H2 is a function defined on Ω, then ∥u∥H2 + ∥p∥H1 ≤ C(∥F∥L2 + ∥Φ∥H2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) (2) if u · n = 0, curl u = ϕ on ∂Ω, where ϕ ∈ H1 is a function defined on Ω, then ∥u∥H2 + ∥p∥H1 ≤ C(∥F∥L2 + ∥ϕ∥H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8 can be simply derived by using the freezing point argument, since we already have the conclusion when µ ≡ constant from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Furthermore, in order to prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3 (see Section 4), we need the following auxiliary lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The purpose for using such result will be explained in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let (v, p) be a strong solution of the following Stokes system, � − div[2µD(v)] + ∇p = F, x ∈ Ω div v = 0, x ∈ Ω (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13) where ∇µ(ρ) ∈ L4, µ is smooth and 0 < µ ≤ µ ≤ µ < ∞, � p = 0 and F ∈ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then there exists a positive constant C depending only on µ, µ and Ω such that (1) if v|∂Ω = Φ, where Φ ∈ H2 is a function defined on Ω, then ∥v∥H2 + ∥p∥H1 ≤ C �� ∥∇µ∥2 L4 + 1 � (∥F∥L2 + ∥∇Φ∥H1) + ∥∇µ∥2 L4 ∥∇v∥L2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14) (2) if v · n = 0, curl v = ϕ on ∂Ω, where ϕ ∈ H1 is a function defined on Ω, then ∥v∥H2 + ∥p∥H1 ≤ C �� ∥∇µ∥2 L4 + 1 � (∥F∥L2 + ∥ϕ∥H1) + ∥∇µ∥2 L4 ∥∇v∥L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We only give the proof for (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' First of all, we can use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4 to find a function R = C[Φ] such that div R = 0 and R|∂Ω = Φ, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13)1 becomes − div[2µ(ρ)D(v − R)] + ∇p = F + div[2µ(ρ)D(R)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using standard energy approach and the fact ∥R∥H1 ≤ C ∥∇Φ∥L2, one has ∥∇v∥L2 + ∥p∥L2 ≤ C(∥F∥H−1 + ∥∇Φ∥L2) ≤ C(∥F∥L2 + ∥∇Φ∥H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16) Next, rewritting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13)1 into the form −∆v + ∇ � p µ(ρ) � = F µ(ρ) + 2µ′∇ρ · D(v) µ(ρ) − pµ′ µ(ρ)2 ∇ρ, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5, we have ∥v∥H2 + ∥p∥H1 ≤ C [∥F∥L2 + ∥∇Φ∥H1 + ∥∇µ(ρ)∥L4 (∥∇v∥L4 + ∥p∥L4)] , which, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16), leads to ∥v∥H2 + ∥p∥H1 ≤ C �� ∥∇µ(ρ)∥2 L4 + 1 � (∥F∥L2 + ∥∇Φ∥H1) + ∥∇µ(ρ)∥2 L4 ∥∇v∥L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Thus, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 12 At last, in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 and Section 6, we need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10 (Simon [31, 34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let X ֒→ B ֒→ Y be three Banach spaces with compact imbedding X ֒→֒→ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Further, let there eixst 0 < θ < 1 and M > 0 such that ∥v∥B ≤ M ∥v∥1−θ X ∥v∥θ Y , for all v ∈ X ∩ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Denote for T > 0, W(0, T ) := W s0,r0(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' X) ∩ W s1,r1(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Y ) with s0, s1 ∈ R, r1, r0 ∈ [1, ∞], and sθ := (1 − θ)s0 + θs1, 1 rθ := 1 − θ r0 + θ r1 , s∗ := sθ − 1 rθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Assume that sθ > 0 and F is a bounded set in W(0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (1) If s∗ ≤ 0, then F is precompact in Lp(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' B) for all 1 ≤ p < − 1 s∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2) If s∗ > 0, then F is precompact in C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 3 A Priori Estimates (I): Case (A) and (B) In this section, we are going to establish the a priori bounds for (ρ, v) which will be used in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Throughout this section, let T ∈ (0, ∞) and (ρ, v) be a smooth solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18) with smooth data (ρ0, v0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Moreover, in order to simplify the notation, we always denote by ε, εi, i ∈ N+, the arbitrarily small number belongs to (0, 1/2], and we use the subscripts Cε, Cεi to emphasize the dependency of the constant C on ε, εi 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 Lower Order Estimates The first lemma is a consequence of the standard maximal principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let α ≤ ρ0 ≤ β and (ρ, v) satisfy the condition (A’) or (B’), one has α ≤ ρ(x, t) ≤ β for x ∈ Ω and all t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We only prove the upper bound, since the lower one can be derived in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17), we convert the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18) into the form ρt + v · ∇ρ − c0∆ log ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) If (ρ, v) satisfies the condition (A’), set k = β and multiply (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) by (ρ − k)+ := max{ρ − k, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' After integrating by parts, we obtain d dt � 1 2(ρ − k)2 + + � c0ρ−1 |∇(ρ − k)+|2 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2) where we have used the identity � v · ∇ρ(ρ − k)+ = � v · ∇(ρ − k)+(ρ − k)+ = 0, since v is divergence-free and v ·n = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Hence, integrating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2) from 0 to T and then, using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7) implies that sup t∈[0,T ] � (ρ − k)2 + ≤ � (ρ0 − k)2 + = 0, which implies that ρ ≤ k = β for all x ∈ Ω and t ∈ [0, T ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The case when (ρ, v) satisfies the condition (B’) can be proved analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Thus, we complete the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Our main purpose in this subsection is establishing the lower order bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We aim to prove the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 13 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let (ρ, v) satisfy the condition (A’) or (B’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Suppose that sup t∈[0,T ] ∥∇ρ∥2 L2 + � T 0 � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � dt ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) There exists a positive constant δ depending on Ω, c0, α, β and ∥v0∥L2 such that, if ∥∇ρ0∥L2 ≤ δ, sup t∈[0,T ] ∥∇ρ∥2 L2 + � T 0 � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � dt ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4) We give the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' First, we estimate the first order derivative of ρ, which is given by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' There exists a positive constant C depending only on c0 and β such that, if (ρ, v) satisfies the condition (A’), sup t∈[0,T ] ∥ρ∥L2 + ∥∇ρ∥L2(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='L2) ≤ C ∥ρ0∥L2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5) if (ρ, v) satisfies the condition (B’), sup t∈[0,T ] ∥ρ − ˜ρ∥L2 + ∥∇ρ∥L2(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='L2) ≤ C ∥ρ0 − ˜ρ∥L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) by ρ (if ρ satisfies the condition (B’), multiply ρ − ˜ρ), integrating over Ω and computing in the same way of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, one has d dt ∥ρ∥2 L2 + 2c0β−1 ∥∇ρ∥2 L2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7) Using Gr¨onwall’s inequality leads to sup t∈[0,T ] ∥ρ∥L2 + � 2c0β−1 ∥∇ρ∥L2(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='L2) ≤ ∥ρ0∥L2 , where we use the fact that ρ ≤ β from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, the following lemma shows that the second order derivative of ρ can be dominated by the norm of v provided ∥∇ρ∥L2 (t) is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let (ρ, v) satisfy the condition (A’) or (B’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then there exist a positive constant δ1 depending on Ω, c0, α, β and a positive constant C depending on Ω, c0, α and β such that, if ∥∇ρ∥L2 (t) ≤ δ1 for all t ∈ [0, T ], sup t∈[0,T ] ∥∇ρ∥2 L2 + � T 0 � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � dt ≤ exp � C � T 0 ∥v∥4 L4 dt � ∥∇ρ0∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Multiplying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18) by (−∆ρ) and integrating over Ω, we obtain d dt � 1 2 |∇ρ|2 + � c0ρ−1 |∆ρ|2 = � (v · ∇ρ)∆ρ − � c0ρ−2 |∇ρ|2 ∆ρ, which implies that, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, d dt � 1 2 |∇ρ|2 + c0β−1 � |∆ρ|2 ≤ C � � |∇ρ|2 + |v| |∇ρ| � |∆ρ| ≤ C � � |∇ρ|4 + |v|2 |∇ρ|2� + c0(2β)−1 � |∆ρ|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Hence, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, we have d dt ∥∇ρ∥2 L2 + ν ∥∆ρ∥2 L2 ≤ C ∥∇ρ∥2 L2 ∥∆ρ∥2 L2 + C ∥v∥4 L4 ∥∇ρ∥2 L2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9) 14 for some positive constant ν = ν(c0, β) and C = C(Ω, c0, α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Thus, if we choose δ1 = ν1/2(2C)−1/2 and set ∥∇ρ∥L2 (t) ≤ δ1 for all t ∈ [0, T ], using the Gr¨onwall’s inequality, we can deduce from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 that sup t∈[0,T ] ∥∇ρ∥2 L2 + ν � T 0 � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � dt ≤ exp � C � T 0 ∥v∥4 L4 dt � ∥∇ρ0∥2 L2 , which concludes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The case when (ρ, v) satisfies the condition (B’) can be com- puted in the same way, since ρt has vanished boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' From the observation of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4, in order to derive the bounds for ρ, we need to control the L4(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L4) norm of v, which is given by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let (ρ, v) satisfy the condition (A’) or (B’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Suppose that condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then there exists a positive constant C depending on Ω, c0, α and β such that sup t∈[0,T ] ∥v∥2 L2 + � T 0 � ∥v∥4 L4 + ∥∇v∥2 L2 � dt ≤ C(1 + ∥v0∥2 L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In order to simplify our proof, we only consider the case when ρ satisfies (B’) and v satisfies (A’), since other cases can be established in the same way and are much easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We first deal with a special case for curl v = −n⊥ · B · v on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We write (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)2, using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17), into the form ρvt + ρu · ∇v − div [2µD(v)] + ∇π1 = c0 div � 2µ∇2ρ−1� − c0 div � ρv ⊗ ∇ρ−1� − c2 0 div � ρ∇ρ−1 ⊗ ∇ρ−1� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) Multiplying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) by v and integrating over Ω, one has d dt � 1 2ρ |v|2 − � div [2µD(v)] · v = � c0 div � 2µ∇2ρ−1� v − � c0 div � ρv ⊗ ∇ρ−1� v − � c2 0 div � ρ∇ρ−1 ⊗ ∇ρ−1� v := 3 � i=1 Ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12) Next, for the last term on the left-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12), we use again Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 to get − � div [2µD(v)] · v = − � 2µ∆v · v − � 2µ′∇ρ · D(v) · v = � 2µ| curl v|2 + � ∂ 2µv · B · v + � 2µ′∇⊥ρ · v(curl v) − � 2µ′∇ρ · D(v) · v, ≥ µ � |curl v|2 − � Cε ∥∇ρ∥4 L4 ∥√ρv∥2 L2 + ε ∥∇v∥2 L2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13) where µ := mins∈[α,β] µ(s) and the last inequality follows from the fact that B is positive semi- definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For I1, we use the similar approach and write it in the component form, I1 = � ∂ −2c0µ∂ijρ−1vinj + � 2c0µ∂ijρ−1∂jvi = � ∂ −4c0µρ−3∂iρ∂jρvinj + � ∂ 2c0µρ−2∂ijρvinj + � 2c0µ∂ijρ−1∂jvi := 3 � i=1 Ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 15 The estimate of J3 can be simply derived by using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, that is, |J3| ≤ C ���� � µ � 2ρ−3∂iρ∂jρ − ρ−2∂ijρ � ∂jvi ���� ≤ Cε(∥∇ρ∥4 L4 + ∥∆ρ∥2 L2) + ε ∥∇v∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14) To the boundary parts J1 and J2, it suffices to estimate J′ 1 = � ∂ φ(ρ)∂iρ∂jρvinj, J′ 2 = � ∂ φ(ρ)∂ijρvinj = − � ∂ φ(ρ)vi∂inj∂jρ, where φ(·) is a positive smooth function defined on (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, one has |J′ 1| = ���� � ∂ φ(ρ)(n · ∇ρ)(v · n⊥)n⊥ · ∇ρ ���� = ���� � ∇⊥[φ(ρ)(n · ∇ρ)] · ∇ρ(v · n⊥) + � φ(ρ)(n · ∇ρ)∇ρ · ∇⊥(v · n⊥) ���� ≤ C � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � + Cε1 ∥∇ρ∥4 L4 ∥√ρv∥2 L2 + ε1 ∥∇v∥2 L2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='15) and |J′ 2| = ���� � ∂ φ(ρ)(v · n⊥)n⊥ · ∇n · ∇ρ ���� = ���� � ∇⊥φ(ρ) · (∇n · ∇ρ)(v · n⊥) − � Ω φ(ρ)∇⊥ · (∇n · ∇ρ)(v · n⊥) dx − � φ(ρ)∇⊥(v · n⊥) · (∇n · ∇ρ) ���� ≤ C � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � + Cε2 ∥∇ρ∥4 L4 ∥√ρv∥2 L2 + ε2 ∥∇v∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16), we deduce that |I1| ≤ C � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � + Cε ∥∇ρ∥4 L4 ∥√ρv∥2 L2 + ε ∥∇v∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17) Similar computation can be applied for I2 and I3, that is, |I2| ≤ Cε3 ∥∇ρ∥4 L4 ∥√ρv∥2 L2 + ε3 ∥∇v∥2 L2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18) |I3| ≤ C � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � + Cε4 ∥∇ρ∥4 L4 ∥√ρv∥2 L2 + ε4 ∥∇v∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='19) Therefore, we go back to the estimate of v, combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='19) and then, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 implies that d dt ∥√ρv∥2 L2 + ν ∥∇v∥2 L2 ≤ C � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � � 1 + ∥√ρv∥2 L2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='20) for some constant C depending on Ω, c0, α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In view of the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3), we obtain the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10) via Gr¨onwall’s inequality and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For the general case when curl v = −n⊥ · B · (v + c0∇ρ−1) on ∂Ω × (0, T ), we can also obtain the desire bounds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10) by calculating the extra boundary term � ∂ 2c0µv · B · ∇ρ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' However, this term is nothing but a special case of J′ 2 with ∇n replaced by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Therefore, following the same computation of J′ 2, we complete the proof for the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Now, we can turn back to prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 16 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5, we obtain the bound of v, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10), under the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10) leads to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8), that is, if ∥∇ρ∥L2 (t) ≤ δ1 for all t ∈ [0, T ], sup t∈[0,T ] ∥∇ρ∥2 L2 + � T 0 � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � dt ≤ C ∥∇ρ0∥2 L2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='21) where C is a positive constant depending on Ω, c0, α, β and ∥v0∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Hence, if we set the constant δ > 0 such that δ = min � C− 1 2 , δ1C− 1 2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='22) and let ∥∇ρ0∥L2 ≤ δ, then it is easy to check that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4) is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Consequently, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' It follows from the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)1 and (∇ρ, v) ∈ L4(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L4) that ρt is bounded in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 Compactness Results Before establishing higher order estimates, we tend to prove the compactness results for (ρ, v), which plays a crucial role in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3, see Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Concerning a sequence of weak solutions (ρn, vn) with π1 replaced by πn 1 satisfying the condition (A’) or (B’) and the initial conditions ρn|t=0 = ρn 0, vn|t=0 = vn 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='23) We assume that (ρn, vn) satisfy, uniformly in n ≥ 1, the a priori bounds that derived in the pre- ceeding section and ∇ρn 0, vn 0 s −−→ ∇ρ0, v0 in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Without loss of generality, extracting subsequences if necessary, we assume \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ρn w∗ −−⇀ ρ in L∞(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1), ρn w −−⇀ ρ in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H2), vn w∗ −−⇀ v in L∞(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2), vn w −−⇀ v in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='24) We may now state our compactness results whose proof is followed by [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Under the hypothesis above, we have, for all p ∈ [1, ∞), ρn s −−→ ρ in C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lp), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='25) vn s −−→ v in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='26) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Since we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='24)2 and ρn t is bounded in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2) from Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='25) can be directly derived by using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' To prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='26), observing that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)2 leads to |⟨(ρnvn)t, φ⟩| ≤ C ∥φ∥L2(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='H1) , for all φ ∈ C∞ c (Ω × [0, T ]) such that div φ = 0, which implies that (ρnvn)t is bounded in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' V −1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' On the other hand, since ρnvn is bounded in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1), it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10 that ρnvn is precompact in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2), that is, ρnvn s −−→ ρv in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='27) Thanks to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='24)4 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='25), we have ρv = ρv, vn s −−→ v in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2), which gives (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3 Higher Order Estimates In this subsection, we will show the a priori estimates for strong solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We still use the assumption at the begining of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Furthermore, throughout this subsection, we always keep ∥∇ρ0∥L2 ≤ δ small enough so that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In a word, we have all the estimates of (ρ, v) from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For convenience, we set F(t) := ∥∇v∥2 L2 + ∥∆ρ∥2 L2 + ∥ρt∥2 L2 , G(t) := ∥∆v∥2 L2 + ∥vt∥2 L2 + ∥∇∆ρ∥2 L2 + ∥∇ρt∥2 L2 , M1(t) := � ∂ µv · B · v, M2(t) := � c0µ∇⊥(v · n⊥) · B · ∇ρ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We now state the proposition we are aimming for in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let (ρ, v) satisfy the condition (A’) or (B’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then sup t∈[0,T ] F(t) + � T 0 � G(t) + ∥π∥2 H1 � dt ≤ C, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='28) where C is a positive constant depending on Ω, α, β, c0, ∥ρ0∥H2 and ∥v0∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In order to prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8, we need several auxiliary lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Under the assumptions at the begining of this section, (1) if (ρ, v) satisfies the condition (A’), then, for all ε1 ∈ (0, 1/2], d dt � ∥∆ρ∥2 L2 + ∥ρt∥2 L2 � + ∥∇∆ρ∥2 L2 + ∥∇ρt∥2 L2 ≤ Cε1A1(t)F(t) + ε1 � ∥v∥2 H2 + ∥vt∥2 L2 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='29) (2) if (ρ, v) satisfies the condition (B’), then ∥∆ log ρ∥2 L2 ≤ C � ∥(log ρ)t∥2 L2 + ∥∇v∥2 L2 � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='30) and for all ε2, ε3 ∈ (0, 1], d dt ∥(log ρ)t∥2 L2 + ∥∇(log ρ)t∥2 L2 ≤ Cε2A2(t) ∥(log ρ)t∥2 L2 + ε2 ∥vt∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='31) ∥∇∆ log ρ∥2 L2 ≤ Cε3A3(t) � ∥∆ log ρ∥2 L2 + ∥∇v∥2 L2 � + Cε3 ∥∇(log ρ)t∥2 L2 + ε3 ∥∆v∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='32) Here, C, Cε1 − Cε3 are positive constants depending on Ω, α, β, c0 with Cε1 − Cε3 extra depending on ε1–ε3 respectively, A1–A3 are all nonnegative integrable functions defined on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We first consider the case when (ρ, v) satisfies the condition (A’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Taking −(∇∆ρ)∇ on the both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)1 and integrating by parts, we have d dt � 1 2 |∆ρ|2 + � c0ρ−1 |∇∆ρ|2 = � ∇∆ρ · ∇v · ∇ρ + � v · ∇2ρ · ∇∆ρ − � 2c0ρ−3 |∇ρ|2 ∇ρ · ∇∆ρ + � c0ρ−2∇(|∇ρ|2) · ∇∆ρ + � c0ρ−2∆ρ∇ρ · ∇∆ρ := 5 � i=1 Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='33) 18 For K1–K5, we use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 to find that \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 |K1| ≤ Cε1,ε2 ∥∇ρ∥4 L4 ∥∇v∥2 L2 + ε1 ∥∇∆ρ∥2 L2 + ε2 ∥∆v∥2 L2 |K2| ≤ Cε3 ∥v∥4 L4 ∥∆ρ∥2 L2 + ε3 ∥∇∆ρ∥2 L2 |K3| ≤ Cε4 ∥∇ρ∥6 L6 + ε4 ∥∇∆ρ∥2 L2 ≤ Cε4 ∥∇ρ∥4 L4 ∥∆ρ∥2 L2 + ε4 ∥∇∆ρ∥2 L2 |K4| ≤ Cε5 ∥∇ρ∥4 L4 ∥∆ρ∥2 L2 + ε5 ∥∇∆ρ∥2 L2 |K5| ≤ Cε6 ∥∇ρ∥4 L4 ∥∆ρ∥2 L2 + ε6 ∥∇∆ρ∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='34) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='33) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='34), we have, for some ν > 0, d dt ∥∆ρ∥2 L2+ν ∥∇∆ρ∥2 L2 ≤ Cε � ∥∇ρ∥4 L4 + ∥v∥4 L4 � ∥∆ρ∥2 L2+Cε ∥∇ρ∥4 L4 ∥∇v∥2 L2+ε ∥∆v∥2 L2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='35) Next, we estimate the bound of ρt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Differentiating (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)1 with respect to t, one has ρtt − c0ρ−1∆ρt = −vt · ∇ρ − v · ∇ρt + 2c0ρ−3ρt |∇ρ|2 − c0ρ−1 � |∇ρ|2� t − c0ρ−2ρt∆ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='36) Multiplying ρt on both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='36) and integrating over Ω, we have d dt � 1 2 |ρt|2 + ν � |∇ρt|2 ≤ � |vt| |∇ρ| |ρt| + C � |ρt|2 |∇ρ|2 + � |ρt| |∇ρt| |∇ρ| + C � |ρt|2 |∆ρ| := 4 � i=1 Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='37) Similarly, we use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 to obtain \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 |L1| ≤ Cε1,ε2 ∥∇ρ∥4 L4 ∥ρt∥2 L2 + ε1 ∥vt∥2 L2 + ε2 ∥∇ρt∥2 L2 , |L2| ≤ Cε3 ∥∇ρ∥4 L4 ∥ρt∥2 L2 + ε3 ∥∇ρt∥2 L2 , |L3| ≤ Cε4 ∥∇ρ∥4 L4 ∥ρt∥2 L2 + ε4 ∥∇ρt∥2 L2 , |L4| ≤ Cε5 ∥∆ρ∥2 L2 ∥ρt∥2 L2 + ε5 ∥∇ρt∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='38) Thus, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='37) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='38), one has, for some ν > 0, d dt ∥ρt∥2 L2 + ν ∥∇ρt∥2 L2 ≤ Cε � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � ∥ρt∥2 L2 + ε ∥vt∥2 L2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='39) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='35) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='39) leads to the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, we trun to the case when (ρ, v) satisfies the condition (B’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The main difficulty in this case is that, although we still have the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='39), we can not use the energy method by integrating by parts to derive the bound of ∇∆ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' To overcome it, we estimate directly from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' More precisely, we first renormalize (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)1 by writting (log ρ)t + v · ∇ log ρ − c0ρ−1∆ log ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='40) Next, differentiating in x on both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='40), one has ∇(log ρ)t + ∇v · ∇ log ρ + v · ∇2 log ρ + c0ρ−2∇ρ∆ log ρ − c0ρ−1∇∆ log ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='41) Then, applying L2-norm for ∇∆ log ρ, then, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 leads to ∥∇∆ log ρ∥L2 ≤ C � ∥∇(log ρ)t∥L2 + ∥|∇v|·|∇ log ρ|∥L2 + ∥ |v|·|∇2 log ρ|∥L2 + ∥ |∇ρ|·|∆ log ρ|∥L2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Thus, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, for all ε1, ε2 ∈ (0, 1/2], there exists a constant Cε1,ε2 depending on Ω, c0, α, β, ε1 and ε2 such that ∥∇∆ log ρ∥2 L2 ≤ Cε1,ε2 � ∥∇(log ρ)t∥2 L2 + ∥v∥4 L4 ∥∆ log ρ∥2 L2 + ∥∇ρ∥4 L4 ∥∆ log ρ∥2 L2 � + Cε2 ∥∇ρ∥4 L4 ∥∇v∥2 L2 + ε1 ∥∇∆ log ρ∥2 L2 + ε2 ∥∆v∥2 L2 , 19 Consequently, ∥∇∆ log ρ∥2 L2 ≤ Cε ∥∇(log ρ)t∥2 L2 + Cε � ∥∇ρ∥4 L4 + ∥v∥4 L4 � ∥∆ log ρ∥2 L2 + Cε ∥∇ρ∥4 L4 ∥∇v∥2 L2 + ε ∥∆v∥2 L2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='42) which gives (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' It remains to show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='30) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='40), we use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 to obtain, ∥∆ log ρ∥2 L2 ≤ C ∥(log ρ)t∥2 L2 + C ∥∇ log ρ∥2 L4 ∥v∥2 L4 ≤ C ∥(log ρ)t∥2 L2 + Cε ∥∇ρ∥2 L2 ∥v∥2 L2 ∥∇v∥2 L2 + ε ∥∆ log ρ∥2 L2 , that is, ∥∆ log ρ∥2 L2 ≤ C � ∥(log ρ)t∥2 L2 + ∥∇v∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='31), we can follow the proof from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='36) to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='39) by applying (log ρ)t∂t on both sided (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='40) and integrating over Ω, that is, d dt � 1 2|(log ρ)t|2 + � c0ρ−1|∇(log ρ)t|2 = − � c0ρ−1∇(log ρ)t · ∇ log ρ(log ρ)t − � vt · ∇ log ρ(log ρ)t + � c0ρ−1|(log ρ)t|2|∇ log ρ|2 := 3 � i=1 Pi, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='43) where, applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 |P1| ≤ C ∥∇ log ρ∥L4 ∥(log ρ)t∥L4 ∥∇(log ρ)t∥L2 ≤ Cε1 ∥∇ρ∥4 L4 ∥(log ρ)t∥2 L2 + ε1 ∥∇(log ρ)t∥2 L2 |P2| ≤ ∥∇ log ρ∥L4 ∥(log ρ)t∥L4 ∥vt∥L2 ≤ Cε2 ∥∇ρ∥4 L4 ∥(log ρ)t∥2 L2 + ε2 ∥vt∥2 L2 |P3| ≤ ∥∇ log ρ∥2 L4 ∥(log ρ)t∥2 L4 ≤ Cε3 ∥∇ρ∥4 L4 ∥(log ρ)t∥2 L2 + ε3 ∥∇(log ρ)t∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='44) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='43) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='44) leads to d dt ∥(log ρ)t∥2 L2 + ν ∥∇(log ρ)t∥2 L2 ≤ Cε ∥∇ρ∥4 L4 ∥(log ρ)t∥2 L2 + ε ∥vt∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' One may find that we estimate log ρ instead of ρ in the proof of (ρ, v) satisfying (B’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' This is based on the observation that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)1 has the dissipative term ∆ log ρ, that is, ρt + v · ∇ρ − c0∆ log ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Thus, such conversion can avoid the occurrence of the nonlinear term |∇ρ|2, otherwise, if we estimate ∆ρ in the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='30), we need additional smallness assumption on ∇ρ0 to handle ��|∇ρ|2�� L2, which is not what we expect (this point will also be seen in Section 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The next lemma shows that v can be bounded by the norm of ρ provided ∥∇ρ0∥L2 is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Under the assumptions at the begining of this section, 20 (1) if (ρ, v) satisfy the condition (A’), then for all ε ∈ (0, 1/2], d dt � M1(t) + ∥√µ curl v∥2 L2 � + ∥vt∥2 L2 + d dtM2(t) ≤ CεA4(t)F(t) + ε � ∥∇ρt∥2 L2 + ∥∇∆ρ∥2 L2 � + A5(t), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='45) and ∥v∥2 H2 + ∥π∥2 H1 ≤ A6(t)F(t) + C � ∥∇∆ρ∥2 L2 + ∥vt∥2 L2 + ∥∇ρt∥2 L2 � + A7(t), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='46) where C and Cε are positive constants depending on Ω, c0, α, β with Cε extra depending on ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' A4–A7 are nonnegative integrable functions defined on [0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (2) if (ρ, v) satisfies the conditin (B’), one still has the estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='45) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='46) with M1(t) = M2(t) = 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' More precisely, d dt ∥√µ|D(v)|∥2 L2 + ∥vt∥2 L2 ≤ CεA8(t) � ∥∇v∥2 L2 + ∥∆ log ρ∥2 L2 � + ε ∥∇∆ log ρ∥2 L2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='47) and ∥v∥2 H2 + ∥π∥2 H1 ≤ CεA9(t) � ∥∇v∥2 L2 + ∥∆ log ρ∥2 L2 � + ε ∥∇∆ log ρ∥2 L2 + C � ∥vt∥2 L2 + ∥∇(log ρ)t∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='48) where C and Cε as in (1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' A8 and A9 are nonnegative integrable functions defined on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Rewrite (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)2 as ρvt − div [2µD(v)] + ∇π1 = −ρu · ∇v + c0 div � 2µ∇2ρ−1� − c0 div � ρv ⊗ ∇ρ−1� − c2 0 div � ρ∇ρ−1 ⊗ ∇ρ−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='49) We first come to the case when (ρ, v) satisfies the condition (A’) and consider the special case when curl v = −n⊥ · B · v on ∂Ω × (0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Multiplying vt on both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='49) and integrating over Ω, one gets � ρ |vt|2 − � div[2µD(v)] · vt = − � ρu · ∇v · vt + � c0 div � 2µ∇2ρ−1� vt − � c0 div � ρv ⊗ ∇ρ−1� vt − � c2 0 div � ρ∇ρ−1 ⊗ ∇ρ−1� vt := 4 � i=1 Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='50) For the second term on the left-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='50), we have − � div[2µD(v)] · vt = − � 2µ∆v · vt − � 2µ′∇ρ · D(v) · vt := 2 � i=1 Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' First, to estimate Q1, we have Q1 = − � 2µ∇⊥(curl v) · vt = − � ∂ 2µ curl v(vt · n⊥) + � µ d dt| curl v|2 + � µ′∇⊥ρ · vt(curl v) = d dt � M1(t) + ∥√µ curl v∥2 L2 � − � ∂ µt(v · B · v) − � µt| curl v|2 + � µ′∇⊥ρ · vt(curl v), 21 where, for the last three terms, we use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 to obtain ���� � ∂ µt(v · B · v) ���� = ���� � ∂ µt(v · n⊥)n⊥ · B · v ���� = ���� � µt∇⊥(v · n⊥) · B · v + � v · n⊥∇⊥ · [µtB · v] ���� ≤ Cε1 � ∥∇ρ∥4 L4 + ∥v∥4 L4 � ∥ρt∥2 L2 + Cε1 ∥v∥4 L4 + ε1 � ∥∇ρt∥2 L2 + ∥v∥2 H2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' ���� � µt |curl v|2 ���� ≤ Cε2 ∥ρt∥2 L2 ∥∇v∥2 L2 + ε2 ∥v∥2 H2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' ���� � µ′∇⊥ρ · vt(curl v) ���� ≤ Cε3 ∥∇ρ∥4 L4 ∥∇v∥2 L2 + ε3 � ∥vt∥2 L2 + ∥v∥2 H2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' which gives Q1 ≥ d dt � M1(t) + ∥√µ curl v∥2 L2 � − Cε4 � ∥ρt∥2 L2 + ∥v∥4 L4 + ∥∇ρ∥4 L4 � � ∥∇v∥2 L2 + ∥ρt∥2 L2 � + Cε4 ∥v∥4 L4 + ε4 � ∥∇ρt∥2 L2 + ∥vt∥2 L2 + ∥v∥2 H2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='51) On the other hand, |Q2| ≤ Cε5 ∥∇ρ∥4 L4 ∥∇v∥2 L2 + ε5 � ∥vt∥2 L2 + ∥v∥2 H2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='52) Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='51) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='52) leads to − � div[2µD(v)] · vt ≥ d dt � M1(t) + ∥√µ curl v∥2 L2 � − Cε6 � ∥ρt∥2 L2 + ∥v∥4 L4 + ∥∇ρ∥4 L4 � � ∥∇v∥2 L2 + ∥ρt∥2 L2 � + Cε6 ∥v∥4 L4 + ε6 � ∥∇ρt∥2 L2 + ∥vt∥2 L2 + ∥v∥2 H2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='53) Next, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 again, we estimate M1 − M4, that is, |M1| ≤ Cε7 � ∥v∥4 L4 + ∥∇ρ∥4 L4 � ∥∇v∥2 L2 + ε7 � ∥vt∥2 L2 + ∥v∥2 H2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='54) |M2| = 2c0 ���� � µ′∂jρ∂ijρ−1(vt)i + � µ(ρ)∂ijjρ−1(vt)i ���� = 2c0 ���� � µ′∂jρ∂ijρ−1(vt)i − � µ′∂iρ∂jjρ−1(vt)i ���� ≤ C � (|∇ρ|3 + |∇ρ| |∇2ρ|) |vt| ≤ Cε8 ∥∇ρ∥4 L4 ∥∆ρ∥2 L2 + ε8 � ∥vt∥2 L2 + ∥∇∆ρ∥2 L2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='55) |M3 + M4| = ����c0 � ∂j (uj∂i log ρ) (vt)i ���� = ����c0 � ∂j (log ρ∂ivj) (vt)i + c2 0 � ∂j � log ρ∂ijρ−1� (vt)i ���� ≤ C � � |∇v| |∇ρ| + |∇ρ|3 + |∇ρ| ��∇2ρ �� � |vt| ≤ Cε9 ∥∇ρ∥4 L4 � ∥∇v∥2 L2 + ∥∆ρ∥2 L2 � + ε9 � ∥vt∥2 L2 + ∥v∥2 H2 + ∥∇∆ρ∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='56) 22 Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='53)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='56), we have, for all ε ∈ (0, 1/2], there exists a positive constant Cε depending on Ω, c0, α, β and ε such that d dt � M1(t) + ∥√µ curl v∥2 L2 � + ∥vt∥2 L2 ≤ Cε � ∥ρt∥2 L2 + ∥v∥4 L4 + ∥∇ρ∥4 L4 � F(t) + Cε ∥v∥4 L4 + ε � ∥∇ρt∥2 L2 + ∥v∥2 H2 + ∥∇∆ρ∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='57) For general boundary case, that is, curl v = −n⊥ · B · (v + c0∇ρ−1), it suffices to calculate the following extra term � ∂ φ(ρ)vt · B · ∇ρ = � ∂ φ(ρ)(vt · n⊥)n⊥ · B · ∇ρ = � (vt · n⊥)∇⊥φ(ρ) · B · ∇ρ + � φ(ρ)(vt · n⊥)∇⊥ · (B · ∇ρ) + � φ(ρ)∇⊥(vt · n⊥) · B · ∇ρ := 3 � i=1 Gi, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='58) where φ(s) := c0µ(s)s−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For the first two terms of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='58), using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, we have |G1 + G2| ≤ Cε1 � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � + ε1 ∥vt∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='59) For G3, since we can not handle the term ∇⊥(vt · n⊥), it shall be converted into G3 = d dtM2(t) − � φ(ρ)t∇⊥(v · n⊥) · B · ∇ρ − � φ(ρ)∇⊥(v · n⊥) · B · ∇ρt ≥ d dtM2(t) − � Cε2 � ∥ρt∥2 L2 + ∥∇ρ∥4 L4 ∥∇v∥2 L2 + ∥∇v∥2 L2 � + ε2 � ∥v∥2 H2 + ∥∇ρt∥2 L2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Thus, combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='60)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='57), we deduce the estimate which is simliar with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='57), that is d dt � M1(t) + ∥√µ curl v∥2 L2 � + ∥vt∥2 L2 + d dtM2(t) ≤ CεA4(t)F(t) + ε � ∥∇ρt∥2 L2 + ∥v∥2 H2 + ∥∇∆ρ∥2 L2 � + A5(t), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='60) for some integrable functions A4 and A5 defined on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We still need estimate ∥v∥H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let us rewrite (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='49) as − µ∆v + ∇π = F, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='61) where F := −ρvt + ∇(log ρ)t − ρu · ∇v + 2µ′∇ρ · D(v) + 2c0 div � µ∇2ρ−1� − c0 div � ρv ⊗ ∇ρ−1� − c2 0 div � ρ∇ρ−1 ⊗ ∇ρ−1� Since µ(ρ(x, t)) is bounded contiuous on QT from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7, it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8 with ϕ = −n⊥ · B · (v + c0∇ρ−1) that ∥v∥H2 + ∥π∥H1 ≤ C (∥F∥L2 + ∥ϕ∥H1) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='62) where ∥ϕ∥H1 ≤ C � ∥∇v∥L2 + ∥∆ρ∥L2 + ∥∇ρ∥2 L4 � , ∥F∥L2 ≤ C � ∥vt∥L2 + ∥∇ρt∥L2 + ∥∇ρ∥2 L4 ∥ρt∥L2 + ∥|v|·|∇v|∥L2 + ∥|∇ρ|·|∇v|∥L2 + ∥∇ρ∥3 L6 + ∥|∇ρ|·|∇2ρ|∥L2 + ∥|v|·|∇2ρ|∥L2 + ∥|v|·|∇ρ|2∥L2� ≤ C (∥vt∥L2 + ∥∇ρt∥L2) + Cε � ∥v∥2 L4 + ∥∇ρ∥2 L4 � (∥∇v∥L2 + ∥∆ρ∥L2 + ∥ρt∥L2) + ε (∥v∥H2 + ∥∇∆ρ∥L2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 23 Hence, using the Poincaré’s inequality, we deduce from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='62) that ∥v∥2 H2 + ∥π∥2 H1 ≤ C � ∥v∥4 L4 + ∥∇ρ∥4 L4 � � ∥∇v∥2 L2 + ∥∆ρ∥2 L2 + ∥ρt∥2 L2 � + C � ∥vt∥2 L2 + ∥∇ρt∥2 L2 � + C � ∥∇∆ρ∥2 L2 + ∥∇v∥2 L2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='63) which gives (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Finally, substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='46) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='60), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='45) and complete the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For (ρ, v) satisfying condition (B’), we first convert (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='49) into ρvt − div [2µD(v)] + ∇π1 = −ρv · ∇v + c0∇ log ρ · ∇v + c0 div � 2µρ−1 � ∇2 log ρ − ∇ log ρ ⊗ ∇ log ρ �� + c0 div (v ⊗ ∇ log ρ) − c2 0 div � ρ−1∇ log ρ ⊗ ∇ log ρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='64) Then, following the calculations from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='49) to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='57) and from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='61) to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='63), we can derive the similar estimates (even much easier, since vt is vanished on the boundary), that is, d dt ∥µD(v)∥2 L2 + ν ∥vt∥2 L2 ≤ Cε � ∥∇ρ∥4 L4 + ∥v∥4 L4 + ∥ρt∥2 L2 � � ∥∇v∥2 L2 + ∥∆ log ρ∥2 L2 � + ε � ∥∇∆ log ρ∥2 L2 + ∥v∥2 H2 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='65) ∥F∥L2 ≤ C (∥vt∥L2 + ∥∇ log ρt∥L2) + Cε � ∥v∥2 L4 + ∥∇ρ∥2 L4 � (∥∇v∥L2 + ∥∆ log ρ∥L2) + ε (∥v∥H2 + ∥∇∆ log ρ∥L2) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='66) and, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8 with Φ = 0, together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='66), ∥v∥2 H2 + ∥π∥2 H1 ≤ C ∥F∥2 L2 ≤ Cε � ∥v∥4 L4 + ∥∇ρ∥4 L4 � � ∥∇v∥2 L2 + ∥∆ log ρ∥2 L2 � + ε ∥∇∆ log ρ∥2 L2 + C � ∥vt∥2 L2 + ∥∇(log ρ)t∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='67) Thus, we complete the proof by plugging (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='67) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Now, combining Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11, we can complete the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We first prove the case when (ρ, v) satisfies the condition (A’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='29) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='45), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='46) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11 leads to d dt � ∥√µ curl v∥2 L2 + ∥∆ρ∥2 L2 + ∥ρt∥2 L2 + M1(t) � + ∥vt∥2 L2 + ∥∇∆ρ∥2 L2 + ∥∇ρt∥2 L2 ≤ − d dtM2(t) + ˜ A1(t)F(t) + ˜ A2(t), ≤ − d dtM2(t) + ˜ A1(t) � ∥√µ curl v∥2 L2 + ∥∆ρ∥2 L2 + ∥ρt∥2 L2 + M1(t) � + ˜ A2(t), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='68) where ˜ A1 and ˜ A2 are positive integrable functions defined on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using Gr¨onwall’s inequality and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 once again, we deduce the bound sup t∈[0,T ] F(t) + � T 0 � ∥∇∆ρ∥2 L2 + ∥∇ρt∥2 L2 + ∥vt∥2 L2 � dt ≤ C(∥v0∥2 H1 + ∥ρ0∥2 H2 + ∥v0∥2 H1 ∥ρ0∥2 H2 + 1) ≤ C, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='69) where we have used, denote by ρt,0 = ρt(x, 0), ∥ρt,0∥L2 ≤ ∥ρ0∥H2 + ∥v0∥H1 ∥ρ0∥H2 , 24 M1 ≥ 0 for B positively semi-definited and �����e � T 0 h(t) dt � T 0 d dtM2(t)e−� t 0 h(s) ds dt ����� ≤ ε sup t∈[0,T ] ∥∇v∥2 L2 + Cε sup t∈[0,T ] ∥∇ρ∥2 L2 , where h(t) is an integrable function on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, integrating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='46) over [0, T ] and using the bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='69) gives � T 0 � ∥v∥2 H2 + ∥π∥2 H1 � dt ≤ C, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='70) which shows (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' To prove the case when (ρ, v) satisfies the condition (B’), we use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='31) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='47) in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' It follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 2 � |D(v)|2 = � |∇v|2 that d dt � ∥∇v∥2 L2 + ∥(log ρ)t∥2 L2 � + ∥vt∥2 L2 + ∥∇(log ρ)t∥2 L2 ≤ ˜ A3(t) � ∥∇v∥2 L2 + ∥(log ρ)t∥2 L2 � + ε ∥∇∆ log ρ∥2 L2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='71) for some nonegative integrable functions ˜ A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using the Gr¨onwall’s inequality, one has the bound sup t∈[0,T ] ∥∇v∥2 L2 + sup t∈[0,T ] ∥(log ρ)t∥2 L2 + � T 0 � ∥vt∥2 L2 + ∥∇(log ρ)t∥2 L2 � dt ≤ C(∥v0∥2 H1 + ∥ρ0∥2 H2 + ∥v0∥2 H1 ∥ρ0∥2 H2 + 1) ≤ C, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='72) With the aid of the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='30), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='32), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='48) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='72), one has sup t∈[0,T ] ∥∆ log ρ∥2 L2 + � T 0 � ∥∇∆ log ρ∥2 L2 + ∥v∥2 H2 + ∥π∥2 H1 � dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='73) At last, noticing that ∆ρ = ρ∆ log ρ + ρ−1|∇ρ|2, and ∇∆ρ = ∇ρ∆ log ρ + ρ∇∆ log ρ − ρ−2∇ρ|∇ρ|2 + 2ρ−1∇ρ · ∇2ρ, we complete the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 4 A Priori Estimates (II): Case (C) In this section, we will prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5 via a different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The main difficulty lines here is that, in this situation, v · n = 0 and v = −c0∇ρ−1 on ∂Ω, which makes one impossible to handle the high order derivatives of ρ appeared in the boundary integrals when we deal with the energy estimates of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' To over come it, we may take a different decomposition on u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' This idea mainly comes from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4, which pushes us to introduce a new function Q = B[c0∆ρ−1] to eliminate the non- divergence-free condition on u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' More precisely, we split u into two parts, u = w + Q, and one can find that w possesses the nice properties, that is, w is divergence-free and w|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Therefore, we can use w to get the energy estimates for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Fortunately, in spite of this difficulty, we still has the estimates on ρ, which has been derived in Section 3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3 such as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='39), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' This is because those estimates only require v · n = 0 on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, using the relation v = u − c0∇ρ−1, one can easily change the norm of v into that of u and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 25 Before giving the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5, we make some comments on the analysis in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In this section, we devote to establish the higher order estimates for (ρ, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' One should notice that, if Q = B[c0∆ρ−1], then Qt = B[c0∆ρ−1 t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' With this fact, using Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4, one has the following estimates, \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 ∥Q∥Lp ≤ C ∥∇ρ∥Lp , ∥Q∥H1 ≤ C � ∥∆ρ∥L2 + ∥∇ρ∥2 L4 � , ∥Qt∥Lp ≤ C (∥∇ρt∥Lp + ∥|ρt||∇ρ|∥Lp) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) for 1 < p < ∞ and some constants C depending only on Ω, c0 and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In addition, the following equivalence will be used frequently, that is, ∥u∥Lp + ∥∇ρ∥Lp ∼ ∥v∥Lp + ∥∇ρ∥Lp , ∥∇u∥Lp + ∥∆ρ∥Lp + ∥∇ρ∥2 L2p ∼ ∥∇v∥Lp + ∥∆ρ∥Lp + ∥∇ρ∥2 L2p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2) Now, we turn to the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The key of the proof is deriving the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using the idea from [22], we first assume the bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) and obtain the a priori estimates of (ρ, u), see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, these bounds lead to smaller ones (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4) provided ∥∇u0∥L2 suitably small, which means that we can close the energy estimates of (ρ, u) and, consequently, we complete the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Suppose that (ρ, u, π) is a smooth solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' There exists a positive constant δ depending only on Ω, α, β and c0 such that, if ∥∇u0∥L2 ≤ δ and sup t∈[0,T ] ∥∇ρ∥L4 ≤ 2, � T 0 � ∥∆ρ∥4 L2 + ∥∇u∥4 L2 � dt ≤ 2 ∥∇u0∥2 L2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) then the following estimates hold sup t∈[0,T ] ∥∇ρ∥L4 ≤ 1, � T 0 � ∥∆ρ∥4 L2 + ∥∇u∥4 L2 � dt ≤ ∥∇u0∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4) In order to prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, we need the following estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Suppose that (ρ, u, π) is a smooth solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' There exists some positive constant C depending on Ω, α, β and c0 such that, for all T ∈ (0, ∞), α ≤ ρ ≤ β and sup t∈[0,T ] ∥ρ − (ρ0)Ω∥2 L2 + � T 0 ∥∇ρ∥2 L2 dt ≤ ∥ρ0 − (ρ0)Ω∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5) Furthermore, if ∥∇u0∥L2 ≤ 1 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) holds, one has sup t∈[0,T ] ∥∇ρ∥2 L2 + � T 0 � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � dt ≤ C ∥∇ρ0∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Suppose that ∥∇u0∥L2 ≤ 1 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) is established, then one has sup t∈[0,T ] F(t) + � T 0 � G(t) + ∥π∥2 H1 � dt ≤ C ∥∇u0∥2 L2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7) where λ and C are positive constants depending on Ω, α, β and c0, F(t) := ∥∇u∥2 L2 + ∥ρt∥2 L2 + ∥∆ρ∥2 L2 , G(t) := ∥ut∥2 L2 + ∥∆u∥2 L2 + ∥∇∆ρ∥2 L2 + ∥∇ρt∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' One should keep in mind that we always have ∥ρ0 − (ρ0)Ω∥L2 ≤ C ∥∇ρ0∥L2 ≤ C ∥u0∥L2 ≤ C ∥∇u0∥L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8) 26 We temporarily assume that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3 are established and prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Since Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3 is established, we have, for some C1 > 0 depending only on Ω, sup t∈[0,T ] ∥∇ρ∥L4 ≤ C1 sup t∈[0,T ] ∥∆ρ∥L2 ≤ C1C ∥∇u0∥L2 ≤ 1 provided ∥∇u0∥L2 ≤ δ1 := (C1C)−1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9) and, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, � T 0 � ∥∆ρ∥4 L2 + ∥∇u∥4 L2 � dt ≤ � sup t∈[0,T ] ∥∆ρ∥2 L2 + sup t∈[0,T ] ∥∇u∥2 L2 � � T 0 � ∥∆ρ∥2 L2 + ∥∇u∥2 L2 � dt ≤ C2 ∥∇u0∥4 L2 ≤ ∥∇u0∥2 L2 provided ∥∇u0∥L2 ≤ δ2 := C−1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10) where C is the constant in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Thus, if we choose ∥∇u0∥L2 ≤ δ := min {1, δ1, δ2} , then the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Now, we trun back to prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Since Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 has already been proved in Section 3, we only give the proof for Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We start with the lower order estimate of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Multiplying w on the both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1)2 and integrating over Ω, one has d dt � 1 2ρ|u|2 + � 2µ|D(u)|2 = � ρut · Q + � ρu · ∇u · Q − � div[2µD(u)] · Q = � ρut · Q + � ρu · ∇u · Q + � 2µD(u) · ∇Q := 3 � i=1 Si, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) where, using estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 |S1| ≤ C ∥Q∥L2 ∥ut∥L2 ≤ Cε1 ∥∇ρ∥2 L2 + ε1 ∥ut∥2 L2 , |S2| ≤ C ∥u∥L4 ∥∇u∥L2 ∥Q∥L4 ≤ Cε2 ∥∆ρ∥4 L2 ∥u∥2 L2 + ε2 ∥∇u∥2 L2 , |S3| ≤ C ∥∇Q∥L2 ∥∇u∥L2 ≤ Cε3 � ∥∆ρ∥2 L2 + ∥∇ρ∥4 L4 � + ε3 ∥∇u∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12) Here, we still use the notation εi ∈ (0, 1/2] and the constant Cεi as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12) leads to, ∃ ν > 0, d dt ∥u∥2 L2 + ν ∥∇u∥2 L2 ≤ Cε � ∥∆ρ∥4 L2 ∥u∥2 L2 + ∥∆ρ∥2 L2 + ∥∇ρ∥4 L4 � + Cε ∥∇ρ∥2 L2 + ε ∥ut∥2 L2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13) For the estimate of ut, multiplying wt on the both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1)2, one has � ρ|ut|2 + d dt � µ|D(u)|2 = − � ρut · Qt − � ρu · ∇u · wt + � µt|D(u)|2 − � div[2µD(u)] · Qt := 4 � i=1 Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14) 27 Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) and Poincaré’s inequality, we have ∥Qt∥2 L2 ≤ C � ∥∇ρt∥2 L2 + ∥ρt∥2 L4 ∥∇ρ∥2 L4 � ≤ C ∥∇ρt∥2 L2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='15) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 |U1| ≤ C ∥Qt∥L2 ∥ut∥L2 ≤ Cε1 ∥∇ρt∥2 L2 + ε1 ∥ut∥2 L2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' |U2| ≤ C ∥u∥L4 ∥∇u∥L4 ∥wt∥L2 ≤ Cε2 ∥u∥2 L4 ∥∇u∥2 L4 + ε2 ∥ut∥2 L2 + C ∥∇ρt∥2 L2 ≤ Cε2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='ε3 ∥∇u∥4 L2 ∥∇u∥2 L2 + C ∥∇ρt∥2 L2 + ε2 ∥ut∥2 L2 + ε3 ∥∆u∥2 L2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' |U3| ≤ C ∥ρt∥L2 ∥∇u∥2 L4 ≤ Cε4 ∥∇u∥2 L2 ∥ρt∥2 L2 + ε4 ∥∆u∥2 L2 ≤ Cε4 ∥∇u∥4 L2 ∥ρt∥2 L2 + ε4 � ∥∇ρt∥2 L2 + ∥∆u∥2 L2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' |U4| ≤ C � ∥∇ρ∥L4 ∥∇u∥L4 + ∥∇2u∥L2� ∥∇ρt∥L2 ≤ Cε5 ∥∆ρ∥4 L2 ∥∇u∥2 L2 + Cε5 ∥∇ρt∥2 L2 + ε5 ∥∆u∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16) Thus, combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16), one has d dt∥√µD(u)∥2 L2 + ν ∥ut∥2 L2 ≤ Cε � ∥∇u∥4 L2 + ∥∆ρ∥4 L2 � ∥∇u∥2 L2 + Cε ∥∇u∥4 L2 ∥ρt∥2 L2 + Cε ∥∇ρt∥2 L2 + ε ∥∆u∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17) From the observation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17), one have to derive the estimate of ∆u, or that of ∆v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Unfortu- nately, we can not directly use, for example, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='63) in the Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The main obstacle here is that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='63) strongly depends on the conitnuity of ρ and we have not closed the lower bounds of v yet, so that we can not apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8 (notice that Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7 requiring v ∈ Ls(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lr)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Consequently, we apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9 with Φ = −c0∇ρ−1 on − div[2µD(v)] + ∇π = F, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18) where F = −ρut − ρu · ∇u + c0 div � 2µ∇2ρ−1� and, using condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and Poincaré’s inequality, ∥F∥L2 ≤ C ∥ut∥L2 + Cε2 ∥u∥2 L4 ∥∇u∥L2 + C ∥∇ρ∥2 L4 ∥∆ρ∥L2 + ε2 ∥∆u∥L2 + C ∥∇∆ρ∥L2 ≤ C ∥ut∥L2 + Cε2 � ∥u∥2 L4 + ∥∇ρ∥2 L4 � (∥∇u∥L2 + ∥∆ρ∥L2) + ε2 ∥v∥H2 + C ∥∇∆ρ∥L2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='19) where we have applied the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2) and ∥∆u∥L2 ≤ C � ∥∆v∥L2 + ��∇∆ρ−1�� L2 � ≤ C � ∥∆v∥L2 + ∥∇ρ∥2 L4 ∥∆ρ∥L2 + ∥∇∆ρ∥L2 � use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) ≤ C (∥∆v∥L2 + ∥∇∆ρ∥L2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='20) Then, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9 and Poincaré’s inequality, combining the condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) and the estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='19), we can derive a similar estimate of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='63), that is, ∥v∥2 H2 + ∥∇π∥2 L2 ≤ C ∥ut∥2 L2 + C � ∥u∥4 L4 + ∥∇ρ∥4 L4 � � ∥∇u∥2 L2 + ∥∆ρ∥2 L2 � + C ∥∇∆ρ∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using again (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='20) we derive that ∥∆u∥2 L2 + ∥∇π∥2 L2 ≤ C ∥ut∥2 L2 + C � ∥u∥4 L4 + ∥∇ρ∥4 L4 � � ∥∇u∥2 L2 + ∥∆ρ∥2 L2 � + C ∥∇∆ρ∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='21) 28 Then, substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='21) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17), ∃ ν > 0, d dt ∥∇u∥2 L2 + ν ∥∆u∥2 L2 + ν ∥ut∥2 L2 ≤ Cε � ∥∇u∥4 L2 + ∥∆ρ∥4 L2 � F(t) + Cε ∥∇ρt∥2 L2 + ε ∥∇∆ρ∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='22) Next, in order to close the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='22), we turn to get the bounds of ∇ρt and ∇∆ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' From the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='39) and ∥vt∥L2 ≤ C(∥ut∥L2 + ��∇ρ−1 t �� L2) ≤ C � ∥ut∥L2 + ∥∇ρt∥L2 + ∥∇ρ∥2 L4 ∥ρt∥L2 � use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) ≤ C(∥ut∥L2 + ∥∇ρt∥L2) we have d dt ∥ρt∥2 L2 + ν ∥∇ρt∥2 L2 ≤ Cε � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � ∥ρt∥2 L2 + ε ∥ut∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='23) On the other hand, for ∇∆ρ, since the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='35) we derived in the Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3 is still valid, replacing v by u and ∇ρ, one has d dt ∥∆ρ∥2 L2 + ν ∥∇∆ρ∥2 L2 ≤ Cε � ∥∆ρ∥4 L2 + ∥∇u∥4 L2 � � ∥∆ρ∥2 L2 + ∥∇u∥2 L2 � + ε ∥∆u∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='24) Using this inequality together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='21), we eliminate term ∆u and, then, we subsititute it, alonging with(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='23), into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='22) to deduce that d dtF(t) + νG(t) ≤ C � ∥∇u∥4 L2 + ∥∆ρ∥4 L2 + ∥∆ρ∥2 L2 � F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='25) Since we have ∥∇ρ0∥H1 ≤ ∥∇u0∥L2 from Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11, applying Gr¨onwall’s inequality for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='25) and using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2, we have sup t∈[0,T ] F(t) + � T 0 G(t) dt ≤ C � ∥∇u0∥2 L2 + ∥∇u0∥4 L2 � ≤ C ∥∇u0∥2 L2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='26) which, turning back to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='21) to get the bound for π, implies the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Therefore, we complete the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 5 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6 In this section, we devote to accomplish the proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6 in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Our proofs are basically relying on the approach in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, we are going to solve the linearized system and give some basic uniform estimates, which is critical for the existence proofs in next few subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3, we will construct an approximate system and use the contraction mapping theorem to show that it admits a unique smooth solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Finally, in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4, we will prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 Linearized Problem Consider the following linearized problem \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ρt + Φ · ∇ρ − div(ϕ−1∇ρ) = 0, ρut + ρ(Φ + ∇ϕ−1) · ∇u − div(2µD) + ∇p = 0, div u = c0∆ρ−1, ρ|t=0 = ρ0, u|t=0 = u0, α ≤ ρ0 ≤ β, (ρ0, u0) ∈ [C∞(Ω)]4 satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11), (ρ, u) satisfies one of the bundary conditions (A) − (C), (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=') where � p = 0, µ = µ(x, t) ∈ H1(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' C∞(Ω)) is a positive function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 29 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 (Linearized problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Assume that the hypotheses of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6 are satisfied by the data (ρ0, u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' If Φ and ϕ satisifes the following conditions \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Φ ∈ C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H2), Φt ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2), ϕ ∈ C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H2) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H3), ϕt ∈ C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1), c−1 ≤ ϕ ≤ c, div Φ = 0 in Ω, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) then there exists a unique global strong solution (ρ, u, p) to the problem (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=') satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We only give the a priori estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The unique sovablity is obvious, since we can first solve (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=')1 by the theories of linear parabolic equations, see [25] and, then, derive u from (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=')2, see [11, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Firstly, the sup-bound and lower order estimates of ρ has been proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' See Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3, that is, α ≤ ρ ≤ β, ∥ρ∥2 L2 + � t 0 ∥∇ρ∥2 L2 ds ≤ C(Ω, c) (or C(Ω, c, ˜ρ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2) Next, we multiply −∆ρ on (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=')1 and integrate over Ω, then, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, we have d dt ∥∇ρ∥2 L2 + ν ∥∆ρ∥2 L2 ≤ C � (|Φ| + |∇ϕ|) |∇ρ||∆ρ| ≤ C � ∥Φ∥4 L4 + ∥∇ϕ∥4 L4 � ∥∇ρ∥2 L2 + ν 2 ∥∆ρ∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Thus, d dt ∥∇ρ∥2 L2 + ν ∥∆ρ∥2 L2 ≤ C � ∥Φ∥4 L4 + ∥∇ϕ∥4 L4 � ∥∇ρ∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) By virtue of Gr¨onwall’s inequality, we derive ∥∇ρ∥2 L2 (t) + ν � t 0 ∥∆ρ∥2 L2 ds ≤ ∥∇ρ0∥2 L2 exp � C � t 0 � ∥Φ∥4 L4 + ∥∇ϕ∥4 L4 � ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4) To get the higher order bounds, if n · ∇ρ = 0 on ∂Ω, we apply −∇∆ρ∇ on both sides of (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=')1 and integrate over Ω, then d dt � 1 2 |∆ρ|2 + � ϕ−1 |∇∆ρ|2 = � ∇∆ρ · ∇Φ · ∇ρ + � Φ · ∇2ρ · ∇∆ρ − � 2ϕ−3|∇ϕ|2∇ρ · ∇∆ρ + � ϕ−2∇ϕ · ∇2ρ · ∇∆ρ + � ϕ−2∇ρ · ∇2ϕ · ∇∆ρ − � ϕ−2∆ρ∇ϕ · ∇∆ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and Poincar e’s inequality, we have d dt ∥∆ρ∥2 L2 + ν ∥∇∆ρ∥2 L2 ≤ (∥∇Φ∥L4 ∥∇ρ∥L4 + ∥Φ∥L4 ∥∆ρ∥L4) ∥∇∆ρ∥L2 + C ∥∇ϕ∥2 L8 ∥∇ρ∥L4 ∥∇∆ρ∥L2 + C � ∥∇ϕ∥L4 ∥∆ρ∥L4 + ��∇2ϕ �� L4 ∥∇ρ∥L4 � ∥∇∆ρ∥L2 ≤ C � ∥Φ∥4 W 1,4 + ∥ϕ∥4 W 2,4 + ∥∇ϕ∥8 L8 � ∥∆ρ∥2 L2 + ν 2 ∥∇∆ρ∥2 L2 , that is d dt ∥∆ρ∥2 L2 + ν ∥∇∆ρ∥2 L2 ≤ C � ∥Φ∥4 W 1,4 + ∥ϕ∥4 W 2,4 + ∥∇ϕ∥8 L8 � ∥∆ρ∥2 L2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5) which, using Gr¨onwall’s inequality, leads to ∥∆ρ∥2 L2 (t) + ν � t 0 ∥∇∆ρ∥2 L2 ds ≤ C ∥∆ρ0∥2 L2 exp �� t 0 � ∥Φ∥4 W 1,4 + ∥ϕ∥4 W 2,4 + ∥∇ϕ∥8 L8 � ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6) 30 For ρ satisfying the non-homogeneous Dirichlet condition, that is, ρ|∂Ω = ˜ρ, taking ρt∂t on (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=')1, one has d dt � 1 2 |ρt|2 + ν � |∇ρt|2 ≤ � |Φt| |∇ρ| |ρt| + C � |ϕt| |ρt||∇ϕ||∇ρ| + C � |ρt| |∇ρt| |∇ϕ| + C � |ρt| |∇ϕt| |∇ρ| + C � |ϕt||ρt| |∆ρ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Simiarly, it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 that d dt ∥ρt∥2 L2 + ν ∥∇ρt∥2 L2 ≤ C (∥Φt∥L2 + ∥∇ϕ∥L4 ∥ϕt∥L4) ∥∇ρ∥L4 ∥ρt∥L4 + C (∥∇ϕ∥L4 ∥∇ρt∥L2 + ∥∇ρ∥L4 ∥∇ϕt∥L2) ∥ρt∥L4 + C ∥∆ρ∥L2 ∥ϕt∥L4 ∥ρt∥L4 ≤ � ∥Φt∥2 L2 + ∥ϕt∥4 L4 + ∥∇ϕ∥4 L4 + ∥∇ϕt∥2 L2 � ∥ρt∥2 L2 + ν 2 ∥∇ρt∥2 L2 + C � ∥Φt∥2 L2 + ∥ϕt∥4 L4 + ∥∇ϕ∥4 L4 + ∥∇ϕt∥2 L2 � ∥∇ρ∥2 L2 + C ∥∆ρ∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' that is, d dt ∥ρt∥2 L2 + ν ∥∇ρt∥2 L2 ≤ C � ∥Φt∥2 L2 + ∥ϕt∥4 L4 + ∥∇ϕ∥4 L4 + ∥∇ϕt∥2 L2 � � ∥ρt∥2 L2 + ∥∇ρ∥2 L2 � + C ∥∆ρ∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7) Then, using Gr¨onwall’s inequality and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4), one has ∥ρt∥2 L2 (t) + ν � t 0 ∥∇ρt∥2 L2 ds ≤ C(Ω, c, α, β, Φ, ϕ, u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8) Noticing that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8) also holds for the Neumann case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, we take L2-norm on (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=')1 and use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 to get ∥∆ρ∥2 L2 ≤ C ∥ρt∥2 L2 + C � ∥Φ∥4 L4 + ∥∇ϕ∥4 L4 � ∥∇ρ∥2 L2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9) and take ∇ on both sides of (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=')1 to get ∥∇∆ρ∥2 L2 ≤ C ∥∇ρt∥2 L2 + C � ∥Φ∥4 W 1,4 + ∥∇ϕ∥8 L8 + ∥∇ϕ∥4 L4 � ∥∆ρ∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10) Thus, we use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10), alonging with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7), to deduce that ∥∆ρ∥2 L2 (t) + � t 0 ∥∇∆ρ∥2 L2 ds ≤ C(Ω, c, α, β, Φ, ϕ, u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) In conclusion, for both cases, it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) that ∥ρt∥2 L2 (t) + ∥∇ρ∥2 H1 (t) + � t 0 � ∥ρt∥2 H1 + ∥∇ρ∥2 H2 � ds ≤ C(Ω, c, α, β, Φ, ϕ, u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12) The next part is estimating v (for case (A) or (B)) or u (for case (C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We first treat the case for u satisfying (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Note that (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=')1 is equivalent to ρt + div[ρ(Φ + ∇ϕ−1)] = ∆(ρϕ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Thus, if we multiply (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=')2 by w := u − Q with Q = B[c0∆ρ−1] and integrate over Ω, we have d dt � 1 2ρ|u|2 + � 2µ|D(u)|2 = � ∆ � ρϕ−1� |u|2 2 + � ρut · Q + � ρ(Φ + ∇ϕ−1) · ∇u · Q + � 2µD(u) · ∇Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 31 Then, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' we have d dt ∥√ρu∥2 L2 + ν ∥∇u∥2 L2 ≤ C � ∥∆ρ∥L2 + ∥∇ρ∥L4 ∥∇ϕ∥L4 + ∥∇ϕ∥2 L4 + ∥∆ϕ∥L2 � ∥u∥2 L4 + C ��Φ + ∇ϕ−1�� L4 ∥Q∥L4 ∥∇u∥L2 + C ∥∇Q∥L2 ∥∇u∥L2 + C ∥Q∥L2 ∥ut∥L2 ≤ C � ∥∆ρ∥2 L2 + ∥∇ρ∥4 L4 + ∥∇ϕ∥4 L4 + ∥∆ϕ∥2 L2 � ∥u∥2 L2 + C ��Φ + ∇ϕ−1��4 L4 ∥Q∥2 L2 + C ∥∇Q∥2 L2 + ν 2 ∥∇u∥2 L2 + Cε ∥Q∥2 L2 + ε ∥ut∥2 L2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' d dt ∥√ρu∥2 L2 + ν ∥∇u∥2 L2 ≤ C � ∥∆ρ∥2 L2 + ∥∇ρ∥4 L4 + ∥∇ϕ∥4 L4 + ∥∆ϕ∥2 L2 � ∥u∥2 L2 + C ��Φ + ∇ϕ−1��4 L4 ∥∇ρ∥2 L2 + Cε ∥Q∥2 H1 + ε ∥ut∥2 L2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' ≤ C � ∥∆ρ∥2 L2 + ∥∇ρ∥4 L4 + ∥∇ϕ∥4 L4 + ∥∆ϕ∥2 L2 � ∥u∥2 L2 + C ��Φ + ∇ϕ−1��4 L4 ∥∇ρ∥2 L2 + Cε � ∥∇ρ∥4 L4 + ∥∇ρ∥2 H1 � + ε ∥ut∥2 L2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13) Next, for the estimate of ut, multiplying wt = ut − Qt on the both sides of (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=')2, one has � ρ|ut|2 + d dt � µ|D(u)|2 = − � ρut · Qt − � ρ(Φ + ∇ϕ−1) · ∇u · wt + � µt|D(u)|2 − � div[2µD(u)] · Qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using again Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, applying Poincaré’s inequality and the fact that ∥Qt∥2 L2 ≤ C � ∥∇ρt∥2 L2 + ∥ρt∥2 L4 ∥∇ρ∥2 L4 � , we obtain ∥ut∥2 L2 + d dt ∥√µD(u)∥2 L2 ≤ C ���Φ + ∇ϕ−1��2 L4 + ∥µt∥L2 + ∥∇µ∥2 L4 � ∥∇u∥2 L4 + C ∥Qt∥2 L2 + C ∥Qt∥L2 ∥∆u∥L2 ≤ Cε ���Φ + ∇ϕ−1��4 L4 + ∥µt∥2 L2 + ∥∇µ∥4 L4 + ∥∇ρ∥4 L4 � ∥∇u∥2 L2 + ε ∥∆u∥2 L2 + Cε � ∥∇ρ∥4 L4 ∥ρt∥2 L2 + ∥∇ρt∥2 L2 � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14) To estimate ∆u, we change (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=')2 into the form −µ∆v + ∇p = 2∇µ · D(u) − ρut − ρ(Φ + ∇ϕ−1) · ∇u + 2c0µ∇∆ρ−1, which, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8, leads to ∥v∥2 H2 + ∥p∥2 H1 ≤ C ∥ut∥2 L2 + C ��∇∆ρ−1��2 L2 + C � ∥∇µ∥2 L4 + ��Φ + ∇ϕ−1��2 L4 � ∥∇u∥2 L4 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='15) that is, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, ∥∆u∥2 L2 + ∥p∥2 H1 ≤ C ∥ut∥2 L2 + C ��∇∆ρ−1��2 L2 + C � ∥∇µ∥4 L4 + ��Φ + ∇ϕ−1��4 L4 � ∥∇u∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16) 32 Then, using this bound together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14), we have ∥ut∥2 L2 + d dt ∥√µD(u)∥2 L2 ≤ Cε ���Φ + ∇ϕ−1��4 L4 + ∥µt∥2 L2 + ∥∇µ∥4 L4 + ∥∇ρ∥4 L4 � ∥∇u∥2 L2 + ε ��∇∆ρ−1��2 L2 + C ∥∇ρ∥4 L4 ∥ρt∥2 L2 + C ∥∇ρt∥2 L2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17) Finally, combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17), then, using Gr¨onwall’s inequality and the bound (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12), we obtain the a priori estimates for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For case (A) or (B), as we have said at the end of Section 1, we convert (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=') into \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ρt + Φ · ∇ρ − div(ϕ−1∇ρ) = 0, � ρvt + ρ(Φ + ∇ϕ−1) · ∇v − div(2µD(v)) + ∇p = c0∇(log ρ)t − c0ρ(Φ + ∇ϕ−1) · ∇2ρ−1 + c0 div(2µ∇2ρ−1), div v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18) Then, we can apply the energy arguements analogous to the case (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' More precisely, multiplying v on both sides of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)2 and integrating over Ω, we have, for all ε ∈ (0, 1/2], d dt � 1 2ρ |v|2 − � div [2µD(v)] · v = � ∆(ρϕ−1)|v|2 2 + � c0 div � 2µ∇2ρ−1� v − � c0ρ(Φ + ∇ϕ−1) · ∇2ρ−1 · v ≤ C � ∥∆ρ∥L2 + ∥∇ρ∥L4 ∥∇ϕ∥L4 + ∥∇ϕ∥2 L4 + ∥∆ϕ∥L2 � ∥v∥2 L4 + C ∥∇µ∥L4 ��∇2ρ−1�� L2 ∥v∥L4 + C ��∇∆ρ−1�� L2 ∥v∥L2 + C (∥Φ∥L4 + ∥∇ϕ∥L4) ��∇2ρ−1�� L2 ∥v∥L4 ≤ Cε � ∥∆ρ∥2 L2 + ∥∇ρ∥4 L4 + ∥∇µ∥4 L4 + ∥Φ∥4 L4 + ∥∇ϕ∥4 L4 + ∥∆ϕ∥2 L2 � ∥v∥2 L2 + ε � ∥∇v∥2 L2 + ��∇∆ρ−1��2 L2 � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='19) where we have used Poincaré’s inequality for the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For the term − � div [2µ(ρ)D(v)]·v, we directly use the results in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5, that is, for case (B’), − � div [2µD(v)] · v = � 2µ|D(v)|2 ≥ ν ∥∇v∥2 L2 , (use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='20) while, for case (A’), − � div [2µD(v)] · v ≥ ν ∥∇v∥2 L2 − � Cε ∥∇µ∥4 L4 ∥√ρv∥2 L2 + Cε ��∆ρ−1��2 L2 + ε ∥∇v∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='21) Thus, combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='19)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='21), in both cases, d dt ∥√ρv∥2 L2 + ν ∥∇v∥2 L2 ≤ C � ∥∆ρ∥2 L2 + ∥∇ρ∥4 L4 + ∥∇µ∥4 L4 + ∥Φ∥4 L4 + ∥∇ϕ∥4 L4 + ∥∆ϕ∥2 L2 � ∥v∥2 L2 + C ��∇∆ρ−1��2 L2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='22) which, using Gr¨onwall’s inequality and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12), gives ∥v∥2 L2 (t) + � t 0 ∥∇v∥2 L2 ds ≤ C(Ω, c, α, β, Φ, ϕ, ρ0, v0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='23) 33 Next, multiplying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)2 by vt and integrating over Ω, one has, using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, � ρ |vt|2 − � div[2µD(v)] · vt = − � ρ(Φ + ∇ϕ−1) · ∇v · vt + � c0 div � 2µ∇2ρ−1� vt − � c0ρ(Φ + ∇ϕ−1) · ∇2ρ−1 · vt ≤ C (∥Φ∥L4 + ∥∇ϕ∥L4) ∥∇v∥L4 ∥vt∥L2 + C � ∥∇µ∥L4 ��∇2ρ−1�� L4 + ��∇∆ρ−1�� L2 � ∥vt∥L2 + C (∥Φ∥L4 + ∥∇ϕ∥L4) ��∇2ρ−1�� L4 ∥vt∥L2 ≤ Cε � ∥Φ∥4 L4 + ∥∇ϕ∥4 L4 � ∥∇v∥2 L2 + Cε ��∇∆ρ−1��2 L2 + Cε � ∥Φ∥4 L4 + ∥∇ϕ∥4 L4 + ∥∇µ∥4 L4 � ��∆ρ−1��2 L2 + ε � ∥vt∥2 L2 + ∥v∥2 H2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='24) For the term − � div[2µD(v)] · vt, if (ρ, v) satisfies the condition (A’), we use the proof from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='53) to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='60), − � div[2µD(v)] · vt ≥ d dt � M1(t) + ∥√µ curl v∥2 L2 � + d dtM2(t) − Cε � ∥µt∥2 H1 + ∥∇µ∥4 L4 + 1 � ∥v∥2 H1 − Cε ∥∇µ∥4 L4 ��∇ρ−1��2 L2 − Cε ��∆ρ−1��2 L2 − ε � ∥vt∥2 L2 + ∥v∥2 H2 + ��∇ρ−1 t ��2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='25) Recalling that M1(t) = � ∂ µv · B · v, M2(t) = � c0µ∇⊥(v · n⊥) · B · ∇ρ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For case (B’), it is much easier, − � div[2µD(v)] · vt = � µ d dt|D(v)|2 = d dt � µ|D(v)|2 − � µt|D(v)|2 ≥ d dt � µ|D(v)|2 − � Cε ∥µt∥2 L2 ∥∇v∥2 L2 + ε ∥v∥2 H2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='26) Furthermore, from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='15), we have ∥v∥2 H2 + ∥p∥2 H1 ≤ C ∥vt∥2 L2 + C ∥∇ log ρt∥2 L2 + C ��∆ρ−1��2 L2 + C � ∥∇µ∥4 L4 + ∥Φ∥4 L4 + ∥∇ϕ∥4 L4 � � ∥∇v∥2 L2 + ��∆ρ−1��2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='27) Therefore, combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='24)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='27), without loss of generality, one has ∥vt∥2 L2 + d dt � M1(t) + ∥√µ curl v∥2 L2 � + d dtM2(t) ≤ Cε � ∥Φ∥4 L4 + ∥∇ϕ∥4 L4 + ∥µt∥2 H1 + ∥∇µ∥4 L4 + 1 � ∥v∥2 H1 + Cε ��∇∆ρ−1��2 L2 + Cε � ∥Φ∥4 L4 + ∥∇ϕ∥4 L4 + ∥∇µ∥4 L4 � ��∆ρ−1��2 L2 + ε � ∥∇ log ρt∥2 L2 + ��∇ρ−1 t ��2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='28) Finally, using Gr¨onwall’s inequality, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='23), we deduce that ∥∇v∥2 L2 (t) + � t 0 ∥vt∥2 L2 ds ≤ C(Ω, c, α, β, Φ, ϕ, ρ0, v0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='29) Therefore, we complete the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 Preliminary Reductions We claim that it is enough to prove the existence results for smooth initial data (ρ0, u0) satisfying the compatiblity conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Once this is established, for general data (ρ0, u0), we can build a sequence of smooth initial data (ρn 0, un 0) such that it converges to (ρ0, u0) in some appropriate functional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, we can obtain a corresponding sequence of solutions (ρn, vn, πn) (or (ρn, un, πn)), which is uniformly bounded with respect of n, satisfying the initial data (ρn 0, vn 0 ) (or (ρn 0, un 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We may use the weak convergence method and compactness reults to deduce that (ρn, vn, πn) (or (ρn, un, πn)) converges to (ρ, v, π) (or (ρ, u, π)) in some functional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' As a result, (ρ, v, π) (or (ρ, u, π)) will be the solution we expect, which proves our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Now, we explain how we obtain such smooth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We begin with α ≤ ρ0 ≤ β, u0 ∈ H1 ω (the case u0 ∈ H1 nd or H1 0 can be done analogously).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' First, as we have said in Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11, we can derive that ρ0 ∈ H2 from the compatiability condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) � ∆ρ−1 0 = c−1 0 div u0, x ∈ Ω, n · ∇ρ−1 0 = 0, x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='30) Consequently, we get v0 ∈ H1 by setting v0 = u0 − c0∇ρ−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, we can construct a smooth sequence (ˆρn 0, ˆvn 0 ) ∈ [C∞(Ω)]4 via flatten method and partition of unity such that ˆρn 0 s −−→ ρ0 in H2, ˆvn 0 s −−→ v0 in H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='31) For details, see [15] Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' However, the sequence (ˆρn 0 , ˆvn 0 ) may be failed to satisfy the boundary conditions and divergence- free condition, which means that we need further construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' First of all, we solve the following ellptic problem \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ∆ρn 0 = ∆ˆρn 0 , x ∈ Ω, n · ∇ρn 0 = 0, x ∈ ∂Ω, (ρn 0 )Ω = (ρ0)Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Of course, for each n ≥ 1, ρn 0 ∈ C∞(Ω) is unique and ∥∇(ρn 0 − ρm 0 )∥H1 ≤ C ∥∇(ˆρn 0 − ˆρm 0 )∥H1 → 0, as n, m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='32) It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='31) that {ρn 0} is a Cauchy sequence, and, thus, ρn 0 s −−→ ρ0 in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using Sobolev embedding theorem, H2 ֒→ C(Ω), we deduce that ρn 0 converges uniformly to ρ0 and thus, without loss of generality, we may assume that ρn 0 ∈ [α, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, to construct vn 0 , we borrow from the construction method in [27], Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' More precisely, consider following Stokes problem of vn 0 \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∆vn 0 + ∇pn = −∆ˆvn 0 , x ∈ Ω, div vn 0 = 0, x ∈ Ω, vn 0 · n = 0, curl vn 0 = −n⊥ ·B ·[vn 0 + c0∇(ρn 0 )−1], x ∈ ∂Ω, where � pn = 0 and {ρn 0} is the smooth sequence we just obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In view of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5, there exists a unique smooth solution (vn 0 , pn) ∈ [C∞(Ω)]4 such that ∥vn 0 ∥H1 + ∥pn∥L2 ≤ C(∥ˆvn 0 ∥H1 + ∥ρn 0∥H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='33) Thus, we obtain a Cauchy sequence ∥vn 0 − vm 0 ∥H1 + ∥pn − pm∥L2 ≤ C(∥ˆvn 0 − ˆvm 0 ∥H1 + ∥ρn 0 − ρm 0 ∥H2) −→ 0, as n, m → ∞, because of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='31) and the strong covergence of {ρn 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Without loss of generality, let vn 0 s −−→ v0 in H1 and pn s −−→ p in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 35 Then, V0 := v0 − v0 solves \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −∆V0 + ∇p = 0, x ∈ Ω, div V0 = 0, x ∈ Ω, V0 · n = 0, curl V0 = −n⊥ · B · V0, x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' It follows form the uniqueness of Stokes equations that V0 ≡ 0, that is, v0 = v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Thus, we have found a smooth divergence-free sequence vn 0 , which satisfies the condtion (A’), that converges strongly to v0 in H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' If we treat the case (C), we just turn back to un 0 by setting un 0 := vn 0 + c0∇(ρn 0 )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, it is easy to check that un 0 ∈ C∞(Ω), un 0|∂Ω = 0 and (ρn 0 , un 0) satisfies the compatiablity condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3 Approximate System In order to get the existence for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1), we first try to establish the smooth solutions for the following system: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ρt + vη · ∇ρ − c0 div � ρ−1 η ∇ρ � = 0, ρut + ρuη · ∇u − div[2µǫD(u)] + ∇π = 0, div u = c0∆ρ−1, ǫ, η ∈ (0, 1], ρ|t=0 = ρ0, u|t=0 = u0, α ≤ ρ0 ≤ β, (ρ0, u0) ∈ [C∞(Ω)]4 satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11), u0, (ρ, u) satisfies one of the bundary conditions (A) − (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=') Let us give an explaination about the new elements in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We define uη := vη + ρη, µǫ := µ(ρǫ), and ρǫ, ρη, vη are constructed as we did in preceeding subsection, that is, ρǫ, ρη, vη ∈ C∞(Ω), div vη = 0, ρǫ, ρη ∈ [α, β] and ρǫ, ρη, vη satisfying corresponding boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For convenience, we collect some bounds here which will be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Obviously, we always have µǫ ∈ C∞(Ω) for every fixed t, ǫ and, for all 1 ≤ r ≤ ∞, k ∈ N, ∥µǫ∥Lr ≤ C(r, Ω) ∥ρ∥L∞ , ∥∇µǫ∥Lr ≤ C(r, ǫ, Ω) ∥ρ∥L∞ , ��∇kρη �� Lr ≤ C(k, r, η, Ω) ∥ρ∥H1 , ��∇kρǫ �� Lr ≤ C(k, r, ǫ, Ω) ∥ρ∥H1 , ��∇kvη �� Lr ≤ C(k, r, η, Ω) ∥v∥L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='34) Also, we have the following uniform controls, for all 1 ≤ q < ∞, ∥vη∥W ℓ,q ≤ C ∥v∥W ℓ,q , ℓ = 0, 1, ∥ρη∥W ℓ,q ≤ C ∥ρ∥W ℓ,q , ℓ = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='35) Our aim is proving the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For every fixed ǫ, η ∈ (0, 1], the problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=') admits an unique smooth solution on QT1 for some positive time T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Our proof is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In the first part, we use iteration arguements and contraction mapping theorem to establish the unique smooth solution of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=') for every fixed η and ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, we recover the original system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) by letting η, ǫ tend to 0 in turn with help of the uniform estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 Uniform Bounds \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ρn t + vn−1 η ∇ρn − c0 div � (ρn−1 η )−1∇ρn� = 0, ρnun t + ρnun−1 η ∇un − div[2µn ǫ D(un)] + ∇πn = 0, div un = c0∆(ρn)−1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='36) with the initial-boundary conditions (ρn, un)(x, 0) = (ρ0, u0), in Ω, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='37) n · ∇ρn = 0, un = 0 on ∂Ω × (0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='38) where we use the following notations µn = µ(ρn), Qn = B[c0∆(ρn)−1], vn = un + c0∇(ρn)−1, wn = un − Qn To prove the existence for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='36), we construct approximate solutions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We first define (ρ0, u0) = (C, 0) and, then, assume that (ρn−1, un−1) was defined for n ≥ 1, let (ρn, un, πn) be the unique global strong solution to the problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' To prove the uniform bounds for the approximate solutions, we introduce the function HN(t) defined by HN(t) := \uf8f1 \uf8f2 \uf8f3 max1≤n≤N � 1 + ∥ρn∥2 H2 + ∥vn∥2 H1 + ∥ρn t ∥2 L2 � , case (A) or (B) max1≤n≤N � 1 + ∥ρn∥2 H2 + ∥un∥2 H1 + ∥ρn t ∥2 L2 � , case (C) Observe that, in all cases, it follows from the maximal principle and energy estimates that α ≤ ρn ≤ β, sup t∈[0,T ] ∥ρn∥2 L2 + � T 0 ∥∇ρn∥2 L2 ≤ C, for all T ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='39) Moreover, let N be a fixed large number, we have Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' There exists a positive constant C depending on Ω, c0, α, β and ρ0 such that ∥∇ρn∥2 L2 (t) + � t 0 ∥∆ρn∥2 L2 ds ≤ C + C � t 0 HN(s)3 ds, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='40) for all n, 1 ≤ n ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3), d dt ∥∇ρn∥2 L2 + ν ∥∆ρn∥2 L2 ≤ C ���vn−1 η ��4 L4 + ��∇ρn−1 η ��4 L4 � ∥∇ρn∥2 L2 ≤ C ���un−1��4 L4 + ��∇ρn−1��4 L4 � ∥∇ρn∥2 L2 ≤ CHN(t)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, we integrate from 0 to t with respect of time and finish the proof of lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The next Lemma concerns with the uniform bounds for case (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let (ρ, u) satisfy the condition (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' There exists a positive constant C depending on Ω, c0, α, β and u0 such that � ∥un∥2 H1 (t) + ∥∇ρn∥2 H1 (t) + ∥ρn t ∥2 L2 (t) � + � t 0 � ∥un∥2 H2 + ∥∆ρn∥2 H1 + ∥ρn t ∥2 H1 � ds ≤ C + C � t 0 HN(s)4 ds, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='41) for all n, 1 ≤ n ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 37 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For the higher regularity, we apply −∇∆ρn∇ on both sides of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='36) and, then, integrate over Ω to derive the analogue of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='33) d dt � 1 2 |∆ρn|2 + � c0 ρn−1 η |∇∆ρn|2 = � ∇∆ρn · ∇vn−1 η ∇ρn + � vn−1 η ∇2ρn · ∇∆ρn − � 2c0 (ρn−1 η )3 ��∇ρn−1 η ��2 ∇ρn · ∇∆ρn + � c0 (ρn−1 η )2 ∇(∇ρn · ∇ρn−1 η ) · ∇∆ρn + � c0 ρn−1 η ∆ρn∇ρn−1 η ∇∆ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, we can obtain the following inequality which is similar with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='35), d dt ∥∆ρn∥2 L2 + ν ∥∇∆ρn∥2 L2 ≤ Cε ���∇ρn−1��4 L4 + ∥∇ρn∥4 L4 + ��vn−1��4 L4 � × ���∆ρn−1��2 L2 + ∥∆ρn∥2 L2 + ��∇vn−1��2 L2 � + ε ��∇vn−1��2 H1 ≤ Cε ���∇ρn−1��4 L4 + ∥∇ρn∥4 L4 + ��un−1��4 L4 � × ���∆ρn−1��2 L2 + ��∇ρn−1��4 L4 + ∥∆ρn∥2 L2 + ��∇un−1��2 L2 � + ε ��∇vn−1��2 H1 ≤ CεHN(t)4 + ε ���∆un−1��2 L2 + ��∇∆ρn−1��2 L2 + ��∇ρn−1��4 L4 ��∆ρn−1��2 L2 � , which gives d dt ∥∆ρn∥2 L2 + ν ∥∇∆ρn∥2 L2 ≤ Cε1HN(t)4 + ε1 ��∆un−1��2 L2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='42) Moreover, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='39),we also have d dt ∥ρn t ∥2 L2 + ∥∇ρn t ∥2 L2 ≤ Cε2HN(t)3 + ε2 ��un−1 t ��2 L2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='43) To get the bounds for un, noticing that the mass equation can be written as ρn t + div(ρnun−1 η ) = c0∆(ρn/ρn−1 η ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, we follow the proof from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13) to get, for all n ≥ 2 d dt ��√ρnun��2 L2 + ν ∥∇un∥2 L2 ≤ Cε3 ∥∇ρn∥2 L2 + C ���un−1��4 L4 + ��∇ρn−1��4 L4 + ∥∇ρn∥4 L4 + ∥∆ρn∥2 L2 � + C � ∥∇ρn∥4 L4 + ��∇ρn−1��4 L4 + ∥∆ρn∥2 L2 + ��∆ρn−1��2 L2 � ∥un∥2 L2 + ε3 ∥un t ∥2 L2 ≤ Cε3HN(t)3 + ε3 ∥un t ∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='44) Similarly, for un t , it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14) that ∥un t ∥2 L2 + d dt ��� µnǫ D(un) ��2 L2 ≤ C ���un−1��2 L4 + ∥ρn t ∥L2 + ∥∇ρn∥2 L4 � ∥∇un∥2 L4 + C ∥Qn t ∥2 L2 + C ∥Qn t ∥L2 ∥∆un∥L2 ≤ Cε4 � HN(t)3 + ∥∇ρn t ∥2 L2 � + ε4 ∥∆un∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='45) Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='44)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='45), we obtain � ∥un t ∥2 L2 + ∥∇un∥2 L2 � + d dt ���� µnǫ D(un) ��2 L2 + ��√ρnun��2 L2 � ≤ Cε5 � HN(t)3 + ∥∇ρn t ∥2 L2 � + ε5 ∥∆un∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='46) 38 Moreover, it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16) that ∥∆un∥2 L2 + ∥πn∥2 H1 ≤ C � HN(t)4 + ∥∇∆ρn∥2 L2 + ∥un t ∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='47) Plugging this into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='42) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='46) and, then, combining two of them, we have d dt ���� µnǫ D(un) ��2 L2 + ∥∆ρn∥2 L2 � + ν � ∥∇∆ρn∥2 L2 + ∥un t ∥2 L2 � ≤ C ∥∇ρn t ∥2 L2 + Cε4HN(t)4 + ε4 ���∇∆ρn−1��2 L2 + ��un−1 t ��2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Alonging with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='43) and choosing ε2 small enough and ε4 = 1/2, one has d dt ���� µnǫ D(un) ��2 L2 + ∥∆ρn∥2 L2 + 2C ∥ρn t ∥2 L2 � + � ∥∇∆ρn∥2 L2 + ∥un t ∥2 L2 + ∥∇ρn t ∥2 L2 � ≤ CHN(t)4 + 1 2 ���∇∆ρn−1��2 L2 + ��un−1 t ��2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='48) For simplicity, we denote by the above d dtPn(t) + � ∥∇∆ρn∥2 L2 + ∥un t ∥2 L2 � ≤ CHN(t)4 + 1 2 ���∇∆ρn−1��2 L2 + ��un−1 t ��2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, integrating over [0, t], one has Pn(t) + � t 0 � ∥∇∆ρn∥2 L2 + ∥un t ∥2 L2 � ds ≤ C � 1 + � t 0 HN(s)4 ds � + 1 2 � t 0 ���∇∆ρn−1��2 L2 + ��un−1 t ��2 L2 � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using this recursive inequality for � t 0 � ∥∇∆ρn∥2 L2 + ∥un t ∥2 L2 � ds, we obtain � t 0 � ∥∇∆ρn∥2 L2 + ∥un t ∥2 L2 � ds ≤ � 1 + 1 2 + · · · + 1 2n � C � 1 + � t 0 HN(s)4 ds � ≤ 2C � 1 + � t 0 HN(s)4 ds � and hence, turning back to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='48), we get Pn(t) + � t 0 � ∥∇∆ρn∥2 L2 + ∥un t ∥2 L2 + ∥∇ρn t ∥2 L2 � ds ≤ C � 1 + � t 0 HN(s)4 ds � , for all 2 ≤ n ≤ N and, thus, for all 1 ≤ n ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Finally, using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='47), we get the bounds for ∆un and πn which concludes the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, we give the uniform estimates for condition (A) or (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let (ρ, v) satisfy the condition (A’) or (B’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' There exists a positive constant C depending on Ω, c0, α, β, ρ0 and v0 such that � ∥vn∥2 H1 (t) + ∥∇ρn∥2 H1 (t) + ∥ρn t ∥2 L2 (t) � + � t 0 � ∥vn∥2 H2 + ∥∆ρn∥2 H1 + ∥ρn t ∥2 H1 � ds ≤ C + C � t 0 HN(s)8 ds, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='49) for all n, 1 ≤ n ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 39 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We still only give the proof for case (A’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='22) and the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11, we get d dt ��√ρnvn��2 L2 + ν ∥∇vn∥2 L2 ≤ C � ∥∆ρn∥2 L2 + ∥∇ρn∥4 L4 + ∥∇µn ǫ ∥4 L4 + ��vn−1 η ��4 L4 + ��∇ρn−1 η ��4 L4 + ��∆ρn−1 η ��2 L2 � ∥vn∥2 L2 + C ��∇∆(ρn)−1��2 L2 ≤ CHN(t)3 + C ∥∇∆ρn∥2 L2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='50) and ∥vn t ∥2 L2 + d dt � Mn 1(t) + ∥√µ curl v∥2 L2 � + d dtMn 2(t) ≤ CHN(t)3 + CHN(t)2 ∥∇vn∥2 L4 ≤ CHN(t)3 + CHN(t)2 ∥∇vn∥L2 ∥v∥H2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='51) while, for ∥v∥H2, we apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9 for − div[2µn ǫ D(vn)] + ∇πn = −ρnvn t + c0∇ log ρn t − ρn[vn−1 η + c0∇(ρn−1)−1] · ∇[vn η + c0∇(ρn)−1] + div[2µn ǫ ∇2(ρn)−1] := F n (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='52) to obtain ∥v∥H2 + ∥π∥H1 ≤ C � ∥∇µn ǫ ∥2 L4 + 1 � � ∥F n∥L2 + ��∆(ρn)−1�� L2 � + ∥∇µn ǫ ∥2 L4 ∥∇vn∥L2 ≤ CHN(t) � ∥vn t ∥L2 + ∥∇ log ρn t ∥L2 + ��∇∆(ρn)−1�� L2 � + CHN(t) 3 2 (∥∇vn∥L4 + ��∆(ρn)−1�� L4) + CHN(t) 3 2 ≤ CHN(t) � ∥vn t ∥L2 + ∥∇ρn t ∥L2 + ∥∇∆ρn∥L2 + HN(t) 3 2 � + C � HN(t)4 + HN(t) 3 2 � + 1 2 ∥vn∥H2 , which leads to ∥v∥H2 + ∥π∥H1 ≤ CHN(t) (∥vn t ∥L2 + ∥∇ρn t ∥L2 + ∥∇∆ρn∥L2) + CHN(t)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='53) Substituting this into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='51), we have ∥vn t ∥2 L2 + d dt � Mn 1(t) + ∥√µ curl v∥2 L2 � + d dtMn 2(t) ≤ CHN(t)3 + CHN(t)4 (∥vn t ∥L2 + ∥∇ρn t ∥L2 + ∥∇∆ρn∥L2) + CHN(t)6 ≤ Cε1H8 N(t) + ε1 � ∥vn t ∥2 L2 + ∥∇ρn t ∥2 L2 + ∥∇∆ρn∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='54) On the other hand, following the proofs of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='42)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='43), one has d dt ∥∆ρn∥2 L2 + ν ∥∇∆ρn∥2 L2 ≤ CHN(t)4 + C ∥∇ρn∥2 L4 ��∇vn−1��2 L4 ≤ CHN(t)4 + C ∥∇ρn∥2 L4 ��∇vn−1�� L2 ��vn−1�� H2 ≤ Cε2HN(t)6 + ε2 ���vn−1 t ��2 L2 + ��∇ρn−1 t ��2 L2 + ��∇∆ρn−1��2 L2 � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='55) and d dt ∥ρn t ∥2 L2 + ∥∇ρn t ∥2 L2 ≤ Cε3HN(t)3 + ε3 ���vn−1 t ��2 L2 + ��∇ρn−1 t ��2 L2 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='56) Therefore, combining Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='50) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='54)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='56) and, then, using the same recursive arguements at the end of the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4, we can obtain the desire bound (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 40 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='47) in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4 that the constant C in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='41) depends on ǫ ∈ (0, 1], which indicates that we can only obtain the local existence for the case (C) with µ = µǫ, in particular, µ being a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' However, from the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5, since we used Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9 to get the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='53), the constant C in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='49) is independent with ǫ and that is why we could extend the local existence for cases (A) and (B) to general viscosity coefficient µ(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In conclusion, we have the bounds HN(t) ≤ C � 1 + � t 0 HN(s)q ds � , for some q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='57) Thanks to this integral inequality, we can easily show that there exists a time T1 ∈ (0, T ) depending only on Ω, c0, α, β and u0 such that sup t∈[0,T1] HN(t) ≤ C0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='58) for some C0 independing with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Therefore, we obtain the bounds, for all n ≥ 1, sup t∈[0,T 1] � ∥un∥2 H1 + ∥ρn∥2 H2 + ∥ρn t ∥2 L2 � + � T1 0 � ∥un∥2 H2 + ∥ρn∥2 H3 + ∥ρn t ∥2 H1 � ds ≤ C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='59) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 Convergence We next show that the whole sequence (ρn, un) converges to a solution to (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=') in a sufficiently strong sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let σn+1 := ρn+1 − ρn, an+1 := un+1 − un, bn+1 := vn+1 − vn, cn+1 := Qn+1 − Qn and Yn(t) := � ∥an∥2 H1 + ∥σn∥2 H2 + ∥σn t ∥2 L2 , case (C) ∥bn∥2 H1 + ∥σn∥2 H2 + ∥σn t ∥2 L2 , case (A) or (B) Zn(t) := � ∥an t ∥2 L2 + ∥σn∥2 H3 + ∥σn t ∥2 H1 , case (C) ∥bn t ∥2 L2 + ∥σn∥2 H3 + ∥σn t ∥2 H1 , case (A) or (B) In addition, we always let In(t) and Bn(t) be generic functions associated with the bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='59) such that � T1 0 In(t) dt + sup t∈[0,T1] Bn(t) ≤ C0, where C0 is the constant as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Case (C): It follows from the linearized mass equation that σn+1 t + vn η · ∇σn+1 − c0 div � 1 ρnη ∇σn+1 � = −bn η · ∇ρn − c0 div � σn η ρn−1 η ρnη ∇ρn � := Gn (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='60) where ∥Gn∥H−1 ≤ C ∥∇ρn∥L4 ���bn η �� L4 + ��σn η �� L4 � ≤ C ∥ρn∥H2 (∥an∥H1 + ∥σn∥H1) ∥Gn∥L2 ≤ ∥∇ρn∥L4 ��bn η �� L4 + C ∥∆ρn∥L2 ��σn η �� L∞ + C ��σn η �� W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4 ∥∇ρn∥L4 ≤ C ∥ρn∥H2 (∥an∥H1 + ∥σn∥H2) 41 ∥∇Gn∥L2 ≤ ∥∇ρn∥L4 ��∇bn η �� L4 + ∥∆ρn∥L2 ��bn η �� L∞ + C ∥∇ρn∥L4 ��∆σn η �� L4 + C ∥∆ρn∥L2 ��∇σn η �� L∞ + C ��(|∇ρn η| + |∇ρn−1 η |)|∇ρn| �� L4 ��∇σn η �� L4 + C ��� |∇ρn η|2 + |∇ρn−1 η |2 + |∇2ρn−1 η | + |∇2ρn η| � |∇ρn| �� L2 ��σn η �� L∞ + C ��(|∇ρn η| + |∇ρn−1 η |)|∇2ρn| �� L2 ��σn η �� L∞ + C ∥∇∆ρn∥L2 ��σn η �� L∞ ≤ Cη � ∥ρn∥H2 + ∥ρn∥2 W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4 � (∥an∥L2 + ∥σn∥H2) + Cη ∥∇∆ρn∥L2 ∥σn∥H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='34) Then, using the simplified notations, the above bounds can be written as follows ∥Gn∥H−1 + ∥Gn∥L2 ≤ CBn(t)Yn(t), ∥∇Gn∥L2 ≤ CηBn(t)Yn(t) + Cη � In(t) ∥σn∥H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='61) Next, we are going to establish the bounds for σn+1 and an+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Multiplying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='60) by σn+1 and integrating over Ω, we obtain d dt ��σn+1��2 L2 + ν ��∇σn+1��2 L2 ≤ C ∥Gn∥H−1 ��σn+1�� H1 , then, using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='61), we deduce that d dt ��σn+1��2 L2 + ν ��∇σn+1��2 L2 ≤ C ∥Gn∥2 H−1 ≤ CBn(t)Yn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='62) Similar with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3), multiplying −∆σn+1 on both sides of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='60) and integrating over Ω, one has d dt ��∇σn+1��2 L2 + ν ��∆σn+1��2 L2 ≤ C ���vn η ��4 L4 + ��∇ρn η ��4 L4 � ��∇σn+1��2 L2 + C ∥Gn∥2 L2 ≤ CIn(t) ��∇σn+1��2 L2 + CBn(t)Yn(t), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='63) where we have used (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='61) for the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' If we integrate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='63) between [0, t], t < T1, we have ��∇σn+1��2 L2 (t) + � t 0 ��∆σn+1��2 L2 ds ≤ C � t 0 Yn(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='64) For ρn satisfying the Neumann condition, we copy the proof from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5) by applying −∇∆σn+1∇ on (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='60), integrating over Ω and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='61), that is d dt ��∆σn+1��2 L2 + ν ��∇∆σn+1��2 L2 ≤ C ���vn η ��4 W 1,4 + ��ρn η ��4 W 2,4 + ��∇ρn η ��8 L8 � ��∆σn+1��2 L2 + C ∥∇Gn∥2 L2 ≤ CIn(t) ��∆σn+1��2 L2 + CBn(t)Yn(t) + CIn(t) ∥σn∥2 H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' This, alonging with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='64), implies that d dt ��∆σn+1��2 L2 + ν ��∇∆σn+1��2 L2 ≤ CIn(t) ��∆σn+1��2 L2 + CBn(t)Yn(t) + CIn(t) � t 0 Yn(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='65) For σn+1 t , we multiply σn+1 t on the both sides of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='60) and integrating over Ω, it follows analogously from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7) that d dt ��σn+1 t ��2 L2 + ν ��∇σn+1 t ��2 L2 ≤ Cε � ∥un t ∥2 L2 + ∥ρn t ∥4 L4 + ∥∇ρn∥4 L4 + ∥∇ρn t ∥2 L2 � ���σn+1 t ��2 L2 + ��∇σn+1��2 L2 � + ε ���∆σn+1��2 L2 + ∥an t ∥2 L2 � + C ∥∆ρn∥2 L2 ∥σn∥2 H1 ≤ CεIn(t) ���σn+1 t ��2 L2 + ��σn+1��2 H1 � + CBn(t)Yn(t) + ε ���∆σn+1��2 L2 + ∥an t ∥2 L2 � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='66) 42 On the other hand, we differeniate the equations between those of un+1 and un to get ρn+1an+1 t + ρn+1un η · ∇an+1 − div[2µn+1 η D(an+1)] + ∇(πn+1 − πn) = −σn+1(un t + un η · ∇un) − ρnan η · ∇un + div[2(µn+1 ǫ − µn ǫ )D(un)] := Kn, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='67) where ∥Kn∥H−1 ≤ ��un t + un η · ∇un�� L2 ��σn+1�� L∞ + ∥∇un∥L2 ��an η �� L∞ + ∥∇un∥L2 ��σn+1 η �� L∞ ∥Kn∥L2 ≤ ��un t + un η · ∇un�� L2 ��σn+1�� L∞ + ∥∇un∥L2 ��an η �� L∞ + ∥∆un∥L2 ��σn+1 η �� L∞ + C ∥∇un∥L4 ��∇ρn η �� L4 ��σn+1 η �� L∞ + C ∥∇un∥L4 ��∇σn+1 η �� L4 , that is, ∥Kn∥H−1 + ∥Kn∥L2 ≤ C � In(t) ��σn+1�� H2 + CηBn(t) ∥an∥L2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='68) Next, following the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13), we multilpy an+1 − cn+1 on both sides of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='67), integrate over Ω and use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='68) to obtain d dt ��� � ρn+1an+1��� 2 L2 + ν ��∇an+1��2 L2 ≤ C ���∆ρn+1��2 L2 + ∥∆ρn∥2 L2 + ��∇ρn+1��4 L4 + ∥∇ρn∥4 L4 � ��an+1��2 L2 + C ��un η ��2 L4 ��cn+1��2 L4 + Cε ��cn+1��2 H1 + ε ��an+1 t ��2 L2 + C ∥Kn∥2 H−1 ≤ CεIn(t) ���an+1��2 L2 + ��σn+1��2 H2 � + CBn(t) ∥an∥2 L2 + C ��∆σn+1��2 L2 + ε ��an+1 t ��2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='69) Here, for the last inequality, we have used Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and ��cn+1�� Lp ≤ C ��σn+1�� W 1,p , ��cn+1�� H1 ≤ C ���∇ρn+1��2 L4 + ∥∇ρn∥2 L4 + ∥∇ρn∥4 L8 + ∥∆ρn∥2 L4 � ��σn+1�� H1 + C ��∆σn+1�� L2 ≤ C � In(t) ��σn+1�� H1 + C ��∆σn+1�� L2 To get the higher bound for an, multiplying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='67) by an+1 t − cn+1 t , it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14) that d dt ��∇an+1��2 L2 + ν ��an+1 t ��2 L2 ≤ C ���un η ��2 L4 + ��ρn+1 η,t �� L2 + ��∇ρn+1 η ��2 L4 � ��∇an+1��2 L4 + C ��cn+1 t ��2 L2 + C ��cn+1 t �� L2 ��∆an+1�� L2 + C ∥Kn∥2 L2 ≤ CεIn(t) ���∇an+1��2 L2 + ��σn+1��2 H2 + ��σn+1 t ��2 L2 � + Cε ��∇σn+1 t ��2 L2 + CBn(t) ∥an∥2 L2 + ε ��∆an+1��2 L2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='70) where we have used the fact that ��cn+1 t �� L2 ≤ C ��∇σn+1 t �� L2 + C (∥∇ρn t ∥L2 + ∥ρn t ∥L4 ∥∇ρn∥L4) ��σn+1�� L∞ + C ∥ρn t ∥L4 ��∇σn+1�� L4 + C ��∇ρn+1�� L4 ��σn+1 t �� L4 ≤ C ��∇σn+1 t �� L2 + C � In(t) ���σn+1�� H2 + ��σn+1 t �� L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' At last, in order to get the estimate of ∆an+1, we use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16) with the additional term Kn, ��∆an+1��2 L2 + ��πn+1 − πn��2 H1 ≤ C ��an+1 t ��2 L2 + C ��∇∆[(ρn+1)−1 − (ρn)−1] ��2 L2 + C ���∇ρn η ��4 L4 + ��un η ��4 L4 � ��∇an+1��2 L2 + C ∥Kn∥2 L2 ≤ C ��an+1 t ��2 L2 + C ��∇∆σn+1��2 L2 use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='68) + CIn(t) ���∇an+1��2 L2 + ��σn+1��2 H2 � + CBn(t) ∥an∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='71) 43 We substitute above into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='70) to get d dt ��∇an+1��2 L2 + ν ��an+1 t ��2 L2 ≤ CεIn(t) ���∇an+1��2 L2 + ��σn+1��2 H2 + ��σn+1 t ��2 L2 � + Cε ��∇σn+1 t ��2 L2 + CBn(t) ∥an∥2 L2 + ε ��∇∆σn+1��2 L2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='72) Therefore, combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='62)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='63), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='65)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='66), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='69) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='72), we eventually get d dtYn+1(t) + νZn+1(t) ≤ C � In(t)Yn+1(t) + Bn(t)Yn(t) + In(t) � t 0 Yn(s) ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='73) Then, applying the Gr¨onwall’s inequality and recalling that Yn(0) = 0 and the definitions of In(t), Bn(t), one has, for all t ∈ (0, T1), Yn+1(t) ≤ C0 � t 0 � CBn(s)Yn(s) ds + CIn(s) � s 0 Yn(τ) dτ � ds ≤ C � t 0 Yn(s) ds, which reduces to the Volterra-type integral equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' After a simple recursive argument, we can show that sup t∈[0,T1] Yn+1(t) ≤ C (CT1)n−1 (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' � T1 0 Y1(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='74) Applying the contraction mapping theorem and using this inequality together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='73), we show that the sequence (ρn, un) converges strongly to an unique limit (ρ, u) and, as a consequence, πn converges strongly to a function π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' More precisely, we have ρn s −−→ ρ in C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H2) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H3), un s −−→ u in C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H2), un t s −−→ ut in L2([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2), πn s −−→ π in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Of course, (ρ, u, π) is the unique strong solution in Ω×(0, T1) for (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Furthermore, we can show (ρ, u, π) is acually smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Indeed, sicne u ∈ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H2) ∩ H1(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2), vη, ρη ∈ H1(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' With this regularity on vη, ρη, using the regularity theories of parabolic equations for (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' )1, we can derive that ρ ∈ H2([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, applying the Lp-theory ([35]) for (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' )2, we get u ∈ H2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H∞) and, hence, we can bootstrap and gain more time regualrity on vη, ρη then ρ, which implies that (ρ, u) ∈ C∞(QT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Therefore, we finish the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Case (A) or (B): We only consider the case (A) here and case (B) can be proved identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Firstly, it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='68) that ∥Kn∥H−1 + ∥Kn∥L2 ≤ C � In(t) ��σn+1�� H2 + CηBn(t)(∥bn∥L2 + ∥σn∥H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='75) Then, applying the estimates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='22) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='28) with ϕ = ρn η and Φ = vn η and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='75), we can 44 obtain d dt ��� � ρn+1bn+1��� 2 L2 + ν ��∇bn+1��2 L2 ≤ C ���∆ρn+1��2 L2 + ��∇ρn+1��4 L4 + ��∇µn+1 ǫ ��4 L4 � ��bn+1��2 L2 + C ���vn η ��4 L4 + ��∇ρn η ��4 L4 + ��∆ρn η ��2 L2 � ��bn+1��2 L2 + C ��∇∆[(ρn+1)−1 − (ρn)−1] ��2 L2 + C ∥Kn∥2 H−1 ≤ CIn(t) ���bn+1��2 L2 + ��σn+1��2 H2 � + C ��∇∆σn+1��2 L2 + CBn(t) � ∥bn∥2 L2 + ∥σn∥2 H1 � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='76) and ��bn+1 t ��2 L2 + d dt � Mn 1(t) + ��∇bn+1��2 L2 � + d dtMn 2(t) ≤ Cε ���vn η ��4 L4 + ��∇ρn η ��4 L4 + ��µn+1 ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='t ��2 H1 + ��∇µn+1 ǫ ��4 L4 + 1 � ��bn+1��2 H1 + Cε ���vn η ��4 L4 + ��∇ρn η ��4 L4 + ��∇µn+1 ǫ ��4 L4 � ��∆[(ρn+1)−1 − (ρn)−1] ��2 L2 + Cε ��∇∆[(ρn+1)−1 − (ρn)−1] ��2 L2 + ε ���∇(log ρn+1 − log ρn)t ��2 L2 + ��∇[(ρn+1)−1 − (ρn)−1]t ��2 L2 � + C ∥Kn∥2 L2 ≤ CεIn(t) ���bn+1��2 H1 + ��σn+1��2 H2 � + Cε ��∇∆σn+1��2 L2 + ε ��∇σn+1 t ��2 L2 + CBn(t) � ∥bn∥2 L2 + ∥σn∥2 H1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='77) where Mn 1(t) := � ∂ µn+1 ǫ bn+1 · B · bn+1, Mn 2(t) := � c0µn+1 ǫ ∇⊥(bn+1 · n⊥) · B · ∇ � (ρn+1)−1 − (ρn)−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Therefore, combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='62)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='63), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='65)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='66), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='76)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='77), we have d dtYn+1(t) + d dtMn 2(t) + νZn+1(t) ≤ C � In(t)Yn+1(t) + Bn(t)Yn(t) + In(t) � t 0 Yn(s) ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='78) Here, we have used the fact that Mn 1(t) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Noticing that |Mn 2(t)| ≤ ε ��bn+1��2 H1 (t) + Cε ���∇σn+1��2 L2 (t) + ∥∇ρn∥4 L4 (t) ��σn+1��2 L2 (t) � ≤ ε ��bn+1��2 H1 (t) + Cε ��σn+1��2 H1 (t) ≤ ε ��bn+1��2 H1 (t) + Cε � t 0 Yn(t) use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='64) Applying Gr¨onwall’s inequality and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='78), we finally get the Volterra-type integral equation Yn+1(t) ≤ C � t 0 Yn(s) ds, t ∈ (0, T1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='79) and, hence, following the proof of case (C), we complete the proof for the case (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In conclusion, we finish the proof for Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 45 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4 Proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6: Recover ǫ and η We temporarily fix ǫ ∈ (0, 1] to let η → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We still first consider the case (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The recovering process is standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2, we get a smooth sequence (ρǫ,η, uǫ,η, πǫ,η) ∈ C∞(QT1) which solves the problem (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=') for each ǫ, η ∈ (0, 1] (for simplicity, we use the notation (ρη, uη, πη)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, we can follow the proofs in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4 step by step and the uniform bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='35) to obtain the following control d dtFη(t) + νGη(t) ≤ CFη(t)3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='80) where Fη(t) := ∥uη∥2 L2 + ∥ρη t ∥2 L2 + ∥ρη∥2 H2 , Gη(t) := ∥uη t ∥2 L2 + ∥uη∥2 H2 + ∥ρη∥2 H3 + ∥ρη t ∥2 H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' and C is a constant which is not depend on η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using the inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='80), we can easily deduce that there eixsts a positive time T2 such that sup t∈[0,T2] Fη(t) + � T2 0 Gη(t) dt ≤ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='81) Therefore, using the above uniform esitmate and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10, we can derive that (ρη, uη, πη) converges in some proper sense to the limit (ρ, u, π) such that \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ρt + div(ρu) = 0, ρut + ρu · ∇u − div[2µǫD(u)] + ∇π = 0, div u = c0∆ρ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='82) The convergence is easy to check, we left it to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Of course, as a special case, we can let µǫ be a constant µ and, thus, we have proved the uniqueness and existence of the local strong solutions for the case (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For the case (A) or (B), the proof is basically the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' However, the difference lies in this case is that we can recover ǫ → 0+ because of the uniform estimates of (ρǫ, uǫ, πǫ), ǫ ∈ (0, 1], see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The convergence is easy to check and we omit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Thus, we have completed the proof of the existence results for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' It remains to check the uniqueness for the case (A) or (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' However, this can be done by following the proof in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Indeed, for example, if we consider the case (A) (another case can be proved analogously), let (ρi, ui, πi), i = 1, 2, be two strong solutions on Ω × (0, T1) with same initial data and set σ := ρ1 − ρ2, a := u1 − u2, b := v1 − v2, c := Q1 − Q2, Y(t) := ∥a∥2 H1 + ∥σ∥2 H2 + ∥σt∥2 L2 , Z(t) := ∥at∥2 L2 + ∥σ∥2 H3 + ∥σt∥2 H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, we can derive the similar type of equations to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='60) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='67), that is, � σt + v2 · ∇σ − c0 div � ρ−1 2 ∇σ � = −b · ∇ρ2 − c0 div � σρ−1 1 ρ−1 2 ∇ρ2 � , ρ1at + ρ1u1 · ∇a − div[2µD(a)] + ∇(π1 − π2) = −σ(u1,t + u1 · ∇u1) − ρ2a · ∇u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Applying the same discussions from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2, we can get the following type inequality d dtY(t) + Z(t) ≤ CI(t)Y(t), where I stands for some integrable functions on time interval (0, T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Thus, using Gr¨onwall’s inequality and the fact that Y(0) = 0, we can easily deduce that Y(t) ≡ 0, which yields the uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 46 6 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5 Since we have already show the existence and uniqueness of strong solution on Ω×(0, T1) for some positive times T1, the proof of global ones is quite standard with the a priori estiamtes we obtained in Section 3–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' One thing we should mention is that there is a gap between the local existence and the global one when (ρ, u) satisfies the condition (C) in that we only established the unique local strong solution for µ = µǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' In every case that follows, one should first recover ǫ → 0+ to get the global existence and, then, show their uniqueness under the smallness assumption ∥∇u0∥L2 ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Fourtunately, the proof of either is simpe and indentical with that in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The only thing one should notice is that, under the restriction ∥∇u0∥L2 ≤ δ, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 holds and, thus, we always have sup t∈[0,T ] ∥∇ρ∥L4 ≤ 1, which allows us to use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9 (in such case, there is no difference between the estimates of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8 and those of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9) and get the uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3 Following the construction process in subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2, one can find a smooth sequence (ρn 0 , vn 0 ) such that ρn 0 s −−→ ρ0 in H1, α ≤ ρn 0 ≤ β, vn 0 s −−→ v0 in L2, div vn 0 = 0, vn 0 · n = 0 on ∂Ω, (ρn 0 , vn 0 ) satisfying (A’) or (B’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) If we define un 0 := vn 0 + c0∇(ρn 0 )−1, it is easy to check that un 0 is smooth and (ρn 0, un 0) satisfies all the conditions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Thus, by using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4, there exists a sequence of global strong solutions (ρn, un) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) with initial data (ρn 0 , un 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, using the uniform bounds we get from subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, extracting subsequences if necessary, we can derive a weak convergent subsequence satisfying \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ρn w∗ −−⇀ ρ in L∞(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1), ρn w −−⇀ ρ in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H2), ρn t w −−⇀ ρt in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2), un w∗ −−⇀ u in L∞(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2), un w −−⇀ u in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2) Next, we can apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7 to obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='25)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' With these hold in hand, one can imme- diately get un s −−→ u in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) Indeed, since vn s −−→ v in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2), it suffices to show the strong convergence for ∇(ρn)−1, that is, ∇(ρn)−1 s −−→ ∇ρ−1 in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4) However, ∇(ρn)−1 = −(ρn)−2∇ρn and ρn s −−→ ρ in L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1), since we have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2)2–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2)3 and, then, use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Therefore, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4) is an easy consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='25) and Egorov theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Finally, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='25), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3), we can recover the weak solutions (ρ, u) for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) and complete the prove of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 47 7 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7 In the last section, we come to prove the blowup criterion for (ρ, u) satisfying one of three conditions (A), (B) and (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let (ρ, u, π) be a local strong solution as being described in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6 and suppose that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14) was false, that is, for some r and s satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='15), lim T →T ∗ ∥∇ρ∥Ls(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='Lr) ≤ M0 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) or lim T →T ∗ ∥u∥Ls(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='Lr) ≤ M0 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2) We also let ˜C be a positive generic constant depending on Ω, c0, α, β, T ∗, M0 and ∥u0∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, our goal is proving the following estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Suppose that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) holds for (ρ, u) satisfying the condition (A) or (B) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2) holds for (ρ, u) satisfying the condition (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, one has, for all T ∈ (0, T ∗), sup t∈[0,T ] � ∥ρt∥2 L2 + ∥ρ∥2 H2 + ∥u∥2 H1 � + � T 0 � ∥ρt∥2 H1 + ∥ρ∥2 H3 + ∥u∥2 H2 � dt ≤ ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3) Before proving the proposition, let us show how to derive the blowup criterion in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6 from Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' For simplicity, we give the prove for the case when (ρ, u) satisfies the con- dition (A), since other cases can be proved identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Note that ˜C, in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3), is uniformly bounded for all T ≤ T ∗, so (ρ, u)(x, T ∗) := lim t→T ∗(ρ, u)(x, t) in the sense of H2 × H1 satisfying the conditions imposed on the initial data, that is, α ≤ ρ0 ≤ β, u0 ∈ H1 ω, at the time t = T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Furthermore, � div u|t=T ∗ = c0∆ρ−1|t=T ∗, x ∈ Ω u|t=T ∗ · n = n · ∇ρ−1|t=T ∗, x ∈ ∂Ω Thus, (ρ, u)(x, T ∗) satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Therefore, we can take (ρ, u)(x, T ∗) as the initial data and apply the existence result in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6 to extend the local strong solution beyond T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' This contradicts the maximality of T ∗ and, hence, we finish the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 Case for (ρ, u) satisfying (A) or (B) In this subsection, we always let (ρ, u) satisfy the condition (A) or (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Recall that it is also equivalent to require (ρ, v) satisfying the condition (A’) or (B’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The proof for the first part of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 will be separated into the following few steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The key of the proof is obtaining the lower order estimates for (ρ, v), that is, (∇ρ, v) ∈ C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' L2) ∩ L2(0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' H1), then, following the proof in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3, the weak solution is automatically a strong one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The first lemma is just the combination of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3, we give it here for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' The following bounds hold for all T ∈ (0, T ∗), that is, α ≤ ρ ≤ β, sup t∈[0,T ] ∥ρ∥2 L2 + ν � T 0 ∥∇ρ∥2 L2 dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4) The next crucial lemma gives the lower bounds of (ρ, v), that is, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Suppose that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1) holds and (ρ, v) satisfies the condition (A’) or (B’), then one has sup t∈[0,T ] � ∥∇ρ∥2 L2 + ∥v∥2 L2 � + � T 0 � ∥∆ρ∥2 L2 + ∥∇v∥2 L2 � dt ≤ ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5) 48 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We first follow the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4, applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2, to get d dt ∥∇ρ∥2 L2 + ν ∥∆ρ∥2 L2 ≤ C � � |∇ρ|2 + |v| |∇ρ| � |∆ρ| ≤ C ∥∇ρ∥Lr � ∥∇ρ∥ L 2r r−2 + ∥v∥ L 2r r−2 � ∥∆ρ∥L2 ≤ C ∥∇ρ∥ 2r r−2 Lr � ∥∇ρ∥2 L2 + ∥v∥2 L2 � + ν 2 ∥∆ρ∥L2 , which implies that d dt ∥∇ρ∥2 L2 + ν ∥∆ρ∥2 L2 ≤ C(∥∇ρ∥s Lr + 1) � ∥∇ρ∥2 L2 + ∥v∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6) On the onther hand, as we did in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12), multiplying v on both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)2 and integrating over Ω, d dt � 1 2ρ |v|2 − � div [2µD(v)] · v = 3 � i=1 Ii, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7) where Ii, i = 1, 2, 3, as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13), applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2, the second term on the left-hand side can be controlled by − � div [2µD(v)] · v ≥ µ � |curl v|2 − C � ∥∇ρ∥Lr ∥√ρv∥ L 2r r−2 ∥∇v∥L2 � ≥ ν ∥∇v∥2 L2 − � Cε(∥∇ρ∥s Lr + 1) ∥√ρv∥2 L2 + ε ∥∇v∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8) Following the proof from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14) to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17), since |J′ 1| = ���� � ∂ φ(ρ)(n · ∇ρ)(v · n⊥)n⊥ · ∇ρ ���� = ���� � ∇⊥[φ(ρ)(n · ∇ρ)] · ∇ρ(v · n⊥) + � φ(ρ)(n · ∇ρ)∇ρ · ∇⊥(v · n⊥) ���� ≤ Cε1 ∥∇ρ∥2 Lr � ∥√ρv∥2 L 2r r−2 + ∥∇ρ∥2 L 2r r−2 � + ε1 � ∥∇v∥2 L2 + ∥∆ρ∥2 L2 � ≤ Cε1(∥∇ρ∥s Lr + 1) � ∥√ρv∥2 L2 + ∥∇ρ∥2 L2 � + ε1 � ∥∇v∥2 L2 + ∥∆ρ∥2 L2 � (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9) |J′ 2| = ���� � ∂ φ(ρ)(v · n⊥)n⊥ · ∇n · ∇ρ ���� = ���� � ∇⊥φ(ρ) · (∇n · ∇ρ)(v · n⊥) − � Ω φ(ρ)∇⊥ · (∇n · ∇ρ)(v · n⊥) dx − � φ(ρ)∇⊥(v · n⊥) · (∇n · ∇ρ) ���� ≤ Cε2 � ∥v∥2 L2 + ∥∇ρ∥2 L2 � + Cε2 ∥∇ρ∥2 Lr ∥√ρv∥2 L 2r r−2 + ε2 � ∥∇v∥2 L2 + ∥∆ρ∥2 L2 � ≤ Cε2 ∥∇ρ∥2 L2 + Cε2(∥∇ρ∥s Lr + 1) ∥√ρv∥2 L2 + ε2 � ∥∇v∥2 L2 + ∥∆ρ∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10) and |J3| = ���� � 2c0µ(ρ)∂ijρ−1∂jvi ���� = ���� � ∂ 2c0µ(ρ)∇ρ−1 · ∇v · n − � 2c0µ′∇ρ−1 · ∇v · ∇ρ ���� = ����− � ∂ 2c0µ(ρ)∇ρ−1 · ∇n · v − � 2c0µ′∇ρ−1 · ∇v · ∇ρ ���� ≤ Cε2(∥∇ρ∥s Lr + 1) � ∥√ρv∥2 L2 + ∥∇ρ∥2 L2 � + ε2 � ∥∇v∥2 L2 + ∥∆ρ∥2 L2 � , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) 49 we deduce that |I1| ≤ Cε(∥∇ρ∥s Lr + 1) � ∥√ρv∥2 L2 + ∥∇ρ∥2 L2 � + ε � ∥∇v∥2 L2 + ∥∆ρ∥2 L2 � , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12) Similarly, for I2–I3, one has |I2| ≤ Cε(∥∇ρ∥s Lr + 1) ∥√ρv∥2 L2 + ε ∥∇v∥2 L2 , |I3| ≤ Cε(∥∇ρ∥s Lr + 1) ∥∇ρ∥2 L2 + ε � ∥∇v∥2 L2 + ∥∆ρ∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='13) Substituting (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='12)–(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14) into (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7) and, then, alonging with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6) leads to d dt � ∥√ρv∥2 L2 + ∥∇ρ∥2 L2 � + ν � ∥∇v∥2 L2 + ∥∆ρ∥2 L2 � ≤ C(∥∇ρ∥s Lr + 1) � ∥∇ρ∥2 L2 + ∥√ρv∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14) Finally, applying the Gr¨onwall’s inequality to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14), we finish the proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Now, we can prove the first part of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' It follows from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='3 that sup t∈[0,T ] � ∥ρ∥2 H1 + ∥v∥2 L2 � + � T 0 � ∥ρ∥2 H2 + ∥v∥2 H1 � dt ≤ ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='15) Thus, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, we get the bounds � T 0 ∥(∇ρ, v)∥4 L4 dt ≤ ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' This, together with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='15), allows us to follow the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8 step by step, since the lower order bounds are enough to deduce the higher ones, according to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11 (noticing that, the proofs of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='9 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11 are merely based on the smallness assumption we derived from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2, that is, ∥∇ρ0∥L2 ≤ δ, without any additional restriction, see also Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We omit the remaining proof here and leave it to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 Case for (ρ, u) satisfying (C) Now, we assume that (ρ, u) satisfies the condition (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' One should notice that condition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2) is also equivalent with lim T →T ∗ � ∥v∥Ls(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='Lr) + ∥∇ρ∥Ls(0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='Lr) � ≤ ˜ M0 < ∞, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16) since ρ is bounded from above and below and the identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Our aim is proving the rest of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 under (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' First, we give the following lemma, which concludes some results we need later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' This nothing but a directly application of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='7 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Let (ρ, u, π) be a local strong solution as being described in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2 still holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Moreover, under the condition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2) (or, equivalently, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16)), one has ρ ∈ Cγ, γ 2 (QT ) for some γ ∈ (0, 1) and for all T ∈ (0, T ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Next, with help of the Serrin’s condition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2), one can get the lower bound of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Suppose that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16) holds and (ρ, u) satisfies (C), then sup t∈[0,T ] ∥∇ρ∥2 L2 + � T 0 � ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 � dt ≤ ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17) 50 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' As we did in Lemma (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4), applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, d dt ∥∇ρ∥2 L2 + ν ∥∆ρ∥2 L2 ≤ Cε � ∥∇ρ∥2 Lr + ∥v∥2 Lr � ∥∇ρ∥2 L 2r r−2 + ε ∥∆ρ∥2 L2 ≤ Cε (∥∇ρ∥s Lr + ∥v∥s Lr + 1) ∥∇ρ∥2 L2 + ε ∥∆ρ∥2 L2 , that is, d dt ∥∇ρ∥2 L2 + ν ∥∆ρ∥2 L2 ≤ C (∥∇ρ∥s Lr + ∥v∥s Lr + 1) ∥∇ρ∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18) Thus, using Gr¨onwall’s inequality and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, we conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' With this Lemma (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5) and condition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='16), we deduce from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='18)1 that � T 0 ∥ρt∥2 L2 dt ≤ ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='19) Now, we can prove Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' We start with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11) d dt � 1 2ρ|u|2 + � 2µ(ρ)|D(u)|2 := 3 � i=1 Si, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='20) where Si as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 |S1| ≤ C ∥Q∥L2 ∥ut∥L2 ≤ Cε1 ∥∇ρ∥2 L2 + ε1 ∥ut∥2 L2 , |S2| ≤ C ∥Q∥Lr ∥u∥ L 2r r−2 ∥∇u∥L2 ≤ Cε2 (∥∇ρ∥s Lr + 1) ∥u∥2 L2 + ε2 ∥∇u∥2 L2 , |S3| ≤ C ∥∇Q∥L2 ∥∇u∥L2 ≤ Cε3 � ∥∆ρ∥2 L2 + ∥∇ρ∥4 L4 � + ε3 ∥∇u∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='21) Combining (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='20) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='21) leads to, d dt ∥u∥2 L2 + ν ∥∇u∥2 L2 ≤ Cε (∥∇ρ∥s Lr + 1) ∥u∥2 L2 + Cε � ∥∆ρ∥2 L2 + ∥∇ρ∥4 L4 � + ε ∥ut∥2 L2 , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='22) Similarly, we deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='14)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17) that d dt∥ � µ(ρ)|D(u)|∥2 L2 + ν ∥ut∥2 L2 ≤ Cε � ∥u∥s Lr + ∥∇ρ∥4 L4 + ∥ρt∥2 L2 + 1 � ∥∇u∥2 L2 + Cε ∥∇ρt∥2 L2 + ε ∥∆u∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='23) By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4, µ(ρ) ∈ C(QT ), hence, we can apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='8 for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='61) with Φ = −c0∇ρ−1 and, then, use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='4 to deduce that ∥v∥2 H2 + ∥π∥2 H1 ≤ C � ∥F∥2 L2 + ��∇∆ρ−1��2 L2 � ≤ C � ∥v∥s Lr + ∥∇ρ∥4 L4 + 1 � � ∥∇v∥2 L2 + ∥∆ρ∥2 L2 + ∥ρt∥2 L2 � + C � ∥vt∥2 L2 + ∥∇ρt∥2 L2 � + C ∥∇∆ρ∥2 L2 , which gives ∥∆u∥2 L2 + ∥π∥2 H1 ≤ C � ∥u∥s Lr + ∥∇ρ∥4 L4 + 1 � � ∥∇u∥2 L2 + ∥∆ρ∥2 L2 + ∥ρt∥2 L2 � + C � ∥ut∥2 L2 + ∥∇ρt∥2 L2 � + C ∥∇∆ρ∥2 L2 , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='24) Plugging (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='24) into (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='23) and choosing ε sufficiently small, we have, for some positive constant ν depending on Ω, c0, α and β, d dt∥ � µ(ρ)|D(u)|∥2 L2 + ν � ∥∆u∥2 L2 + ∥ut∥2 L2 � ≤ C � ∥u∥s Lr + ∥∇ρ∥4 L4 + ∥ρt∥2 L2 + 1 � ∥∇u∥2 L2 + C � ∥∇ρt∥2 L2 + ∥∇∆ρ∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='25) 51 On the other hand, following the proof from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='33) to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='35), replacing ∥v∥4 L4 by ∥v∥s Lr, then, replacing v by u via (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='17) and applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1, one has d dt ∥∆ρ∥2 L2 + ν ∥∇∆ρ∥2 L2 ≤ Cε � ∥∇ρ∥4 L4 + ∥u∥s Lr + 1 � � ∥∆ρ∥2 L2 + ∥∇u∥2 L2 � + ε ∥∆u∥2 L2 , Alonging with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='23), we deduce that d dt � ∥∆ρ∥2 L2 + ∥ρt∥2 L2 � + ∥∇∆ρ∥2 L2 + ∥∇ρt∥2 L2 ≤ Cε � ∥u∥s Lr + ∥∇ρ∥4 L4 + ∥∆ρ∥2 L2 + 1 � � ∥∆ρ∥2 L2 + ∥ρt∥2 L2 + ∥∇u∥2 L2 � + ε � ∥∆u∥2 L2 + ∥ut∥2 L2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='26) Combining (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='22), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='25) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='26), then, applying the Gr¨onwall’s inequality, condition (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='2) and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='5, we get, for all T ∈ (0, T ∗), sup t∈[0,T ] � ∥u∥2 H1 + ∥ρt∥2 L2 + ∥∆ρ∥2 L2 � + � T 0 � ∥∇u∥2 H1 + ∥∇ρt∥2 L2 + ∥∇∆ρ∥2 L2 � dt ≤ ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Then, we can turn back to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='24) to get � T 0 ∥π∥2 H1 dt ≤ ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Therefore, we complete the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Abidi and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Global well-posedness of 3-D density-dependent Navier-Stokes system with variable viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Science China Mathematics, 58:1129–1150, 2015.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' Journal of Differential Equations, 263(8):4978–4996, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} +page_content=' 54' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE1T4oBgHgl3EQfLgMI/content/2301.02976v1.pdf'} diff --git a/e9E4T4oBgHgl3EQfqQ0g/content/tmp_files/2301.05198v1.pdf.txt b/e9E4T4oBgHgl3EQfqQ0g/content/tmp_files/2301.05198v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e1768685075e7ce012a3cdbebf4bd1eb6a50f3e6 --- /dev/null +++ b/e9E4T4oBgHgl3EQfqQ0g/content/tmp_files/2301.05198v1.pdf.txt @@ -0,0 +1,459 @@ +The Keyword Explorer Suite: A Toolkit for Understanding +Online Populations +Philip Feldman, Shimei Pan, James R. Foulds +2023-01-13 +Abstract +We have developed a set of Python applications that use large language models to identify +and analyze data from social media platforms relevant to a population of interest. Our pipeline +begins with using OpenAI’s GPT-3 to generate potential keywords for identifying relevant text +content from the target population. The keywords are then validated, and the content down- +loaded and analyzed using GPT-3 embedding and manifold reduction. Corpora are then created +to fine-tune GPT-2 models to explore latent information via prompt-based queries. These tools +allow researchers and practitioners to gain valuable insights into population subgroups online. +(a) KeywordExplorer +(b) TweetDownloader +(c) TweetEmbedExplorer +Figure 1: Three Applications from the Keyword Explorer Suite. +1 +Introduction +Imagine a researcher who is trying to understand how a particular population subgroup on social +media might react to an event that hasn’t happened yet and that they are not currently discussing. +This task has a variety of applications such as estimating public opinion, planning marketing cam- +paigns, and identifying potential risks and opportunities. This is not a straightforward task. The +researcher must first find the group in question, and they typically do not know the terms that the +group uses to describe themselves or their interests. However, large language models like GPT-3 +1 +arXiv:2301.05198v1 [cs.HC] 12 Jan 2023 + +KeywordExplorer (v 8.15.22) +一 +口 +File +Experiment name: +Phil_KeywordExplorer_2022-08-18-17-02-51 +GPT- +GPT Params +Prompt: +Here's a short list of simpsons characters: +Tokens +32 +Default style: +1) +(128) +64 +Here's a list of X: +128 +1) +Engines +davinci +(davinci) +curie +Response +Homer Simpson +babbage +2) Marge Simpson +3) +Bart simpson +4) +Lisa simpson +Maggie Simpson +Apu Nahasapeemapetilon +Moe Szyslak +8) +Chief Wiggum +(s +Krusty the Clown +10) +Sideshow Mel +Max chars: +30 +Parse regex +(n[0-9]+ V /μn[0-9]+ [0-9]+ V] +Actions +New prompt +Extend prompt +Parse response +Twitter +Twitter Params +Test Keyword(s) +Homer Simpson + Sample +day +(day) +week +Marge Simpson +Bart Simpson +month +Lisa simpson +Maggie Simpson +Krusty the Clown +Mr. Burns +Smithers +Start Date +August 08, 2022 +End Date +August 18, 2022 +Actions +Clear +Test Keyword +Plot +Save +Launch Twitter +Console +[3] +tokens = 128, +engine = davinci +prompt = I +Here's a short list of +simpsons +characters: +1)TweetDownloader (v 9.2.22) +口 +X +File +Experiment name: +Phil_ TweetDownloader_2022-08-12-16-55-20 +Twitter +Twitter Params +Test Keyword(s) +ivermectin +Sample (10-500): +500 +hydroxychloroquine oR HCq +Clamp: +1000 +chlorine dioxide +paxlovid +Percent: +100 +monoclonal +antibodies + Randomize +remdesivir +Nirmatrelvir +Options + Stream to DB + Stream to CSV +Corpus Size: +2000 +August 09, 2022 +Lowest/Day: +chlorine dioxide: 24 +Start Date +Highest/Day: +paxlovid: 7,783 +End Date +August 12, 2022 +Cur Date +August 09, 2022 +Duration: +m +set start +set end +Collect: +Balanced +Percent +Analytics: +Calc rates +Browser +Console +[15] +[Nirmatrelvir]100.0%: 159 keywords/day = 13.6 days for 2.0k +[14] +[13] +[12] +[paxlovid]100.0%: 7,783 keywords/day = 1.3 days for 2.0k +[11] +[10] +skep 'z = Aep/spiomAay s9t't :%o'oot[boh Ho 2utnboiotuaxoipau] +3 for 2.0k +[9] +[ivermectin]100.0%: 1,6l5 keywords/day = 2.2 days for 2.0kEmbeddingsExplorer (v 10.13.22) +口 +X +File +Experiment name: +Phil EmbeddingsExplorer 2022-11-01-08-25-54 +Experiment: +1: ivermectin, paxlovid +1: ivermectin, paxlovid +Keyword: +paxlovid OR Nirmatrel +paxlovid OR Nirmatrel +Num Entries: +Canvas +Corpora +PCA Dim: +10 +EPS: +8 +Min Samples: +5 +Perplex: +80 +Limit: +5432 +Commands +Retreive +Reduce +Cluster + Plot +Explore +Topics +Interactive embedding explorer (Mouse wheel to zoom) +< +Ex clude Cluster: +Exclude +Console +[101 +Finished +creating points +6 +Explore: num nodes +5432 +8 +loaded +5000 +of +5432 +reeoros +loaded +000F +5432 +[9] +loaded +3000 +5432 +reeoroscan generate potential keywords for identifying content created by the group in question since these +models are trained on vast amounts of text data and are likely to have ingested interactions from +the group. +Once the researcher has identified potential terms, they must be able to verify them against +actual social media conversations to see if they are being used by the group or “hallucinated” +by the model. If the terms are relevant, the researcher needs to download, store and clean the +appropriate posts, removing any irrelevant or redundant information. +Finally, small language models like GPT-2 can be finetuned on these posts to create a model +tailored to the group’s language and context. Because the desired opinions and beliefs are not +explicitly in the text, but are latent in the model, the model can be used to generate artificial text +simulating how the population might react to the hypothetical event. +This scenario illustrates how language models can identify and understand population subgroups +on social media, even when the desired information is not explicitly present in the text. Such models +can provide valuable insights into the thoughts and behaviors of these groups, allowing researchers +to make more informed decisions and predictions. +To address this scenario, we have developed a Python toolkit for using large language models +to identify and analyze relevant data from social media platforms. +Our applications facilitate +understanding and analyzing population subgroups online, enabling researchers and practitioners +to gain insights that would not be possible through traditional methods. +2 +Background +Query term identification is the process of identifying relevant terms and phrases for describing +a particular topic or concept. However, this process is often ad-hoc and deeply reflects the re- +searcher’s awareness and bias [11, 2]. Other approaches rely on query logs and cannot be used for +recommending important words based on a domain where the user has little prior knowledge or +experience [9]. This can lead to a lack of consistency and reproducibility. +In recent years, there have been attempts to create tools to help researchers determine the +optimal keywords for search in social media in disciplines such as information retrieval. These tools +often use various techniques, such as natural language processing [4] and machine learning [13], to +analyze large datasets and identify relevant keywords. +For example, some researchers have used topic modeling algorithms, such as Latent Dirichlet +Allocation (LDA), to identify the most common themes and topics in a dataset [5]. These themes +can then be used as keywords to search for relevant content. Other researchers have used senti- +ment analysis to identify the sentiment associated with specific keywords, which can be useful for +understanding how different groups of people feel about a particular topic [13]. +There have also been efforts to develop keyword generation tools specifically for social media +platforms, such as Twitter [1]. These tools often rely on machine learning algorithms to identify +patterns in the language and behavior of users, and use this information to suggest relevant keywords +for search [12]. +While these tools have had some success in helping researchers identify relevant keywords, many +of these tools rely on supervised learning techniques, which require large amounts of labeled data for +training [10]. This can be difficult to obtain, particularly in domains where there is limited available +data or where the language used is highly specialized. Additionally, these tools (e.g. LDA) may +be limited in their ability to capture the complexity and nuance of human language and behavior, +2 + +which can be important for understanding social media conversations and other phenomena. +Alternatively, transformer language models such as BERT and GPT are trained on vast amounts +of data from a wide range of sources, including books, articles, and social media posts, and as a +result, they have a broad understanding of language and context. This can make them useful for +sociology research, such as addressing query term identification by generating lists of slang terms +or other specialized vocabulary that may be difficult to find using other means [8]. +These large language models could potentially further be used to analyze the retrieved data, but +they may not be tailored to the specific needs and concerns of examining a particular subgroup. +To address this, smaller language models such as GPT-2 can be quickly fine-tuned on corpora that +are specific to a target domain [6]. This allows the models to capture the language and context +of particular social subgroups, enabling a new form of computational sociology. +By repeatedly +prompting these finetuned models to produce a large volume of responses, researchers can gain +insight into the language and behaviors of these groups in a more nuanced and detailed manner +than via traditional means [7]. +3 +Application Pipeline Overview +The Keyword Explorer Suite is a toolkit for understanding online populations, consisting of a set +of Python applications that work together in a pipeline, where each app produces outputs that are +used in subsequent applications. The suite includes a graphical user interface (GUI) that allows +the user to explore the data in an interactive environment. +The pipeline consists of: keyword exploration and validation, continues with data collection, then +data cleaning and refinement, and concludes with model training and exploration. These parts of +the pipeline are discussed in detail below: +3.1 +Keyword Exploration and Validation +Keyword exploration uses the KeywordExplorer app (Figure 1a), which lets the user prompt the +GPT-3 using the OpenAI API to produce lists of keywords. Prompts are generally of the form: +“xxxx: ¡newline¿ 1)”. Here, ¡newline¿ is a line break and xxxx is a string such as List slang terms +that have been used to refer to COVID-19:, or Create a list of hashtags that are important on Black +Twitter. As an illustrative example we will be using Here’s a short list of exotic pets. Each time the +model is prompted, slightly different responses will be generated, and can be evaluated. For this +query, the responses started with “1) Bats, 2) Monkeys, 3) Snakes,” and went to “9) Tarantulas, +10) Scorpions, and 11) Sugar Gliders.” +When prompted in this form, such responses from the GPT-3 are easily parsed using a regex. +Once applied, an unadorned list is produced that can be passed to Twitter or Wikipedia to evaluate +the keywords based on the usage of each keyword over time. With the default parameters we retrieve +count data for 10 days in the past to the current date, though these are easily edited. Submitting +these words to Twitter returns the total usage over that period (Dec. 17-27, 2022) to be Monkeys +(36,772), Snakes (29,830), Bats (21,156), Alligators (3,258), Tarantulas (689), and Sugar Gliders +(196). To further evaluate the applicability of the keywords, the app can launch a series of tabs +in the default browser for each of the keywords across the defined timeframe. This can help to +find times when the context switches, as in the case for “Birds” during 2014, when the Baltimore +Orioles baseball team, referred to as “The Birds,” had a particularly good season. +3 + +3.2 +Data Collection +Once the keywords are validated, they are passed into the TweetDownloader app (Figure 1b). This +app allows the user to submit the keywords individually to the Twitter API and sample the relevant +tweets over a defined period. The number of tweets per keyword per day can be adjusted so that +they are the same for each day or proportional with respect to the largest number of tweets for +any particular keyword. The TweetDownloader app also allows the user to also filter for language, +location, and other criteria. The results are saved to a MariaDB relational database, which allows +sophisticated queries and analysis. The database stores data and time information, which supports +downstream chronological sampling. +Users can apply daily and overall limits for each keyword +corpus. +3.3 +Data Cleaning and Refinement +TweetEmbedExplorer (Figure 1c) filters, analyzes, and augments tweet information. Augmented +information can then be used to create a train/test corpus for finetuning language models such as +the GPT-2. For any keyword, user information associated with the ID of each post can be collected +and placed in its own table in the database, allowing more sophisticated queries. Using the OpenAI +text embeddings API, the embeddings for each tweet can be stored in the database. These can +be projected to 2D using a combination of PCA and T-SNE. Once dimension reduction has been +performed, clustering can be performed interactively using DBSCAN [3], and outlier clusters can +be marked “excluded.” +The next step is to produce a corpus for finetuning a GPT-2 model. Finetuned models have +been shown to accurately predict, for example, the vegetarian preferences of Yelp reviewers when all +vegetarian data has been excluded from the test/train data [7]. This means that a model finetuned +on a set of tweets that may not contain the explicit information about a certain subject may still +be able to generate tweets that are likely to contain that information in some cases. +We use a process we call “meta wrapping” that creates text that has metadata in it in addition +to the tweet, e.g.: +[[text: +I know I’m not the prettiest dog but my love for you is unconditional always because +I have a beautiful heart and soul || created: +2022-12-27 07:10:25 || location: +USA || probability: +twenty]] +Each entry in the string is wrapped by [[ and ]]. All elements within the meta wrapped string +are separated by ||. The “text:” element precedes the posting, the “created:” tag precedes the +data the post was created, and the “location:” tag contains the coordinates of the tweet or the +poster if available. For model convergence validation, there is a substring of the form “probability: +xxxx”, where xxxx can be “ten”, “twenty”, “thirty”, or “forty.” The likelihood that a line will have +the appropriate string reflects the probability of the random number generator hitting that value. +As a sanity check, if the generated data does not match these percentages, insufficient finetuning +can clearly be diagnosed. +3.4 +Model Training and Exploration +ModelExplorer is a tool that allows users to interactively explore finetuned GPT-2 models (Figure 2). +These models are finetuned on tweet corpora generated from TweetEmbedExplorer. It allows the +user to provide a set of comma-separated probes along with model hyperparameters and then +generate a series of predictions from the model. Text that is output from the model is parsed, +4 + +Figure 2: ModelExplorer: GPT Parameters and the Results of the Meta Wrapping Probability +Sanity Check. +displayed in the text output area of the tool, and if desired, stored in the database for further +analysis. In the example in Figure 2, probes regarding Ivermectin and Paxlovid were used. The +results show that there was a maximum deviation of 4% with respect to the target percentages, so +the finetuning sanity check was passed. +4 +Conclusions +We presented a Python toolkit for gaining insight into population subgroups.1 Our pipeline begins +with using large language models to generate potential keywords for exploring population subgroups, +which are then validated using historical usage trends on social media platforms. The resulting data +is then analyzed and clustered using a combination of GPT-3 embeddings and manifold reduction. +Finally, the posts are used to generate corpora that can be used to fine-tune language models and +identify latent information. We plan to use our toolkit to study COVID-19 racism. +Acknowledgements +We would like to thank OpenAI for giving us permission to use the GPT-3 in creepy in ways that +are not really per their policy, HuggingFace for their wonderful ecosystem and API, and Twitter +for (as of this writing) not falling apart. +References +[1] Saroj Kr Biswas, Monali Bordoloi, and Jacob Shreya. A graph based keyword extraction model +using collective node weight. Expert Systems with Applications, 97:51–59, 2018. +[2] Cody Buntain, Erin McGrath, and Brandon Behlendorf. Sampling social media: Supporting +information retrieval from microblog data resellers with text, network, and spatial analysis. In +Proceedings of the 51st Hawaii International Conference on System Sciences, 2018. +[3] Timothy DeLise. +Data segmentation via t-sne, dbscan, and random forest. +In Intelligent +Computing, pages 139–151. Springer, 2021. +1Full code available at https://github.com/pgfeldman/KeywordExplorer. +5 + +GPT-2 +Max Length: +128 +Top K: +50 +Top P: +0.95 +Num Seguences: +100 +Batch size: +1 +Total: +244 +Ten: +%6 +Twenty: +16% +Thirty: +34% +Forty: +38% +Flags + Re-use Seed Save to DB + Save to spreadsheet +Probe: +[text: Ivermectin is, [text: Paxlovid is +Actions: +Run[4] Alicia Eads, Alexandra Schofield, Fauna Mahootian, David Mimno, and Rens Wilderom. Sepa- +rating the wheat from the chaff: A topic and keyword-based procedure for identifying research- +relevant text. Poetics, 86:101527, 2021. +[5] Ali Feizollah, Sulaiman Ainin, Nor Badrul Anuar, Nor Aniza Binti Abdullah, and Mohamad +Hazim. Halal products on twitter: Data extraction and sentiment analysis using stack of deep +learning algorithms. IEEE Access, 7:83354–83362, 2019. +[6] Philip Feldman. +Navigating language models with synthetic agents. +arXiv preprint +arXiv:2008.04162, 2020. +[7] Philip Feldman, Aaron Dant, James R Foulds, and Shimei Pan. Polling latent opinions: A +method for computational sociolinguistics using transformer language models. arXiv preprint +arXiv:2204.07483, 2022. +[8] Philip Feldman, Sim Tiwari, Charissa SL Cheah, James R Foulds, and Shimei Pan. Analyzing +covid-19 tweets with transformer-based language models. arXiv preprint arXiv:2104.10259, +2021. +[9] Jenny Felser, Jian Xi, Christoph Demus, Dirk Labudde, and Michael Spranger. Recommenda- +tion of query terms for colloquial texts in forensic text analysis. INFORMATIK 2022, 2022. +[10] Ammar Ismael Kadhim. Survey on supervised machine learning techniques for automatic text +classification. Artificial Intelligence Review, 52(1):273–292, 2019. +[11] Gary King, Patrick Lam, and Margaret E Roberts. Computer-assisted keyword and document +set discovery from unstructured text. American Journal of Political Science, 61(4):971–988, +2017. +[12] Stephen Wai Hang Kwok, Sai Kumar Vadde, and Guanjin Wang. Tweet topics and sentiments +relating to covid-19 vaccination among australian twitter users: machine learning analysis. +Journal of Medical Internet Research, 23(5):e26953, 2021. +[13] Matthew Louis Mauriello, Cody Buntain, Brenna McNally, Sapna Bagalkotkar, Samuel Kush- +nir, and Jon E Froehlich. SMIDGen: An approach for scalable, mixed-initiative dataset gen- +eration from online social networks. HCIL Tech Reports, pages 2018–01, 2018. +6 + diff --git a/e9E4T4oBgHgl3EQfqQ0g/content/tmp_files/load_file.txt b/e9E4T4oBgHgl3EQfqQ0g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b20d5f8a9194b8b5c6fd9df1a05e76640afcf43 --- /dev/null +++ b/e9E4T4oBgHgl3EQfqQ0g/content/tmp_files/load_file.txt @@ -0,0 +1,281 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf,len=280 +page_content='The Keyword Explorer Suite: A Toolkit for Understanding Online Populations Philip Feldman, Shimei Pan, James R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Foulds 2023-01-13 Abstract We have developed a set of Python applications that use large language models to identify and analyze data from social media platforms relevant to a population of interest.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' (a) KeywordExplorer (b) TweetDownloader (c) TweetEmbedExplorer Figure 1: Three Applications from the Keyword Explorer Suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' 1 Introduction Imagine a researcher who is trying to understand how a particular population subgroup on social media might react to an event that hasn’t happened yet and that they are not currently discussing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' This task has a variety of applications such as estimating public opinion, planning marketing cam- paigns, and identifying potential risks and opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' This is not a straightforward task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' The researcher must first find the group in question, and they typically do not know the terms that the group uses to describe themselves or their interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' However, large language models like GPT-3 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='05198v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='HC] 12 Jan 2023 KeywordExplorer (v 8.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='22) 口 X File Experiment name: Phil_ TweetDownloader_2022-08-12-16-55-20 Twitter Twitter Params Test Keyword(s) ivermectin Sample (10-500): 500 hydroxychloroquine oR HCq Clamp: 1000 chlorine dioxide paxlovid Percent: 100 monoclonal antibodies Randomize remdesivir Nirmatrelvir Options Stream to DB Stream to CSV Corpus Size: 2000 August 09,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' 2022 Lowest/Day: chlorine dioxide: 24 Start Date Highest/Day: paxlovid: 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='783 End Date August 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' 2022 Cur Date August 09,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' 2022 Duration: m set start set end Collect: Balanced Percent Analytics: Calc rates Browser Console [15] [Nirmatrelvir]100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='0%: 159 keywords/day = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='6 days for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='0k [14] [13] [12] [paxlovid]100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='0%: 7,783 keywords/day = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='3 days for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content="0k [11] [10] skep 'z = Aep/spiomAay s9t't :%o'oot[boh Ho 2utnboiotuaxoipau] 3 for 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='0k [9] [ivermectin]100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='0%: 1,6l5 keywords/day = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='2 days for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='0kEmbeddingsExplorer (v 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='22) 口 X File Experiment name: Phil EmbeddingsExplorer 2022-11-01-08-25-54 Experiment: 1: ivermectin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' paxlovid 1: ivermectin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' paxlovid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='Keyword: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='paxlovid OR Nirmatrel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='paxlovid OR Nirmatrel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='Num Entries: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='Canvas ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='Corpora ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='PCA Dim: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='EPS: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='Min Samples: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='Perplex: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='Limit: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='5432 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='Commands ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='5432 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='reeoros ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='loaded ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='000F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='5432 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='[9] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='loaded ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='5432 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='reeoroscan generate potential keywords for identifying content created by the group in question since these ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='models are trained on vast amounts of text data and are likely to have ingested interactions from ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Once the researcher has identified potential terms, they must be able to verify them against actual social media conversations to see if they are being used by the group or “hallucinated” by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' If the terms are relevant, the researcher needs to download, store and clean the appropriate posts, removing any irrelevant or redundant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Finally, small language models like GPT-2 can be finetuned on these posts to create a model tailored to the group’s language and context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Because the desired opinions and beliefs are not explicitly in the text, but are latent in the model, the model can be used to generate artificial text simulating how the population might react to the hypothetical event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' This scenario illustrates how language models can identify and understand population subgroups on social media, even when the desired information is not explicitly present in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Such models can provide valuable insights into the thoughts and behaviors of these groups, allowing researchers to make more informed decisions and predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' To address this scenario, we have developed a Python toolkit for using large language models to identify and analyze relevant data from social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Our applications facilitate understanding and analyzing population subgroups online, enabling researchers and practitioners to gain insights that would not be possible through traditional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' 2 Background Query term identification is the process of identifying relevant terms and phrases for describing a particular topic or concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' However, this process is often ad-hoc and deeply reflects the re- searcher’s awareness and bias [11, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Other approaches rely on query logs and cannot be used for recommending important words based on a domain where the user has little prior knowledge or experience [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' This can lead to a lack of consistency and reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' In recent years, there have been attempts to create tools to help researchers determine the optimal keywords for search in social media in disciplines such as information retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' These tools often use various techniques, such as natural language processing [4] and machine learning [13], to analyze large datasets and identify relevant keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' For example, some researchers have used topic modeling algorithms, such as Latent Dirichlet Allocation (LDA), to identify the most common themes and topics in a dataset [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' These themes can then be used as keywords to search for relevant content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Other researchers have used senti- ment analysis to identify the sentiment associated with specific keywords, which can be useful for understanding how different groups of people feel about a particular topic [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' There have also been efforts to develop keyword generation tools specifically for social media platforms, such as Twitter [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' These tools often rely on machine learning algorithms to identify patterns in the language and behavior of users, and use this information to suggest relevant keywords for search [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' While these tools have had some success in helping researchers identify relevant keywords, many of these tools rely on supervised learning techniques, which require large amounts of labeled data for training [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' This can be difficult to obtain, particularly in domains where there is limited available data or where the language used is highly specialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Additionally, these tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' LDA) may be limited in their ability to capture the complexity and nuance of human language and behavior, 2 which can be important for understanding social media conversations and other phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Alternatively, transformer language models such as BERT and GPT are trained on vast amounts of data from a wide range of sources, including books, articles, and social media posts, and as a result, they have a broad understanding of language and context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' This can make them useful for sociology research, such as addressing query term identification by generating lists of slang terms or other specialized vocabulary that may be difficult to find using other means [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' These large language models could potentially further be used to analyze the retrieved data, but they may not be tailored to the specific needs and concerns of examining a particular subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' To address this, smaller language models such as GPT-2 can be quickly fine-tuned on corpora that are specific to a target domain [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' This allows the models to capture the language and context of particular social subgroups, enabling a new form of computational sociology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' By repeatedly prompting these finetuned models to produce a large volume of responses, researchers can gain insight into the language and behaviors of these groups in a more nuanced and detailed manner than via traditional means [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' 3 Application Pipeline Overview The Keyword Explorer Suite is a toolkit for understanding online populations, consisting of a set of Python applications that work together in a pipeline, where each app produces outputs that are used in subsequent applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' The suite includes a graphical user interface (GUI) that allows the user to explore the data in an interactive environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' The pipeline consists of: keyword exploration and validation, continues with data collection, then data cleaning and refinement, and concludes with model training and exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' These parts of the pipeline are discussed in detail below: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='1 Keyword Exploration and Validation Keyword exploration uses the KeywordExplorer app (Figure 1a), which lets the user prompt the GPT-3 using the OpenAI API to produce lists of keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Prompts are generally of the form: “xxxx: ¡newline¿ 1)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Here, ¡newline¿ is a line break and xxxx is a string such as List slang terms that have been used to refer to COVID-19:, or Create a list of hashtags that are important on Black Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' As an illustrative example we will be using Here’s a short list of exotic pets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Each time the model is prompted, slightly different responses will be generated, and can be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' For this query, the responses started with “1) Bats, 2) Monkeys, 3) Snakes,” and went to “9) Tarantulas, 10) Scorpions, and 11) Sugar Gliders.” When prompted in this form, such responses from the GPT-3 are easily parsed using a regex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Once applied, an unadorned list is produced that can be passed to Twitter or Wikipedia to evaluate the keywords based on the usage of each keyword over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' With the default parameters we retrieve count data for 10 days in the past to the current date, though these are easily edited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Submitting these words to Twitter returns the total usage over that period (Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' 17-27, 2022) to be Monkeys (36,772), Snakes (29,830), Bats (21,156), Alligators (3,258), Tarantulas (689), and Sugar Gliders (196).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' To further evaluate the applicability of the keywords, the app can launch a series of tabs in the default browser for each of the keywords across the defined timeframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' This can help to find times when the context switches, as in the case for “Birds” during 2014, when the Baltimore Orioles baseball team, referred to as “The Birds,” had a particularly good season.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='2 Data Collection Once the keywords are validated, they are passed into the TweetDownloader app (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' This app allows the user to submit the keywords individually to the Twitter API and sample the relevant tweets over a defined period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' The number of tweets per keyword per day can be adjusted so that they are the same for each day or proportional with respect to the largest number of tweets for any particular keyword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' The TweetDownloader app also allows the user to also filter for language, location, and other criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' The results are saved to a MariaDB relational database, which allows sophisticated queries and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' The database stores data and time information, which supports downstream chronological sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Users can apply daily and overall limits for each keyword corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='3 Data Cleaning and Refinement TweetEmbedExplorer (Figure 1c) filters, analyzes, and augments tweet information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Augmented information can then be used to create a train/test corpus for finetuning language models such as the GPT-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' For any keyword, user information associated with the ID of each post can be collected and placed in its own table in the database, allowing more sophisticated queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Using the OpenAI text embeddings API, the embeddings for each tweet can be stored in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' These can be projected to 2D using a combination of PCA and T-SNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Once dimension reduction has been performed, clustering can be performed interactively using DBSCAN [3], and outlier clusters can be marked “excluded.” The next step is to produce a corpus for finetuning a GPT-2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Finetuned models have been shown to accurately predict, for example, the vegetarian preferences of Yelp reviewers when all vegetarian data has been excluded from the test/train data [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' This means that a model finetuned on a set of tweets that may not contain the explicit information about a certain subject may still be able to generate tweets that are likely to contain that information in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' We use a process we call “meta wrapping” that creates text that has metadata in it in addition to the tweet, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' : [[text: I know I’m not the prettiest dog but my love for you is unconditional always because I have a beautiful heart and soul || created: 2022-12-27 07:10:25 || location: USA || probability: twenty]] Each entry in the string is wrapped by [[ and ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' All elements within the meta wrapped string are separated by ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' The “text:” element precedes the posting, the “created:” tag precedes the data the post was created, and the “location:” tag contains the coordinates of the tweet or the poster if available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' For model convergence validation, there is a substring of the form “probability: xxxx”, where xxxx can be “ten”, “twenty”, “thirty”, or “forty.” The likelihood that a line will have the appropriate string reflects the probability of the random number generator hitting that value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' As a sanity check, if the generated data does not match these percentages, insufficient finetuning can clearly be diagnosed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='4 Model Training and Exploration ModelExplorer is a tool that allows users to interactively explore finetuned GPT-2 models (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' These models are finetuned on tweet corpora generated from TweetEmbedExplorer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' It allows the user to provide a set of comma-separated probes along with model hyperparameters and then generate a series of predictions from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Text that is output from the model is parsed, 4 Figure 2: ModelExplorer: GPT Parameters and the Results of the Meta Wrapping Probability Sanity Check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' displayed in the text output area of the tool, and if desired, stored in the database for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' In the example in Figure 2, probes regarding Ivermectin and Paxlovid were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' The results show that there was a maximum deviation of 4% with respect to the target percentages, so the finetuning sanity check was passed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' 4 Conclusions We presented a Python toolkit for gaining insight into population subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='1 Our pipeline begins with using large language models to generate potential keywords for exploring population subgroups, which are then validated using historical usage trends on social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' The resulting data is then analyzed and clustered using a combination of GPT-3 embeddings and manifold reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Finally, the posts are used to generate corpora that can be used to fine-tune language models and identify latent information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' We plan to use our toolkit to study COVID-19 racism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Acknowledgements We would like to thank OpenAI for giving us permission to use the GPT-3 in creepy in ways that are not really per their policy, HuggingFace for their wonderful ecosystem and API, and Twitter for (as of this writing) not falling apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' References [1] Saroj Kr Biswas, Monali Bordoloi, and Jacob Shreya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' A graph based keyword extraction model using collective node weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Expert Systems with Applications, 97:51–59, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' [2] Cody Buntain, Erin McGrath, and Brandon Behlendorf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Sampling social media: Supporting information retrieval from microblog data resellers with text, network, and spatial analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' In Proceedings of the 51st Hawaii International Conference on System Sciences, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' [3] Timothy DeLise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Data segmentation via t-sne, dbscan, and random forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' In Intelligent Computing, pages 139–151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Springer, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' 1Full code available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='com/pgfeldman/KeywordExplorer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' 5 GPT-2 Max Length: 128 Top K: 50 Top P: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content='95 Num Seguences: 100 Batch size: 1 Total: 244 Ten: %6 Twenty: 16% Thirty: 34% Forty: 38% Flags Re-use Seed Save to DB Save to spreadsheet Probe: [text: Ivermectin is, [text: Paxlovid is Actions: Run[4] Alicia Eads, Alexandra Schofield, Fauna Mahootian, David Mimno, and Rens Wilderom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Sepa- rating the wheat from the chaff: A topic and keyword-based procedure for identifying research- relevant text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' Poetics, 86:101527, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/e9E4T4oBgHgl3EQfqQ0g/content/2301.05198v1.pdf'} +page_content=' [5] Ali Feizollah, Sulaiman Ainin, Nor Badrul Anuar, Nor Aniza Binti Abdullah, and Mohamad Hazim.' metadata={'source': 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b/j9E3T4oBgHgl3EQfhQpJ/content/tmp_files/2301.04569v1.pdf.txt @@ -0,0 +1,549 @@ +A SHARP BOUND FOR HYPERGEOMETRIC RANK IN DIMENSION THREE +CHRISTINE BERKESCH AND MAR´IA-CRUZ FERN ´ANDEZ-FERN ´ANDEZ +ABSTRACT. We provide a sharp upper bound on the quotient of the rank of an A-hypergeometric +system with a three-dimensional torus action by the normalized volume of A; in this case, the upper +bound is two. +INTRODUCTION +A-hypergeometric D-modules, also known as GKZ-systems, were introduced in [GGZ87, GKZ89] +to generalize classical hypergeometric equations. These systems of linear partial differential equa- +tions in several complex variables are determined by a matrix A = (ai,j) ∈ Zd×n with columns +ak ∈ Zd such that ZA = Za1 + · · · + Zan = Zd and a parameter vector β ∈ Cd. We assume that +A is pointed, i.e., that all columns in A lie in a single linear halfspace of Rd that does not contain +the origin. +Definition 1.1. Let x1, x2, . . . , xn be coordinates on Cn, with corresponding partial derivatives +∂1, ∂2, . . . , ∂n, so that the Weyl algebra D on Cn is generated by x1, . . . , xn, ∂1, . . . , ∂n. Let +IA ..= ⟨∂u − ∂v | u, v ∈ Nn, Au = Av⟩ ⊆ C[∂1, . . . , ∂n] +denote the toric ideal of A, and let Ei ..= �n +j=1 ai,jxj∂j be the ith Euler operator of A. The +A-hypergeometric D-module with parameter β ∈ Cd is the left D-module +MA(β) ..= D/D · ⟨IA, E1 − β1, . . . , Ed − βd⟩. +For any choice of A and β, the module MA(β) is holonomic [GGZ87, Ado94]. Consequently, the +dimension of the space of germs of holomorphic solutions of MA(β) at a nonsingular point, also +known as its (holonomic) rank, is finite. When β ∈ Cd is generic, the rank of MA(β) is equal to +the normalized volume vol(A) of the matrix A [GKZ89, Ado94] inside the lattice Zd, which is the +Euclidean volume of the convex hull ∆A ⊆ Rd of the columns of A and the origin divided by d!; +but in general, this is only a lower bound [SST00, MMW05]. The set +E(A) ..= {β ∈ Cd | rank(MA(β)) > vol(A)} +is called the exceptional arrangement of A, which is an affine subspace arrangement of codimen- +sion at least two that is closely related to the local cohomology modules of the toric ring C[∂]/IA +[MMW05]. A parameter β ∈ E(A) is called a rank jumping parameter. Combinatorial formulas to +compute the rank of MA(β) in terms of the ranking lattices Eβ of A at β appear in [Oku06, Ber11]. +Unfortunately, the presence of alternating signs in these formulas do not yield a strong upper bound +for the rank of MA(β). One exception is the case d = 2, where it was shown in [CDD99] that +2010 Mathematics Subject Classification. 13N10, 32C38, 33C70, 14M25. +CB was partially supported by NSF Grant DMS 2001101. +MCFF was partially supported by projects PID2020-117843GB-I00 (Ministerio de Ciencia e Innovaci´on, Spain), +P20-01056 and US-1262169 (Consejer´ıa de Econom´ıa, Conocimiento, Empresas y Universidad, Junta de Andaluc´ıa +and FEDER). +1 +arXiv:2301.04569v1 [math.AG] 11 Jan 2023 + +2 +CHRISTINE BERKESCH AND MAR´IA-CRUZ FERN ´ANDEZ-FERN ´ANDEZ +rank(MA(β)) ≤ vol(A) + 1 is a sharp bound. For arbitrary d, A, and β, previously known upper +bounds for the holonomic rank of MA(β) are as follows: +rank(MA(β)) ≤ +� +4d · vol(A) +if IA is homogeneous [SST00], +4d+1 · vol(A) +otherwise [BFM18]. +However, it is believed that these upper bounds are much too large. In [BF22], we showed that +when β is generic among rank-jumping parameters (i.e., simple in [Ber11]), then +rank(MA(β)) +vol(A) +≤ (d − 1), +(1.1) +and we showed that this bound is tight by constructing a sequence of examples for which the ratio +rank(MA(β))/vol(A) tends to d − 1. Still, this case neglects the rank-jumping parameters β for +which the highest jumps are possible. In fact, there are families of examples for which the ratio +rank(MA(β))/vol(A) grow exponentially with d [Fer13]. In this note, we show that when d = 3, +the bound (1.1) holds for all parameters β, with a strict inequality. +Theorem 1.2. There is a strict sharp inequality for all A ∈ Z3×n and β ∈ C3: +rank(MA(β)) +vol(A) +< 2. +(1.2) +Outline. We begin with results in Ehrhart theory in §2 and rank jumps in §3. Next, in §4, we +consider rank jumps in the case that ∆A has degree one. Finally, the proof of Theorem 1.2 is +completed in §5. +Acknowledgements. We thank Christian Haase for stating and proving Lemma 5.1, providing the +catalyst for this article. We are also grateful to Laura Felicia Matusevich, Vic Reiner, and Uli +Walther for helpful conversations related to this work. +2. PRELIMINARIES ON EHRHART THEORY +Let ∆ ⊆ Rd be a lattice polytope, that is, the convex hull in Rd of a finite set of points in Zd. We +denote by |∆ ∩ Zd| the cardinality of the set of lattice points in ∆. The function g∆(t) : N −→ N +defined by +g∆(k) := |k∆ ∩ Zd| +counts the number of lattice points in the k-fold dilatation of ∆. Ehrhart proved that this function is +a polynomial in k, which is now called the Ehrhart polynomial of ∆. Moreover, he also proved that +when the polytope ∆ ⊆ Rd is d-dimensional the degree of g∆(k) is d and its leading coefficient is +equal to the normalized volume of ∆ with respect to Zd, that is, the Euclidean volume of ∆ divided +by d! [Ehr62]. The Ehrhart series of ∆ is the generating function +E∆(t) := +� +k≥0 +g∆(k)tk. +The following result is well known (see, for example, [BN07] and the references therein). +Theorem 2.1. For a lattice polytope ∆ ⊆ Rd, there exists +h∗(t) = 1 + h∗ +1t + · · · + h∗ +ktk, +called the h-polynomial of ∆ such that the following properties hold: + +3 +(1) E∆(t) = h∗(t)/(1 − t)d+1, +(2) h∗ +j ∈ Z≥0 for all j = 1, . . . , k, +(3) vol(∆) = 1 + h∗ +1 + · · · + h∗ +k, +(4) h∗ +1 = |∆ ∩ Zd| − d − 1, and +(5) the leading coefficient h∗ +k = |(d + 1 − k)∆◦ ∩ Zd|, where ∆◦ denotes the interior of ∆. +The degree of ∆, denoted deg(∆), is defined to be the degree of the h-polynomial of ∆. An +interesting fact is that +deg(∆) = min{j ∈ N | |i∆◦ ∩ Zd| = ∅, ∀1 ≤ i ≤ d − j}. +It follows from Theorem 2.1 that lattice polytopes of degree zero are basic simplices, that is, lattice +polytopes whose vertices form an affine lattice basis of Zd. Batyrev and Nill classified those +polytopes having degree one [BN07, Theorem 2.5]; they proved that deg(∆) ≤ 1 if and only if ∆ +is an exceptional simplex or a Lawrence prism, which we now define. +First, if F is a face of ∆ such that |∆ ∩ Zd| − |F ∩ Zd| − codim(F) = 0, then we say that ∆ is an +iterated pyramid over F. An exceptional triangle is a 2-dimensional basic simplex multiplied by +2. An exceptional simplex is a simplex that is the (d−2)-fold pyramid over an exceptional triangle. +In other words, ∆ is the convex hull in Rd of +e0, e0 + 2(e1 − e0), e0 + 2(e2 − e0), e3, . . . , ed +where e0, . . . , ed is some affine lattice basis of Zd. Finally, a Lawrence prism of heights b1, . . . , bd ≥ +0, denoted by L(b1, . . . , bs), is the convex hull in Rd of +e0, e0 + b1(ed − e0), e1, e1 + b2(ed − e0), . . . , ed−1, ed−1 + bd(ed − e0), +where e0, . . . , ed is some affine lattice basis of Zd. +A Lawrence prism L(b1, . . . , bd) has degree one if b1 + · · · + bd ≥ 2 [BN07, Proposition 2.4]. +Otherwise, it is a basic simplex of degree zero. +Remark 2.2. The normalized volume of the Lawrence prism L(b1, . . . , bd) is b1 + · · · + bd. +3. PRELIMINARIES ON RANK JUMPS +In this section, we return to the setting of a pointed matrix A ∈ Zd×n with ZA = Zd. We will soon +restrict to the case d = 3, but it is not needed for the following definitions. Recall that ∆A ⊆ Rd +denotes the convex hull of the columns of A and the origin. The set of columns of A will be also +denoted by A. A submatrix F of A (or a subset of its set of columns) is called a face of A, denoted +F ⪯ A, if ∆F is a face of the polytope ∆A ⊆ Rd and A ∩ ∆F = F. In particular, the empty set +and A are faces of A. For a face F ⪯ A, consider the union of the lattice translates +Eβ +F ..= +� +Zd ∩ (β + CF) +� +∖ (NA + ZF) = +� +b∈Bβ +F +(b + ZF), +where Bβ +F ⊆ Zd is a set of lattice translate representatives. Since |Bβ +F| is the number of translates +of ZF appearing in Eβ +F, it is by definition equal to the difference between [Zd ∩ QF : ZF] and the +number of translates of ZF along β + CF that are contained in NA + ZF. + +4 +CHRISTINE BERKESCH AND MAR´IA-CRUZ FERN ´ANDEZ-FERN ´ANDEZ +Given the set J (β) ..= {(F, b) | F ⪯ A, b ∈ Bβ +F}, the ranking lattices of A at β are defined to be +Eβ ..= +� +(F,b)∈J (β) +(b + ZF). +Note that the ranking lattices of A at β are precisely the union of those sets (b + ZF) contained in +Zd \ NA such that β ∈ (b + CF). This is closely related to the set of holes of the affine semigroup +NA, namely the set (Zd ∩ R≥0A) \ NA. +A rank jumping parameter β is simple (for a face G ⪯ A) if the set of maximal pairs (F, b) in J (β) +with respect to inclusion on b + ZF all correspond to a unique face G ⪯ A. +The main result in [Ber11] states how the rank of MA(β) can be computed from the combinatorics +of Eβ and ∆A. An explicit formula for the rank is given when the rank jumping parameter β is +simple for a face G ⪯ A (see [Oku06] for this particular case); in this case, +rank(MA(β)) = vol(A) + |Bβ +G| · (codim(G) − 1) · volZG(G). +(3.1) +Example 3.1. It is shown in [Ber11, Example 6.21] that if β ∈ Cd is such that the ranking lattices +of A at β involve only two faces, F1 and F2, then the rank jump of M at β is +rank(MA(β)) − vol(A) = +2 +� +i=1 +� +|Bβ +Fi| · [codim(Fi) − 1] · vol(Fi) +� ++ |Bβ +G| · Cβ · vol(G), +(3.2) +where G = F1 ∩ F2 and the constant Cβ is given by +Cβ = +�codim(G) +2 +� +− codim(G) + 1 − +�codim(F1) +2 +� +− +�codim(F2) +2 +� ++ +�codim(CF1 + CF2) +2 +� +. +When d = 3, Okuyama [Oku06] provided a formula for the rank of MA(β), as follows. +Theorem 3.2. [Oku06, Theorem 2.6] Let d = 3 and β ∈ C3. Define an equivalence relation on +J (β) by +(F, b) ∼ (G, c) +if and only if +(b + ZF) ∩ (c + ZG) ̸= ∅. +Let FA(β) denote the set of equivalence classes of J (β) under ∼. For each Λ ∈ FA(β), let J β +Λ +denote the order complex on the face poset of faces F with (F, b) ∈ Λ for some b ∈ Z3. Then the +rank jump of A at β ∈ Cd is given by +rank(MA(β)) − vol(A) = +� +Λ∈FA(β) +jΛ(β), +where +jΛ(β) := +� +� +� +� +� +� +� +� +� +� +� +� +� +� +rays F∈J β +Λ +(vol(F) − 1) + m − 1 +if �Hp(J β +Λ ) ∼= 0 for p ̸= 0, �H0(J β +Λ ) ∼= Cm−1 with m > 1, +vol(F) +if �Hp(J β +Λ ) ∼= 0 for all p, so J β +Λ consists of a ray F, +2 +if �H−1(J β +Λ ) ∼= C, �Hp(J β +Λ ) ∼= 0 for p ̸= −1, +0 +otherwise. + +5 +4. RANK JUMPS FOR MA(β) WHEN ∆A HAS DEGREE ONE +Recall that we denote the convex hull of the columns of A and the origin by ∆A. When ∆A +is a lattice polytope of degree one, it is an exceptional simplex or a Lawrence prism of heights +b1, . . . , bd ∈ N. In this section, we compute the possible rank jumps for MA(β) when ∆A is the +former or if it is the latter and d = 3. +Lemma 4.1. If ∆ = ∆A is an exceptional simplex, then MA(β) has no rank-jumping parameters. +Proof. If ∆A is an iterated pyramid over an exceptional triangle, then one of the following cases +holds: +(i) ∆A is an exceptional triangle, +(ii) ∆A is (up to rigid transformation) the convex hull of the origin in R3 and +� 1 1 1 +0 2 0 +0 0 2 +� +, +(iii) A is an iterated pyramid as defined in [SW12, Definition 3.4] over a face F ⪯ A for which +∆F is a polytope of the form of ∆A in cases (i) or (ii). +For (iii), Cd = CF ⊕ CF and rank(MA(β)) = rank(MF(βF)), where β = βF + βF for unique +βF ∈ CF and βF ∈ CF (see [SW12, Lemma 3.7]). Thus, it is enough to handle cases (i) and (ii). +For (i), we can assume for simplicity that e0 is the origin of R2. In this case, the vertices 2e1, 2e2 of +∆A are columns of A. Moreover, since ZA = Zd, at least two elements of the set {e1, e2, e1 + e2} +are also columns of A. If e1, e2 are columns of A, then e1+e2 is in NA and NA is normal. If e1+e2 +and one ei are columns of A, then NA has a one dimensional set of holes, given by ej + N{ej} +with j ̸= i. As this is a codimension-one lattice translate in Z2 and it is the only set of holes in +NA, it does not yield rank-jumping parameters by (3.1). +For (ii), since ZA = Z3, it must be that at least two of the vectors in +� 1 1 1 +1 0 1 +0 1 1 +� +must also be in A. An +exhaustive search reveals that none of these configurations yield any rank jumping parameters. +Thus in all cases where ∆A is the exceptional simplex, MA(β) admits no rank-jumping parame- +ters β. +□ +The final case to consider in this section is that ∆A is a Lawrence prism of heights b1, b2, b3 ∈ N. +Lemma 4.2. If ∆ is isomorphic to a three-dimensional Lawrence polytope of the form L(b1, b2, b3), +then +rank(MA(β)) < 2 · vol(A). +Proof. Since MA(β) is invariant under GL3(Z)-transformation and its rank is invariant under re- +ordering of the columns of A, we first note that if ∆ = ∆A is isomorphic to a Lawrence prism, then +we may assume for simplicity that ∆ = L(b1, b2, b3) as in §2, where e0 is the origin and {e1, e2, e3} +is the standard basis for R3. +If either b2 or b3 are zero, then A is a pyramid over a face of dimension 2 and this implies that +rank(MA(β)) ≤ vol(A) + 1; indeed, the pyramid construction does not increase rank [SW12], +and vol(A) + 1 is the maximal rank possible when d = 2 [CDD99]. Thus, we can assume that +b2, b3 ≥ 1. +If b2, b3 ≥ 1 but b1 = 0, then ∆ has four edges that contain the origin, each of volume 1 and +lattice index 1. Working from Theorem 3.2, there are six nontrivial potential order complexes + +6 +CHRISTINE BERKESCH AND MAR´IA-CRUZ FERN ´ANDEZ-FERN ´ANDEZ +J β +Λ to consider, noting that each β has a unique nontrivial Λ ∈ FA(β). For the possible J β +Λ with +�H0(J β +Λ ) ∼= Cm−1 for m > 1, jA(β) = jΛ(β) = m−1 ≤ 3. In the remaining cases, jA(β) ∈ {1, 2}. +It now follows that +rank(MA(β)) ≤ 3 + vol(A). +(4.1) +If vol(A) is 2 or 3, then ∆A is equal to the convex hull of the origin and the columns of +� 1 1 0 0 +0 0 1 1 +0 1 0 1 +� +or +� 1 1 0 0 +0 0 1 1 +0 2 0 1 +� +, respectively. In both cases, EA = ∅, so jA(β) = 0. Thus, if b1 = 0, b2 ≥ 1, b3 ≥ 1, +and jA(β) > 0, then vol(A) ≥ 4. Therefore by (4.1), rank(MA(β)) < 2 · vol(A) when b1 = 0. +We can assume for the rest of the proof that b1, b2, b3 ≥ 1. Thus, the edges of A are the lines +ℓ1 = Re1 ∩ A, ℓ2 = Re2 ∩ A, and ℓ3 = Re3 ∩ A. However, since the vertices e1 and e2 of ∆ are +necessarily columns of A, volZℓj(ℓj) = 1 for all j = 1, 2 and |Bβ +ℓj| ≤ 1 for all j = 1, 2 and all +β ∈ C3. +Again working from Theorem 3.2, there are seven possible order complexes J β +Λ for a given Λ ∈ +FA(β). We consider these options now. +Let v denote the normalized volume of the convex hull of A ∩ Re3 with the origin inside the lattice +spanned by Z(A ∩ Re3). If the maximal lattice translates under inclusion in Λ are the three rays +corresponding to ℓ1, ℓ2, and ℓ3, then jΛ(β) = 1 + v. If all maximal elements under inclusion in +Λ are facets, then Theorem 3.2 implies that jΛ(β) = 0. The remaining cases involve maximal +elements of FA(β) being a facet and a line, two lines, or one line. In each case, +jΛ(β) = +� +v +if (one of) the line(s) is ℓ3, +1 +if ℓ3 is not among the lines. +In all cases, the only additional Λ available in FA(β) come from translates of ℓ3. Such Λ will each +have jΛ(β) = v, and there are at most [Ze3 : Z(A ∩ Re3)] − 1 such Λ in FA(β). +Comparing the possible sums of jΛ(β) that are allowed in Theorem 3.2 when computing the rank +jump of MA(β) at β, it follows that +rank(MA(β)) − vol(A) ≤ 1 + [Ze3 : Z(A ∩ Re3)] · v += 1 + [Ze3 : Z(A ∩ Re3)] · volZ(A∩Re3)(conv({0, A ∩ Re3})) += 1 + b3. +Finally, we rearrange the inequality to compute the desired result when b1 + b2 ≥ 2, the last +remaining case: +rank(MA(β)) ≤ vol(A) + 1 + b3 +< vol(A) + 2 + b3 +≤ vol(A) + b1 + b2 + b3 += 2 · vol(A). +□ +5. UPPER BOUND FOR RANK IN DIMENSION THREE +In this section, we prove Theorem 1.2. First, we state a lemma shared with us by Christian Haase. + +7 +Lemma 5.1. Let ∆ ⊆ R3 be a convex lattice polytope and ℓ1, . . . , ℓr be the edges of ∆ that contain +a fixed vertex v. Then +r +� +j=1 +vol(ℓj) ≤ vol(∆) + 2. +(5.1) +Moreover, if equality holds in (5.1), then ∆ is a 3-simplex with at least one facet of normalized +volume one. +Proof. Since vol(ℓ) = |Z3 ∩ ℓ| − 1 for any edge ℓ, then by Theorem 2.1, +r +� +j=1 +vol(ℓj) + 1 ≤ |∆ ∩ Z3| = h∗ +1 + 4 ≤ vol(∆) + 3. +(5.2) +This yields the desired inequality. Moreover, if +h∗ +1 + 4 = vol(∆) + 3, +(5.3) +then h∗ +2 = h∗ +3 = 0, so ∆ has degree 1. Batyrev and Nill characterized all polytopes with h- +polynomial of degree one in [BN07, Theorem 2.5]. Within this classification, the only polytopes +for which the first inequality in (5.2) is an equality are precisely the simplices with at least one +facet of volume one. +□ +Proof of Theorem 1.2. By [BF22, Corollary 2.2], we may assume without loss of generality that +β ∈ EA is not simple. From Okuyama’s formula in Theorem 3.2, in the case d = 3, +rank(MA(β)) − vol(A) ≤ +�� +F +volZ3∩CF(F) +� +− 1, +(5.4) +where F runs over all one dimensional faces of A. +By way of contradiction, suppose that there is some A ∈ Z3×n and β ∈ C3 such that +rank(MA(β)) ≥ 2 · vol(A); +in particular, the rank jump of MA(β) at β would be at least vol(A). Then by (5.4), +vol(A) ≤ +�� +F +volZ3∩CF(F) +� +− 1, +(5.5) +where the summation runs over all edges F in ∆ that contain the origin. Combining this with (5.2) +yields +vol(A) + 2 ≤ +�� +F +volZ3∩CF(F) +� ++ 1 ≤ |∆ ∩ Z3| = h∗ +1 + 4 ≤ vol(A) + 3, +(5.6) +where again the summation runs over all edges F in ∆ that contain the origin. Comparing the +outer terms of (5.6), it follows that exactly two of the three inequalities present must be equalities. +We distinguish two cases, based on the third inequality. +First, consider the case in which the third inequality in (5.6) is an equality, so that h∗ +1 +1 = vol(A). +This implies by Theorem 2.1 that h∗ +2 = h∗ +3 = 0. Thus by [BN07, Theorem 2.5], ∆ is either an +(iterated) pyramid over the exceptional triangle or a Lawrence polytope. These cases are handled +in Lemmas 4.1 and 4.2, showing that the inequality (1.2) is strict in this case. + +8 +CHRISTINE BERKESCH AND MAR´IA-CRUZ FERN ´ANDEZ-FERN ´ANDEZ +Finally, we are left to consider the case that the third inequality in (5.6) is strict, so that the first +two inequalities are both equalities. Now (5.6) becomes +vol(∆) + 2 = +�� +F +volZ3∩CF(F) +� ++ 1 = |∆ ∩ Z3| = h∗ +1 + 4 < vol(∆) + 3. +Since the summation is over all edges F of ∆ with the origin as a vertex, the second equality +implies that all lattice points in ∆ lie on an edge of ∆ that has the origin as a vertex and ∆ has +no interior lattice points. Thus every edge of ∆ that does not contain the origin has volume 1. +We will show that no ∆ fitting this case admits a rank-jump higher than one. To begin, note that +h∗ +1 + 2 = vol(∆), so h∗ +2 + h∗ +3 = 1 since h∗ +1 + h∗ +2 + h∗ +3 + 1 = vol(∆). +If h∗ +2 = 0 and h∗ +3 = 1, then ∆ must have an interior lattice point by property (5) in Theorem 2.1, a +contradiction. Thus, we must be in the case that h∗ +2 = 1 and h∗ +3 = 0. Given that d = 3, it follows +that deg(∆) = 2. By [Tre10, Theorem 2], ∆ satisfies one of two possible cases. +First, ∆ could be isomorphic to the convex hull of the origin and the columns of conv +� 3 0 0 +0 3 0 +0 0 1 +� +in +R3, so that vol(∆) = 9 and |∆ ∩ Z3| = 11. In this case, ∆ is a pyramid over a face of dimension +two, so by [CDD99, SW12], +rank(MA(β)) ≤ vol(A) + 1 < 2 · vol(A). +Second, since ∆ is not isomorphic to the convex hull of the origin and the columns of conv +� 3 0 0 +0 3 0 +0 0 1 +� +in R3, then [Tre10, Theorem 2] implies that the following equivalent statements hold: +(1) vol(∆) ≤ 4 · (|(2∆)◦ ∩ Z3| + 1), +(2) |∆ ∩ Z3| ≤ 3 · |(2∆)◦ ∩ Z3| + 7, and +(3) |∆ ∩ Z3| ≤ 3 +4 · vol(∆) + 4. +Also, by [Tre10, Lemma 9], vol(∆) = |∆∩Z3|+|(2∆)◦∩Z3|−3. Combining this with |∆∩Z3| = +vol(∆) + 2 from (5.6) implies that |(2∆)◦ ∩ Z3| = 1. Thus +|∆ ∩ Z3| ≤ 3|(2∆)◦ ∩ Z3| + 7 = 10 +and +vol(∆) = |∆ ∩ Z3| − 2 ≤ 8. +Now the Ehrhart polynomial of ∆, +g∆(t) = g3t3 + g2t2 + g1t + 1, +satisfies the following constraints. First, |∆ ∩ Z3| = vol(∆) − 2, so +g3 + 2 = vol(∆) + 2 = b = g∆(1) = g3 + g2 + g1 + 1, +which implies that g1 = 1 − g2. Next, since ∆ has no interior lattice points, by Ehrhart reciprocity, +which states that g∆◦(t) = (−1)dg∆(−t), +0 = |∆◦ ∩ Z3| = −g∆(−1) = g3 − g2 + g1 − 1. +Finally, since i = |(2∆)◦ ∩Z3| = 1, 1 = −g∆(−2) = 8g3 −4g2 +2g1 −1. However, the equations +g1 = 1 − g2, +0 = g3 − g2 + g1 − 1, +and +1 = 8g3 − 4g2 + 2g1 − 1 +are incompatible, so there is no ∆ that fits this case. Having exhausted all possibilities, we conclude +that if the third inequality in (5.6) is strict, then rank(MA(β)) < 2 · vol(A). +Finally, the sequence of examples constructed in [BF22, Section 3] proves that the upper bound +in (1.2) is sharp. +□ + +9 +REFERENCES +[Ado94] +Alan Adolphson, Hypergeometric functions and rings generated by monomials, Duke Math. J. 73 (1994), +269–290. 1 +[BN07] +Victor Batyrev and Benjamin Nill, Multiples of lattice polytopes without interior lattice points, Moscow +Mathematical Journal 7 (2007), no. 2, 195–207. 2, 3, 7 +[Ber11] +Christine Berkesch, The rank of a hypergeometric system, Compos. Math. 147 (2011), no. 1, 284–318. 1, +2, 4 +[BFM18] +Christine Berkesch, Jens Forg˚ard, and Laura Felicia Matusevich, On the parametric behavior of A- +hypergeometric functions, Trans. Amer. Math. Soc. 370 (2018), no. 6, pp. 4089–4109. 2 +[BF22] +Christine Berkesch and Mar´ıa-Cruz Fern´andez-Fern´andez, On the rank of an A-hypergeometric D-module +versus the normalized volume of A, Bull. London Math. Society, 54 (2022), no. 1, pp. 182–192. 2, 7, 8 +[CDD99] +Eduardo Cattani, Carlos D’Andrea, and Alicia Dickenstein, The A-hypergeometric system associated with +a monomial curve, Duke Math. J. 99 (1999), no. 2, 179-207. 1, 5, 8 +[Ehr62] +Eug`ene Ehrhart, Sur les poly`edres rationnels homoth´etiques `a n dimensions, C. R. Acad. Sci. Paris 254 +(1962), 616–618. 2 +[Fer13] +Mar´ıa Cruz Fern´andez Fern´andez, Exponential growth of rank jumps for A-hypergeometric systems, Rev. +Mat. Iberoam. 29 (2013), no. 4, 1397–1404. 2 +[GGZ87] +I. M. Gel′fand, M. I. Graev, and A. V. Zelevinski˘ı, Holonomic systems of equations and series of hyperge- +ometric type, Dokl. Akad. Nauk SSSR 295 (1987), no. 1, 14–19. 1 +[GKZ89] +I. M. Gel′fand, A. V. Zelevinski˘ı, and M. M. Kapranov, Hypergeometric functions and toric varieties, +Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 12–26. Correction in ibid, 27 (1993), no. 4, 91. 1 +[Hoc72] +M. Hochster, Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes, +Ann. of Math. (2) 96 (1972), 318–337. +[MMW05] Laura Felicia Matusevich, Ezra Miller, and Uli Walther, Homological methods for hypergeometric fami- +lies, J. Amer. Math. Soc. 18 (2005), no. 4, 919–941. 1 +[MW07] +L. F. Matusevich, U. 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A 117 (2010), no. 3, 354–360. 8 +SCHOOL OF MATHEMATICS, UNIVERSITY OF MINNESOTA. +Email address: cberkesc@umn.edu +DEPARTAMENTO DE ´ALGEBRA, UNIVERSIDAD DE SEVILLA. +Email address: mcferfer@algebra.us.es + diff --git a/j9E3T4oBgHgl3EQfhQpJ/content/tmp_files/load_file.txt b/j9E3T4oBgHgl3EQfhQpJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ae59eb38180c16143811248c64e7b46cfdf1ef0 --- /dev/null +++ b/j9E3T4oBgHgl3EQfhQpJ/content/tmp_files/load_file.txt @@ -0,0 +1,406 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf,len=405 +page_content='A SHARP BOUND FOR HYPERGEOMETRIC RANK IN DIMENSION THREE CHRISTINE BERKESCH AND MAR´IA-CRUZ FERN ´ANDEZ-FERN ´ANDEZ ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' We provide a sharp upper bound on the quotient of the rank of an A-hypergeometric system with a three-dimensional torus action by the normalized volume of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' in this case, the upper bound is two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' INTRODUCTION A-hypergeometric D-modules, also known as GKZ-systems, were introduced in [GGZ87, GKZ89] to generalize classical hypergeometric equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' These systems of linear partial differential equa- tions in several complex variables are determined by a matrix A = (ai,j) ∈ Zd×n with columns ak ∈ Zd such that ZA = Za1 + · · · + Zan = Zd and a parameter vector β ∈ Cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' We assume that A is pointed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=', that all columns in A lie in a single linear halfspace of Rd that does not contain the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Let x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , xn be coordinates on Cn, with corresponding partial derivatives ∂1, ∂2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , ∂n, so that the Weyl algebra D on Cn is generated by x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , xn, ∂1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , ∂n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Let IA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='.= ⟨∂u − ∂v | u, v ∈ Nn, Au = Av⟩ ⊆ C[∂1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , ∂n] denote the toric ideal of A, and let Ei .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='.= �n j=1 ai,jxj∂j be the ith Euler operator of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' The A-hypergeometric D-module with parameter β ∈ Cd is the left D-module MA(β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='.= D/D · ⟨IA, E1 − β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , Ed − βd⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' For any choice of A and β, the module MA(β) is holonomic [GGZ87, Ado94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Consequently, the dimension of the space of germs of holomorphic solutions of MA(β) at a nonsingular point, also known as its (holonomic) rank, is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' When β ∈ Cd is generic, the rank of MA(β) is equal to the normalized volume vol(A) of the matrix A [GKZ89, Ado94] inside the lattice Zd, which is the Euclidean volume of the convex hull ∆A ⊆ Rd of the columns of A and the origin divided by d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' but in general, this is only a lower bound [SST00, MMW05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' The set E(A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='.= {β ∈ Cd | rank(MA(β)) > vol(A)} is called the exceptional arrangement of A, which is an affine subspace arrangement of codimen- sion at least two that is closely related to the local cohomology modules of the toric ring C[∂]/IA [MMW05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' A parameter β ∈ E(A) is called a rank jumping parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Combinatorial formulas to compute the rank of MA(β) in terms of the ranking lattices Eβ of A at β appear in [Oku06, Ber11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Unfortunately, the presence of alternating signs in these formulas do not yield a strong upper bound for the rank of MA(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' One exception is the case d = 2, where it was shown in [CDD99] that 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' 13N10, 32C38, 33C70, 14M25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' CB was partially supported by NSF Grant DMS 2001101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' MCFF was partially supported by projects PID2020-117843GB-I00 (Ministerio de Ciencia e Innovaci´on, Spain), P20-01056 and US-1262169 (Consejer´ıa de Econom´ıa, Conocimiento, Empresas y Universidad, Junta de Andaluc´ıa and FEDER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='04569v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='AG] 11 Jan 2023 2 CHRISTINE BERKESCH AND MAR´IA-CRUZ FERN ´ANDEZ-FERN ´ANDEZ rank(MA(β)) ≤ vol(A) + 1 is a sharp bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' For arbitrary d, A, and β, previously known upper bounds for the holonomic rank of MA(β) are as follows: rank(MA(β)) ≤ � 4d · vol(A) if IA is homogeneous [SST00], 4d+1 · vol(A) otherwise [BFM18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' However, it is believed that these upper bounds are much too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' In [BF22], we showed that when β is generic among rank-jumping parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=', simple in [Ber11]), then rank(MA(β)) vol(A) ≤ (d − 1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1) and we showed that this bound is tight by constructing a sequence of examples for which the ratio rank(MA(β))/vol(A) tends to d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Still, this case neglects the rank-jumping parameters β for which the highest jumps are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' In fact, there are families of examples for which the ratio rank(MA(β))/vol(A) grow exponentially with d [Fer13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' In this note, we show that when d = 3, the bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1) holds for all parameters β, with a strict inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' There is a strict sharp inequality for all A ∈ Z3×n and β ∈ C3: rank(MA(β)) vol(A) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2) Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' We begin with results in Ehrhart theory in §2 and rank jumps in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Next, in §4, we consider rank jumps in the case that ∆A has degree one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Finally, the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2 is completed in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' We thank Christian Haase for stating and proving Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1, providing the catalyst for this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' We are also grateful to Laura Felicia Matusevich, Vic Reiner, and Uli Walther for helpful conversations related to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' PRELIMINARIES ON EHRHART THEORY Let ∆ ⊆ Rd be a lattice polytope, that is, the convex hull in Rd of a finite set of points in Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' We denote by |∆ ∩ Zd| the cardinality of the set of lattice points in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' The function g∆(t) : N −→ N defined by g∆(k) := |k∆ ∩ Zd| counts the number of lattice points in the k-fold dilatation of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Ehrhart proved that this function is a polynomial in k, which is now called the Ehrhart polynomial of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Moreover, he also proved that when the polytope ∆ ⊆ Rd is d-dimensional the degree of g∆(k) is d and its leading coefficient is equal to the normalized volume of ∆ with respect to Zd, that is, the Euclidean volume of ∆ divided by d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' [Ehr62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' The Ehrhart series of ∆ is the generating function E∆(t) := � k≥0 g∆(k)tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' The following result is well known (see, for example, [BN07] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' For a lattice polytope ∆ ⊆ Rd, there exists h∗(t) = 1 + h∗ 1t + · · · + h∗ ktk, called the h-polynomial of ∆ such that the following properties hold: 3 (1) E∆(t) = h∗(t)/(1 − t)d+1, (2) h∗ j ∈ Z≥0 for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , k, (3) vol(∆) = 1 + h∗ 1 + · · · + h∗ k, (4) h∗ 1 = |∆ ∩ Zd| − d − 1, and (5) the leading coefficient h∗ k = |(d + 1 − k)∆◦ ∩ Zd|, where ∆◦ denotes the interior of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' The degree of ∆, denoted deg(∆), is defined to be the degree of the h-polynomial of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' An interesting fact is that deg(∆) = min{j ∈ N | |i∆◦ ∩ Zd| = ∅, ∀1 ≤ i ≤ d − j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' It follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1 that lattice polytopes of degree zero are basic simplices, that is, lattice polytopes whose vertices form an affine lattice basis of Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Batyrev and Nill classified those polytopes having degree one [BN07, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' they proved that deg(∆) ≤ 1 if and only if ∆ is an exceptional simplex or a Lawrence prism, which we now define.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' First, if F is a face of ∆ such that |∆ ∩ Zd| − |F ∩ Zd| − codim(F) = 0, then we say that ∆ is an iterated pyramid over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' An exceptional triangle is a 2-dimensional basic simplex multiplied by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' An exceptional simplex is a simplex that is the (d−2)-fold pyramid over an exceptional triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' In other words, ∆ is the convex hull in Rd of e0, e0 + 2(e1 − e0), e0 + 2(e2 − e0), e3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , ed where e0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , ed is some affine lattice basis of Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Finally, a Lawrence prism of heights b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , bd ≥ 0, denoted by L(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , bs), is the convex hull in Rd of e0, e0 + b1(ed − e0), e1, e1 + b2(ed − e0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , ed−1, ed−1 + bd(ed − e0), where e0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , ed is some affine lattice basis of Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' A Lawrence prism L(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , bd) has degree one if b1 + · · · + bd ≥ 2 [BN07, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Otherwise, it is a basic simplex of degree zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' The normalized volume of the Lawrence prism L(b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , bd) is b1 + · · · + bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' PRELIMINARIES ON RANK JUMPS In this section, we return to the setting of a pointed matrix A ∈ Zd×n with ZA = Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' We will soon restrict to the case d = 3, but it is not needed for the following definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Recall that ∆A ⊆ Rd denotes the convex hull of the columns of A and the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' The set of columns of A will be also denoted by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' A submatrix F of A (or a subset of its set of columns) is called a face of A, denoted F ⪯ A, if ∆F is a face of the polytope ∆A ⊆ Rd and A ∩ ∆F = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' In particular, the empty set and A are faces of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' For a face F ⪯ A, consider the union of the lattice translates Eβ F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='.= � Zd ∩ (β + CF) � ∖ (NA + ZF) = � b∈Bβ F (b + ZF), where Bβ F ⊆ Zd is a set of lattice translate representatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Since |Bβ F| is the number of translates of ZF appearing in Eβ F, it is by definition equal to the difference between [Zd ∩ QF : ZF] and the number of translates of ZF along β + CF that are contained in NA + ZF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' 4 CHRISTINE BERKESCH AND MAR´IA-CRUZ FERN ´ANDEZ-FERN ´ANDEZ Given the set J (β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='.= {(F, b) | F ⪯ A, b ∈ Bβ F}, the ranking lattices of A at β are defined to be Eβ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='.= � (F,b)∈J (β) (b + ZF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Note that the ranking lattices of A at β are precisely the union of those sets (b + ZF) contained in Zd \\ NA such that β ∈ (b + CF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' This is closely related to the set of holes of the affine semigroup NA, namely the set (Zd ∩ R≥0A) \\ NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' A rank jumping parameter β is simple (for a face G ⪯ A) if the set of maximal pairs (F, b) in J (β) with respect to inclusion on b + ZF all correspond to a unique face G ⪯ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' The main result in [Ber11] states how the rank of MA(β) can be computed from the combinatorics of Eβ and ∆A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' An explicit formula for the rank is given when the rank jumping parameter β is simple for a face G ⪯ A (see [Oku06] for this particular case);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' in this case, rank(MA(β)) = vol(A) + |Bβ G| · (codim(G) − 1) · volZG(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1) Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' It is shown in [Ber11, Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='21] that if β ∈ Cd is such that the ranking lattices of A at β involve only two faces, F1 and F2, then the rank jump of M at β is rank(MA(β)) − vol(A) = 2 � i=1 � |Bβ Fi| · [codim(Fi) − 1] · vol(Fi) � + |Bβ G| · Cβ · vol(G), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2) where G = F1 ∩ F2 and the constant Cβ is given by Cβ = �codim(G) 2 � − codim(G) + 1 − �codim(F1) 2 � − �codim(F2) 2 � + �codim(CF1 + CF2) 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' When d = 3, Okuyama [Oku06] provided a formula for the rank of MA(β), as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' [Oku06, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='6] Let d = 3 and β ∈ C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Define an equivalence relation on J (β) by (F, b) ∼ (G, c) if and only if (b + ZF) ∩ (c + ZG) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Let FA(β) denote the set of equivalence classes of J (β) under ∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' For each Λ ∈ FA(β), let J β Λ denote the order complex on the face poset of faces F with (F, b) ∈ Λ for some b ∈ Z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Then the rank jump of A at β ∈ Cd is given by rank(MA(β)) − vol(A) = � Λ∈FA(β) jΛ(β), where jΛ(β) := � � � � � � � � � � � � � � rays F∈J β Λ (vol(F) − 1) + m − 1 if �Hp(J β Λ ) ∼= 0 for p ̸= 0, �H0(J β Λ ) ∼= Cm−1 with m > 1, vol(F) if �Hp(J β Λ ) ∼= 0 for all p, so J β Λ consists of a ray F, 2 if �H−1(J β Λ ) ∼= C, �Hp(J β Λ ) ∼= 0 for p ̸= −1, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' RANK JUMPS FOR MA(β) WHEN ∆A HAS DEGREE ONE Recall that we denote the convex hull of the columns of A and the origin by ∆A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' When ∆A is a lattice polytope of degree one, it is an exceptional simplex or a Lawrence prism of heights b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , bd ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' In this section, we compute the possible rank jumps for MA(β) when ∆A is the former or if it is the latter and d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' If ∆ = ∆A is an exceptional simplex, then MA(β) has no rank-jumping parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' If ∆A is an iterated pyramid over an exceptional triangle, then one of the following cases holds: (i) ∆A is an exceptional triangle, (ii) ∆A is (up to rigid transformation) the convex hull of the origin in R3 and � 1 1 1 0 2 0 0 0 2 � , (iii) A is an iterated pyramid as defined in [SW12, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='4] over a face F ⪯ A for which ∆F is a polytope of the form of ∆A in cases (i) or (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' For (iii), Cd = CF ⊕ CF and rank(MA(β)) = rank(MF(βF)), where β = βF + βF for unique βF ∈ CF and βF ∈ CF (see [SW12, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Thus, it is enough to handle cases (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' For (i), we can assume for simplicity that e0 is the origin of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' In this case, the vertices 2e1, 2e2 of ∆A are columns of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Moreover, since ZA = Zd, at least two elements of the set {e1, e2, e1 + e2} are also columns of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' If e1, e2 are columns of A, then e1+e2 is in NA and NA is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' If e1+e2 and one ei are columns of A, then NA has a one dimensional set of holes, given by ej + N{ej} with j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' As this is a codimension-one lattice translate in Z2 and it is the only set of holes in NA, it does not yield rank-jumping parameters by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' For (ii), since ZA = Z3, it must be that at least two of the vectors in � 1 1 1 1 0 1 0 1 1 � must also be in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' An exhaustive search reveals that none of these configurations yield any rank jumping parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Thus in all cases where ∆A is the exceptional simplex, MA(β) admits no rank-jumping parame- ters β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' □ The final case to consider in this section is that ∆A is a Lawrence prism of heights b1, b2, b3 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' If ∆ is isomorphic to a three-dimensional Lawrence polytope of the form L(b1, b2, b3), then rank(MA(β)) < 2 · vol(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Since MA(β) is invariant under GL3(Z)-transformation and its rank is invariant under re- ordering of the columns of A, we first note that if ∆ = ∆A is isomorphic to a Lawrence prism, then we may assume for simplicity that ∆ = L(b1, b2, b3) as in §2, where e0 is the origin and {e1, e2, e3} is the standard basis for R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' If either b2 or b3 are zero, then A is a pyramid over a face of dimension 2 and this implies that rank(MA(β)) ≤ vol(A) + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' indeed, the pyramid construction does not increase rank [SW12], and vol(A) + 1 is the maximal rank possible when d = 2 [CDD99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Thus, we can assume that b2, b3 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' If b2, b3 ≥ 1 but b1 = 0, then ∆ has four edges that contain the origin, each of volume 1 and lattice index 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Working from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2, there are six nontrivial potential order complexes 6 CHRISTINE BERKESCH AND MAR´IA-CRUZ FERN ´ANDEZ-FERN ´ANDEZ J β Λ to consider, noting that each β has a unique nontrivial Λ ∈ FA(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' For the possible J β Λ with �H0(J β Λ ) ∼= Cm−1 for m > 1, jA(β) = jΛ(β) = m−1 ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' In the remaining cases, jA(β) ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' It now follows that rank(MA(β)) ≤ 3 + vol(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1) If vol(A) is 2 or 3, then ∆A is equal to the convex hull of the origin and the columns of � 1 1 0 0 0 0 1 1 0 1 0 1 � or � 1 1 0 0 0 0 1 1 0 2 0 1 � , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' In both cases, EA = ∅, so jA(β) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Thus, if b1 = 0, b2 ≥ 1, b3 ≥ 1, and jA(β) > 0, then vol(A) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Therefore by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1), rank(MA(β)) < 2 · vol(A) when b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' We can assume for the rest of the proof that b1, b2, b3 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Thus, the edges of A are the lines ℓ1 = Re1 ∩ A, ℓ2 = Re2 ∩ A, and ℓ3 = Re3 ∩ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' However, since the vertices e1 and e2 of ∆ are necessarily columns of A, volZℓj(ℓj) = 1 for all j = 1, 2 and |Bβ ℓj| ≤ 1 for all j = 1, 2 and all β ∈ C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Again working from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2, there are seven possible order complexes J β Λ for a given Λ ∈ FA(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' We consider these options now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Let v denote the normalized volume of the convex hull of A ∩ Re3 with the origin inside the lattice spanned by Z(A ∩ Re3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' If the maximal lattice translates under inclusion in Λ are the three rays corresponding to ℓ1, ℓ2, and ℓ3, then jΛ(β) = 1 + v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' If all maximal elements under inclusion in Λ are facets, then Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2 implies that jΛ(β) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' The remaining cases involve maximal elements of FA(β) being a facet and a line, two lines, or one line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' In each case, jΛ(β) = � v if (one of) the line(s) is ℓ3, 1 if ℓ3 is not among the lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' In all cases, the only additional Λ available in FA(β) come from translates of ℓ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Such Λ will each have jΛ(β) = v, and there are at most [Ze3 : Z(A ∩ Re3)] − 1 such Λ in FA(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Comparing the possible sums of jΛ(β) that are allowed in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2 when computing the rank jump of MA(β) at β, it follows that rank(MA(β)) − vol(A) ≤ 1 + [Ze3 : Z(A ∩ Re3)] · v = 1 + [Ze3 : Z(A ∩ Re3)] · volZ(A∩Re3)(conv({0, A ∩ Re3})) = 1 + b3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Finally, we rearrange the inequality to compute the desired result when b1 + b2 ≥ 2, the last remaining case: rank(MA(β)) ≤ vol(A) + 1 + b3 < vol(A) + 2 + b3 ≤ vol(A) + b1 + b2 + b3 = 2 · vol(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' UPPER BOUND FOR RANK IN DIMENSION THREE In this section, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' First, we state a lemma shared with us by Christian Haase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' 7 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Let ∆ ⊆ R3 be a convex lattice polytope and ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' , ℓr be the edges of ∆ that contain a fixed vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Then r � j=1 vol(ℓj) ≤ vol(∆) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1) Moreover, if equality holds in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1), then ∆ is a 3-simplex with at least one facet of normalized volume one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Since vol(ℓ) = |Z3 ∩ ℓ| − 1 for any edge ℓ, then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1, r � j=1 vol(ℓj) + 1 ≤ |∆ ∩ Z3| = h∗ 1 + 4 ≤ vol(∆) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2) This yields the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Moreover, if h∗ 1 + 4 = vol(∆) + 3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='3) then h∗ 2 = h∗ 3 = 0, so ∆ has degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Batyrev and Nill characterized all polytopes with h- polynomial of degree one in [BN07, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Within this classification, the only polytopes for which the first inequality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2) is an equality are precisely the simplices with at least one facet of volume one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' By [BF22, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2], we may assume without loss of generality that β ∈ EA is not simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' From Okuyama’s formula in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2, in the case d = 3, rank(MA(β)) − vol(A) ≤ �� F volZ3∩CF(F) � − 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='4) where F runs over all one dimensional faces of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' By way of contradiction, suppose that there is some A ∈ Z3×n and β ∈ C3 such that rank(MA(β)) ≥ 2 · vol(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' in particular, the rank jump of MA(β) at β would be at least vol(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='4), vol(A) ≤ �� F volZ3∩CF(F) � − 1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='5) where the summation runs over all edges F in ∆ that contain the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Combining this with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2) yields vol(A) + 2 ≤ �� F volZ3∩CF(F) � + 1 ≤ |∆ ∩ Z3| = h∗ 1 + 4 ≤ vol(A) + 3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='6) where again the summation runs over all edges F in ∆ that contain the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Comparing the outer terms of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='6), it follows that exactly two of the three inequalities present must be equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' We distinguish two cases, based on the third inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' First, consider the case in which the third inequality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='6) is an equality, so that h∗ 1 +1 = vol(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' This implies by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1 that h∗ 2 = h∗ 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Thus by [BN07, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='5], ∆ is either an (iterated) pyramid over the exceptional triangle or a Lawrence polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' These cases are handled in Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2, showing that the inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2) is strict in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' 8 CHRISTINE BERKESCH AND MAR´IA-CRUZ FERN ´ANDEZ-FERN ´ANDEZ Finally, we are left to consider the case that the third inequality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='6) is strict, so that the first two inequalities are both equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Now (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='6) becomes vol(∆) + 2 = �� F volZ3∩CF(F) � + 1 = |∆ ∩ Z3| = h∗ 1 + 4 < vol(∆) + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Since the summation is over all edges F of ∆ with the origin as a vertex, the second equality implies that all lattice points in ∆ lie on an edge of ∆ that has the origin as a vertex and ∆ has no interior lattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Thus every edge of ∆ that does not contain the origin has volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' We will show that no ∆ fitting this case admits a rank-jump higher than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' To begin, note that h∗ 1 + 2 = vol(∆), so h∗ 2 + h∗ 3 = 1 since h∗ 1 + h∗ 2 + h∗ 3 + 1 = vol(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' If h∗ 2 = 0 and h∗ 3 = 1, then ∆ must have an interior lattice point by property (5) in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Thus, we must be in the case that h∗ 2 = 1 and h∗ 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Given that d = 3, it follows that deg(∆) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' By [Tre10, Theorem 2], ∆ satisfies one of two possible cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' First, ∆ could be isomorphic to the convex hull of the origin and the columns of conv � 3 0 0 0 3 0 0 0 1 � in R3, so that vol(∆) = 9 and |∆ ∩ Z3| = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' In this case, ∆ is a pyramid over a face of dimension two, so by [CDD99, SW12], rank(MA(β)) ≤ vol(A) + 1 < 2 · vol(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Second, since ∆ is not isomorphic to the convex hull of the origin and the columns of conv � 3 0 0 0 3 0 0 0 1 � in R3, then [Tre10, Theorem 2] implies that the following equivalent statements hold: (1) vol(∆) ≤ 4 · (|(2∆)◦ ∩ Z3| + 1), (2) |∆ ∩ Z3| ≤ 3 · |(2∆)◦ ∩ Z3| + 7, and (3) |∆ ∩ Z3| ≤ 3 4 · vol(∆) + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Also, by [Tre10, Lemma 9], vol(∆) = |∆∩Z3|+|(2∆)◦∩Z3|−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Combining this with |∆∩Z3| = vol(∆) + 2 from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='6) implies that |(2∆)◦ ∩ Z3| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Thus |∆ ∩ Z3| ≤ 3|(2∆)◦ ∩ Z3| + 7 = 10 and vol(∆) = |∆ ∩ Z3| − 2 ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Now the Ehrhart polynomial of ∆, g∆(t) = g3t3 + g2t2 + g1t + 1, satisfies the following constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' First, |∆ ∩ Z3| = vol(∆) − 2, so g3 + 2 = vol(∆) + 2 = b = g∆(1) = g3 + g2 + g1 + 1, which implies that g1 = 1 − g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Next, since ∆ has no interior lattice points, by Ehrhart reciprocity, which states that g∆◦(t) = (−1)dg∆(−t), 0 = |∆◦ ∩ Z3| = −g∆(−1) = g3 − g2 + g1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Finally, since i = |(2∆)◦ ∩Z3| = 1, 1 = −g∆(−2) = 8g3 −4g2 +2g1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' However, the equations g1 = 1 − g2, 0 = g3 − g2 + g1 − 1, and 1 = 8g3 − 4g2 + 2g1 − 1 are incompatible, so there is no ∆ that fits this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Having exhausted all possibilities, we conclude that if the third inequality in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='6) is strict, then rank(MA(β)) < 2 · vol(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Finally, the sequence of examples constructed in [BF22, Section 3] proves that the upper bound in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='2) is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' □ 9 REFERENCES [Ado94] Alan Adolphson, Hypergeometric functions and rings generated by monomials, Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' 73 (1994), 269–290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' 1 [BN07] Victor Batyrev and Benjamin Nill, Multiples of lattice polytopes without interior lattice points, Moscow Mathematical Journal 7 (2007), no.' metadata={'source': 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Algebra Number Theory 6 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' 3, 527–537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' 5, 8 [Tre10] Jason Treutlein, Lattice polytopes of degree 2, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' A 117 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' 3, 354–360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' 8 SCHOOL OF MATHEMATICS, UNIVERSITY OF MINNESOTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Email address: cberkesc@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='edu DEPARTAMENTO DE ´ALGEBRA, UNIVERSIDAD DE SEVILLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content=' Email address: mcferfer@algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} +page_content='es' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E3T4oBgHgl3EQfhQpJ/content/2301.04569v1.pdf'} diff --git a/jNAyT4oBgHgl3EQfX_d5/content/tmp_files/2301.00194v1.pdf.txt b/jNAyT4oBgHgl3EQfX_d5/content/tmp_files/2301.00194v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ea1b7e872db0a8b3f91747c6a081ed7bff2e57d5 --- /dev/null +++ b/jNAyT4oBgHgl3EQfX_d5/content/tmp_files/2301.00194v1.pdf.txt @@ -0,0 +1,1448 @@ +Chordal graphs with bounded tree-width +Jordi Castellv´ı +Michael Drmota +Marc Noy +Cl´ement Requil´e +January 3, 2023 +Abstract +Given t ≥ 2 and 0 ≤ k ≤ t, we prove that the number of labelled k-connected chordal graphs +with n vertices and tree-width at most t is asymptotically cn−5/2γnn!, as n → ∞, for some +constants c, γ > 0 depending on t and k. Additionally, we show that the number of i-cliques +(2 ≤ i ≤ t) in a uniform random k-connected chordal graph with tree-width at most t is normally +distributed as n → ∞. +The asymptotic enumeration of graphs of tree-width at most t is wide open for t ≥ 3. To +the best of our knowledge, this is the first non-trivial class of graphs with bounded tree-width +where the asymptotic counting problem is solved. Our starting point is the work of Wormald +[Counting Labelled Chordal Graphs, Graphs and Combinatorics (1985)], were an algorithm is +developed to obtain the exact number of labelled chordal graphs on n vertices. +1 +Introduction +Tree-width is a fundamental parameter in structural and algorithmic graph theory, as illustrated +for instance in [9]. It can be defined in terms of tree-decompositions or equivalently in terms of +k-trees. A k-tree is defined recursively as either a complete graph on k + 1 vertices or a graph +obtained by adjoining a new vertex adjacent to a k-clique of a k-tree. The tree-width of a graph +Γ is the minimum k such that Γ is a subgraph of a k-tree. In particular, k-trees are the maximal +graphs with tree-width at most k. The number of k-trees on n labelled vertices was independently +shown [3, 18] to be +�n +k +� +(k(n − k) + 1)n−k−2 = +1 +√ +2π k! kk+2 n−5/2 (ek)n n! (1 + o(1)), +(1) +where the estimate holds for k fixed and n → ∞. However, there are relatively few results on the +enumeration of graphs of given tree-width or on properties of random graphs with given tree-width. +Graphs of tree-width one are forests (acyclic graphs) and their enumeration is a classical result, +while graphs of tree-width at most two are series-parallel graphs and were first counted in [6]. The +problem of counting graphs of tree-width three is still open. From now on, we will use t to denote +the tree-width while k will denote the connectivity of a graph. All graphs we consider are labelled. +Given that tree-width is non-increasing under taking minors, the class of graphs with tree-width +at most t is ‘small’ when t is fixed, in the sense that the number gn,t of labelled graphs with n +vertices and tree-wdith at most t grows at most like cnn! for some c > 0 depending on t (see +[19, 14]). The best bounds known for gn,t are, up to lower order terms, +�2ttn +log t +�n +≤ gn,t ≤ (2ttn)n. +1 +arXiv:2301.00194v1 [math.CO] 31 Dec 2022 + +The upper bound follows by considering all possible subgraphs of t-trees, and the lower bound uses +a suitable construction developed in [2]. In the present work we determine the asymptotic number +of labelled chordal graphs with tree-width at most t, following the approach in [15] and [12], and +based on the analysis of systems of equations satisfied by generating functions. +A graph is chordal if every cycle of length greater than three contains at least one chord, that +is, an edge connecting non-consecutive vertices of the cycle. Chordal graphs have been extensively +studied in structural graph theory and graph algorithms (see for instance [16]), but not so much +from the point of view of enumeration. Wormald [21] used generating functions to develop a method +for finding the exact number of chordal graphs with n vertices for a given value of n. It is based +on decomposing chordal graphs into k-connected components for each k ≥ 1. As remarked in [21], +it is difficult to define the k-connected components of arbitrary graphs for k > 3, but for chordal +graphs they are well defined. +Wormald introduces generating functions Ck(x) for k-connected +chordal graphs and finds an equation linking Ck(x) and Ck+1(x) reflecting the decomposition into +k-connected components. This is precisely the starting point of our work. +An important parameter in [21] is the size w of the largest clique. For chordal graphs one can +show that the tree-width is equal to w − 1, hence bounding the tree-width by t is the same as +bounding w by t + 1. The parameter w also plays a substantial rˆole in showing that almost all +chordal graphs are split graphs [4], which in turn implies that the number of chordal graphs with +n vertices is of order 2n2/4(1 + o(1)). +For fixed n, t ≥ 1 and 0 ≤ k ≤ t, let Gt,k,n be the set of k-connected chordal graphs with n +labelled vertices and tree-width at most t. Our two main results are the following. +Theorem 1.1. For t ≥ 1 and 0 ≤ k ≤ t, there exist constants ct,k > 0 and γt,k ∈ (0, 1) such that +|Gt,k,n| = ct,k n−5/2 γn +t,k n! (1 + o(1)) +as n → ∞. +Remark that by setting t = k in Theorem 1.1 one recovers the general form of the asymptotic +estimate of the number of k-trees (1). The fact that γk,k = ek is reproven in Section 3. In principle, +for fixed t and k the constants ct,k and γt,k can be computed, at least approximately. Table 1 in +Section 4 displays approximations of 1/γt,k for 1 ≤ k ≤ t ≤ 7. +Theorem 1.2. Let t ≥ 1, 0 ≤ k ≤ t. For i ∈ {2, . . . , t} let Xn,i denote the number of i-cliques in +a uniform random graph in Gt,k,n, and set Xn = (Xn,2, . . . , Xn,t). Then Xn satisfies a multivariate +central limit theorem, that is, as n → ∞ we have +1 +√n (Xn − E Xn) d→ N(0, Σ), +with +E Xn ∼ αn +and +Cov Xn ∼ Σn, +and where α is a (t−1)-dimensional vector of positive numbers and Σ is a (t−1)×(t−1)-dimensional +positive semi-definite matrix. +Let us mention that more structural asymptotic results can be expected. Notably, the class of +chordal graphs with tree-width at most t is subcritical in the sense of [13], as further discussed in +the concluding Section 4. It follows from [20] that the uniform random connected chordal graph +with tree-width at most t with distances rescaled by 1/√n admits the Continuum Random Tree +(CRT) [1] as a scaling limit, multiplied by a constant that depends on t. +The proofs of Theorems 1.1 and 1.2 are based on a recursive decomposition of chordal graphs +translated in the combinatorial language of generating functions, the so-called symbolic method [15], +2 + +into a system of functional equations explicited in Section 2 and analysed in Section 3 by means of +analytic methods [15]. More precisely, in Section 2.1 we show the unicity of the decomposition of +chordal graphs into their k-connected components. This is translated in Section 2.2 into a system of +funtional equations satisfied by exponential generating functions encoding the numbers of graphs +in each class. In Section 2.3 we prove an alternative encoding of an integral operator, which is +instrumental for computing the numerical values in Table 1. The subsequent asymptotic analysis +is rather delicate as it involves singularity functions depending on several variables. Our notions +of proper and fully movable singularity functions are key ingredients, and are defined along several +technical notions in Section 3.1. Some useful propeties are proven in Section 3.2, before embarking +in the proofs of the main results in Section 3.3. +2 +Decomposition of chordal graphs +All graphs considered in this work will be simple and labelled, that is with vertex-set [n]. Let Γ be +a graph and k ≥ 1. A k-separator of Γ is a subset of k vertices whose removal disconnects Γ. The +graph Γ is said to be k-connected if it contains no i-separator for i ∈ [k − 1]. Notice that with this +definition we consider the complete graph on k vertices to be k-connected, for any k ≥ 1, contrary +to the usual definition of connectivity (see for instance [10]). A k-connected component of Γ is a +k-connected subgraph that is maximal, in term of subgraph containment, with that property. +An essential consequence of chordality is that k-connected chordal graphs admit a unique de- +composition into (k + 1)-connected components through its k-separators. This is the subject of the +next section. +2.1 +Unicity of the components +The following well-known result will play a central role in our definition of the decomposition of a +chordal graph into k-connected components. For completeness we provide a short proof. +Proposition 2.1 (Dirac [11]). In a chordal graph every minimal separator is a clique. +Proof. Let Γ be a chordal graph and let S be a minimal separator with at least two vertices. +Suppose for contradiction that there are u, v ∈ S such that uv is not an edge. Let A and B be +different components of Γ − S, and consider shortest paths P and Q between x and y whose inner +vertices are, respectively, in A and B. Then the concatenation of P and Q is a chordless cycle of +length at least four, contradicting the fact that Γ is chordal. +For the rest of this section, we fix k ≥ 1. +Definition 2.2. Let Γ be a k-connected chordal graph with a k-separator S, and, for m ≥ 1, let +C1, . . . , Cm be the (possibly empty) connected components of Γ − S. Then, for i ∈ [m], the induced +subgraphs Γi = Γ[V (Ci) ∪ S] will be called the slices of Γ, obtained after cutting Γ through S. +Remark that by Proposition 2.1, S is a clique of Γ. Furthermore, as each slice Γi (i ∈ [m]) +contains a copy of S, Γ can be obtained by identifying together these m copies of S. This operation +will be called gluing through S. The next Proposition 2.3 implies that the slices Γi are k-connected. +Proposition 2.3. Let Γ be a k-connected chordal graph with a k-separator S inducing the slices +Γ1, . . . , Γm. If, for some i ∈ [m], Γi has a separator T, then T is also a separator of Γ. Furhtermore, +T ̸= S is a k-separator of Γ if it is a separator of the slice Γi it belongs to. +3 + +Proof. If S ⊆ T, the first claim is direct. Suppose now that v ∈ S \ T. Then, T separates v from +some other vertex w ∈ Γi, because otherwise every vertex would be reachable from v and T would +not be a separator. Since the vertices in S form a clique, w is in fact separated from all vertices in +S \ T. Then, in Γ, the same set T also separates w from S \ T. +For the second claim, observe that since all the vertices and edges in Γ are also in some Γi and +vice-versa, a set of vertices T ̸= S is a subclique of Γ if and only if it is a subclique of some slice +Γi. Now suppose to a contradiction that Γi − T is connected. Since, for j ̸= i, T is not entirely +contained in any other slice Γj, and Γj is k-connected, it follows that Γj − T is also connected. In +particular, every vertex in Γ − T is reachable from some vertex v ∈ S \ T, a contradiction. +Consider now the set of slices obtained after cutting Γ through all its k-separators. Because they +contain no k-separators, all these slices are (k+1)-connected and form in fact the (k+1)-connected +components of Γ. They are well defined thanks to the following result. +Proposition 2.4. Let Γ be a k-connected chordal graph with k-separators S and T. Then the slices +Γ1, . . . , Γm obtained after cutting Γ first through S then through T can be characterised as follows: +(i) Let i ∈ [m]. For each connected component Ci of Γ − (S ∪ T) there will be a slice Γi with +vertex set Vi, in such a way that Γi = Γ[Vi]. The set Vi contains the vertices in Ci, and also +contains the vertices of S (resp. T) if there is some vertex in S \ T (resp. T \ S) that has a +neighbour in Ci. +(ii) If none of the slices described in (i) contains the vertices in both S and T, there will be an +additional slice Γm+1 = Γ[S ∪ T], and these are all the possible slices. +In particular, doing the cuts first through T and then through S results in the same slices. +Proof. We first consider the slices obtained after cutting only through S. For 1 ≤ i ≤ mS, each of +the slices ΓS +i corresponds to a connected component CS +i of Γ − S. Among these slices there is only +one containing T as a subclique, say ΓS +1 , because the vertices in T \ S necessarily belong to the +same component CS +1 . Therefore ΓS +1 is the unique slice that will be cut through T while the others +stay unchanged. For 2 ≤ i ≤ mS, each of the other slices ΓS +i contains the vertices of CS +i , which +is certainly a connected component of Γ − (S ∪ T), and the vertices in S, but contains no vertex +in T \ S. This agrees with (i) because all the vertices in S have a neighbour in CS +i , since S is a +minimal separator, while the vertices in T \ S have no neighbours in CS +i . +We now consider the slices obtained after cutting ΓS +1 through T. Again, for 1 ≤ i ≤ mT each +of these slices corresponds to a connected component CT +i of ΓS +1 − T, denoted by ΓT +i . Among these +slices, there is only one containing S as a subclique, say ΓT +1 , because the vertices in T \S necessarily +belong to the same component CT +1 . The rest of the slices contain no vertex in S \ T. In fact, they +are analogous to the slices ΓS +i , 2 ≤ i ≤ mS, and they agree with (i) for the same reasons. +There are two possibilities for CT +1 : either it contains vertices other than the ones in S \ T or +it does not. In the first case, observe that CT +1 − S is connected. Indeed, if this was not the case +ΓS +1 would have S ∪ T as a separator, while neither S nor T are separators. But then there would +be a minimal k-separator of ΓS +1 that is not a clique, which is not possible. Therefore, CT +1 − S is a +connected component of Γ − (S ∪ T), which has some neighbour in S \ T and also in T \ S, since +CS +1 is connected. This also agrees with (i). On the other hand, if CT +1 contains no vertices other +than the ones in S \ T, then it is not a component of Γ − (S ∪ T) and we are in the case (ii). +Since this characterisation does not depend on the order of the cuts, the last claim follows. +4 + +The k-connected components of Γ are thus the maximal k-connected subgraphs, since, for +i ∈ [k −1], there is a single way of cutting through all the i-separators. And we can uniquely define +the 2-connected components of a connected chordal graph, the 3-connected components of these +2-connected components, the 4-connected components of these 3-connected components, and so on. +An illustration is given in Figure 1. This decomposition is the generalisation of the well-known +1 +1 +2 +2 +3 +3 +3 +1 +1 +2 +2 +4 +4 +4 +5 +5 +5 +1 +1 +2 +2 +3 +3 +3 +3 +3 +2 +3 +Figure 1: Decomposition of a connected graph with tree-width 3 into its 2, 3 and 4-connected components. Vertices with +the same label are identified. +decomposition of a connected graph into blocks (i.e. maximal 2-connected components). And it +induces, as shown in the next section, a system of functional equations satisfied by the generating +function counting chordal graphs of tree-width at most k. +2.2 +Functional equations for the generating functions +For the rest of this section, we fix some t ≥ 1 and let G be the family of chordal graphs with tree- +width at most t. For a graph Γ ∈ G and j ∈ [t], let us denote by nj(Γ) the number of j-cliques of Γ. +In the rest of the paper, we will write x as a short-hand for x1, . . . , xt, and define the multivariate +(exponential) generating function associated to G to be +G(x) = G(x1, . . . , xt) = +� +Γ∈G +1 +n1(Γ)! +t� +j=1 +xnj(Γ) +j +, +5 + +Let gn denote the number of chordal graphs with n vertices and tree-width at most t. Then, +G(x, 1, . . . , 1) = +� +n≥1 +gn +n! xn. +For 0 ≤ k ≤ t + 1, let Gk be the family of k-connected chordal graphs with tree-width at most t +and Gk(x) be the associated generating function. In particular, we have +Gt+1(x) = +1 +(t + 1)! +� +j∈[t] +x(t+1 +j ) +j +. +(2) +For other values of k, we need to consider graphs rooted at a clique. Rooting the graph Γ ∈ Gk +at an i-clique means distinguishing one i-clique K of Γ and choosing an ordering of (the labels of) +the vertices of K. In order to avoid over-counting, we will discount the subcliques of K. Let i ∈ [k] +and define G(i) +k +to be the family of k-connected chordal graphs with tree-width at most t and rooted +at an i-clique. Let then G(i) +k (x) be the associated generating function, where now for 1 ≤ j ≤ i the +variables xj mark the number of j-cliques that are not subcliques of the root. +Kk +G(k) +k+1 +Kk +Kk +Kk +Kk +Kk +Kk +G(k) +k+1 +G(k) +k+1 +G(k) +k +G(k) +k +G(k) +k +Figure 2: Recursive decomposition of a k-connected chordal graph into (k + 1)-connected components. +Lemma 2.5. Let k ∈ [t]. Then the following equations hold: +G(k) +k+1(x) = k! +k−1 +� +j=1 +x +−(k +j) +j +∂ +∂xk +Gk+1(x), +(3) +G(k) +k (x) = exp +� +G(k) +k+1 +� +x1, . . . , xk−1, xkG(k) +k (x), xk+1, . . . , xt +�� +, +(4) +Gk(x) = 1 +k! +k−1 +� +j=1 +x(k +j) +j +� +G(k) +k (x) dxk. +(5) +Proof. As per the symbolic method, taking the derivative of a generating function with respect to +the variable xi amounts to rooting a graph at an i-clique, while taking the integral will correspond +to “unrooting”. Thus, both Equations (3) and (5) follow from the definition. +On the other hand, Equation (4) is derived from the decomposition of k-connected chordal +graphs into their (k+1)-connected components, and is illustrated in Figure 2. Indeed, a k-connected +6 + +kK +7kchordal graph rooted at a k-clique K is obtained by gluing through K a set of (k + 1)-connected +graphs Γ1, . . . , Γm containg K, and then further gluing some k-connected chordal graph at each +k-clique of Γi, other than K, for all i ∈ [m]. Recall that the k-clique is itself considered to be +k-connected. The substitution of xk by xkG(k) +k +in Equation (4) reflects the recursive process of +gluing through k-cliques. Finally, the fact that the graphs are rooted at ordered cliques ensures +that the gluing process is made in all possible ways. +Finally, the fact that a graph is the set of its connected components can be translated as +G(x) = G0(x) = exp(G1(x)), +Observe then that one can derive G0(x) from Gt+1(x) by successively using Identities (3), (4) and +(5) from Lemma 2.5, as illustrated in Figure 3. Furthermore, in the steps where Identity (5) is +used, one needs to compute a formal integral. An alternative is to use the dissymmetry theorem for +tree-decomposable classes, as presented in Proposition 2.6. This is the purpose of the next section. +Gt+1 → G(t) +t+1 +↓ +G(t) +t +→ Gt → G(t−1) +t +↓ +G(t−1) +t−1 +→ Gt−1 → G(t−2) +t−1 +↓ +... +↓ +G(2) +2 +→ G2 → G(1) +2 +↓ +G(1) +1 +→ G1 +exp +−→ G0 +Figure 3: The schema to derive G0(x) from Gt+1(x). +2.3 +Combinatorial integration +In this section, we prove an alternative equation for (5) which does not involve an integral operator. +It is useful when computing the numerical values in Table 1. +A class of graphs A is said to be tree-decomposable if for each graph Γ ∈ A we can associate +in a unique way a tree τ(Γ) whose nodes are distinguishable, for instance by using the labels. +Let A• denote the class of graphs in A where a node of τ(Γ) is distinguished. Similarly, A•−• is +the class of graphs in A where an edge of τ(Γ) is distinguished, and A•→• those where an edge +τ(Γ) is distinguished and given a direction. As presented in [8], the dissymmetry theorem for tree- +decomposable classes is a generalisation of the well-known dissymetry theorem for trees of [5], and +allows one to express the class of unrooted graphs in A in terms of the rooted classes. +7 + +b +c +b +c +b +b +c +b +c +b +Figure 4: Tree-decomposition (right) associated to a 2-connected chordal graph (left) of tree-width 3. +Proposition 2.6 (Dissymmetry Theorem [8]). Let A be a tree-decomposable class, then +A + A•→• ≃ A• + A•−•, +where ≃ is a bijection preserving the number of nodes. In particular, if the encoding trees have no +adjacent nodes of the same type then we have +A ≃ A• − A•−•. +An example of the decomposition of a chordal graph Γ of bounded tree-width and its associated +tree τ(Γ) is depicted in Figure 4. Next, we make use of this decomposition to obtain, via the above +Proposition, the generating function of unrooted chordal graphs of bounded tree-width. +Lemma 2.7. Let k ∈ [t]. Then the following equation holds: +Gk(x) = Gk+1 +� +x1, . . . , xk−1, xkG(k) +k (x), xk+1, . . . , xt +� ++ 1 +k! +� +j∈[k] +x(k +j) +j +G(k) +k (x) +� +1 − G(k) +k+1 +� +x1, . . . , xk−1, xkG(k) +k (x), xk+1, . . . , xt +�� +. +(6) +Proof. To each Γ ∈ Gk different from the comple graph on k vertices, we associate a unique tree +τ(Γ) as follows. The tree τ(Γ) admits two different types of nodes, namely b and c. Nodes of type +b represent the (k + 1)-connected components of Γ, while those of type c represent the k-cliques of +Γ through which the (k + 1)-connected components are glued together. +Let Bk and Ck be the generating functions counting the trees τ(Γ) (Γ ∈ Gk) rooted at nodes of +type b and c, respectively, and Ek be the generating function of those trees rooted at an undirected +edge between nodes of types b and c. They can also be respectively seen as the generaring functions +counting k-connected graphs with a distinguished (k + 1)-connected component, a distinguished +k-clique that belongs to more than one (k + 1)-connected components, or a distinguished (k + 1)- +connected component C together with a distinguished k-clique in C that belongs to at least another +8 + +(k + 1)-connected component C′. They are specified next using the symbolic method: +Bk(x) = Gk+1 +� +x1, . . . , xk−1, xkG(k) +k (x), xk+1, . . . , xt +� +, +Ck(x) = 1 +k! +� +j∈[k] +x(k +j) +j +� +G(k) +k (x) − +� +1 + G(k) +k+1 +� +x1, . . . , xk−1, xkG(k) +k (x), xk+1, . . . , xt +��� +, +Ek(x) = 1 +k! +� +j∈[k] +x(k +j) +j +G(k) +k+1 +� +x1, . . . , xk−1, xkG(k) +k (x), xk+1, . . . , xt +� � +G(k) +k (x) − 1 +� +. +The equation defining Bk(x) follows directly from the decomposition discussed in the previous +section, while the equation for Ck(x) is obtained from Equation (4) by substracting the first two +terms of the exponential (because there are at least two (k+1)-connected components glued through +the k-clique). The equation for Ek(x) can be derived by considering a (k + 1)-connected chordal +graph Γ rooted at a k-clique K and gluing through it a k-connected chordal graph Γ′ rooted at K, +and containing at least one (k + 1)-connected component, then further gluing some k-connected +chordal graph to other k-cliques of Γ′. The correcting factors in the last two equations are there to +mark all the subcliques of the root k-clique and forget the order of its vertices. +Finally, recall that we consider the complete graph on k vertices to be k-connected. So that +Proposition 2.6 directly implies that the unrooted graphs are counted by +Gk(x) = 1 +k! +� +j∈[k] +x(k +j) +j ++ Bk(x) + Ck(x) − Ek(x). +By translating this equation in light of the above three equations, one concludes the proof. +3 +Asymptotic analysis +Fix t ≥ 1. +In this section we prove Theorems 1.1 and 1.2. +We use rather classical methods +from [15], which consist in deriving asymptotic estimates from local expansions of the generating +functions from Section 2 at their singularities. Those expansions are in turn derived from successive +applications of the implicit system of equations described in Lemma 2.5, to “transfer” the local +expansion of Gt+1(x) to G0(x1, 1, . . . , 1), as illustrated by the schema in Figure 3. +We will follow the method developed in [12, Chapter 2], but will need to extend some of the +tools and notions present there in order to deal with multivariate generating functions and the fact +that the local expansions are with respect to different variables from one step to the next. This is +the purpose of the next section. +3.1 +Proper singularity functions and singular expansions +Let ρ : U → C be an analytic function defined on an open set. For u ∈ U and δ, η > 0, a ∆-domain +at ρ(u) is a complex region of the form +∆(ρ(u), δ, η) = ∆(ρ(u)) = {z ∈ C : |z| < ρ(u) + η and | arg(z/ρ(u) − 1)| > δ}. +Our main tool is a “transfer theorem”. The proof can be found in [12] (see also [15, Chapter VI.3]). +9 + +Proposition 3.1. (Transfer Theorem [12, Lemma 2.18]). Let f(z, u) be a power series in z and a +parameter u ∈ U, and suppose that it admits an expansion of the form +f(z, u) = C(u) +� +1 − +z +ρ(u) +�−α(u) ++ O +�� +1 − +z +ρ(u) +�−β(u)� +, +that is uniform for u ∈ U and z ∈ ∆(ρ(u)), and with functions C(u), ρ(u), α(u) and β(u) that +remain bounded and satisfy β(u) < ℜ(α(u)) for all u ∈ U. +Then the following estimate holds uniformly for u ∈ U and as n → ∞ +[zn]f(z, u) = C(u) nα(u)−1 +Γ(α(u))ρ(u)−n + O +� +ρ(u)−n nmax(ℜ(α(u))−2, β(u)−1)� +. +By setting u = 1 in Proposition 3.1, one recovers the “classical” transfer theorem for univariate +analytic functions, see for instance [12, Lemma 2.15]. +Next we introduce several definitions which will allow us to extend the notion of local expansion +of an analytic function at an algebraic singularity to our multivariate setting. First is the notion +of fully movable proper singularity function. +Definition 3.2. We say that a function ρ(x2, . . . , xt) is a proper singularity function if it +satisfies the following conditions: +(i) It is defined in a (t − 1)-dimensional proper complex neighbourhood of Rt−1 ++ , where it is also +analytic. +(ii) It is positive and real if x2, . . . , xt are positive and real, and it is strictly decreasing with +negative derivatives in all t − 1 positive real variables. +Furthermore we say it is fully movable with respect to the variables x2, . . . , xk if the following +condition holds: +(iii) ρ(x2, . . . , xt) → 0 (resp. ∞) if one of the variables x2, . . . , xk tends to ∞ (resp. 0), whereas +all the other variables including xk+1, . . . , xt are fixed positive real numbers. +With this notion at hand, we can define that of a positive function with a proper α-singularity. +Definition 3.3. Let α ∈ R \ Z. We say that a function G(x) is a positive function with a +proper α-singularity if the following properties hold: +(i) G(x) is a power series in x = (x1, . . . , xt) with non-negative coefficents. +(ii) There exists a proper singularity function ρ(x2, . . . , xt) such that for every fixed choice of +x2, . . . , xt ∈ R+, ρ(x2, . . . , xt) is the radius of convergence of the power series x1 �→ G(x). +(iii) For every choice of X0, X1 ∈ R with 0 < X0 < X1, there exist δ > 0 and analytic func- +tions g1(x), g2(x), that are defined and non-zero for X0 < |x2|, . . . , |xt| < X1 and |x1 − +ρ(x2, . . . , xt)| < δ with x1, . . . , xt sufficiently close to the positive real axis, such that in this +range, provided that arg(x1 − ρ(x2, . . . , xt)) ̸= 0, we have +G(x) = g1(x) + g2(x) +� +1 − +x1 +ρ(x2, . . . , xt) +�α +. +(7) +In this case we say that x1 is the leading variable of G(x). +10 + +Finally, in order to apply Proposition 3.1 to a positive function with a proper α-singularity, we +need some notion of analytic continuation to a ∆-domain. +Definition 3.4. A positive function G(x) with a proper α-singularity (α ∈ R \ Z) and proper +singularity function ρ(x2, . . . , xt) is said to be aperiodic and analytically continuable with +respect to the variable x1 if the following holds: +(i) For every fixed choice of x2, . . . , xt ∈ R+, ρ(x2, . . . , xt) is the unique singularity of the function +x1 �→ G(x) on the circle |x1| = ρ(x2, . . . , xt). +(ii) There exists δ > 0 such that x1 �→ G(x) can be analytically continued to the region +|x1| < |ρ(x2, . . . , xt)| + δ/2 +and +|x1 − ρ(x2, . . . , xt)| > δ. +(8) +In particular this function cannot be represented as a function of the form xa +1f(xb +1) for some positive +integers a, b, where b > 1. +Fix k ∈ {2, . . . , t} and observe that setting xi = 1 for i ̸= k in Definition 3.4(ii) implies that +G(x1, 1, . . . , 1, xk, 1, . . . , 1) can be analytically continued to a domain of the form ∆(xk). +3.2 +Transfer properties of proper singular expansions +We now prove certain “transfer properties” of proper α-singular expansions in the neighbourhood +of a proper singularity function. Our main tools here will be the Implicit Function Theorem (IFT) +for analytic functions, and its refinement known as the Weierstrass Preparation Theorem (WPT). +For a statement and a proof of those famous theorems, we refer the reader to [17]. +The first property generalises [12, Lemma 2.28] to proper singularity functions. +The proof +follows the same line and we only sketch it here. +Lemma 3.5. Let k ∈ {2, . . . , t−1} and suppose that ρ(x2, x3, . . . , xt) is a proper singularity function +that is fully movable with respect to the variables x2, . . . , xk. Then there exists a proper singularity +function κ(x1, . . . , xk−1, xk+1, . . . , xt) that is fully movable with respect to x1, . . . , xk−1 such that +x1 = ρ(x2, . . . , xk−1, κ(x1, . . . , xk−1, xk+1, . . . , xt), xk+1, . . . , xt). +(9) +Furthermore there exists a function K(x) that is analytic and non-zero on a t-dimensional +complex neighbourhood of Rt ++ such that +x1 − ρ(x2, . . . , xt) = K(x) (xk − κ(x1, . . . , xk−1, xk+1, . . . , xt)) . +(10) +Proof. Suppose first that x1, . . . , xt are positive real variables. Since ρ is strictly decreasing and +tends to 0 (resp. ∞), if one of the variables x2, . . . , xk tends to ∞ (resp. 0) then it immediately +follows from the continuity of ρ and the IFT that a function κ = κ(x1, . . . , xk−1, xk+1, . . . , xt) +satisfying (9) exists. Furthermore, κ is strictly decreasing and tends to 0 (resp. ∞) if one of the +variables x1, . . . , xk−1 tends to ∞ (resp. 0). +Next, fix x2, . . . , xt ∈ R+ and set x1 = ρ(x2, . . . , xt). Since from Definition 3.2 ρ is analytic and +satisfies +∂ +∂xk ρ < 0 for 2 ≤ k ≤ t, it follows, by applying the IFT to (9), that the function κ can be +(uniquely) analytically continued to a complex neighbourhood of (x1, . . . , xk−1, xk+1, . . . , xt). In +fact, using the WPT in the degree one case, it can further be shown that there exists a function +K(x) that is analytic and non-zero in a complex neighbourhood of x such that (10) holds. +Finally, a standard analytic continuation argument shows that both κ and K can be globally +defined so that (10) holds in the proposed range, that is, a complex neighbourhood of Rt ++. +11 + +An important consequence of Lemma 3.5 is that for k ∈ [t] the representation (7) can be +rewritten into +G(x) = g1(x) + g2(x) +� +1 − xk +κ +�α +, +with κ = κ(x1, . . . , xk−1, xk+1, . . . , xt) and where the analytic function +g2(x) = g2(x) +� +K(x)κ +ρ(x2, . . . , xt) +�α +is defined and non-zero for X0 < |x2|, . . . , |xt| < X1. This means that any of the variables x1, . . . , xk +can be the leading one in the definition of a positive function with a proper α-singularity, provided +that the proper singularity function ρ(x2, . . . , xt) is fully movable with respect to x2, . . . , xk. Fur- +thermore κ is certainly a singularity of the mapping xk �→ G(x). And by the monotonicity property +in Definition 3.2(ii) there is no smaller positive real singularity. Thus, κ is the radius of convergence +of the mapping xk �→ G(x), provided that x1, . . . , xk−1, xk+1, . . . , xt ∈ R+. +Next, we extend [12, Lemma 2.27] to the context of positive function with a proper α-singularity. +Lemma 3.6. For k ∈ [t] and α ∈ R\Z, let G(x) be a positive function with a proper α-singularity, +aperiodic and analytically continuable with respect to x1, and with a proper singularity function +ρ(x2, . . . , xt) that is fully movable with respect to x2, . . . , xk. Set +D(x) = +∂ +∂xk +G(x) +and +H(x) = +� xk +0 +G(x1, . . . , xk−1, y, xk+1, . . . , xt) dy. +(11) +Then D(x) is a positive function with a proper (α − 1)-singularity, while and H(x) is a positive +function with a proper (α + 1)-singularity, and both are aperiodic and analytically continuable with +respect to x1. Furthermore the proper singularity functions of G(x), D(x) and H(x) coincide. +Proof. Fix δ > 0. First, the analytic continuability of both mappings x1 �→ D(x) and x1 �→ H(x) +to a region of the form (8) is immediate by assumption on G(x) and properties of the derivative +and the integral. Second, if |x1 − ρ(x2, . . . , xt)| < δ then Lemma 3.5 implies that there exist δ′ > 0 +and a proper singularity function κ = κ(x1, . . . , xk−1, xk+1, . . . , xt) such that for |xk −κ| < δ′, G(x) +can be represented as +G(x) = g1(x) + g2(x) +� +1 − xk +κ +�α +. +(12) +Set d1(x) = (∂/∂xk)g1(x) and d2(x) = g2(x)/(2κ) + (∂/∂xk)g2(x) (1 − xk/κ). Then taking the +partial derivative of (12) with respect to xk gives +D(x) = +∂ +∂xk +g1(x) + +∂ +∂xk +g2(x) +� +1 − xk +κ +�α ++ g2(x) +2κ +� +1 − xk +κ +�α−1 += d1(x) + d2(x) +� +1 − xk +κ +�α−1 +. +Now, in order to compute the integral of (12) we first compute the Taylor expansions of the +functions g1(x) and g2(x) at xk ∼ κ and obtain a representation of the form +G(x) = +� +ℓ≥0 +Gℓ(x1, . . . , xk−1, xk+1, . . . , xt) +� +1 − xk +κ +�αℓ +(13) +12 + +that is certainly convergent for |xk − κ| < δ′. Next we split up the integral in (11) into three parts +I1(x1, . . . , xk−1, xk+1, . . . , xt) := +� (1−η)κ +0 +G(x1, . . . , xk−1, y, xk+1, . . . , xt) dy, +I2(x1, . . . , xk−1, xk+1, . . . , xt) := +� κ +(1−η)κ +G(x1, . . . , xk−1, y, xk+1, . . . , xt) dy, +I3(x) := +� xk +κ +G(x1, . . . , xk−1, y, xk+1, . . . , xt) dy, +where η > 0 is chosen in such a way that η < δ′/|κ|. The first integral is certainly an analytic +function in x1, . . . , xk−1, xk, . . . , xt, as a definite integral with respect to y in a range where G is +analytic. The second integral can be directly computed by the series expansion (13) +I2(x1, . . . , xk−1, xk+1, . . . , xt) = +κ +� +(1−η)κ +G(x1, . . . , xk−1, y, xk+1, . . . , xt) dy += +� +ℓ≥0 +Gℓ(x1, . . . , xk−1, xk+1, . . . , xt) +κ +� +(1−η)κ +(1 − y/κ)αℓ dy += κ +� +ℓ≥0 +Gℓ(x1, . . . , xk−1, xk+1, . . . , xt) +αℓ + 1 +ηαℓ+1. +This series is absolutely convergent and represents an analytic function in x1, . . . , xk−1, xk+1, . . . , xt. +Finally, for the third integral we use again the series expansion (13) and obtain +I3(x) = +� xk +κ +G(x1, . . . , xk−1, y, xk+1, . . . , xt) dy += +� +ℓ≥0 +Gℓ(x1, . . . , xk−1, xk+1, . . . , xt) +xk +� +κ +(1 − y/κ)αℓ dy += −κ +� +ℓ≥0 +Gℓ(x1, . . . , xk−1, xk+1, . . . , xt) +αℓ + 1 +(1 − xk/κ)αℓ+1. +This series representation can be rewritten into +I3(x) = h1(x) + h2(x) +� +1 − xk +κ +�α+1 +. +Next, note that since +g2(x1, . . . , xk−1, κ, xk+1, . . . , xt) = −α + 1 +κ +h2(x1, . . . , xk−1, κ, xk+1, . . . , xt), +then both g2 and h2 are non-zero, even if |xk − κ| < δ′′ for a sufficiently small δ′′ > 0. Moreover, +since the coefficients of G(x) and H(x) are non-negative we have g1(x) > 0 for xj ∈ R+ and +h1(x) = h1(x) + I1(x1, . . . , xk−1, xk+1, . . . , xt) + I2(x1, . . . , xk−1, xk+1, . . . , xt) > 0. +13 + +Finally, by another application of Lemma 3.5, we can rewrite D(x) and H(x) into +D(x) = d1(x) + d2(x) +� +1 − +x1 +ρ(x2, . . . , xt) +�α−1 +and H(x) = h1(x) + h2(x) +� +1 − +x1 +ρ(x2, . . . , xt) +�α+1 +with d2, h2 ̸= 0. This completes the proof. +Finally, we generalise [12, Theorem 2.21]. This theorem states that if G(x, u) is a univariate +function, with parameter u = (x2, . . . , xt), defined implicitely in terms of another function F(x, u, y) +(i.e. such that F(x, u, G(x, u)) = 0) that admits a 1/2-singular expansion at some R > 0, then +G(x, u) also admits a 1/2-singular expansion at some ρ < R. The next result extends this to the +case where F has a proper 1/2-singularity. The proof follows the same lines and we sketch it next. +Lemma 3.7. Suppose that F(x, y) is a positive function in t + 1 variables with a proper 1/2- +singularity and singularity function R(x2, . . . , xt; y) that is fully movable with respect to the variables +x2, . . . , xk and y for some 2 ≤ k ≤ t. Furthermore assume that F(x1, x2, . . . , xt, y) = 0 if one of +the variables x1, . . . , xk are zero. Then the functional equation +G = exp(F(x, G)) +(14) +has a unique solution G = G(x) with G(0) = 1 which is a positive function with a proper 1/2- +singularity, too. Its singularity function ρ(x2, . . . , xt) is fully movable with respect to the variables +x2, . . . , xk and satisfies +ρ(x2, . . . , xt) < R +� +x2, . . . , xt; G(ρ(x2, . . . , xt), x2, . . . , xt) +� +. +Moreover, if F(x, y) is periodic and analytically continuable with respect to the variable x1 then the +same property holds for G(x). +Proof. First, by iteration (or by the IFT), Equation (14) admits a unique power series solution +with G(0) = 1 and non-negative coefficients. +Next we define a singularity function ρ(x2, . . . , xt). For this purpose we fix x2, . . . , xt ∈ R+ and +vary x1. We claim that there exists a unique x1 > 0 such that for x = (x1, x2, . . . , xt) we have +x1 < R(x2, . . . , xt, G(x)) and satisfying +G(x) = exp(F(x, G(x))), +(15) +and +1 = exp(F(x, G(x)))∂F +∂y (x, G(x)). +(16) +Since all the coefficients of G are non-negative, the solution function G is strictly increasing in x1. +Consequently, the factor exp(F(x, G(x))) in (16) is also strictly increasing in x1. +Now we study the factor (∂F/∂y)(x, G(x)) in (16). By assumption we have +∂F +∂y (0, x2, . . . , xt, G(0, x2, . . . , xt)) = 0. +14 + +Moreover, since F is a positive function with a proper 1/2-singularity, by Lemma 3.6 ∂F/∂y is a +positive function with a proper (−1/2)-singularity, that is, when x1 ∼ R(x2, . . . , xt, y) it can be +represented as +∂F +∂y (x, y) = f1(x, y) + f2(x, y) +� +1 − +x1 +R(x2, . . . , xt, y) +�−1/2 +, +where f1 and f2 are analytic and f2 ̸= 0. Since G is strictly increasing in x1 and R is a proper +singularity function fully movable in y, R(x2, . . . , xt, G(x)) is strictly decreasing in x1 and goes to 0 +as x1 → ∞. Therefore, (1 − x1/R(x2, . . . , xt, G(x)))−1/2 is strictly increasing in x1 and unbounded +while x1 < R(x2, . . . , xt, G(x)). The same is true for (∂F/∂y)(x, G(x)) because f1(x, G(x)) and +f2(x, G(x)) are strictly increasing functions in x1 when x1 < R(x2, . . . , xt, G(x)). Our claim follows. +From [12, Theorem 2.19], which amounts to evaluating the parameter u in R+ in [12, The- +orem 2.21], this implies that for x2, . . . , xt ∈ R+ the univariate function x1 → G(x) has a 1/2- +singularity at x1 = x1. Therefore we set +ρ(x2, . . . , xt) := x1. +The system formed by equations (15) and (16) can be used to get more information on ρ. Notice +first that, given x2, . . . , xt, the system determines x1 = ρ(x2, . . . , xt) and G(x). Then, since the +determinant +�������� +−eF ∂F +∂x1 +1 − eF ∂F +∂y +−eF ∂F +∂x1 +∂F +∂y − eF ∂2F +∂y∂x1 +−eF +�∂F +∂y +�2 +− eF ∂2F +∂y2 +�������� += e2F +�������� +∂F +∂x1 +0 +∂F +∂x1 +∂F +∂y + ∂2F +∂y∂x1 +�∂F +∂y +�2 ++ ∂2F +∂y2 +�������� += e2F ∂F +∂x1 +��∂F +∂y +�2 ++ ∂2F +∂y2 +� +is positive, it follows by the IFT that the function ρ(x2, . . . , xt) can be locally analytically continued. +Fix now some 2 ≤ j ≤ k. By differentiating (15) with respect to xj, one obtains +0 = ∂[G(x)] +∂xj +− exp(F(x, G(x))) +� ∂F +∂x1 +(x, G(x)) ∂ρ +∂xj ++ ∂F +∂xj +(x, G(x)) + ∂F +∂y (x, G(x))∂[G(x)] +∂xj +� += − exp(F(x, G(x))) +� ∂F +∂x1 +(x, G(x)) ∂ρ +∂xj ++ ∂F +∂xj +(x, G(x)) +� +. +In other words, +∂ρ +∂xj += − +∂F +∂xj +(x, G(x)) +∂F +∂x1 +(x, G(x)) +< 0. +This means that ρ is strictly decreasing in all variables, provided they are real and positive. Let us +finally consider the behaviour of ρ as xj tends to 0 or +∞. Suppose first that x1 = ρ is bounded +away from 0 when xj → ∞. Then G(x) → +∞, and R → 0. However, this is impossible since +15 + +x1 < R. Thus ρ → 0 as xj → ∞. On the other hand, suppose that x1 = ρ stays bounded when +xj → 0. In this case, G(x) stays bounded and so by assumptions on the zeros of F, F(x, G(x))) → 0 +and (∂F/∂y)(x, G(x))) → 0 as xj → 0. However, this is not possible by (16). Summing up, this +means from Definition 3.2 that the function ρ is a proper singularity function that is fully movable +with respect to x2, . . . , xk. +Furthermore, it follows from [12, Theorem 2.21] that we also get an expansion of the form (7) +with α = 1/2 for G(x). And it remains to check that for x2, . . . , xt ∈ R+, the mapping x1 �→ G(x) +admits an analytic continuation away from ρ, in a region of the form (8). To that end, we now +consider (14) as a functional equation for the function x1 �→ G(x). Since G(x) can be written as +G(x) = 1 + ˜G(x), where ˜G is a power series with non-negative coefficients, we have that +G(x) = exp(F(x, G(x))) = exp(F(x, 1 + ˜G(x))) += exp(F(x, 1) + ˜F(x, ˜G(x))) += 1 + F(x, 1) + ˜F(x, ˜G(x)) + +� +n≥2 +(F(x, 1) + ˜F(x, ˜G(x)))n +n! +, +where ˜F(x, y) is a power series with non-negative coefficients. Now, since F(x, y) is aperiodic in +x1, it follows that G(x) has to also be aperiodic with respect to x1. This implies that ρ(x2, . . . , xt) +is the unique dominant singularity of the function x1 �→ G(x) and we conclude by a standard +compactness argument that it can be analytically continued to a region of the form (8). +3.3 +Proofs of the main results +Fix t ≥ 1 and recall that starting with Gt+1, which is an explicit monomial, one can recursively +obtain the generating functions Gt, Gt−1, . . . , G1, and finally G0 = exp(G1). Let us next discuss +the first step of this induction, from Gt+1 to Gt, since it is slightly different from the general step. +Proposition 3.8. Let t ≥ 1, x2, . . . , xt ∈ R+. Then there exists two functions h1(x) and h2(x), +that are analytic and non-zero at x = 1/e, such that for x ∼ 1/e we have +Gt(x) = +�t +j=1 x(t +j) +j +t! +� +h1(tX) + h2(tX)(1 − etX)3/2� +, +where X = +t� +j=1 +x( +t +j−1) +j +. +(17) +Proof. From Equation (2) and by (3) we directly get +G(t) +t+1(x) = +t� +j=1 +x( +t +j−1) +j +. +Consequently, from the relation (4) the function G(t) +t += G(t) +t (x) satisfies the equation +G(t) +t += exp +� +� +t� +j=1 +x( +t +j−1) +j +[G(t) +t ]t +� +� . +16 + +Let T(z) denote the tree function, i.e. that satisfies the equation T(z) = z exp(T(z)). Then, +using the change of variable z = tX with X = �t +j=1 x( +t +j−1) +j +, we can represent G(t) +t (x) as +G(t) +t (x) = +�T(tX) +tX +�1/t += exp (T(tX)/t) . +With the help of (6) and the relation T(x) = x exp(T(x)), this also leads to +Gt(x) = 1 +t! +t� +j=1 +x(t +j) +j +G(t) +t (x) − +t +(t + 1)! +t� +j=1 +x(t+1 +j ) +j +� +G(t) +t (x) +�t+1 += +�t +j=1 x(t +j) +j +t! +exp +�T(tX) +t +� � +1 − T(tX) +t + 1 +� +. +(18) +It is well known (see for instance [15]) that T(z) has its dominant singularity at z0 = 1/e and a +local Puiseux expansion at z ∼ z0 of the form +T(z) = 1 − +√ +2 +√ +1 − ez + 2 +3(1 − ez) − 11 +√ +2 +36 (1 − ez)3/2 + O +� +(1 − ez)2� +. +Furthermore, z0 = 1/e is the only singularity on the circle |z| = 1/e and T(z) can be analytically +continued to a region of the form |z| < 1/e + δ/2, |z − 1/e| > δ for some δ > 0. +Hence we get +exp +�T(x) +t +� � +1 − T(x) +t + 1 +� += te1/t +t + 1 +� +1 − 1 +t2 (1 − ex) + 2 +√ +2(t + 1) +3t3 +(1 − ex)3/2 + O +� +(1 − ex)2� +� += h1(x) + h2(x)(1 − ex)3/2, +where h1(x) and h2(x) are functions that are analytic and non-zero at x ∼ 1/e. This directly leads +to the claimed local representation of Gt(x). +Note that the appearance of the dominant singularity (1 − etX)3/2 is not unexpected since +G(t) +t (x) has a dominant singularity of the form +√ +1 − etX and Gt(x) and is as per (5) – more or +less – the integral of G(t) +t (x). +Furthermore, one can deduce from Proposition 3.8 the case k = t of Theorem 1.1 by a direct +application of Proposition 3.1 (setting u = 1). Similarly, a central limit theorem for the case k = t +of Theorem 1.2 follows immediately from Proposition 3.8 by an application of [12, Theorem 2.25]. +For k < t, Theorems 1.1 and 1.2 can also be deduced from local representations of a form similar +to (17), modulo some technical conditions. Thus the main step of the proofs is to show that the +above representation for Gt(x) implies corresponding representation for Gt−1(x), Gt−2(x), . . . , G1(x). +This is the object of the next proposition. Before stating it, let us remark the following. +Remark 3.9. For k ∈ [t + 1], the function Gk(x) admits � +j∈[k] x(k +j) +j +as a factor. If k = t + 1, this +is Equation (2). For k ≤ t, it follows from Equation (5). And by (3) this implies that the function +G(k−1) +k +(x) has � +j∈[k] x(k−1 +j−1) +j +as a factor. In particular, G(k−1) +k +(x) is zero if one of the variables +x1, . . . , xk are zero. +17 + +Proposition 3.10. Suppose that 2 ≤ k ≤ t and let Gk(x) be a positive function with a proper +3/2-singularity that is aperiodic and analytically continuable with respect to x1, and with a proper +singularity function ρk(x2, . . . , xt) that is fully movable with respect to x2, . . . , xk. +Then the function Gk−1(x) is also a positive function with a proper 3/2-singularity that is aperi- +odic and analytically continuable with respect to x1, where the singularity function ρk−1(x2, . . . , xt) +is again fully movable with respect to x2 . . . , xk. Moreover, if x2, . . . , xt ∈ R+ then +ρk−1(x2, . . . , xt) < ρk(x2, . . . , xt). +(19) +Proof. In a first step, using Lemma 3.5 we replace ρk(x2, . . . , xt) by the proper singularity function +κk = κk(x1, . . . , xk−2, xk, . . . , xt), that is fully movable with respect to x1, . . . , xk−2, so that we can +represent Gk(x) as +Gk(x) = g1(x) + g2(x) +� +1 − xk−1 +κk +�3/2 +. +Next, we deduce from Lemma 3.6 and the relation (3) that G(k−1) +k +(x) is a positive function with +a proper 1/2-singularity and admits the same proper singularity function κk as Gk(x), in particular +it is fully movable with respect to x1, . . . , xk−2. +From there we apply Lemma 3.7 to the relation (4), noting that +F(x, y) = G(k−1) +k +(x1, . . . , xk−2, xk−1y, xk, . . . , xt) +is a positive function with a proper 1/2-singularity and that by Remark 3.9 F(x, y) has zeros +x1, . . . , xk. Furthermore, it admits a proper singularity function given by +R(x1, . . . , xk−2, xk, . . . , xt, y) = 1 +yκk(x1, . . . , xk−2, xk, . . . , xt). +Clearly R is fully movable in x1, . . . , xk−2, xk and y. Consequently, using Lemma 3.7 the solution +function y = G(k−1) +k−1 (x) is a positive function with a proper 1/2-singularity, and leading variable +xk−1, for which the singularity function κk−1 = κk−1(x1, . . . , xk−2, xk, . . . , xt) satisfies +κk−1(x1, . . . , xk−2, xk, . . . , xt) < κk(x1, . . . , xk−2, xk, . . . , xt) +G(k−1) +k−1 (x) +. +Note that G(k−1) +k−1 (0) = 1. Hence, G(k−1) +k−1 (x) > 1, and it follows that +κk−1(x1, . . . , xk−2, xk, . . . , xt) < κk(x1, . . . , xk−2, xk, . . . , xt). +Finally, we apply Lemma 3.6 on relation (5) and obtain that Gk−1(x) is a positive function +with a proper 3/2-singularity and leading variable xk−1. By another application of Lemma 3.5 +we see that we can change it back to the leading variable x1, such that the corresponding proper +singularity function ρk−1(x2, . . . , xk) satisfies (19). +We are now in a position to prove the two main results of the paper. +18 + +Proof of Theorem 1.1. +Proposition 3.8 implies that Gt(x) is a positive function with a proper +3/2-singularity, and is aperiodic and analytically continuable with respect to x1. In particular, +compare (7) with (17) and note that x1 appears in X only in the first power x(t +0) +1 . In this case, the +proper singularity function is explicitly given by +ρt(x2, . . . , xt) = 1 +et +t� +j=2 +x +−( +t +j−1) +j +. +From there, successive applications of Proposition 3.10 imply that the function Gk(x) also has +these properties for each k ∈ [t]. Since the exponential is an entire function this also holds for +G(x) = G0(x) = exp(G1(x)). And we conclude the proof by setting x2 = · · · = xt = 1 then +applying Proposition 3.1. +Proof of Theorem 1.2. +Suppose that (x2, . . . , xt) is in a sufficiently small complex neigh- +bourhood U of (1, . . . , 1) in Ct−1. +From Proposition 3.8 then k − 1 succesive applications of +Proposition 3.10, we derive a local representation of the form (7) holds for G(x) = Gk(x), with +ρ(x2, . . . , xk) = ρk(x2, . . . , xk) and α = 3/2, when x1 is close to ρk(x2, . . . , xk). Furthermore, by +continuity there exists δ > 0 such that the function x1 �→ Gk(x) is still analytically continuable to a +region of the form (8), with ρ = ρk. And we deduce from Proposition 3.1 the following asymptotic +estimate for the coefficients of x1 in Gk(x) +[xn +1] Gk(x) = Ck(x2, . . . , xk) n−5/2 ρk(x2, . . . , xt)−n (1 + o(1)) +as n → ∞, +for some non-zero function Ck(x2, . . . , xk) analytic in U. This leads to a quasi-power situation for +the probability generating function +E +� +xX2 +2 +· · · xXt +t +� += +[xn +1] Gk(x) +[xn +1] Gk(x1, 1, . . . , 1) ∼ Ck(x2, . . . , xk) +Ck(1, . . . , 1) +� ρk(1, . . . , 1) +ρk(x2, . . . , xt) +�n +. +Finally, setting xj = euj and λn = n in [12, Theorem 2.22] implies the claimed joint central limit +theorem. Note that one could alternatively apply [12, Theorem 2.25]. Furthermore, the relation +G0(x) = exp (G1(x)) implies, as above, that G(x) = G0(x) has the same singularities and singular +expansion as G1(x), up to a multiplicative constant. This concludes the proof. +4 +Concluding remarks +With the help of a computer algebra system, making use of the representation in Lemma 2.7, we +have been able to compute the following table of numerical values for the singularities ρt,k. We +have stopped at t = 7 since the size of the system of functional equations needed to determine ρt,k +grows too fast. +Let us mention a recent result giving an estimate cn−5/2γnn! for the number of labelled planar +chordal graphs with γ ≈ 11.89 [7]. Is is easy to see that the class of chordal graphs with tree-width +at most three is exactly the same as the class of chordal graphs not containing K5 as a minor, +whose asymptotic growth is, according to Theorem 1.1 and the table above, of the form cn−5/2δnn! +with δ = 1/ρ3,1 ≈ 12.98. +19 + +k = 1 +k = 2 +k = 3 +k = 4 +k = 5 +k = 6 +k = 7 +t = 1 +0.36788 +t = 2 +0.14665 +0.18394 +t = 3 +0.07703 +0.08421 +0.12263 +t = 4 +0.04444 +0.04662 +0.05664 +0.09197 +t = 5 +0.02657 +0.02732 +0.03092 +0.04152 +0.07358 +t = 6 +0.01608 +0.01635 +0.01773 +0.02184 +0.03214 +0.06131 +t = 7 +0.00974 +0.00984 +0.01038 +0.01204 +0.01614 +0.02583 +0.05255 +Table 1: Approximations of the radii of convergence of the generating functions counting k-connected chordal graphs with +tree-width at most t for small values of t and k. +Furthermore, notice that if we denote by C(x) and B(x), respectively, the generating functions +of connected and 2-connected graphs in Gt,0, then Equation 4 reads for k = 1 +C′(x) = exp(B′(xC′(x)). +If ρC and ρB are the singularities of C(x) and B(x), respectively, the condition for being subcritical +is that ρCC′(ρC) < ρB, so that the singularity of C(x) arises as a branch-point in the former +equation and is not inherited by that of B(x); in our case this condition is safistied because of +Lemma 3.7. +Since the number of all chordal graphs grows like 2n2/4, we know that the singularity ρt = ρt,1 +of chordal graphs with tree-wdith at most t goes to 0 as t → ∞. The question is at which rate +ρt → 0 as t → ∞. Since the exponential growth of t-trees is (etn)n, we have ρt = O(1/t). And since +the growth of all graphs of tree-width at most t is at most (2ttn)n, we also have ρt = Ω(1/(t2t)). +We leave as an open problem to narrow the gap between the upper and lower bound. Heuristic +arguments suggest that ρt decreases exponentially in t. +As a final question, we consider letting t = t(n) grow with n. Recall that a class of labelled +graphs is small when the number of graphs in the class grows at most like cnn! for some c > 0, and +large otherwise. We know that the class of all chordal graphs is large, while the class of chordal +graphs with tree-width at most t is small for fixed t. Let us see that if t = (1 + ϵ)(log n) then the +class is large for each ϵ > 0. A graph is split if the vertex set can be partitioned into a clique and +an independent set. It is well-known and easy to prove that split graphs are chordal. Consider split +graphs with a clique of size t = (1 + ϵ) log n and the complement an independent set, so that he +largest clique is of size at most t + 1 and the tree-width at most t. Every edge between the clique +and the complement can be chosen independently, hence there are at least +2(1+ϵ) log n(n−(1+ϵ) log n) +such graphs, a quantity that grows faster than cnn! for every c > 0. We leave as an open problem +to determine at which order of magnitude between t = O(1) and t = log n the class ceases to be +small. +Acknowledgements +We gratefully acknowledge earlier discussions with Juanjo Ru´e and Dimitrios Thilikos on the prob- +lem of counting chordal graphs with bounded tree-width. +20 + +The authors acknowledge support from the Marie Curie RISE research network “RandNet” +MSCA-RISE-2020-101007705. Moreover, M.D. was supported by the Special Research Program +SFB F50-02 “Algorithmic and Enumerative Combinatorics”, and by the project P35016 “Infinite +Singular Systems and Random Discrete Objects” of the FWF (Austrian Science Fund). +Addi- +tionally, M.N. and C.R. acknowledge the financial support of the Spanish State Research Agency +through projects MTM2017-82166-P and PID2020-113082GB-I00, while M.N. acknowledges sup- +port from the Severo Ochoa and Mar´ıa de Maeztu Program for Centers and Units of Excellence +(CEX2020-001084-M), and C.R. acknowledges support from the grant Beatriu de Pin´os BP2019, +funded by the H2020 COFUND project No 801370 and AGAUR (the Catalan agency for manage- +ment of university and research grants). +References +[1] D. Aldous. The Continuum Random Tree. I. The Annals of Probability, 19(1):1 – 28, 1991. +[2] J. Baste, M. Noy, and I. Sau. On the number of labeled graphs of bounded treewidth. European +Journal of Combinatorics, 71:12–21, 2018. +[3] L. 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The Electronic Journal of Combinatorics, +15(P148), 2008. +[9] M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, +and S. Saurabh. Parameterized algorithms. Springer, 2015. +[10] R. Diestel. Graph Theory. Springer Berlin, Heidelberg, Fifth edition, 2016. +[11] G. A. Dirac. On rigid circuit graphs. Abhandlungen aus dem Mathematischen Seminar der +Universit¨at Hamburg, 25:71–76, 1961. +[12] M. Drmota. Random Trees. Springer-Verlag Wien, 2009. +[13] M. Drmota, E. Fusy, M. Kang, V. Kraus, and J. Ru´e. Asymptotic study of subcritical graph +classes. SIAM Journal on Discrete Mathematics, 25(4):1615–1651, 2011. +21 + +[14] Z. Dvoˇr´ak and S. Norine. Small graph classes and bounded expansion. Journal of Combina- +torial Theory. Series B, 100(2):171–175, 2010. +[15] P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press, Cam- +bridge, 2009. +[16] M. C. Golumbic. +Algorithmic Graph Theory and Perfect Graphs, volume 57 of Annals of +Discrete Mathematics. +Elsevier Science B.V., Amsterdam, second edition, 2004. +With a +foreword by Claude Berge. +[17] L. Kaup and B. Kaup. Holomorphic Functions of Several Variables, volume 3 of de Gruyter +Studies in Mathematics. Walter de Gruyter, Berlin New York, 1983. Translated by Michael +Bridgiand. +[18] J. W. Moon. The number of labeled k-trees. Journal of Combinatorial Theory, 6(2):196–199, +1969. +[19] S. Norine, P. Seymour, R. Thomas, and P. Wollan. Proper minor-closed families are small. +Journal of Combinatorial Theory. Series B, 96(5):754–757, 2006. +[20] K. Panagiotou, B. Stufler, and K. Weller. Scaling limits of random graphs from subcritical +classes. The Annals of Probability, 44(5):3291–3334, 2016. +[21] N. C. Wormald. Counting labelled chordal graphs. Graphs and Combinatorics, 1(2):193–200, +1985. +Jordi Castellv´ı +Departament de Matem`atiques de la Universitat Polit`ecnica de Catalunya (UPC), Barcelona, Spain. +E-mail: jordi.castellvi@upc.edu +Michael Drmota +Institute for Discrete Mathematics and Geometry of the Technische Universit¨at Wien, Austria. +E-mail: michael.drmota@tuwien.ac.at +Marc Noy +Departament de Matem`atiques and Institut de Matem`atiques (IMTech) de la Universitat Polit`ecnica +de Catalunya (UPC), and Centre de Recerca Matem`atica (CRM), Barcelona, Spain. +E-mail: marc.noy@upc.edu +Cl´ement Requil´e +Departament de Matem`atiques and Institut de Matem`atiques (IMTech) de la Universitat Polit`ecnica +de Catalunya (UPC), Barcelona, Spain. +E-mail: clement.requile@upc.edu +22 + diff --git a/jNAyT4oBgHgl3EQfX_d5/content/tmp_files/load_file.txt b/jNAyT4oBgHgl3EQfX_d5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d3d850d5e504f5065e7a881e09f757cff70e1253 --- /dev/null +++ b/jNAyT4oBgHgl3EQfX_d5/content/tmp_files/load_file.txt @@ -0,0 +1,1337 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf,len=1336 +page_content='Chordal graphs with bounded tree-width Jordi Castellv´ı Michael Drmota Marc Noy Cl´ement Requil´e January 3, 2023 Abstract Given t ≥ 2 and 0 ≤ k ≤ t, we prove that the number of labelled k-connected chordal graphs with n vertices and tree-width at most t is asymptotically cn−5/2γnn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=', as n → ∞, for some constants c, γ > 0 depending on t and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Additionally, we show that the number of i-cliques (2 ≤ i ≤ t) in a uniform random k-connected chordal graph with tree-width at most t is normally distributed as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The asymptotic enumeration of graphs of tree-width at most t is wide open for t ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' To the best of our knowledge, this is the first non-trivial class of graphs with bounded tree-width where the asymptotic counting problem is solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Our starting point is the work of Wormald [Counting Labelled Chordal Graphs, Graphs and Combinatorics (1985)], were an algorithm is developed to obtain the exact number of labelled chordal graphs on n vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 1 Introduction Tree-width is a fundamental parameter in structural and algorithmic graph theory, as illustrated for instance in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' It can be defined in terms of tree-decompositions or equivalently in terms of k-trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' A k-tree is defined recursively as either a complete graph on k + 1 vertices or a graph obtained by adjoining a new vertex adjacent to a k-clique of a k-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The tree-width of a graph Γ is the minimum k such that Γ is a subgraph of a k-tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In particular, k-trees are the maximal graphs with tree-width at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The number of k-trees on n labelled vertices was independently shown [3, 18] to be �n k � (k(n − k) + 1)n−k−2 = 1 √ 2π k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' kk+2 n−5/2 (ek)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (1 + o(1)), (1) where the estimate holds for k fixed and n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' However, there are relatively few results on the enumeration of graphs of given tree-width or on properties of random graphs with given tree-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Graphs of tree-width one are forests (acyclic graphs) and their enumeration is a classical result, while graphs of tree-width at most two are series-parallel graphs and were first counted in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The problem of counting graphs of tree-width three is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' From now on, we will use t to denote the tree-width while k will denote the connectivity of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' All graphs we consider are labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Given that tree-width is non-increasing under taking minors, the class of graphs with tree-width at most t is ‘small’ when t is fixed, in the sense that the number gn,t of labelled graphs with n vertices and tree-wdith at most t grows at most like cnn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' for some c > 0 depending on t (see [19, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The best bounds known for gn,t are, up to lower order terms, �2ttn log t �n ≤ gn,t ≤ (2ttn)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='00194v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='CO] 31 Dec 2022 The upper bound follows by considering all possible subgraphs of t-trees, and the lower bound uses a suitable construction developed in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In the present work we determine the asymptotic number of labelled chordal graphs with tree-width at most t, following the approach in [15] and [12], and based on the analysis of systems of equations satisfied by generating functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' A graph is chordal if every cycle of length greater than three contains at least one chord, that is, an edge connecting non-consecutive vertices of the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Chordal graphs have been extensively studied in structural graph theory and graph algorithms (see for instance [16]), but not so much from the point of view of enumeration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Wormald [21] used generating functions to develop a method for finding the exact number of chordal graphs with n vertices for a given value of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' It is based on decomposing chordal graphs into k-connected components for each k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' As remarked in [21], it is difficult to define the k-connected components of arbitrary graphs for k > 3, but for chordal graphs they are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Wormald introduces generating functions Ck(x) for k-connected chordal graphs and finds an equation linking Ck(x) and Ck+1(x) reflecting the decomposition into k-connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This is precisely the starting point of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' An important parameter in [21] is the size w of the largest clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For chordal graphs one can show that the tree-width is equal to w − 1, hence bounding the tree-width by t is the same as bounding w by t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The parameter w also plays a substantial rˆole in showing that almost all chordal graphs are split graphs [4], which in turn implies that the number of chordal graphs with n vertices is of order 2n2/4(1 + o(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For fixed n, t ≥ 1 and 0 ≤ k ≤ t, let Gt,k,n be the set of k-connected chordal graphs with n labelled vertices and tree-width at most t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Our two main results are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For t ≥ 1 and 0 ≤ k ≤ t, there exist constants ct,k > 0 and γt,k ∈ (0, 1) such that |Gt,k,n| = ct,k n−5/2 γn t,k n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (1 + o(1)) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Remark that by setting t = k in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1 one recovers the general form of the asymptotic estimate of the number of k-trees (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The fact that γk,k = ek is reproven in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In principle, for fixed t and k the constants ct,k and γt,k can be computed, at least approximately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Table 1 in Section 4 displays approximations of 1/γt,k for 1 ≤ k ≤ t ≤ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let t ≥ 1, 0 ≤ k ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , t} let Xn,i denote the number of i-cliques in a uniform random graph in Gt,k,n, and set Xn = (Xn,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , Xn,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then Xn satisfies a multivariate central limit theorem, that is, as n → ∞ we have 1 √n (Xn − E Xn) d→ N(0, Σ), with E Xn ∼ αn and Cov Xn ∼ Σn, and where α is a (t−1)-dimensional vector of positive numbers and Σ is a (t−1)×(t−1)-dimensional positive semi-definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let us mention that more structural asymptotic results can be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Notably, the class of chordal graphs with tree-width at most t is subcritical in the sense of [13], as further discussed in the concluding Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' It follows from [20] that the uniform random connected chordal graph with tree-width at most t with distances rescaled by 1/√n admits the Continuum Random Tree (CRT) [1] as a scaling limit, multiplied by a constant that depends on t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The proofs of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2 are based on a recursive decomposition of chordal graphs translated in the combinatorial language of generating functions, the so-called symbolic method [15], 2 into a system of functional equations explicited in Section 2 and analysed in Section 3 by means of analytic methods [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' More precisely, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1 we show the unicity of the decomposition of chordal graphs into their k-connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This is translated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2 into a system of funtional equations satisfied by exponential generating functions encoding the numbers of graphs in each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='3 we prove an alternative encoding of an integral operator, which is instrumental for computing the numerical values in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The subsequent asymptotic analysis is rather delicate as it involves singularity functions depending on several variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Our notions of proper and fully movable singularity functions are key ingredients, and are defined along several technical notions in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Some useful propeties are proven in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2, before embarking in the proofs of the main results in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 2 Decomposition of chordal graphs All graphs considered in this work will be simple and labelled, that is with vertex-set [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let Γ be a graph and k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' A k-separator of Γ is a subset of k vertices whose removal disconnects Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The graph Γ is said to be k-connected if it contains no i-separator for i ∈ [k − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Notice that with this definition we consider the complete graph on k vertices to be k-connected, for any k ≥ 1, contrary to the usual definition of connectivity (see for instance [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' A k-connected component of Γ is a k-connected subgraph that is maximal, in term of subgraph containment, with that property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' An essential consequence of chordality is that k-connected chordal graphs admit a unique de- composition into (k + 1)-connected components through its k-separators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This is the subject of the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1 Unicity of the components The following well-known result will play a central role in our definition of the decomposition of a chordal graph into k-connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For completeness we provide a short proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1 (Dirac [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In a chordal graph every minimal separator is a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let Γ be a chordal graph and let S be a minimal separator with at least two vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Suppose for contradiction that there are u, v ∈ S such that uv is not an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let A and B be different components of Γ − S, and consider shortest paths P and Q between x and y whose inner vertices are, respectively, in A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then the concatenation of P and Q is a chordless cycle of length at least four, contradicting the fact that Γ is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For the rest of this section, we fix k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let Γ be a k-connected chordal graph with a k-separator S, and, for m ≥ 1, let C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , Cm be the (possibly empty) connected components of Γ − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then, for i ∈ [m], the induced subgraphs Γi = Γ[V (Ci) ∪ S] will be called the slices of Γ, obtained after cutting Γ through S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Remark that by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1, S is a clique of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Furthermore, as each slice Γi (i ∈ [m]) contains a copy of S, Γ can be obtained by identifying together these m copies of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This operation will be called gluing through S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The next Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='3 implies that the slices Γi are k-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let Γ be a k-connected chordal graph with a k-separator S inducing the slices Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , Γm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' If, for some i ∈ [m], Γi has a separator T, then T is also a separator of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Furhtermore, T ̸= S is a k-separator of Γ if it is a separator of the slice Γi it belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' If S ⊆ T, the first claim is direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Suppose now that v ∈ S \\ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then, T separates v from some other vertex w ∈ Γi, because otherwise every vertex would be reachable from v and T would not be a separator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Since the vertices in S form a clique, w is in fact separated from all vertices in S \\ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then, in Γ, the same set T also separates w from S \\ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For the second claim, observe that since all the vertices and edges in Γ are also in some Γi and vice-versa, a set of vertices T ̸= S is a subclique of Γ if and only if it is a subclique of some slice Γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Now suppose to a contradiction that Γi − T is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Since, for j ̸= i, T is not entirely contained in any other slice Γj, and Γj is k-connected, it follows that Γj − T is also connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In particular, every vertex in Γ − T is reachable from some vertex v ∈ S \\ T, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Consider now the set of slices obtained after cutting Γ through all its k-separators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Because they contain no k-separators, all these slices are (k+1)-connected and form in fact the (k+1)-connected components of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' They are well defined thanks to the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let Γ be a k-connected chordal graph with k-separators S and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then the slices Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , Γm obtained after cutting Γ first through S then through T can be characterised as follows: (i) Let i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For each connected component Ci of Γ − (S ∪ T) there will be a slice Γi with vertex set Vi, in such a way that Γi = Γ[Vi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The set Vi contains the vertices in Ci, and also contains the vertices of S (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' T) if there is some vertex in S \\ T (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' T \\ S) that has a neighbour in Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (ii) If none of the slices described in (i) contains the vertices in both S and T, there will be an additional slice Γm+1 = Γ[S ∪ T], and these are all the possible slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In particular, doing the cuts first through T and then through S results in the same slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' We first consider the slices obtained after cutting only through S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For 1 ≤ i ≤ mS, each of the slices ΓS i corresponds to a connected component CS i of Γ − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Among these slices there is only one containing T as a subclique, say ΓS 1 , because the vertices in T \\ S necessarily belong to the same component CS 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Therefore ΓS 1 is the unique slice that will be cut through T while the others stay unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For 2 ≤ i ≤ mS, each of the other slices ΓS i contains the vertices of CS i , which is certainly a connected component of Γ − (S ∪ T), and the vertices in S, but contains no vertex in T \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This agrees with (i) because all the vertices in S have a neighbour in CS i , since S is a minimal separator, while the vertices in T \\ S have no neighbours in CS i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' We now consider the slices obtained after cutting ΓS 1 through T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Again, for 1 ≤ i ≤ mT each of these slices corresponds to a connected component CT i of ΓS 1 − T, denoted by ΓT i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Among these slices, there is only one containing S as a subclique, say ΓT 1 , because the vertices in T \\S necessarily belong to the same component CT 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The rest of the slices contain no vertex in S \\ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In fact, they are analogous to the slices ΓS i , 2 ≤ i ≤ mS, and they agree with (i) for the same reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' There are two possibilities for CT 1 : either it contains vertices other than the ones in S \\ T or it does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In the first case, observe that CT 1 − S is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Indeed, if this was not the case ΓS 1 would have S ∪ T as a separator, while neither S nor T are separators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' But then there would be a minimal k-separator of ΓS 1 that is not a clique, which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Therefore, CT 1 − S is a connected component of Γ − (S ∪ T), which has some neighbour in S \\ T and also in T \\ S, since CS 1 is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This also agrees with (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' On the other hand, if CT 1 contains no vertices other than the ones in S \\ T, then it is not a component of Γ − (S ∪ T) and we are in the case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Since this characterisation does not depend on the order of the cuts, the last claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 4 The k-connected components of Γ are thus the maximal k-connected subgraphs, since, for i ∈ [k −1], there is a single way of cutting through all the i-separators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' And we can uniquely define the 2-connected components of a connected chordal graph, the 3-connected components of these 2-connected components, the 4-connected components of these 3-connected components, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' An illustration is given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This decomposition is the generalisation of the well-known 1 1 2 2 3 3 3 1 1 2 2 4 4 4 5 5 5 1 1 2 2 3 3 3 3 3 2 3 Figure 1: Decomposition of a connected graph with tree-width 3 into its 2, 3 and 4-connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Vertices with the same label are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' decomposition of a connected graph into blocks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' maximal 2-connected components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' And it induces, as shown in the next section, a system of functional equations satisfied by the generating function counting chordal graphs of tree-width at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2 Functional equations for the generating functions For the rest of this section, we fix some t ≥ 1 and let G be the family of chordal graphs with tree- width at most t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For a graph Γ ∈ G and j ∈ [t], let us denote by nj(Γ) the number of j-cliques of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In the rest of the paper, we will write x as a short-hand for x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt, and define the multivariate (exponential) generating function associated to G to be G(x) = G(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) = � Γ∈G 1 n1(Γ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' t� j=1 xnj(Γ) j , 5 Let gn denote the number of chordal graphs with n vertices and tree-width at most t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then, G(x, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , 1) = � n≥1 gn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For 0 ≤ k ≤ t + 1, let Gk be the family of k-connected chordal graphs with tree-width at most t and Gk(x) be the associated generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In particular, we have Gt+1(x) = 1 (t + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' � j∈[t] x(t+1 j ) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (2) For other values of k, we need to consider graphs rooted at a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Rooting the graph Γ ∈ Gk at an i-clique means distinguishing one i-clique K of Γ and choosing an ordering of (the labels of) the vertices of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In order to avoid over-counting, we will discount the subcliques of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let i ∈ [k] and define G(i) k to be the family of k-connected chordal graphs with tree-width at most t and rooted at an i-clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let then G(i) k (x) be the associated generating function, where now for 1 ≤ j ≤ i the variables xj mark the number of j-cliques that are not subcliques of the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Kk G(k) k+1 Kk Kk Kk Kk Kk Kk G(k) k+1 G(k) k+1 G(k) k G(k) k G(k) k Figure 2: Recursive decomposition of a k-connected chordal graph into (k + 1)-connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let k ∈ [t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then the following equations hold: G(k) k+1(x) = k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' k−1 � j=1 x −(k j) j ∂ ∂xk Gk+1(x), (3) G(k) k (x) = exp � G(k) k+1 � x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xkG(k) k (x), xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt �� , (4) Gk(x) = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' k−1 � j=1 x(k j) j � G(k) k (x) dxk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' As per the symbolic method, taking the derivative of a generating function with respect to the variable xi amounts to rooting a graph at an i-clique, while taking the integral will correspond to “unrooting”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Thus, both Equations (3) and (5) follow from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' On the other hand, Equation (4) is derived from the decomposition of k-connected chordal graphs into their (k+1)-connected components, and is illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Indeed, a k-connected 6 kK 7kchordal graph rooted at a k-clique K is obtained by gluing through K a set of (k + 1)-connected graphs Γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , Γm containg K, and then further gluing some k-connected chordal graph at each k-clique of Γi, other than K, for all i ∈ [m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Recall that the k-clique is itself considered to be k-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The substitution of xk by xkG(k) k in Equation (4) reflects the recursive process of gluing through k-cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Finally, the fact that the graphs are rooted at ordered cliques ensures that the gluing process is made in all possible ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Finally, the fact that a graph is the set of its connected components can be translated as G(x) = G0(x) = exp(G1(x)), Observe then that one can derive G0(x) from Gt+1(x) by successively using Identities (3), (4) and (5) from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='5, as illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Furthermore, in the steps where Identity (5) is used, one needs to compute a formal integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' An alternative is to use the dissymmetry theorem for tree-decomposable classes, as presented in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This is the purpose of the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Gt+1 → G(t) t+1 ↓ G(t) t → Gt → G(t−1) t ↓ G(t−1) t−1 → Gt−1 → G(t−2) t−1 ↓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' ↓ G(2) 2 → G2 → G(1) 2 ↓ G(1) 1 → G1 exp −→ G0 Figure 3: The schema to derive G0(x) from Gt+1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='3 Combinatorial integration In this section, we prove an alternative equation for (5) which does not involve an integral operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' It is useful when computing the numerical values in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' A class of graphs A is said to be tree-decomposable if for each graph Γ ∈ A we can associate in a unique way a tree τ(Γ) whose nodes are distinguishable, for instance by using the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let A• denote the class of graphs in A where a node of τ(Γ) is distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Similarly, A•−• is the class of graphs in A where an edge of τ(Γ) is distinguished, and A•→• those where an edge τ(Γ) is distinguished and given a direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' As presented in [8], the dissymmetry theorem for tree- decomposable classes is a generalisation of the well-known dissymetry theorem for trees of [5], and allows one to express the class of unrooted graphs in A in terms of the rooted classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 7 b c b c b b c b c b Figure 4: Tree-decomposition (right) associated to a 2-connected chordal graph (left) of tree-width 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='6 (Dissymmetry Theorem [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let A be a tree-decomposable class, then A + A•→• ≃ A• + A•−•, where ≃ is a bijection preserving the number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In particular, if the encoding trees have no adjacent nodes of the same type then we have A ≃ A• − A•−•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' An example of the decomposition of a chordal graph Γ of bounded tree-width and its associated tree τ(Γ) is depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Next, we make use of this decomposition to obtain, via the above Proposition, the generating function of unrooted chordal graphs of bounded tree-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let k ∈ [t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then the following equation holds: Gk(x) = Gk+1 � x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xkG(k) k (x), xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt � + 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' � j∈[k] x(k j) j G(k) k (x) � 1 − G(k) k+1 � x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xkG(k) k (x), xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' To each Γ ∈ Gk different from the comple graph on k vertices, we associate a unique tree τ(Γ) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The tree τ(Γ) admits two different types of nodes, namely b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Nodes of type b represent the (k + 1)-connected components of Γ, while those of type c represent the k-cliques of Γ through which the (k + 1)-connected components are glued together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let Bk and Ck be the generating functions counting the trees τ(Γ) (Γ ∈ Gk) rooted at nodes of type b and c, respectively, and Ek be the generating function of those trees rooted at an undirected edge between nodes of types b and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' They can also be respectively seen as the generaring functions counting k-connected graphs with a distinguished (k + 1)-connected component, a distinguished k-clique that belongs to more than one (k + 1)-connected components, or a distinguished (k + 1)- connected component C together with a distinguished k-clique in C that belongs to at least another 8 (k + 1)-connected component C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' They are specified next using the symbolic method: Bk(x) = Gk+1 � x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xkG(k) k (x), xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt � , Ck(x) = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' � j∈[k] x(k j) j � G(k) k (x) − � 1 + G(k) k+1 � x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xkG(k) k (x), xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt ��� , Ek(x) = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' � j∈[k] x(k j) j G(k) k+1 � x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xkG(k) k (x), xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt � � G(k) k (x) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The equation defining Bk(x) follows directly from the decomposition discussed in the previous section, while the equation for Ck(x) is obtained from Equation (4) by substracting the first two terms of the exponential (because there are at least two (k+1)-connected components glued through the k-clique).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The equation for Ek(x) can be derived by considering a (k + 1)-connected chordal graph Γ rooted at a k-clique K and gluing through it a k-connected chordal graph Γ′ rooted at K, and containing at least one (k + 1)-connected component, then further gluing some k-connected chordal graph to other k-cliques of Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The correcting factors in the last two equations are there to mark all the subcliques of the root k-clique and forget the order of its vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Finally, recall that we consider the complete graph on k vertices to be k-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' So that Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='6 directly implies that the unrooted graphs are counted by Gk(x) = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' � j∈[k] x(k j) j + Bk(x) + Ck(x) − Ek(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' By translating this equation in light of the above three equations, one concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 3 Asymptotic analysis Fix t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In this section we prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' We use rather classical methods from [15], which consist in deriving asymptotic estimates from local expansions of the generating functions from Section 2 at their singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Those expansions are in turn derived from successive applications of the implicit system of equations described in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='5, to “transfer” the local expansion of Gt+1(x) to G0(x1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , 1), as illustrated by the schema in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' We will follow the method developed in [12, Chapter 2], but will need to extend some of the tools and notions present there in order to deal with multivariate generating functions and the fact that the local expansions are with respect to different variables from one step to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This is the purpose of the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1 Proper singularity functions and singular expansions Let ρ : U → C be an analytic function defined on an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For u ∈ U and δ, η > 0, a ∆-domain at ρ(u) is a complex region of the form ∆(ρ(u), δ, η) = ∆(ρ(u)) = {z ∈ C : |z| < ρ(u) + η and | arg(z/ρ(u) − 1)| > δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Our main tool is a “transfer theorem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The proof can be found in [12] (see also [15, Chapter VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 9 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (Transfer Theorem [12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let f(z, u) be a power series in z and a parameter u ∈ U, and suppose that it admits an expansion of the form f(z, u) = C(u) � 1 − z ρ(u) �−α(u) + O �� 1 − z ρ(u) �−β(u)� , that is uniform for u ∈ U and z ∈ ∆(ρ(u)), and with functions C(u), ρ(u), α(u) and β(u) that remain bounded and satisfy β(u) < ℜ(α(u)) for all u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then the following estimate holds uniformly for u ∈ U and as n → ∞ [zn]f(z, u) = C(u) nα(u)−1 Γ(α(u))ρ(u)−n + O � ρ(u)−n nmax(ℜ(α(u))−2, β(u)−1)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' By setting u = 1 in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1, one recovers the “classical” transfer theorem for univariate analytic functions, see for instance [12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Next we introduce several definitions which will allow us to extend the notion of local expansion of an analytic function at an algebraic singularity to our multivariate setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' First is the notion of fully movable proper singularity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' We say that a function ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) is a proper singularity function if it satisfies the following conditions: (i) It is defined in a (t − 1)-dimensional proper complex neighbourhood of Rt−1 + , where it is also analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (ii) It is positive and real if x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt are positive and real, and it is strictly decreasing with negative derivatives in all t − 1 positive real variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Furthermore we say it is fully movable with respect to the variables x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk if the following condition holds: (iii) ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) → 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' ∞) if one of the variables x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk tends to ∞ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 0), whereas all the other variables including xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt are fixed positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' With this notion at hand, we can define that of a positive function with a proper α-singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let α ∈ R \\ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' We say that a function G(x) is a positive function with a proper α-singularity if the following properties hold: (i) G(x) is a power series in x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) with non-negative coefficents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (ii) There exists a proper singularity function ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) such that for every fixed choice of x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt ∈ R+, ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) is the radius of convergence of the power series x1 �→ G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (iii) For every choice of X0, X1 ∈ R with 0 < X0 < X1, there exist δ > 0 and analytic func- tions g1(x), g2(x), that are defined and non-zero for X0 < |x2|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , |xt| < X1 and |x1 − ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt)| < δ with x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt sufficiently close to the positive real axis, such that in this range, provided that arg(x1 − ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt)) ̸= 0, we have G(x) = g1(x) + g2(x) � 1 − x1 ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) �α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (7) In this case we say that x1 is the leading variable of G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 10 Finally, in order to apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1 to a positive function with a proper α-singularity, we need some notion of analytic continuation to a ∆-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' A positive function G(x) with a proper α-singularity (α ∈ R \\ Z) and proper singularity function ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) is said to be aperiodic and analytically continuable with respect to the variable x1 if the following holds: (i) For every fixed choice of x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt ∈ R+, ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) is the unique singularity of the function x1 �→ G(x) on the circle |x1| = ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (ii) There exists δ > 0 such that x1 �→ G(x) can be analytically continued to the region |x1| < |ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt)| + δ/2 and |x1 − ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt)| > δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (8) In particular this function cannot be represented as a function of the form xa 1f(xb 1) for some positive integers a, b, where b > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Fix k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , t} and observe that setting xi = 1 for i ̸= k in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='4(ii) implies that G(x1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , 1, xk, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , 1) can be analytically continued to a domain of the form ∆(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2 Transfer properties of proper singular expansions We now prove certain “transfer properties” of proper α-singular expansions in the neighbourhood of a proper singularity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Our main tools here will be the Implicit Function Theorem (IFT) for analytic functions, and its refinement known as the Weierstrass Preparation Theorem (WPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For a statement and a proof of those famous theorems, we refer the reader to [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The first property generalises [12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='28] to proper singularity functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The proof follows the same line and we only sketch it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , t−1} and suppose that ρ(x2, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) is a proper singularity function that is fully movable with respect to the variables x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then there exists a proper singularity function κ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) that is fully movable with respect to x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1 such that x1 = ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, κ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt), xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (9) Furthermore there exists a function K(x) that is analytic and non-zero on a t-dimensional complex neighbourhood of Rt + such that x1 − ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) = K(x) (xk − κ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Suppose first that x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt are positive real variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Since ρ is strictly decreasing and tends to 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' ∞), if one of the variables x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk tends to ∞ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 0) then it immediately follows from the continuity of ρ and the IFT that a function κ = κ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) satisfying (9) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Furthermore, κ is strictly decreasing and tends to 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' ∞) if one of the variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1 tends to ∞ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Next, fix x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt ∈ R+ and set x1 = ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Since from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2 ρ is analytic and satisfies ∂ ∂xk ρ < 0 for 2 ≤ k ≤ t, it follows, by applying the IFT to (9), that the function κ can be (uniquely) analytically continued to a complex neighbourhood of (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In fact, using the WPT in the degree one case, it can further be shown that there exists a function K(x) that is analytic and non-zero in a complex neighbourhood of x such that (10) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Finally, a standard analytic continuation argument shows that both κ and K can be globally defined so that (10) holds in the proposed range, that is, a complex neighbourhood of Rt +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 11 An important consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='5 is that for k ∈ [t] the representation (7) can be rewritten into G(x) = g1(x) + g2(x) � 1 − xk κ �α , with κ = κ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) and where the analytic function g2(x) = g2(x) � K(x)κ ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) �α is defined and non-zero for X0 < |x2|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , |xt| < X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This means that any of the variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk can be the leading one in the definition of a positive function with a proper α-singularity, provided that the proper singularity function ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) is fully movable with respect to x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Fur- thermore κ is certainly a singularity of the mapping xk �→ G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' And by the monotonicity property in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2(ii) there is no smaller positive real singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Thus, κ is the radius of convergence of the mapping xk �→ G(x), provided that x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Next, we extend [12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='27] to the context of positive function with a proper α-singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For k ∈ [t] and α ∈ R\\Z, let G(x) be a positive function with a proper α-singularity, aperiodic and analytically continuable with respect to x1, and with a proper singularity function ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) that is fully movable with respect to x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Set D(x) = ∂ ∂xk G(x) and H(x) = � xk 0 G(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, y, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (11) Then D(x) is a positive function with a proper (α − 1)-singularity, while and H(x) is a positive function with a proper (α + 1)-singularity, and both are aperiodic and analytically continuable with respect to x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Furthermore the proper singularity functions of G(x), D(x) and H(x) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Fix δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' First, the analytic continuability of both mappings x1 �→ D(x) and x1 �→ H(x) to a region of the form (8) is immediate by assumption on G(x) and properties of the derivative and the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Second, if |x1 − ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt)| < δ then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='5 implies that there exist δ′ > 0 and a proper singularity function κ = κ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) such that for |xk −κ| < δ′, G(x) can be represented as G(x) = g1(x) + g2(x) � 1 − xk κ �α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (12) Set d1(x) = (∂/∂xk)g1(x) and d2(x) = g2(x)/(2κ) + (∂/∂xk)g2(x) (1 − xk/κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then taking the partial derivative of (12) with respect to xk gives D(x) = ∂ ∂xk g1(x) + ∂ ∂xk g2(x) � 1 − xk κ �α + g2(x) 2κ � 1 − xk κ �α−1 = d1(x) + d2(x) � 1 − xk κ �α−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Now, in order to compute the integral of (12) we first compute the Taylor expansions of the functions g1(x) and g2(x) at xk ∼ κ and obtain a representation of the form G(x) = � ℓ≥0 Gℓ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) � 1 − xk κ �αℓ (13) 12 that is certainly convergent for |xk − κ| < δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Next we split up the integral in (11) into three parts I1(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) := � (1−η)κ 0 G(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, y, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) dy, I2(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) := � κ (1−η)κ G(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, y, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) dy, I3(x) := � xk κ G(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, y, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) dy, where η > 0 is chosen in such a way that η < δ′/|κ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The first integral is certainly an analytic function in x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt, as a definite integral with respect to y in a range where G is analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The second integral can be directly computed by the series expansion (13) I2(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) = κ � (1−η)κ G(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, y, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) dy = � ℓ≥0 Gℓ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) κ � (1−η)κ (1 − y/κ)αℓ dy = κ � ℓ≥0 Gℓ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) αℓ + 1 ηαℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This series is absolutely convergent and represents an analytic function in x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Finally, for the third integral we use again the series expansion (13) and obtain I3(x) = � xk κ G(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, y, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) dy = � ℓ≥0 Gℓ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) xk � κ (1 − y/κ)αℓ dy = −κ � ℓ≥0 Gℓ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) αℓ + 1 (1 − xk/κ)αℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This series representation can be rewritten into I3(x) = h1(x) + h2(x) � 1 − xk κ �α+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Next, note that since g2(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, κ, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) = −α + 1 κ h2(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, κ, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt), then both g2 and h2 are non-zero, even if |xk − κ| < δ′′ for a sufficiently small δ′′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Moreover, since the coefficients of G(x) and H(x) are non-negative we have g1(x) > 0 for xj ∈ R+ and h1(x) = h1(x) + I1(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) + I2(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−1, xk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 13 Finally, by another application of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='5, we can rewrite D(x) and H(x) into D(x) = d1(x) + d2(x) � 1 − x1 ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) �α−1 and H(x) = h1(x) + h2(x) � 1 − x1 ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) �α+1 with d2, h2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Finally, we generalise [12, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This theorem states that if G(x, u) is a univariate function, with parameter u = (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt), defined implicitely in terms of another function F(x, u, y) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' such that F(x, u, G(x, u)) = 0) that admits a 1/2-singular expansion at some R > 0, then G(x, u) also admits a 1/2-singular expansion at some ρ < R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The next result extends this to the case where F has a proper 1/2-singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The proof follows the same lines and we sketch it next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Suppose that F(x, y) is a positive function in t + 1 variables with a proper 1/2- singularity and singularity function R(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' y) that is fully movable with respect to the variables x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk and y for some 2 ≤ k ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Furthermore assume that F(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt, y) = 0 if one of the variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then the functional equation G = exp(F(x, G)) (14) has a unique solution G = G(x) with G(0) = 1 which is a positive function with a proper 1/2- singularity, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Its singularity function ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) is fully movable with respect to the variables x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk and satisfies ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) < R � x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' G(ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt), x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Moreover, if F(x, y) is periodic and analytically continuable with respect to the variable x1 then the same property holds for G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' First, by iteration (or by the IFT), Equation (14) admits a unique power series solution with G(0) = 1 and non-negative coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Next we define a singularity function ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For this purpose we fix x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt ∈ R+ and vary x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' We claim that there exists a unique x1 > 0 such that for x = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) we have x1 < R(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt, G(x)) and satisfying G(x) = exp(F(x, G(x))), (15) and 1 = exp(F(x, G(x)))∂F ∂y (x, G(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (16) Since all the coefficients of G are non-negative, the solution function G is strictly increasing in x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Consequently, the factor exp(F(x, G(x))) in (16) is also strictly increasing in x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Now we study the factor (∂F/∂y)(x, G(x)) in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' By assumption we have ∂F ∂y (0, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt, G(0, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 14 Moreover, since F is a positive function with a proper 1/2-singularity, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='6 ∂F/∂y is a positive function with a proper (−1/2)-singularity, that is, when x1 ∼ R(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt, y) it can be represented as ∂F ∂y (x, y) = f1(x, y) + f2(x, y) � 1 − x1 R(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt, y) �−1/2 , where f1 and f2 are analytic and f2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Since G is strictly increasing in x1 and R is a proper singularity function fully movable in y, R(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt, G(x)) is strictly decreasing in x1 and goes to 0 as x1 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Therefore, (1 − x1/R(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt, G(x)))−1/2 is strictly increasing in x1 and unbounded while x1 < R(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt, G(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The same is true for (∂F/∂y)(x, G(x)) because f1(x, G(x)) and f2(x, G(x)) are strictly increasing functions in x1 when x1 < R(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt, G(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Our claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' From [12, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='19], which amounts to evaluating the parameter u in R+ in [12, The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='21], this implies that for x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt ∈ R+ the univariate function x1 → G(x) has a 1/2- singularity at x1 = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Therefore we set ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) := x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The system formed by equations (15) and (16) can be used to get more information on ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Notice first that, given x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt, the system determines x1 = ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) and G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then, since the determinant �������� −eF ∂F ∂x1 1 − eF ∂F ∂y −eF ∂F ∂x1 ∂F ∂y − eF ∂2F ∂y∂x1 −eF �∂F ∂y �2 − eF ∂2F ∂y2 �������� = e2F �������� ∂F ∂x1 0 ∂F ∂x1 ∂F ∂y + ∂2F ∂y∂x1 �∂F ∂y �2 + ∂2F ∂y2 �������� = e2F ∂F ∂x1 ��∂F ∂y �2 + ∂2F ∂y2 � is positive, it follows by the IFT that the function ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) can be locally analytically continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Fix now some 2 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' By differentiating (15) with respect to xj, one obtains 0 = ∂[G(x)] ∂xj − exp(F(x, G(x))) � ∂F ∂x1 (x, G(x)) ∂ρ ∂xj + ∂F ∂xj (x, G(x)) + ∂F ∂y (x, G(x))∂[G(x)] ∂xj � = − exp(F(x, G(x))) � ∂F ∂x1 (x, G(x)) ∂ρ ∂xj + ∂F ∂xj (x, G(x)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In other words, ∂ρ ∂xj = − ∂F ∂xj (x, G(x)) ∂F ∂x1 (x, G(x)) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This means that ρ is strictly decreasing in all variables, provided they are real and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let us finally consider the behaviour of ρ as xj tends to 0 or +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Suppose first that x1 = ρ is bounded away from 0 when xj → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then G(x) → +∞, and R → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' However, this is impossible since 15 x1 < R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Thus ρ → 0 as xj → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' On the other hand, suppose that x1 = ρ stays bounded when xj → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In this case, G(x) stays bounded and so by assumptions on the zeros of F, F(x, G(x))) → 0 and (∂F/∂y)(x, G(x))) → 0 as xj → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' However, this is not possible by (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Summing up, this means from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2 that the function ρ is a proper singularity function that is fully movable with respect to x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Furthermore, it follows from [12, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='21] that we also get an expansion of the form (7) with α = 1/2 for G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' And it remains to check that for x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt ∈ R+, the mapping x1 �→ G(x) admits an analytic continuation away from ρ, in a region of the form (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' To that end, we now consider (14) as a functional equation for the function x1 �→ G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Since G(x) can be written as G(x) = 1 + ˜G(x), where ˜G is a power series with non-negative coefficients, we have that G(x) = exp(F(x, G(x))) = exp(F(x, 1 + ˜G(x))) = exp(F(x, 1) + ˜F(x, ˜G(x))) = 1 + F(x, 1) + ˜F(x, ˜G(x)) + � n≥2 (F(x, 1) + ˜F(x, ˜G(x)))n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , where ˜F(x, y) is a power series with non-negative coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Now, since F(x, y) is aperiodic in x1, it follows that G(x) has to also be aperiodic with respect to x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This implies that ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) is the unique dominant singularity of the function x1 �→ G(x) and we conclude by a standard compactness argument that it can be analytically continued to a region of the form (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='3 Proofs of the main results Fix t ≥ 1 and recall that starting with Gt+1, which is an explicit monomial, one can recursively obtain the generating functions Gt, Gt−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , G1, and finally G0 = exp(G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let us next discuss the first step of this induction, from Gt+1 to Gt, since it is slightly different from the general step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let t ≥ 1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then there exists two functions h1(x) and h2(x), that are analytic and non-zero at x = 1/e, such that for x ∼ 1/e we have Gt(x) = �t j=1 x(t j) j t!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' � h1(tX) + h2(tX)(1 − etX)3/2� , where X = t� j=1 x( t j−1) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (17) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' From Equation (2) and by (3) we directly get G(t) t+1(x) = t� j=1 x( t j−1) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Consequently, from the relation (4) the function G(t) t = G(t) t (x) satisfies the equation G(t) t = exp � � t� j=1 x( t j−1) j [G(t) t ]t � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 16 Let T(z) denote the tree function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' that satisfies the equation T(z) = z exp(T(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then, using the change of variable z = tX with X = �t j=1 x( t j−1) j , we can represent G(t) t (x) as G(t) t (x) = �T(tX) tX �1/t = exp (T(tX)/t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' With the help of (6) and the relation T(x) = x exp(T(x)), this also leads to Gt(x) = 1 t!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' t� j=1 x(t j) j G(t) t (x) − t (t + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' t� j=1 x(t+1 j ) j � G(t) t (x) �t+1 = �t j=1 x(t j) j t!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' exp �T(tX) t � � 1 − T(tX) t + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (18) It is well known (see for instance [15]) that T(z) has its dominant singularity at z0 = 1/e and a local Puiseux expansion at z ∼ z0 of the form T(z) = 1 − √ 2 √ 1 − ez + 2 3(1 − ez) − 11 √ 2 36 (1 − ez)3/2 + O � (1 − ez)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Furthermore, z0 = 1/e is the only singularity on the circle |z| = 1/e and T(z) can be analytically continued to a region of the form |z| < 1/e + δ/2, |z − 1/e| > δ for some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Hence we get exp �T(x) t � � 1 − T(x) t + 1 � = te1/t t + 1 � 1 − 1 t2 (1 − ex) + 2 √ 2(t + 1) 3t3 (1 − ex)3/2 + O � (1 − ex)2� � = h1(x) + h2(x)(1 − ex)3/2, where h1(x) and h2(x) are functions that are analytic and non-zero at x ∼ 1/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This directly leads to the claimed local representation of Gt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Note that the appearance of the dominant singularity (1 − etX)3/2 is not unexpected since G(t) t (x) has a dominant singularity of the form √ 1 − etX and Gt(x) and is as per (5) – more or less – the integral of G(t) t (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Furthermore, one can deduce from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='8 the case k = t of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1 by a direct application of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1 (setting u = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Similarly, a central limit theorem for the case k = t of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2 follows immediately from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='8 by an application of [12, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For k < t, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2 can also be deduced from local representations of a form similar to (17), modulo some technical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Thus the main step of the proofs is to show that the above representation for Gt(x) implies corresponding representation for Gt−1(x), Gt−2(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , G1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This is the object of the next proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Before stating it, let us remark the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For k ∈ [t + 1], the function Gk(x) admits � j∈[k] x(k j) j as a factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' If k = t + 1, this is Equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' For k ≤ t, it follows from Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' And by (3) this implies that the function G(k−1) k (x) has � j∈[k] x(k−1 j−1) j as a factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In particular, G(k−1) k (x) is zero if one of the variables x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 17 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Suppose that 2 ≤ k ≤ t and let Gk(x) be a positive function with a proper 3/2-singularity that is aperiodic and analytically continuable with respect to x1, and with a proper singularity function ρk(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) that is fully movable with respect to x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Then the function Gk−1(x) is also a positive function with a proper 3/2-singularity that is aperi- odic and analytically continuable with respect to x1, where the singularity function ρk−1(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) is again fully movable with respect to x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Moreover, if x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt ∈ R+ then ρk−1(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) < ρk(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' (19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In a first step, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='5 we replace ρk(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) by the proper singularity function κk = κk(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−2, xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt), that is fully movable with respect to x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−2, so that we can represent Gk(x) as Gk(x) = g1(x) + g2(x) � 1 − xk−1 κk �3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Next, we deduce from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='6 and the relation (3) that G(k−1) k (x) is a positive function with a proper 1/2-singularity and admits the same proper singularity function κk as Gk(x), in particular it is fully movable with respect to x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' From there we apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='7 to the relation (4), noting that F(x, y) = G(k−1) k (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−2, xk−1y, xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) is a positive function with a proper 1/2-singularity and that by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='9 F(x, y) has zeros x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Furthermore, it admits a proper singularity function given by R(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−2, xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt, y) = 1 yκk(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−2, xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Clearly R is fully movable in x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−2, xk and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Consequently, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='7 the solution function y = G(k−1) k−1 (x) is a positive function with a proper 1/2-singularity, and leading variable xk−1, for which the singularity function κk−1 = κk−1(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−2, xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) satisfies κk−1(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−2, xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) < κk(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−2, xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) G(k−1) k−1 (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Note that G(k−1) k−1 (0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Hence, G(k−1) k−1 (x) > 1, and it follows that κk−1(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−2, xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) < κk(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk−2, xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Finally, we apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='6 on relation (5) and obtain that Gk−1(x) is a positive function with a proper 3/2-singularity and leading variable xk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' By another application of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='5 we see that we can change it back to the leading variable x1, such that the corresponding proper singularity function ρk−1(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk) satisfies (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' We are now in a position to prove the two main results of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 18 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='8 implies that Gt(x) is a positive function with a proper 3/2-singularity, and is aperiodic and analytically continuable with respect to x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In particular, compare (7) with (17) and note that x1 appears in X only in the first power x(t 0) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' In this case, the proper singularity function is explicitly given by ρt(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) = 1 et t� j=2 x −( t j−1) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' From there, successive applications of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='10 imply that the function Gk(x) also has these properties for each k ∈ [t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Since the exponential is an entire function this also holds for G(x) = G0(x) = exp(G1(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' And we conclude the proof by setting x2 = · · · = xt = 1 then applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Suppose that (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) is in a sufficiently small complex neigh- bourhood U of (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , 1) in Ct−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='8 then k − 1 succesive applications of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='10, we derive a local representation of the form (7) holds for G(x) = Gk(x), with ρ(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk) = ρk(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk) and α = 3/2, when x1 is close to ρk(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Furthermore, by continuity there exists δ > 0 such that the function x1 �→ Gk(x) is still analytically continuable to a region of the form (8), with ρ = ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' And we deduce from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1 the following asymptotic estimate for the coefficients of x1 in Gk(x) [xn 1] Gk(x) = Ck(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk) n−5/2 ρk(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt)−n (1 + o(1)) as n → ∞, for some non-zero function Ck(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk) analytic in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This leads to a quasi-power situation for the probability generating function E � xX2 2 · · xXt t � = [xn 1] Gk(x) [xn 1] Gk(x1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , 1) ∼ Ck(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xk) Ck(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , 1) � ρk(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , 1) ρk(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' , xt) �n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Finally, setting xj = euj and λn = n in [12, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='22] implies the claimed joint central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Note that one could alternatively apply [12, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Furthermore, the relation G0(x) = exp (G1(x)) implies, as above, that G(x) = G0(x) has the same singularities and singular expansion as G1(x), up to a multiplicative constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 4 Concluding remarks With the help of a computer algebra system, making use of the representation in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='7, we have been able to compute the following table of numerical values for the singularities ρt,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' We have stopped at t = 7 since the size of the system of functional equations needed to determine ρt,k grows too fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let us mention a recent result giving an estimate cn−5/2γnn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' for the number of labelled planar chordal graphs with γ ≈ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='89 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Is is easy to see that the class of chordal graphs with tree-width at most three is exactly the same as the class of chordal graphs not containing K5 as a minor, whose asymptotic growth is, according to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='1 and the table above, of the form cn−5/2δnn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' with δ = 1/ρ3,1 ≈ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 19 k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 t = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='36788 t = 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='14665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='18394 t = 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='07703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='08421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='12263 t = 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='04444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='04662 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='05664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='09197 t = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='02657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='02732 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='03092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='04152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='07358 t = 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='01608 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='01635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='01773 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='02184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='03214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='06131 t = 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='00974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='00984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='01038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='01204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='01614 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='02583 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='05255 Table 1: Approximations of the radii of convergence of the generating functions counting k-connected chordal graphs with tree-width at most t for small values of t and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Furthermore, notice that if we denote by C(x) and B(x), respectively, the generating functions of connected and 2-connected graphs in Gt,0, then Equation 4 reads for k = 1 C′(x) = exp(B′(xC′(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' If ρC and ρB are the singularities of C(x) and B(x), respectively, the condition for being subcritical is that ρCC′(ρC) < ρB, so that the singularity of C(x) arises as a branch-point in the former equation and is not inherited by that of B(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' in our case this condition is safistied because of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Since the number of all chordal graphs grows like 2n2/4, we know that the singularity ρt = ρt,1 of chordal graphs with tree-wdith at most t goes to 0 as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' The question is at which rate ρt → 0 as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Since the exponential growth of t-trees is (etn)n, we have ρt = O(1/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' And since the growth of all graphs of tree-width at most t is at most (2ttn)n, we also have ρt = Ω(1/(t2t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' We leave as an open problem to narrow the gap between the upper and lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Heuristic arguments suggest that ρt decreases exponentially in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' As a final question, we consider letting t = t(n) grow with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Recall that a class of labelled graphs is small when the number of graphs in the class grows at most like cnn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' for some c > 0, and large otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' We know that the class of all chordal graphs is large, while the class of chordal graphs with tree-width at most t is small for fixed t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Let us see that if t = (1 + ϵ)(log n) then the class is large for each ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' A graph is split if the vertex set can be partitioned into a clique and an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' It is well-known and easy to prove that split graphs are chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Consider split graphs with a clique of size t = (1 + ϵ) log n and the complement an independent set, so that he largest clique is of size at most t + 1 and the tree-width at most t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Every edge between the clique and the complement can be chosen independently, hence there are at least 2(1+ϵ) log n(n−(1+ϵ) log n) such graphs, a quantity that grows faster than cnn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' for every c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' We leave as an open problem to determine at which order of magnitude between t = O(1) and t = log n the class ceases to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Acknowledgements We gratefully acknowledge earlier discussions with Juanjo Ru´e and Dimitrios Thilikos on the prob- lem of counting chordal graphs with bounded tree-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' 20 The authors acknowledge support from the Marie Curie RISE research network “RandNet” MSCA-RISE-2020-101007705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Moreover, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' was supported by the Special Research Program SFB F50-02 “Algorithmic and Enumerative Combinatorics”, and by the project P35016 “Infinite Singular Systems and Random Discrete Objects” of the FWF (Austrian Science Fund).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' Addi- tionally, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' acknowledge the financial support of the Spanish State Research Agency through projects MTM2017-82166-P and PID2020-113082GB-I00, while M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNAyT4oBgHgl3EQfX_d5/content/2301.00194v1.pdf'} +page_content=' acknowledges sup- port from the Severo Ochoa and Mar´ıa de Maeztu Program for Centers and Units of Excellence (CEX2020-001084-M), and C.' metadata={'source': 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sha256:116a347c10ad886c66012f40dffd416b8efdf092a57c2cc6f2f88e6b2334cff1 +size 177536 diff --git a/nNE2T4oBgHgl3EQfJgZo/content/tmp_files/2301.03692v1.pdf.txt b/nNE2T4oBgHgl3EQfJgZo/content/tmp_files/2301.03692v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a97347593e41bd25ff55c1bb13c1d56fbb8fc71f --- /dev/null +++ b/nNE2T4oBgHgl3EQfJgZo/content/tmp_files/2301.03692v1.pdf.txt @@ -0,0 +1,692 @@ +Fourier Coefficients of Asynchronous Collective Motions in Heavy-ion Collisions +Zhiwan Xu,1, ∗ Gang Wang,1, † Aihong Tang,2 and Huan Zhong Huang1, 3 +1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA +2Brookhaven National Laboratory, Upton, New York 11973 +3Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), +and Institute of Modern Physics, Fudan University, Shanghai-200433, People’s Republic of China +Various modes of collective motions in high-energy heavy-ion collisions could behave like asyn- +chronous processes. +For example, directed flow, elliptic flow and out-of-plane charge separation +originate from different physics mechanisms, and may be not coordinated with each other, whether +they are developed simultaneously or not. If we employ a distinct Fourier expansion with one har- +monic to describe how each asynchronous collective motion affects particle emission, the particle +azimuthal distribution should be the product of all these expansions. Consequently, cross terms +between different harmonics will appear, and the experimental observables based on a linear Fourier +series will deviate from the true coefficients in the factorized form. In this work, we assume the chiral +magnetic effect (CME) and elliptic flow are asynchronous, and investigate how their superposition +impacts the experimental signatures of the CME and the shear-induced CME, respectively. We will +trial this idea using a realistic model of Event-By-Event Anomalous-Viscous Fluid Dynamics. +keywords: CME; elliptic flow; heavy-ion collision +In high-energy heavy-ion collisions, the emission pat- +tern of final-state particles reveals different collectivity +modes of the created nuclear medium. Figure 1 demon- +strates a few examples of collective motions at mid- +rapidities in non-central collisions. +A particular con- +cept is the reaction plane, spanned by impact param- +eter (x-axis) and beam momenta (z-axis). +(a) In the +reaction plane, a slightly tilted participant region [1] +violates the boost invariance, and leads to a rapidity- +odd release of produced particles, known as directed flow +(v1). (b) When viewed along the beam line, the almond- +shaped overlap zone is translated by a hydrodynamic +expansion [2] into a rapidity-even degeneracy between +in-plane and out-of-plane emissions, called elliptic flow +(v2). (c) The chiral magnetic effect (CME) [3] induces +an out-of-plane electric charge separation (a± +1 ), provided +that a quark chirality imbalance emerges from the chiral +anomaly [4], and an intense magnetic field ( ⃗B) is gener- +ated by nuclear fragments [5]. +(d) Recently, a higher- +order effect, the shear-induced CME (siCME) [6] has +been proposed, in which the combination of magnetic +field and hydrodynamic shear creates a charge-dependent +triangular flow (a± +3 ). +To quantify the collective motions for a given kinematic +region, it is convenient to expresses the azimuthal angle +(ϕ) distribution of produced particles in each collision +with a Fourier expansion: +2π +N ± +dN ± +dϕ += 1+ +∞ +� +n=1 +2a± +n sin n∆ϕ+ +∞ +� +n=1 +2v± +n cos n∆ϕ, (1) +where ∆ϕ is the azimuthal angle of a particle relative to +the reaction plane, and the superscripts +, − indicate +∗ zhiwanxu@physics.ucla.edu +† gwang@physics.ucla.edu +x +y +x +z +y +x +⃗ +B +- +- ++ ++ +j +(a) +(b) +(c) +(d) +y +x +⃗ +B +j +Bσ ++ +- ++ +- ++ +- ++ +- +z +z +y +× +z +FIG. 1. Illustration of collective motions in heavy-ion colli- +sions: (a) directed flow, (b) elliptic flow, (c) the CME-induced +electric current j along the ⃗B field, and (d) the siCME- +induced charge-dependent triangular emission. The dashed +arrows represent produced-particle momenta. Bσ denotes the +new component of the ⃗B field caused by its interplay with +shear flow. +the charge sign. For simplicity, we will omit these su- +perscripts in the following discussions, and restore them +only where necessary. The coefficients an ≡ ⟨sin n∆ϕ⟩ +and vn ≡ ⟨cos n∆ϕ⟩ are observables obtained by averag- +ing over particles of interest and over events. However, if +the collectivity modes happen asynchronously, each mode +should be characterized by its own single-harmonic (˜an +or ˜vn) Fourier expansion, and it is the product of these +short expansions that details the particle azimuthal dis- +tribution. Then the an (vn) measured from a long linear +arXiv:2301.03692v1 [nucl-th] 9 Jan 2023 + +2 +Fourier expansion may not fully match the true ˜an (˜vn) +of the pertinent physics process. Consider a special sce- +nario: the initial magnetic field decays so fast that the +CME almost stops at proper time τ = 0.6 fm/c, but +the collision system becomes well equilibrated only after +τ = 0.6 fm/c to start the hydrodynamic evolution. In this +sequence, ˜a1 and ˜v2 should appear in two separate Fourier +expansions, (1+2˜a1 sin ∆ϕ) and (1+2˜v2 cos 2∆ϕ), respec- +tively, and their product specifies the final-state particle +distribution. Compared with Eq. 1, the factorized form +provides an extra cross term: +4˜a1˜v2 sin ∆ϕ cos 2∆ϕ += −2˜a1˜v2 sin ∆ϕ + 2˜a1˜v2 sin 3∆ϕ. +(2) +In this case, the a1 and a3 manifested in the linear Fourier +expansion deviate from the true ˜a1 and ˜a3, respectively. +Admittedly, the aforementioned scenario of the pre- +hydro CME may be impractical, as magnetic field could +be prolonged by the medium induction so that the CME +overlaps with the formation of elliptic flow. +However, +the sequential processes are not required to define “asyn- +chronous”. In general, asynchronous processes, such as +distinct physics mechanisms, could occur simultaneously, +but do not rely on each other’s existence to evolve. For +example, elliptic flow can be developed regardless of the +CME, and vice versa. +As long as the collective mo- +tions have separate agendas to affect particle emission, +we consider them to be not coordinated with each other +and adopt the factorized form of Fourier expansions. In +the experimental extraction of Fourier coefficients, con- +cerns of factorization or rather lack thereof have been +raised from the viewpoint of nonflow [7–9] or decorre- +lation [10, 11], but none of them involves asynchronous +processes. If we make an extreme assumption that all +the collective motions are asynchronous, the particle az- +imuthal distribution can be expressed as +2π +N ± +dN ± +dϕ += +∞ +� +n=1 +(1 + 2˜a± +n sin n∆ϕ) +∞ +� +n=1 +(1 + 2˜v± +n cos n∆ϕ). +(3) +Note that Eq. (3) is not a replacement of Eq. (1), but they +are two representations of the same distribution with dif- +ferent emphases. The coefficients in the former expansion +represent the genuine strengths of the collective motions, +whereas those in the latter are experimental observables. +In reality, the difference between an and ˜an or between +vn and ˜vn is negligible for many harmonics. For example, +the magnitude of ˜an˜am could be much smaller than ˜vn+m +or ˜v|n−m|. We will limit our discussion to only three co- +efficients: ˜a1, ˜a3 and ˜v2, and study how they are related +to their counterparts, a1, a3 and v2. Directed flow is not +included, because v1 is a rapidity-odd function for sym- +metric collisions, and its rapidity-integrated contribution +to other coefficients will be zero in most cases. +Now, +Eq. (3) takes a specific form: +2π +N ± +dN ± +dϕ +∝ (1 + 2˜a± +1 sin ∆ϕ) × (1 + 2˜a± +3 sin 3∆ϕ) +×(1 + 2˜v± +2 cos 2∆ϕ). +(4) +By comparing Eqs. (1) and (4), we find the following +connections between phenomena and noumena, +a1 = ˜a1 − ˜a1˜v2 + ˜a3˜v2, +(5) +a3 = ˜a3 + ˜a1˜v2, +(6) +v2 = ˜v2 + ˜a1˜a3. +(7) +Here we ignore any higher-order term involving ˜a1˜a3˜v2. +Since the magnitude of ˜a1˜a3 is typically lower than +that of v2 by a few decades, v2 and ˜v2 are almost the +same. We will abandon ˜v2, and only use v2 in the fol- +lowing discussions. +Given that the siCME-induced ˜a3 +is much smaller than the CME-induced ˜a1, Eq. (5) in- +dicates that the observed a1 roughly equals ˜a1(1 − v2). +Furthermore, Eq. (6) asserts that the observed a3 con- +tains a contribution of ˜a1v2 on top of the primordial ˜a3, +if any. For completeness, we also express ˜a1 and ˜a3 in +terms of experimental observables: +˜a1 = a1 − a3v2 +1 − v2 − v2 +2 +, +(8) +˜a3 = a3 − a1v2 − a3v2 +1 − v2 − v2 +2 +. +(9) +We use the Event-by-Event Anomalous-Viscous Fluid +Dynamics (EBE-AVFD) model [12–14] to test our infer- +ences from asynchronous processes via relations derived +in Eqs. (5) and (6). +The EBE-AVFD event generator +simulates the dynamical CME transport for u, d and +s quarks in addition to the hydrodynamically expand- +ing viscous medium in heavy-ion collisions, and properly +handles local charge conservation and resonance decays. +We have analyzed 5.8 × 107 events of Au+Au collisions +at √sNN = 200 GeV in the 30–40% centrality range, us- +ing the same settings and input parameters as adopted +in Ref. [15]. For simplicity, we will use the true reaction +plane to perform the simulation analysis, and ignore the +possible fluctuation effects concerning the observables of +elliptic flow and the CME. +The initial conditions for entropy density (s) profiles +and for electromagnetic field vary in accordance with +the event-by-event nucleon configuration from the Monte +Carlo Glauber simulations [16]. The chirality charge den- +sity (n5) is implemented in the form of n5/s, which con- +trols the strength of the CME transport. In this study, +we take a modest value of n5/s = 0.1, the same as used +in Ref. [6]. +The medium expansion is managed by the VISH2+1 +simulation package [17], which is a boost-invariant hydro- +dynamics framework. Consequently, directed flow van- +ishes, and elliptic flow is a major collectivity mode. In +these EBE-AVFD calculations, magnetic field is set to +last long enough, so that for a substantial time period the + +3 +0.2 +− +0.1 +− +0 +0.1 +0.2 +) +- +(h +2 +v +20 +− +10 +− +0 +) +- +(h +1 +a + +× + +3 +10 +) +2 +v +0.01 +± +(1 - 1.26 +× +-3 +10 +× +0.01 +± + = -7.87 +1 +a +) +2 +v +0.03 +± +(1 - 1.08 +× +-3 +10 +× +0.06 +± + = -13.95 +1 +a +) +2 +v +0.02 +± +(1 - 1.14 +× +-3 +10 +× +0.03 +± + = -9.69 +1 +a +(b) +| < 1 +η| + < 2 GeV/c +T +0.2 < p +π +K +p +0.2 +− +0.1 +− +0 +0.1 +0.2 +) ++ +(h +2 +v +0 +10 +20 +) ++ +(h +1 +a + +× + +3 +10 +) +2 +v +0.01 +± +(1 - 1.15 +× +-3 +10 +× +0.01 +± + = 8.28 +1 +a +) +2 +v +0.03 +± +(1 - 1.04 +× +-3 +10 +× +0.06 +± + = 15.68 +1a +) +2 +v +0.02 +± +(1 - 1.03 +× +-3 +10 +× +0.03 +± + = 9.92 +1 +a +(a) +/s = 0.1 +5 +n +200 GeV Au+Au (AVFD): 30-40% +FIG. 2. +Correlations between a1 and v2 calculated on an +event-by-event basis from EBE-AVFD simulations of 30–40% +Au+Au collisions at √sNN = 200 GeV for (a) π+, K+ and +p, and for (b) π−, K− and ¯p. +The fit function, a1(v2) = +a1(0) × (1 + Cv2), returns C close to −1 for all the cases. +CME does coexist with the hydrodynamic formation of +elliptic flow. However, the dynamical CME transport is +governed by anomalous hydrodynamic equations as a lin- +ear perturbation on top of the medium flow background, +since the back-reaction of finite chiral quark densities is +negligible in collisions at √sNN = 200 GeV [12]. There- +fore, the two modes of collective motions featuring ˜a± +1 +and v2, respectively, develop independently of each other, +and satisfy the generalized criterion for asynchronous +processes. We do not invoke the siCME in these sim- +ulations, and thus ˜a± +3 is zero. Accordingly, Eqs. (5) and +(6) become as simple as +a1 = ˜a1(1 − v2), +(10) +a3 = ˜a1v2. +(11) +Figure 2 shows the observed a1 vs v2 from EBE-AVFD +simulations of Au+Au collisions at √sNN = 200 GeV in +the centrality interval of 30–40% for (a) π+, K+ and p, +and for (b) π−, K− and ¯p. Both a1 and v2 are calcu- +lated on an event-by-event basis within the pseudorapid- +ity range of |η| < 1 and the transverse momentum range +of 0.2 < pT < 2 GeV/c. +For all the particle species, +a linear correlation is present between a1 and v2. The +fit function (dashed lines), a1(v2) = a1(0) × (1 + Cv2), +renders the parameter C close to −1 for all the cases, sup- +portive of Eq. (10) and hence the idea of asynchronous +processes. There appears to be an anti-correlation be- +tween the ˜a1 magnitude itself and v2, especially for pi- +ons, whose C values are around −1.2. This higher-order +effect may arise from resonance decays, since secondary +pions will smear and dilute the ˜a1 value averaged over all +pions [12], while inheriting a larger v2 from resonances +at higher pT [18, 19]. The a1(0) or ˜a1 values retrieved +from the fits also exhibit a particle-species dependence, +which could be partially explained by the mean pT ef- +fect. Similar to v2, the ˜a1 magnitude increases with pT +at the low-pT region [12], and pions, kaons, and protons +form an ascending order of mean pT . The quark coales- +cence mechanism [20] could also play a role by making +the ˜a1 magnitude larger for baryons than for mesons at +the intermediate-pT region. +0 +1 +2 +3 +5 +− +0 +5 +(p) +3 +a + +× + +3 +10 +(c) +p +p +0 +1 +2 +3 + (GeV/c) +T +p +5 +− +0 +5 +) +π +( +3 +a + +× + +3 +10 +(a) +/s = 0.1 +5 +n +200 GeV Au+Au (AVFD): 30-40% ++ +π +-π +3 +a +) +2 +/(1-v +2 +v +1 +a +0 +1 +2 +3 + (GeV/c) +T +p +5 +− +0 +5 +(K) +3 +a + +× + +3 +10 +(b) ++ +K +- +K +| < 1 +η| + (GeV/c) +T +p +FIG. 3. EBE-AVFD simulations of a3 as a function of pT in +30–40% Au+Au collisions at 200 GeV for (a) π+ and π−, for +(b) K+ and K−, and for (c) p and ¯p. +In comparison, the +values of a1v2/(1 − v2) are shown with shaded bands. +Figure 3 shows EBE-AVFD calculations of the ob- +served a3 as a function of pT in 30–40% Au+Au collisions +at 200 GeV for (a) π+ and π−, for (b) K+ and K−, and +for (c) p and ¯p. Although the a3 magnitudes for pions +and protons are comparable to the corresponding predic- +tions for the siCME-induced a3 (as shown in the upper +panel of Fig. 3 in Ref. [6]), our simulations do not entail +the siCME. On the contrary, we depict a1v2/(1 − v2) or +˜a1v2 with shaded bands, which describe the trend and +the magnitude of a3 reasonably well for all the particle +species under study. Therefore, the simulations support +Eq. (11), and verify another imprint of the asynchronous +collective motions. At pT > 1 GeV/c, a3 seems to have +a smaller magnitude than ˜a1v2, which could be partially + +4 +explained by the aforementioned anti-correlation between +˜a1 and v2. The gradual breakdown of hydrodynamics to- +wards higher pT could also add to this discrepancy, since +collective motions start to collapse. +In summary, we have explored some consequences of +asynchronous collective motions in high-energy heavy-ion +collisions. With a generalized definition of asynchronous +processes, we express the particle azimuthal distribution +in a factorized form rather than a long linear Fourier +series, to reflect the genuine strength of each collectivity +mode. As a concrete example, we focus on the extra cross +terms between the CME-induced a1, the siCME-induced +a3, and elliptic flow v2, and have confirmed two signifi- +cant outcomes using the EBE-AVFD model. First, the +observed a1 roughly equals the true ˜a1 scaled by (1−v2). +Therefore, the presence of positive elliptic flow will re- +duce the true CME signal, and this effect is more im- +portant at higher pT , where v2 is larger. +Since most +CME-sensitive observables contain a2 +1, the corresponding +reduction factor should be about (1 − 2v2). Second, the +observed a3 receives a sizeable contribution from ˜a1v2, +which complicates the interpretation of this siCME hunt- +ing observable. Nevertheless, a finite a3, if confirmed, al- +ways evidences the CME, whether it originates from ˜a3 +or ˜a1v2 or both. +While the search for the CME and the siCME is +still ongoing, we propose a feasible approach to test +the idea of asynchronous collective motions involving +only the vn coefficients. +If directed flow and ellip- +tic flow are asynchronous in Nature, the product of +(1+2˜v1 cos ∆ϕ) and (1+2˜v2 cos 2∆ϕ) yields a cross term, +4˜v1˜v2 cos ∆φ cos 2∆φ = 2˜v1˜v2(cos ∆φ + cos 3∆φ), simi- +lar to Eq. (2). Therefore, the v1 observed with a linear +Fourier expansion will be the true ˜v1 scaled by a factor of +(1 + ˜v2), and the observed v3 will contain a rapidity-odd +component of ˜v1˜v2 on top of the existing rapidity-even +˜v3, if any. With respect to the reaction plane, ˜v3 is likely +to be zero because the triangular anisotropy in the ini- +tial collision geometry is dominated by event-by-event +fluctuations, and essentially decouples from the reaction +plane [21]. Thus, the observation of a vodd +3 +component es- +tablishes a signature of the factorized Fourier expansions +and hence asynchronous processes. +ACKNOWLEDGMENTS +The authors thank Shuzhe Shi and Jinfeng Liao for +providing the EBE-AVFD code and for the inspirations. +We also thank Yufu Lin for generating the EBE-AVFD +events. +We are especially grateful to Zhongling Ji for +the fruitful discussions. Z. Xu, G. Wang and H. Huang +are supported by the U.S. Department of Energy under +Grant No. +DE-FG02-88ER40424 and by the National +Natural Science Foundation of China under Contract +No.1835002. A.H. Tang is supported by the US Depart- +ment of Energy under Grants No. DE-AC02-98CH10886, +DE-FG02-89ER40531. +[1] P. Bozek and I. Wyskiel, Phys. Rev. C 81, 054902 (2010). +[2] U. Heinz and R. Snellings, Ann. Rev. Nucl. Part. Sci. 63, +123 (2013). +[3] D. E. Kharzeev, L. D. McLerran and H. J. Warringa, +Nucl. Phys. A 803, 227 (2008). +[4] D. Kharzeev, R. D. Pisarski, M. H. G. Tytgat, Phys. 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C 82, 039903 (2010). + diff --git a/nNE2T4oBgHgl3EQfJgZo/content/tmp_files/load_file.txt b/nNE2T4oBgHgl3EQfJgZo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2a7affeb37d3572bd37ae25d1ed5fb29edae575 --- /dev/null +++ b/nNE2T4oBgHgl3EQfJgZo/content/tmp_files/load_file.txt @@ -0,0 +1,373 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf,len=372 +page_content='Fourier Coefficients of Asynchronous Collective Motions in Heavy-ion Collisions Zhiwan Xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' ∗ Gang Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' † Aihong Tang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='2 and Huan Zhong Huang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' 3 1Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Los Angeles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' California 90095,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' USA 2Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' New York 11973 3Key Laboratory of Nuclear Physics and Ion-beam Application (MOE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' and Institute of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Shanghai-200433,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' People’s Republic of China Various modes of collective motions in high-energy heavy-ion collisions could behave like asyn- chronous processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' For example, directed flow, elliptic flow and out-of-plane charge separation originate from different physics mechanisms, and may be not coordinated with each other, whether they are developed simultaneously or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' If we employ a distinct Fourier expansion with one har- monic to describe how each asynchronous collective motion affects particle emission, the particle azimuthal distribution should be the product of all these expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Consequently, cross terms between different harmonics will appear, and the experimental observables based on a linear Fourier series will deviate from the true coefficients in the factorized form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' In this work, we assume the chiral magnetic effect (CME) and elliptic flow are asynchronous, and investigate how their superposition impacts the experimental signatures of the CME and the shear-induced CME, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' We will trial this idea using a realistic model of Event-By-Event Anomalous-Viscous Fluid Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' keywords: CME;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' elliptic flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' heavy-ion collision In high-energy heavy-ion collisions, the emission pat- tern of final-state particles reveals different collectivity modes of the created nuclear medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Figure 1 demon- strates a few examples of collective motions at mid- rapidities in non-central collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' A particular con- cept is the reaction plane, spanned by impact param- eter (x-axis) and beam momenta (z-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (a) In the reaction plane, a slightly tilted participant region [1] violates the boost invariance, and leads to a rapidity- odd release of produced particles, known as directed flow (v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (b) When viewed along the beam line, the almond- shaped overlap zone is translated by a hydrodynamic expansion [2] into a rapidity-even degeneracy between in-plane and out-of-plane emissions, called elliptic flow (v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (c) The chiral magnetic effect (CME) [3] induces an out-of-plane electric charge separation (a± 1 ), provided that a quark chirality imbalance emerges from the chiral anomaly [4], and an intense magnetic field ( ⃗B) is gener- ated by nuclear fragments [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (d) Recently, a higher- order effect, the shear-induced CME (siCME) [6] has been proposed, in which the combination of magnetic field and hydrodynamic shear creates a charge-dependent triangular flow (a± 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' To quantify the collective motions for a given kinematic region, it is convenient to expresses the azimuthal angle (ϕ) distribution of produced particles in each collision with a Fourier expansion: 2π N ± dN ± dϕ = 1+ ∞ � n=1 2a± n sin n∆ϕ+ ∞ � n=1 2v± n cos n∆ϕ, (1) where ∆ϕ is the azimuthal angle of a particle relative to the reaction plane, and the superscripts +, − indicate ∗ zhiwanxu@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='edu † gwang@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='edu x y x z y x ⃗ B + + j (a) (b) (c) (d) y x ⃗ B j Bσ + + + + z z y × z FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Illustration of collective motions in heavy-ion colli- sions: (a) directed flow, (b) elliptic flow, (c) the CME-induced electric current j along the ⃗B field, and (d) the siCME- induced charge-dependent triangular emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' The dashed arrows represent produced-particle momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Bσ denotes the new component of the ⃗B field caused by its interplay with shear flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' the charge sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' For simplicity, we will omit these su- perscripts in the following discussions, and restore them only where necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' The coefficients an ≡ ⟨sin n∆ϕ⟩ and vn ≡ ⟨cos n∆ϕ⟩ are observables obtained by averag- ing over particles of interest and over events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' However, if the collectivity modes happen asynchronously, each mode should be characterized by its own single-harmonic (˜an or ˜vn) Fourier expansion, and it is the product of these short expansions that details the particle azimuthal dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Then the an (vn) measured from a long linear arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='03692v1 [nucl-th] 9 Jan 2023 2 Fourier expansion may not fully match the true ˜an (˜vn) of the pertinent physics process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Consider a special sce- nario: the initial magnetic field decays so fast that the CME almost stops at proper time τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='6 fm/c, but the collision system becomes well equilibrated only after τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='6 fm/c to start the hydrodynamic evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' In this sequence, ˜a1 and ˜v2 should appear in two separate Fourier expansions, (1+2˜a1 sin ∆ϕ) and (1+2˜v2 cos 2∆ϕ), respec- tively, and their product specifies the final-state particle distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Compared with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' 1, the factorized form provides an extra cross term: 4˜a1˜v2 sin ∆ϕ cos 2∆ϕ = −2˜a1˜v2 sin ∆ϕ + 2˜a1˜v2 sin 3∆ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (2) In this case, the a1 and a3 manifested in the linear Fourier expansion deviate from the true ˜a1 and ˜a3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Admittedly, the aforementioned scenario of the pre- hydro CME may be impractical, as magnetic field could be prolonged by the medium induction so that the CME overlaps with the formation of elliptic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' However, the sequential processes are not required to define “asyn- chronous”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' In general, asynchronous processes, such as distinct physics mechanisms, could occur simultaneously, but do not rely on each other’s existence to evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' For example, elliptic flow can be developed regardless of the CME, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' As long as the collective mo- tions have separate agendas to affect particle emission, we consider them to be not coordinated with each other and adopt the factorized form of Fourier expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' In the experimental extraction of Fourier coefficients, con- cerns of factorization or rather lack thereof have been raised from the viewpoint of nonflow [7–9] or decorre- lation [10, 11], but none of them involves asynchronous processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' If we make an extreme assumption that all the collective motions are asynchronous, the particle az- imuthal distribution can be expressed as 2π N ± dN ± dϕ = ∞ � n=1 (1 + 2˜a± n sin n∆ϕ) ∞ � n=1 (1 + 2˜v± n cos n∆ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (3) Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (3) is not a replacement of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (1), but they are two representations of the same distribution with dif- ferent emphases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' The coefficients in the former expansion represent the genuine strengths of the collective motions, whereas those in the latter are experimental observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' In reality, the difference between an and ˜an or between vn and ˜vn is negligible for many harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' For example, the magnitude of ˜an˜am could be much smaller than ˜vn+m or ˜v|n−m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' We will limit our discussion to only three co- efficients: ˜a1, ˜a3 and ˜v2, and study how they are related to their counterparts, a1, a3 and v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Directed flow is not included, because v1 is a rapidity-odd function for sym- metric collisions, and its rapidity-integrated contribution to other coefficients will be zero in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Now, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (3) takes a specific form: 2π N ± dN ± dϕ ∝ (1 + 2˜a± 1 sin ∆ϕ) × (1 + 2˜a± 3 sin 3∆ϕ) ×(1 + 2˜v± 2 cos 2∆ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (4) By comparing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (1) and (4), we find the following connections between phenomena and noumena, a1 = ˜a1 − ˜a1˜v2 + ˜a3˜v2, (5) a3 = ˜a3 + ˜a1˜v2, (6) v2 = ˜v2 + ˜a1˜a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (7) Here we ignore any higher-order term involving ˜a1˜a3˜v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Since the magnitude of ˜a1˜a3 is typically lower than that of v2 by a few decades, v2 and ˜v2 are almost the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' We will abandon ˜v2, and only use v2 in the fol- lowing discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Given that the siCME-induced ˜a3 is much smaller than the CME-induced ˜a1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (5) in- dicates that the observed a1 roughly equals ˜a1(1 − v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Furthermore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (6) asserts that the observed a3 con- tains a contribution of ˜a1v2 on top of the primordial ˜a3, if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' For completeness, we also express ˜a1 and ˜a3 in terms of experimental observables: ˜a1 = a1 − a3v2 1 − v2 − v2 2 , (8) ˜a3 = a3 − a1v2 − a3v2 1 − v2 − v2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (9) We use the Event-by-Event Anomalous-Viscous Fluid Dynamics (EBE-AVFD) model [12–14] to test our infer- ences from asynchronous processes via relations derived in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' The EBE-AVFD event generator simulates the dynamical CME transport for u, d and s quarks in addition to the hydrodynamically expand- ing viscous medium in heavy-ion collisions, and properly handles local charge conservation and resonance decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' We have analyzed 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='8 × 107 events of Au+Au collisions at √sNN = 200 GeV in the 30–40% centrality range, us- ing the same settings and input parameters as adopted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' For simplicity, we will use the true reaction plane to perform the simulation analysis, and ignore the possible fluctuation effects concerning the observables of elliptic flow and the CME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' The initial conditions for entropy density (s) profiles and for electromagnetic field vary in accordance with the event-by-event nucleon configuration from the Monte Carlo Glauber simulations [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' The chirality charge den- sity (n5) is implemented in the form of n5/s, which con- trols the strength of the CME transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' In this study, we take a modest value of n5/s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='1, the same as used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' The medium expansion is managed by the VISH2+1 simulation package [17], which is a boost-invariant hydro- dynamics framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Consequently, directed flow van- ishes, and elliptic flow is a major collectivity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' In these EBE-AVFD calculations, magnetic field is set to last long enough, so that for a substantial time period the 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='2 ) (h 2 v 20 − 10 − 0 ) (h 1 a × 3 10 ) 2 v 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='01 ± (1 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='26 × 3 10 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='01 ± = -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='87 1 a ) 2 v 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='03 ± (1 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='08 × 3 10 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='06 ± = -13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='95 1 a ) 2 v 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='02 ± (1 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='14 × 3 10 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='03 ± = -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='69 1 a (b) | < 1 η| < 2 GeV/c T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='2 < p π K p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='1 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='2 ) + (h 2 v 0 10 20 ) + (h 1 a × 3 10 ) 2 v 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='01 ± (1 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='15 × 3 10 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='01 ± = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='28 1 a ) 2 v 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='03 ± (1 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='04 × 3 10 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='06 ± = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='68 1a ) 2 v 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='02 ± (1 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='03 × 3 10 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='03 ± = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='92 1 a (a) /s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='1 5 n 200 GeV Au+Au (AVFD): 30-40% FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Correlations between a1 and v2 calculated on an event-by-event basis from EBE-AVFD simulations of 30–40% Au+Au collisions at √sNN = 200 GeV for (a) π+, K+ and p, and for (b) π−, K− and ¯p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' The fit function, a1(v2) = a1(0) × (1 + Cv2), returns C close to −1 for all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' CME does coexist with the hydrodynamic formation of elliptic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' However, the dynamical CME transport is governed by anomalous hydrodynamic equations as a lin- ear perturbation on top of the medium flow background, since the back-reaction of finite chiral quark densities is negligible in collisions at √sNN = 200 GeV [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' There- fore, the two modes of collective motions featuring ˜a± 1 and v2, respectively, develop independently of each other, and satisfy the generalized criterion for asynchronous processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' We do not invoke the siCME in these sim- ulations, and thus ˜a± 3 is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Accordingly, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (5) and (6) become as simple as a1 = ˜a1(1 − v2), (10) a3 = ˜a1v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (11) Figure 2 shows the observed a1 vs v2 from EBE-AVFD simulations of Au+Au collisions at √sNN = 200 GeV in the centrality interval of 30–40% for (a) π+, K+ and p, and for (b) π−, K− and ¯p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Both a1 and v2 are calcu- lated on an event-by-event basis within the pseudorapid- ity range of |η| < 1 and the transverse momentum range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='2 < pT < 2 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' For all the particle species, a linear correlation is present between a1 and v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' The fit function (dashed lines), a1(v2) = a1(0) × (1 + Cv2), renders the parameter C close to −1 for all the cases, sup- portive of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (10) and hence the idea of asynchronous processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' There appears to be an anti-correlation be- tween the ˜a1 magnitude itself and v2, especially for pi- ons, whose C values are around −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' This higher-order effect may arise from resonance decays, since secondary pions will smear and dilute the ˜a1 value averaged over all pions [12], while inheriting a larger v2 from resonances at higher pT [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' The a1(0) or ˜a1 values retrieved from the fits also exhibit a particle-species dependence, which could be partially explained by the mean pT ef- fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Similar to v2, the ˜a1 magnitude increases with pT at the low-pT region [12], and pions, kaons, and protons form an ascending order of mean pT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' The quark coales- cence mechanism [20] could also play a role by making the ˜a1 magnitude larger for baryons than for mesons at the intermediate-pT region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' 0 1 2 3 5 − 0 5 (p) 3 a × 3 10 (c) p p 0 1 2 3 (GeV/c) T p 5 − 0 5 ) π ( 3 a × 3 10 (a) /s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content='1 5 n 200 GeV Au+Au (AVFD): 30-40% + π π 3 a ) 2 /(1-v 2 v 1 a 0 1 2 3 (GeV/c) T p 5 − 0 5 (K) 3 a × 3 10 (b) + K K | < 1 η| (GeV/c) T p FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' EBE-AVFD simulations of a3 as a function of pT in 30–40% Au+Au collisions at 200 GeV for (a) π+ and π−, for (b) K+ and K−, and for (c) p and ¯p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' In comparison, the values of a1v2/(1 − v2) are shown with shaded bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Figure 3 shows EBE-AVFD calculations of the ob- served a3 as a function of pT in 30–40% Au+Au collisions at 200 GeV for (a) π+ and π−, for (b) K+ and K−, and for (c) p and ¯p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Although the a3 magnitudes for pions and protons are comparable to the corresponding predic- tions for the siCME-induced a3 (as shown in the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' 3 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' [6]), our simulations do not entail the siCME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' On the contrary, we depict a1v2/(1 − v2) or ˜a1v2 with shaded bands, which describe the trend and the magnitude of a3 reasonably well for all the particle species under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Therefore, the simulations support Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (11), and verify another imprint of the asynchronous collective motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' At pT > 1 GeV/c, a3 seems to have a smaller magnitude than ˜a1v2, which could be partially 4 explained by the aforementioned anti-correlation between ˜a1 and v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' The gradual breakdown of hydrodynamics to- wards higher pT could also add to this discrepancy, since collective motions start to collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' In summary, we have explored some consequences of asynchronous collective motions in high-energy heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' With a generalized definition of asynchronous processes, we express the particle azimuthal distribution in a factorized form rather than a long linear Fourier series, to reflect the genuine strength of each collectivity mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' As a concrete example, we focus on the extra cross terms between the CME-induced a1, the siCME-induced a3, and elliptic flow v2, and have confirmed two signifi- cant outcomes using the EBE-AVFD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' First, the observed a1 roughly equals the true ˜a1 scaled by (1−v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Therefore, the presence of positive elliptic flow will re- duce the true CME signal, and this effect is more im- portant at higher pT , where v2 is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Since most CME-sensitive observables contain a2 1, the corresponding reduction factor should be about (1 − 2v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Second, the observed a3 receives a sizeable contribution from ˜a1v2, which complicates the interpretation of this siCME hunt- ing observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Nevertheless, a finite a3, if confirmed, al- ways evidences the CME, whether it originates from ˜a3 or ˜a1v2 or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' While the search for the CME and the siCME is still ongoing, we propose a feasible approach to test the idea of asynchronous collective motions involving only the vn coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' If directed flow and ellip- tic flow are asynchronous in Nature, the product of (1+2˜v1 cos ∆ϕ) and (1+2˜v2 cos 2∆ϕ) yields a cross term, 4˜v1˜v2 cos ∆φ cos 2∆φ = 2˜v1˜v2(cos ∆φ + cos 3∆φ), simi- lar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Therefore, the v1 observed with a linear Fourier expansion will be the true ˜v1 scaled by a factor of (1 + ˜v2), and the observed v3 will contain a rapidity-odd component of ˜v1˜v2 on top of the existing rapidity-even ˜v3, if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' With respect to the reaction plane, ˜v3 is likely to be zero because the triangular anisotropy in the ini- tial collision geometry is dominated by event-by-event fluctuations, and essentially decouples from the reaction plane [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Thus, the observation of a vodd 3 component es- tablishes a signature of the factorized Fourier expansions and hence asynchronous processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors thank Shuzhe Shi and Jinfeng Liao for providing the EBE-AVFD code and for the inspirations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' We also thank Yufu Lin for generating the EBE-AVFD events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' We are especially grateful to Zhongling Ji for the fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE2T4oBgHgl3EQfJgZo/content/2301.03692v1.pdf'} +page_content=' Z.' metadata={'source': 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b/otE1T4oBgHgl3EQfOwPh/content/tmp_files/2301.03020v1.pdf.txt @@ -0,0 +1,1623 @@ +arXiv:2301.03020v1 [math.DG] 8 Jan 2023 +STABLE ANISOTROPIC CAPILLARY HYPERSURFACES +IN THE HALF-SPACE +JINYU GUO AND CHAO XIA +Abstract. In this paper, we study stability problem of anisotropic capillary hypersurfaces in +an Euclidean half-space. We prove that any compact immersed anisotropic capillary constant +anisotropic mean curvature hypersurface in the half-space is weakly stable if and only if it is +a truncated Wulff shape. On the other hand, we prove a Bernstein-type theorem for stable +anisotropic capillary minimal surfaces in the three dimensional half-space under Euclidean area +growth assumption. +1. Introduction +Capillary phenomena appear in the study of the equilibrium shape of liquid drops and crystals +in a given solid container. The mathematical model has been established through the work of +Young, Laplace, Gauss and others, as a variational problem on minimizing a free energy func- +tional under a volume constraint. For more detailed description on the isotropic and anisotropic +capillary phenomena, we refer to [18] and [14]. +A capillary hypersurface in a domain B of a Riemannian manifold is an immersed constant +mean curvature (CMC) hypersurface in B which intersects ∂B at a constant contact angle. Cap- +illary hypersurfaces are the stationary points of an energy functional under volume preserving +variation. As a special and important case, free boundary CMC hypersurfaces are the station- +ary points of an area functional under volume preserving variation. A capillary hypersurface +is called weakly stable if it is local energy-minimizing under volume preserving variation. The +stability of free boundary CMC or capillary hypersurfaces was initiated by Ros-Vergasta [42] +and Ros-Souam [43]. The classification problem for stable compact free boundary CMC or cap- +illary hypersurfaces in an Euclidean ball and in an Euclidean half-space Rn+1 ++ +have been studied +intensively, see for instance [38, 39, 4, 1, 46, 35]. The classification has been completed eventu- +ally by Wang and the second-named author [50] for the Euclidean ball case and Souam [47] for +the Euclidean half-space case respectively. This generalizes the classical result of Barbosa-do +Carmo [5] on the classfication of stable closed CMC hypersurfaces in Rn+1. See also recent work +[22, 55]. +A modern formulation of Gauss’ model of capillary phenomena includes a possibly anisotropic +surface tension density, which we are interested in this paper. +Let F : Sn → R+ be a positive smooth function such that the matrix (D2F + FId) is positive +definite, where D2F is the Hessian of F and Id denotes the identity matrix. F determines a +unique strictly convex hypersurface WF in Rn+1, which is called the Wulff shape. For a closed +hypersurface Σ immersed in Rn+1, the anisotropic area functional is given by +AF(Σ) = +� +Σ +F(ν)dA. +Key words and phrases. anisotropic capillary hypersurfaces, Wulff shape, stability, Bernstein’s theorem. +JG is supported by Shuimu Tsinghua Scholar Program (No.2022SM046) and China Postdoctoral Science Foun- +dation (No.2022M720079). CX is supported by the NSFC (Grant No.11871406, 12271449). +1 + +2 +JINYU GUO AND CHAO XIA +The well-known Wulff theorem (see for example [19, 49]) says that the Wulff shape (up to trans- +lation and homothety) is the global minimizer to the anisotropic area functional under fixed +volume constraint. From variational point of view, the stationary points for the anisotropic area +functional under volume-preserving variations are closed hypersurfaces with constant anisotropic +mean curvature (CAMC). Palmer [40] (see also Winklmann [52]) proved that Wulff shape (up +to translation and homothety) is the only stable CAMC hypersurfaces, which is the anisotropic +counterpart of Barbosa-do Carmo’s [5] result. For more rigidity results on closed CAMC hyper- +surfaces and related problems, we refer to [20, 24, 25, 29, 30]. +For a compact, orientable hypersurface Σ immersed in some container B ⊂ Rn+1 with bound- +ary ∂Σ, which intersects ∂B transversely, the anisotropic energy functional is given by +EF (Σ) = AF (Σ) + ω0AW(Σ), +where ω0 ∈ R is a real number, AW(Σ) is so-called wetting area. We remark that throughout +this paper, Σ will be always orientable. +The global minimizer of EF under fixed volume constraint has been characterized by Winter- +bottom [54] (see also [31]) as a truncated Wulff shape (it is also called a Winterbottom shape), +which can be viewed as the capillary counterpart of Wulff shape. For anisotropic free energy +functionals involving a gravitational potential energy term, the existence, the regularity and +boundary regularity for global or local minimizers have been studied by De Giorgi [13], Almgren +[2], Taylor [48] and De Philippis-Maggi [14, 15]. For the symmetry and uniqueness of global +minimizers, we refer to the work of Baer [3] for a class of F with certain symmetry and the work +of Gonzalez [21] in the isotropic case via a symmetrization technique. +From variational point of view, the stationary points of EF under fixed volume constraint are +the anisotropic capillary CAMC hypersurfaces, which are of CAMC and satisfy an anisotropic +capillary condition (see (3.9) below). In a series of papers [31, 32, 33], Koiso-Palmer studied the +anisotropic capillary hypersurfaces in a slab (the domain bounded by two parallel hyperplane) +and their stabilities. In Koiso-Palmer’s paper, the second variation of EF is derived for a class +of anisotropy satisfying certain symmetric condition in two dimensions. In this paper, we first +compute the second variation of EF for any anisotropic F in any dimensions. Hence we give a +clear characterization for stability. +Proposition 1.1. An anisotropic capillary CAMC immersion x : Σ → B ⊂ Rn+1 is weakly +stable if and only if +− +� +Σ +(divΣ(AF ∇f) + ⟨AF ◦ dν, dν⟩f)f dA + +� +∂Σ +(⟨AF ∇f, µ⟩ − qFf) f ds ≥ 0, +(1.1) +for any f ∈ C∞(Σ) satisfying +� +Σ f dA = 0. Here qF is given in (3.14) below. +For some notations involved in the above stability inequality (1.1), we refer to Section 3. +Recently, Koiso [28, 27] studied the stability problem of anisotropic capillary CAMC hypersur- +faces in a wedge. In particular, she proved that a compact stable immersed anisotropic capillary +CAMC hypersurface Σ in Rn+1 ++ +with boundary ∂Σ must be a truncated Wulff shape, provided +∂Σ ⊂ Rn is embedded for n = 2 and is convex for n ≥ 3. This extends the result of Choe-Koiso +[8] in the isotropic case. As we mentioned, in the isotropic case, Souam [47] classified stable +capillary hypersurfaces in Rn+1 ++ +without any additional assumption. It is natural to ask whether +the embeddedness condition for n = 2 and the convexity condition for n ≥ 3 in Koiso’s result +[28, 27] can be removed for the classification. In this paper we will give an affirmative answer +and our main result is the following. + +STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE +3 +Theorem 1.1 (Theorem 4.1). A compact, immersed anisotropic capillary CAMC hypersurface +in Rn+1 ++ +is weakly stable if and only if it is a truncated Wulff shape, up to translation and +homothety. +As a special case, we have a classification for stable anisotropic free boundary CAMC hyper- +surfaces. +Corollary 1.1. A compact, immersed anisotropic free boundary CAMC hypersurface in Rn+1 ++ +is weakly stable if and only if it is a truncated Wulff shape, up to translation and homothety. +The proof of Theorem 1.1 is based on the stability inequality (1.1) and the following Minkowski- +type formula +(1.2) +� +Σ +� +n(F(ν) + ω0⟨EF +n+1, ν⟩) − HF⟨x, ν⟩ +� +dA = 0, +where EF +n+1 ∈ Rn+1 is a constant vector given in (4.3) below. Formula (1.2) has been proved by +Jia, Wang, Zhang and the second-named author [26, Theorem 1.3], which was used to prove an +Alexandrov-type theorem for embedded anisotropic capillary hypersurfaces. It is an standard +approach to apply Minkowski-type formula involving no boundary terms to handle the stability +for free boundary or capillary problems, see for example [1, 22, 47, 50]. +In the second part of this paper, we are interested in (strongly) stable anisotropic capillary +minimal surfaces in the half-space. The second variational formula gives the following charac- +terization of strong stability for anisotropic capillary minimal hypersurfaces. +Proposition 1.2. An anisotropic capillary minimal immersion x : Σ → B ⊂ Rn+1 is (strongly) +stable if and only if +− +� +Σ +(divΣ(AF ∇f) + ⟨AF ◦ dν, dν⟩f)f dA + +� +∂Σ +(⟨AF ∇f, µ⟩ − qFf) f ds ≥ 0, +for any f ∈ C∞ +c (Σ). +A classical Bernstein theorem, proved Fischer-Colbrie-Schoen [17], do Carmo-Peng [12] and +Pogorelov [41] independently, says that the only complete stable minimal surfaces in R3 are flat. +Quite recently, Chodosh-Li [9] resolved a well-known conjecture of Schoen [7, Conjecture 2.12] +that the only complete stable minimal hypersurfaces in R4 are flat, see also Catino-Mastrolia- +Roncoroni [6] for another proof. Schoen-Simon-Yau [44] have shown that any complete sta- +ble minimal hypersurfaces in Rn+1 with n + 1 ≤ 6 with Euclidean area growth must be flat. +Bernstein-type theorem for the anisotropic case in R3 has been proved by White [51] under +the Euclidean area growth assumption, by Lin [36] when F is C2-close to 1. Under similar +assumptions, Bernstein-type theorem for the anisotropic case has been studied by Simon [45], +Winklmann [53], Chodosh-Li [9]. It is still open question whether these extra assumptions could +be removed, even in R3. +We refer to Chodosh-Li’s paper [10] on recent progress for stable +anisotropic minimal hypersurfaces. +Initiated by the min-max construction for capillary minimal surfaces, Bernstein-type theorem +for capillary minimal surfaces in R3 ++ has recently attracted much attentions. In particular, Li- +Zhou-Zhu [34] and De Masi-De Philippis [16], independently, proved that any properly immersed +stable capillary minimal surfaces in R3 ++ with quadratic area growth must be a half-plane. By us- +ing Fischer-Colbrie-Schoen’s technique, Hong-Saturnino [23] proved the Bernstein-type theorem +without Euclidean area growth assumption. +It is natural to consider the case of anisotropic capillary minimal surfaces. Here we prove the +following Bernstein-type theorem for stable anisotropic capillary minimal surfaces in R3 ++. + +4 +JINYU GUO AND CHAO XIA +Theorem 1.2 (Theorem 5.1). Let Σ be an immersed anisotropic capillary minimal surface +in R3 ++. Assume that Σ has Euclidean area growth, that is, there exists some C > 0 such that +(1.3) +Area (Σ ∩ Br(0)) < Cr2 +for any r > 0. Then Σ is stable if and only if Σ is a half-plane. +In case F ≡ 1, Theorem 1.2 reduces to [34, Theorem 0.2] or [16, Theorem 6.3]. +The remaining part of this paper is organized as follows. In Section 2 we review some def- +initions and notations about anisotropic geometry. In Section 3 we calculate the first and the +second variation formula of anisotropic energy functional in a general domain. In Section 4 we +present some useful geometric formulas for anisotropic capillary hypersurfaces in Rn+1 ++ +and prove +Theorem 1.1. In Section 5 we discuss the stability of noncompact anisotropic capillary minimal +surface in R3 ++ and prove Theorem 1.2. +2. Notations and Preliminaries +Let F : Sn → R+ be a positive smooth function. Denote by DF and D2F the gradient and +Hessian of F on Sn. Then we require the matrix +(2.1) +AF := (D2F + FId)|x > 0 +for any x ∈ Sn, +where Id denotes the identity on TxSn and “ > ” means the matrix is positive definite. +We define the map +Φ : Sn +−→ +Rn+1 +(2.2) +x +�→ +F(x)x + DF(x) +whose image WF = Φ(Sn) is a smooth strictly convex hypersurface in Rn+1 called the Wulff +shape. +We may regard F as a convex function on Rn+1 by one-homogenous extension of F. Precisely, +set F(x) = |x|F( x +|x|) when x ̸= 0 and F(0) = 0, the new F : Rn+1 → R is a one-homogenous +function on Rn+1 which is C2 ∈ (Rn+1 \ {0}). We use ¯∇ to denote the Euclidean corvariant +derivative of F. It is standard to see that for x ∈ Sn and V, W ∈ TxSn, we have +¯∇F(x) = DF(x) + F(x)x = Φ(x), +(2.3) +¯∇2F(x)(V, W) = (D2F + FId)(x)(V, W) = AF (x)(V, W). +(2.4) +Let B be a closed region in an (n+1)-dimensional Euclidean space Rn+1 with smooth boundary +∂B. Let x : Σ → B be an isometric immersion from a n-dimensional smooth manifold Σ such +that ∂Σ ⊂ ∂B. We denote by ¯∇, ¯∆ and ¯∇2 the gradient, the Laplacian and the Hessian on +Rn+1 respectively, while by ∇, ∆ and ∇2 the gradient, the Laplacian and the Hessian on Σ +respectively. Let T(∂Σ) and N(∂Σ) be the tangent bundle and the normal bundle of ∂Σ as a +co-dimensional two submainfolds in Rn+1. We will use the following terminology for four normal +vector fields. We choose one of the unit normal vector field along x and denote it by ν. We +denote by ¯N the unit outward normal to ∂B in B and µ be the unit outward normal to ∂Σ in +Σ. Let ¯ν be the unit normal to ∂Σ in ∂B such that the bases {ν, µ} and {¯ν, ¯N} have the same +orientation in N(∂Σ). Hence, in N(∂Σ), the following relations hold: +µ = −⟨ν, ¯N⟩¯ν + ⟨µ, ¯N⟩ ¯N, +(2.5) +ν = ⟨µ, ¯N⟩¯ν + ⟨ν, ¯N⟩ ¯N. +(2.6) + +STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE +5 +Equivalently, +¯ν = −⟨ν, ¯N⟩µ + ⟨µ, ¯N⟩ν, +(2.7) +¯N = ⟨µ, ¯N⟩µ + ⟨ν, ¯N⟩ν. +(2.8) +We always assume that Σ interesects ∂B transversally, so that ⟨µ, ¯N⟩ ̸= 0. For a vector field +Y on Rn+1, we denote Y Σ and Y ∂Σ to be the tangential projection of Y on TΣ and on T(∂Σ) +respectively. +Denote by h and H the second fundamental form and the mean curvature of the immersion +x respectively. Precisely, h(X, Y ) = ⟨ ¯∇Xν, Y ⟩ for X, Y ∈ TΣ and H = trg(h). Denote by h∂B +the second fundamental form of ∂B in B, that is, h∂B(X, Y ) = ⟨ ¯∇X ¯N, Y ⟩ for X, Y ∈ T(∂B). +Let νF be the anisotropic normal of Σ given by +(2.9) +νF = Φ(ν) := DF(ν) + F(ν)ν. +Hence DF(ν) = νF − ⟨νF , ν⟩ν = νΣ +F ∈ TΣ. The anisotropic principal curvatures +� +κF +i +�n +i=1 of Σ +are given by the eigenvalues of the anisotropic Weingarten map +(2.10) +dνF = AF (ν) ◦ dν : TΣ → TΣ. +The eigenvalues are real since (AF) is positive definite and symmetric. The anisotropic second +fundamental form and anisotropic mean curvature are denoted respectively by +hF (X, Y ) = ⟨ ¯∇XνF, Y ⟩ = (AF ◦ h)(X, Y ), +(2.11) +HF = trg(dνF ) = trg(AF (ν) ◦ dν) = divΣ(DF(ν)) + HF(ν). +(2.12) +When F ≡ 1, we see AF = IdSn and hence hF and HF are the usual second fundamental form +h and mean curvature H respectively. +3. The first and second variational formula +Let x : Σ → B be an isometric immersion such that x(∂Σ) = x(Σ) ∩ ∂B. By a compactly +supported admissible variation of x, we mean a differentiable map x : (−ǫ, ǫ)×Σ → B such that +x(t, ·) : Σ → B is an immersion satisfying x(t, intΣ) ⊂ intB, x(t, ∂Σ) ⊂ ∂B and the support of +∂ +∂tx(t, ·) is compact for every t ∈ (−ǫ, ǫ) and x(0, ·) = x. +The anisotropic area functional AF : (−ǫ, ǫ) → R and the oriented volume functional V : +(−ǫ, ǫ) → R are given by +AF(t) = +� +˜Σ +F(ν)dAt, +V(t) = +� +[0,t]×˜Σ +x(t, ·)∗dV, +where ˜Σ ⊆ Σ is the support of the variation +∂ +∂tx(t, ·), dAt is the area element of Σ with respect +to the metric induced by x(t, ·) and dV is the volume element in B. When Σ is compact, then +˜Σ = Σ. An admissible variation is said to be volume-preserving if V(t) = V(0) = 0 for each +t ∈ (−ǫ, ǫ). +The wetting area functional AW(t) : (−ǫ, ǫ) → R are defined by +(3.1) +AW(t) = +� +[0,t]×(∂Σ∩˜Σ) +x(t, ·)∗dA∂B, +where dA∂B is the area element of ∂B. + +6 +JINYU GUO AND CHAO XIA +Fix a real number ω0 ∈ R. The anisotropic energy functional EF(t) : (−ǫ, ǫ) → R is defined +by +(3.2) +EF (t) = AF(t) + ω0AW(t). +It is well known that the first variation formulas of V and AW for a compactly supported +admissible variation with variation vector field Y = +∂ +∂tx(t, ·)|t=0 ∈ C∞ +c (Σ) such that Y |∂Σ ∈ +T(∂B) are given by +V′(0) = +� +Σ +⟨Y, ν⟩dA, +(3.3) +A′ +W(0) = +� +∂Σ +⟨Y, µ⟩ds, +(3.4) +where dA and ds are the area element of Σ and ∂Σ respectively. +Next we derive the first +variational formula for EF . +Proposition 3.1. Let x(·, t) be a compactly supported admissible variation with variational +vector field Y . Then +(3.5) +E′ +F (0) = +� +Σ +HF⟨Y, ν⟩dA + +� +∂Σ +⟨Y, µF + ω0¯ν⟩ds, +where +µF := ⟨νF , ν⟩µ − ⟨νF , µ⟩ν = F(ν)µ − ⟨νF , µ⟩ν ∈ N(∂Σ). +(3.6) +The first variational formula (3.5) is known to experts. For completeness, we will give a proof +of Proposition 3.1 in Appendix A. +An interesting property for µF is as follows. +Proposition 3.2. Along ∂Σ, we have +⟨µF , ¯ν⟩ += +−⟨νF, ¯N⟩, +(3.7) +⟨µF, ¯N⟩ += +⟨νF , ¯ν⟩. +(3.8) +Proof. It follows directly by the definition of µF, νF and also the relations (2.7) and (2.8). +□ +Proposition 3.3. Stationary points of EF under compactly supported volume preserving admis- +sible variation are hypersurfaces with constant anisotropic mean curvature (CAMC) satisfying +the anisotropic capillary boundary condition +(3.9) +⟨νF , ¯N⟩ = ω0 +along ∂Σ. +Similarly, stationary points of EF under any compactly supported admissible variation are hy- +persurfaces with zero anisotropic mean curvature and satisfying (3.9). +Proof. From the first variation formulas (3.5) and (3.3), it is clear x has constant anisotropic +mean curvature. Next we show (3.9). From the first variation formulas (3.5) and admissible +condition, we have +⟨Y, µF + ω0¯ν⟩ = 0 for any Y ∈ T(∂B). +(3.10) +Note that µF + ω0¯ν ∈ N(∂Σ). Since any Y ∈ T(∂B) can be split as Y = ⟨Y, ¯ν⟩¯ν + Y ∂Σ, where +Y ∂Σ ∈ T(∂Σ), we see that (3.10) is equivalent that +⟨¯ν, µF ⟩ = −ω0 +along ∂Σ. +(3.11) +From (3.7), we see (3.11) is equivalent to (3.9). +□ + +STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE +7 +Definition 3.1. An immersion x : Σ → B is said to be anisotropic capillary CAMC if it +has CAMC and satisfies boundary condition (3.9). In particular, Σ is called anisotropic free +boundary CAMC hypersurface if ω0 = 0. +Definition 3.2. An immersion x : Σ → B is said to be anisotropic capillary minimal if HF = 0 +and ⟨νF , ¯N⟩ = ω0 along Σ ∩ ∂B. +For a volume-preserving admissible variation Y , by splitting Y = Y Σ + fν, we see +(3.12) +� +Σ +f dA = 0. +Conversely, we have the following fact. +Proposition 3.4. ([1, Proposition 2.1]) Let x : Σ → B be a compact immersed hypersurface with +∂Σ ⊆ ∂B. Then for a given f ∈ C∞(Σ) satisfying +� +Σ f dA = 0, there exists a volume-preserving +admissible variation of x with variational vector field having fν as its normal part. +Next we give the second variational formula for the energy functional EF. This formula is +new in the anisotropic capillary case, even for n = 2. +Proposition 3.5. Let x : Σ → B be an immersed anisotropic capillary CAMC (anisotropic +capillary minimal, respectively) hypersurface and x(·, t) be a volume-preserving (compactly sup- +ported, respectively) admissible variation with variational vector field Y having fν as its normal +part. Then +E′′ +F(0) = − +� +Σ +(divΣ(AF ∇f) + ⟨AF ◦ dν, dν⟩f)f dA + +� +∂Σ +(⟨AF ∇f, µ⟩ − qFf) f ds, +(3.13) +where +qF := +1 +⟨µ, ¯N⟩2 +� +⟨µF , ¯N⟩h∂B(¯ν, ¯ν) + h∂B(¯ν, (νF )∂Σ) +� +− ⟨ν, ¯N⟩ +⟨µ, ¯N⟩hF (µ, µ). +(3.14) +We postpone the proof of Proposition 3.5 to Appendix A. +Definition 3.3. An anisotropic capillary CAMC immersion x : Σ → B is called weakly stable +if E′′ +F(0) ≥ 0 for any volume-preserving admissible variation. +An anisotropic capillary minimal immersion x : Σ → B is called (strongly) stable if E′′ +F (0) ≥ 0 +for any compactly supported admissible variation. +As a consequence of Proposition 3.5 and Proposition 3.4, we get the following +Proposition 3.6. An anisotropic capillary CAMC immersion x : Σ → B is weakly stable if and +only if +− +� +Σ +(divΣ(AF ∇f) + ⟨AF ◦ dν, dν⟩f)f dA + +� +∂Σ +(⟨AF ∇f, µ⟩ − qFf) f ds ≥ 0, +(3.15) +for any f satisfying +� +Σ f dA = 0. +An anisotropic capillary minimal immersion x : Σ → B is (strongly) stable if and only if +(3.15) is satisfied for any f ∈ C∞ +c (Σ). + +8 +JINYU GUO AND CHAO XIA +4. Rigidity for stable anisotropic capillary hypersurfaces in Rn+1 ++ +In this and next section, we consider the anisotropic capillary hypersurfaces in Rn+1 ++ +, that is +the domain is B = Rn+1 ++ +. In this setting, h∂B ≡ 0 and ¯N = −En+1. Abuse of notation, we use +µn+1 and νn+1 to denote ⟨µ, En+1⟩ and ⟨ν, En+1⟩ on Σ respectively. +In this case, qF in the stability inequality (3.15) is given by +qF = − νn+1 +µn+1 +hF (µ, µ) = +1 +F(ν) +� ω0 +µn+1 ++ ⟨νF , µ⟩ +� +hF (µ, µ). +(4.1) +The following proposition is a fundamental fact for anisotropic capillary hypersurfaces in +Rn+1 ++ +. +Proposition 4.1. Let x : Σ → Rn+1 ++ +be an anisotropic capillary immersion. Then µ is an +anisotropic principal direction of ∂Σ in Σ, that is, hF (e, µ) = 0 for any e ∈ T(∂Σ). +Proof. For any e ∈ T(∂Σ), by using (3.9) and (2.8), we have +0 += +∇e⟨νF , En+1⟩ = ⟨ ¯∇eνF , En+1⟩ = ⟨ ¯∇eνF, µn+1µ + νn+1ν⟩ += +µn+1⟨ ¯∇eνF , µ⟩ + νn+1⟨ ¯∇eνF, ν⟩ = µn+1hF (e, µ). +Since µn+1 ̸= 0 along ∂Σ, we get the assertion. +□ +In [26], the following anisotropic Minkowski-type formula has been proved. +Proposition 4.2. ([26, Theorem 1.3]) Let x : Σ → Rn+1 ++ +be a compact anisotropic capillary +immersion. Then +(4.2) +� +Σ +� +n(F(ν) + ω0⟨EF +n+1, ν⟩) − HF ⟨x, ν⟩ +� +dA = 0 +where EF +n+1 is a constant vector field defined by +EF +n+1 = +� +Φ(En+1) +F (En+1) +if ω0 < 0, +− Φ(−En+1) +F (−En+1) +if ω0 ≥ 0. +(4.3) +Note that EF +n+1 satisfies ⟨EF +n+1, En+1⟩ = 1. From boundary condition (3.9) and the anisotropic +Cauchy-Schwarz inequality (see for example [25, Proposition 2.4]), we know that +(4.4) +ω0 ∈ (−F(En+1), F(−En+1)). +It was proved in [26, Proposition 3.4] that +(4.5) +F(z) + ω0 +� +EF +n+1, z +� +> 0, +for any z ∈ Sn. +Since Sn is compact, we have +(4.6) +0 < C1 ≤ F(z) + ω0 +� +EF +n+1, z +� +≤ C2, +for any z ∈ Sn, +where C1 and C2 are constant only depending on ω0. +Next we derive the equations for several geometric quantities. Denote the F-Jacobi operator +JF := divΣ(AF ∇·) + ⟨AF ◦ dν, dν⟩. +Proposition 4.3. The following identities holds on Σ: +JF F(ν) += +⟨∇HF, DF|ν⟩ + tr(h2 +F ), +(4.7) +JF ⟨EF +n+1, ν⟩ += +⟨EF +n+1, ∇HF⟩, +(4.8) +JF ⟨x, ν⟩ += +⟨x, ∇HF ⟩ + HF. +(4.9) + +STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE +9 +Proof. The above formulas have been shown in [37] (also see [7, 52]). For the convenience of +reader, we give a direct computation. +For a fixed p ∈ Σ, let {ei}n +i=1 at p be the local orthonormal basis and ∇e1ej|p = 0. In the +following we calculate at p. We have +divΣ(AF ∇(F(ν))) += +(AijFk ◦ νhkj),i = (Aij ◦ ν),i(Fk ◦ ν)hkj + Aij((Fk ◦ ν)hkj),i += +AijphpiFkhkj + Aij(F;kshsihkj + Fkhkji) += +(Aijphpihkj + Aijhkji)Fk + AijhsihkjF;ks += +(Aijhij),kFk + Aijhsihkj(Ask − Fδks) += +⟨∇HF, DF|ν⟩ + tr(h2 +F) − tr(AFh2)F(ν). +divΣ(AF ∇⟨EF +n+1, ν⟩) += +(Aij∇j⟨EF +n+1, ν⟩),i = (Aij⟨EF +n+1, ¯∇jν⟩),i += +Aijkhki⟨EF +n+1, ¯∇jν⟩ + Aij⟨EF +n+1, ¯∇i ¯∇jν⟩ += +(Aijkhkihjp + Aijhjpi)⟨EF +n+1, ep⟩ − Aijhjkhki⟨EF +n+1, ν⟩ += +(Aijhij),p⟨EF +n+1, ep⟩ − tr(AFh2)⟨EF +n+1, ν⟩ += +⟨EF +n+1, ∇HF⟩ − tr(AFh2)⟨EF +n+1, ν⟩. +divΣ(AF ∇⟨x, ν⟩) += +(Aij∇j⟨x, ν⟩),i = (Aij ◦ ν⟨x, ¯∇jν⟩),i += +Aijkhki⟨x, ¯∇jν⟩ + Aij(hij + ⟨x, ¯∇i ¯∇jν⟩) += +(Aijkhkihjp + Aijhjpi)⟨x, ep⟩ + HF − Aijhjkhki⟨x, ν⟩ += +(Aijhij),p⟨x, ep⟩ + HF − tr(AF h2)⟨x, ν⟩ += +⟨x, ∇HF ⟩ + HF − tr(AF h2)⟨x, ν⟩. +In the above computation we have used Codazzi equation hijk = hikj. +□ +Next we verify the boundary equations for the geometric quantities. +Proposition 4.4. Let x : Σ → Rn+1 ++ +be an anisotropic capillary immersion. Then along ∂Σ, +we have +⟨AF ∇⟨x, ν⟩, µ⟩ = qF⟨x, ν⟩, +(4.10) +⟨AF ∇[F(ν) + ω0⟨EF +n+1, ν⟩], µ⟩ = qF(F(ν) + ω0⟨EF +n+1, ν⟩). +(4.11) +Proof. In this proof we always take value along ∂Σ. From Proposition 4.1 and (2.8), we have +⟨AF ∇⟨x, ν⟩, µ⟩ += +⟨AF ◦ dν(x), µ⟩ = hF (µ, µ)⟨x, µ⟩ += +− νn+1 +µn+1 +hF (µ, µ)⟨x, ν⟩ = qF⟨x, ν⟩. +Here we have used the fact xn+1 = 0 on ∂Rn+1 ++ +. +Similarly, by Proposition 4.1 and (2.8), we have +⟨AF ∇[F(ν) + ω0⟨EF +n+1, ν⟩], µ⟩ += +⟨AF ◦ dν( ¯∇F(ν) + ω0EF +n+1), µ⟩ += +hF (µ, µ)⟨ ¯∇F(ν) + ω0EF +n+1, µ⟩ += +hF (µ, µ) +� +¯∇F(ν) + ω0EF +n+1, +1 +µn+1 +En+1 − νn+1 +µn+1 +ν +� += +qF(F(ν) + ω0⟨EF +n+1, ν⟩). + +10 +JINYU GUO AND CHAO XIA +where in the last equality we use boundary condition ⟨νF , En+1⟩ = ⟨ ¯∇F(ν), En+1⟩ = −ω0 and +the fact ⟨EF +n+1, En+1⟩ = 1. +□ +Initiated by Minkowski-type formula (4.2), we define +(4.12) +ϕ := n(F(ν) + ω0⟨EF +n+1, ν⟩) − HF ⟨x, ν⟩. +Proposition 4.5. Let x : Σ → Rn+1 ++ +be a compact anisotropic capillary CAMC immersion. +Then there hold: +JF ϕ += +ntr(h2 +F ) − H2 +F, +(4.13) +⟨AF ∇ϕ, µ⟩ += +qFϕ, +(4.14) +� +Σ +ϕ dA += +0. +(4.15) +Proof. (4.13) follows from Proposition 4.3 and (4.14) from Proposition 4.4. (4.15) is the Minkowski- +type formula (4.2). +□ +Theorem 4.1. A compact, immersed anisotropic capillary CAMC hypersurface in Rn+1 ++ +is +weakly stable if and only if it is a truncated Wulff shape, up to translation and homothety. +Proof. We first notice that a truncated Wulff shape in Rn+1 ++ +is energy-minimizing ([54, 31]), +hence it is weakly stable. +Next, we let x : Σ → Rn+1 ++ +be a compact weakly stable anisotropic capillary CAMC immersion +and show it is a truncated Wulff shape. From (4.15), we know that ϕ is an admissible test +function in (3.15). Therefore, by (4.14), we have +0 +≤ +− +� +Σ +ϕJF ϕ + +� +∂Σ +ϕ(⟨AF ∇ϕ, µ⟩ − qF ϕ) +(4.16) += +− +� +Σ +[n(F(ν) + ω0⟨EF +n+1, ν⟩) − HF⟨x, ν⟩]JF ϕ += +− +� +Σ +n(F(ν) + ω0⟨EF +n+1, ν⟩)JF ϕ + HF +� +Σ +⟨x, ν⟩JF ϕ. +We compute the last term of (4.16) by Green’s formula. By (4.9), (4.10), (4.14) and (4.15), we +get +� +Σ +⟨x, ν⟩JF ϕ += +� +Σ +ϕJF ⟨x, ν⟩ + +� +∂Σ +⟨x, ν⟩⟨AF ∇ϕ, µ⟩ − ϕ⟨AF ∇⟨x, ν⟩, µ⟩ +(4.17) += +HF +� +Σ +ϕ + +� +∂Σ +⟨x, ν⟩(qF ϕ) − ϕ(qF ⟨x, ν⟩), += +0. +Hence, inserting (4.13) and (4.17) into (4.16), we see +� +Σ +(F(ν) + ω0⟨EF +n+1, ν⟩)(ntr(h2 +F ) − H2 +F )dA ≤ 0. +(4.18) +By (4.5) and the fact that ntr(h2 +F ) ≥ H2 +F , we see the above inequality is in fact an equality. It +follows that ntr(h2 +F ) = H2 +F and in turn Σ is anisotropic umbilical. By [40], Σ is a part of the +Wulff shape (up to translation and homothety). +□ + +STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE +11 +5. Bernstein-type theorem for anisotropic capillary minimal surfaces +In this section we prove the following Bernstein-type theorem for properly immersed, anisotropic +capillary minimal surfaces in R3 ++. +Theorem 5.1. Let Σ be an immersed anisotropic capillary minimal surface in R3 ++. Assume +that Σ has Euclidean area growth, that is, there exists some C > 0 such that +(5.1) +Area (Σ ∩ Br(0)) < Cr2 +for any r > 0. Then Σ is stable if and only if Σ is a half-plane. +Proof. If Σ is a half-plane, then qF = 0 and it is clear that for f ∈ C∞ +c (Σ), +E′′ +F (0) = +� +Σ +AF (∇f, ∇f)dA ≥ 0. +Hence it is stable. +Next let Σ be stable. Let ψ := F(ν) + ω0⟨EF +n+1, ν⟩. From (4.7)-(4.8) and (4.11) we see that +(5.2) +� +JF ψ = trh2 +F +in Σ, +⟨AF ∇ψ, µ⟩ = qFψ +on ∂Σ, +For f ∈ C∞ +c (Σ), we put ψf into stability inequality (3.15): +(5.3) +0 ≤ E′′ +F (0) = − +� +Σ +ψfJF (ψf)dA + +� +∂Σ +ψf[⟨AF ∇(ψf), µ⟩ − qFψf]ds, +we now calculate the first term of (5.3) by integration by parts, +− +� +Σ +ψfJF(ψf)dA +(5.4) += +− +� +Σ +ψf[fJFψ + 2⟨AF ∇ψ, ∇f⟩ + ψdiv(AF ∇f)]dA += +− +� +Σ +ψf 2trh2 +F + 1 +2⟨AF ∇ψ2, ∇f 2⟩ + ψ2fdiv(AF ∇f)dA += +− +� +Σ +ψf 2trh2 +F + ψ2fdiv(AF ∇f)dA + 1 +2 +� +Σ +ψ2div(AF ∇f 2)dA − 1 +2 +� +∂Σ +ψ2⟨AF ∇f 2, µ⟩ds += +− +� +Σ +ψf 2trh2 +F − ψ2⟨AF ∇f, ∇f⟩dA − +� +∂Σ +ψ2f⟨AF∇f, µ⟩ds. +Next we compute the boundary term of (5.3), +� +∂Σ +ψf[⟨AF ∇(ψf), µ⟩ − qFψf]ds += +� +∂Σ +ψf[f(⟨AF ∇ψ, µ⟩ − qF ψ) + ψ⟨AF ∇f, µ⟩]ds +(5.5) += +� +∂Σ +ψ2f⟨AF∇f, µ⟩ds +Inserting (5.4)-(5.5) into (5.3), we get +(5.6) +� +Σ +ψf 2trh2 +F dA ≤ +� +Σ +ψ2⟨AF ∇f, ∇f⟩dA +From (4.6) we obtain that +(5.7) +� +Σ +f 2trh2 +FdA ≤ C2 +2C−1 +1 +� +Σ +⟨AF ∇f, ∇f⟩dA ≤ C2 +2C−1 +1 Λ +� +Σ +|∇f|2dA. +where Λ is maximal eigenvalue of AF on S2. + +12 +JINYU GUO AND CHAO XIA +Using the area growth condition (1.3) and choosing a standard logarithmic cutoff function +(see for example [11, Proposition 1.37]) for f in (5.7), we deduce that trh2 +F = 0. We conclude +that Σ is a half-plane in R3 ++. +□ +Appendix A. Proof of the first and second variational formulas +The appendix is devoted to prove the first and second variational formulas, that is Proposition +3.1 and Proposition 3.5. +Let Y ∈ T(∂B) such that +(A.1) +Y = Y Σ + fν = Y ∂Σ + ⟨Y, µ⟩µ + fν. +We have from (2.8) that +0 = ⟨Y, ¯N⟩ = ⟨µ, ¯N⟩⟨Y, µ⟩ + ⟨ν, ¯N⟩⟨Y, ν⟩ +on ∂Σ, +Therefore, +(A.2) +⟨Y, µ⟩ = − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩f +on ∂Σ. +On the other hand, from (A.1), (A.2) and (2.7), we see Y can be also expressed as follows +(A.3) +Y = Y ∂Σ − +f +⟨µ, ¯N⟩(⟨ν, ¯N⟩µ − ⟨µ, ¯N⟩ν) = Y ∂Σ + +f +⟨µ, ¯N⟩ ¯ν. +It follows that, +(A.4) +⟨Y, ¯ν⟩ = +1 +⟨µ, ¯N⟩f +on ∂Σ. +We use a prime to denote the time derivative at t = 0 in the following. +Lemma A.1. +Let ∇∂Σ denote the gradient on ∂Σ. Let SΣ, S∂B denote the shape operator of +Σ in B with respect to ν and ∂B in B with respect to ¯N respectively. Let S1 and S2 denote the +shape operator of ∂Σ in Σ with respect to µ, and of ∂Σ in ∂B with respect to ¯ν respectively. +Then we have +ν′ += +SΣ(Y Σ) − ∇f, +(A.5) +µ′ += +(−h(Y Σ, µ) + ∇µf)ν + S1(Y ∂Σ) + ⟨ν, ¯N⟩ +⟨µ, ¯N⟩ ∇∂Σf +(A.6) ++ ⟨ν, ¯N⟩2 +⟨µ, ¯N⟩2 f +� +SΣ(µ) − h(µ, µ)µ +� ++ +1 +⟨µ, ¯N⟩2 f +� +S∂B(¯ν) − h∂B(¯ν, ¯ν)¯ν +� +, +¯ν′ += +S2(Y ∂Σ) − +1 +⟨µ, ¯N⟩∇∂Σf − h∂B(Y, ¯ν) ¯N +(A.7) +− ⟨ν, ¯N⟩ +⟨µ, ¯N⟩2 f +� +SΣ(µ) − h(µ, µ)µ + S∂B(¯ν) − h∂B(¯ν, ¯ν)¯ν +� +. + +STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE +13 +Proof. Let {ei}n +i=1 be an orthonormal basis of TpΣ for some p ∈ Σ and denote ei(t) = (x(t, ·))∗(ei). +Using the fact ⟨ei(t), ν(t)⟩ = 0 and [ei(t), Y (t)] = 0, we have +ν′ = +n +� +i=1 +⟨ν′, ei⟩ei = − +n +� +i=1 +⟨ν, e′ +i⟩ei += − +n +� +i=1 +⟨ν, ¯∇eiY ⟩ei = − +n +� +i=1 +⟨ν, ¯∇ei(Y Σ + fν)⟩ei += +n +� +i=1 +⟨SΣ(Y Σ), ei⟩ei − +n +� +i=1 +df(ei)ei += SΣ(Y Σ) − ∇f. +This is (A.5). As a consequence of (A.5) we get +(A.8) +⟨µ′, ν⟩ = −⟨µ, ν′⟩ = −h(Y Σ, µ) + ∇µf. +Let now {τα}n−1 +α=1 be an orthonormal basis of Tp(∂Σ) and denote τα(t) = (x(t, ·))∗(τα). Using +(A.1), (A.2) and the fact [τα(t), Y (t)] = 0, we have +⟨µ′, τα⟩ += +−⟨µ, τ ′ +α⟩ = −⟨µ, ¯∇ταY ⟩ = − +� +µ, ¯∇τα +� +Y ∂Σ − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩fµ + fν +�� +(A.9) += +−⟨µ, ¯∇ταY ∂Σ⟩ + d +� ⟨ν, ¯N⟩ +⟨µ, ¯N⟩f +� +(τα) − f⟨µ, ¯∇ταν⟩ += +−⟨µ, ¯∇ταY ∂Σ⟩ + +� +⟨µ, ¯N⟩d⟨ν, ¯ +N⟩(τα) − ⟨ν, ¯N⟩d⟨µ, ¯ +N⟩(τα) +� +f +⟨µ, ¯N⟩2 ++ ⟨ν, ¯N⟩ +⟨µ, ¯N⟩df(τα) − fh(µ, τα). +From (2.7), we see +⟨µ, ¯N⟩d⟨ν, ¯N⟩(τα) − ⟨ν, ¯N⟩d⟨µ, ¯N⟩(τα) +(A.10) += +⟨µ, ¯N⟩⟨ ¯∇ταν, ¯N⟩ − ⟨ν, ¯N⟩⟨ ¯∇ταµ, ¯N⟩ + d ¯N(¯ν, τα) += +⟨µ, ¯N⟩2⟨ ¯∇ταν, µ⟩ − ⟨ν, ¯N⟩2⟨ ¯∇ταµ, ν⟩ + h∂B(τα, ¯ν) += +h(µ, τα) + h∂B(¯ν, τα). +Replacing (A.10) in (A.9), we get +⟨µ′, τα⟩ = −⟨µ, ¯∇ταY ∂Σ⟩ + ⟨ν, ¯N⟩ +⟨µ, ¯N⟩df(τα) + ⟨ν, ¯N⟩2 +⟨µ, ¯N⟩2 h(µ, τα)f + +1 +⟨µ, ¯N⟩2 h∂B(¯ν, τα)f. +(A.11) +It follows from (A.8) and (A.11) that +µ′ = ⟨µ′, µ⟩µ + ⟨µ′, ν⟩ν + +n−1 +� +α=1 +⟨µ′, τα⟩τα +(A.12) += (−h(Y Σ, µ) + ∇µf)ν + S1(Y ∂Σ) + ⟨ν, ¯N⟩ +⟨µ, ¯N⟩ ∇∂Σf ++ ⟨ν, ¯N⟩2 +⟨µ, ¯N⟩2 f +� +SΣ(µ) − h(µ, µ)µ +� ++ +1 +⟨µ, ¯N⟩2 f +� +S∂B(¯ν) − h∂B(¯ν, ¯ν)¯ν +� +. +This is (A.6). + +14 +JINYU GUO AND CHAO XIA +Lastly, using [τα(t), Y (t)] = 0 again and (A.3), we have +⟨¯ν′, τα⟩ = −⟨¯ν, τ ′ +α⟩ = −⟨¯ν, ¯∇ταY ⟩ +(A.13) += −⟨¯ν, ¯∇ταY ∂Σ⟩ − d +� +f +⟨µ, ¯N⟩ +� +(τα) += ⟨S2(Y ∂Σ), τα⟩ − +1 +⟨µ, ¯N⟩df(τα) − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩2 f +� +h(µ, τα) + h∂B(¯ν, τα) +� +. +Here the last equality we used (2.5) and (2.8). +Now (A.7) follows from (A.13) and the fact ⟨¯ν′, ¯N⟩ = −h∂B(Y, ¯ν). +□ +Lemma A.2. Along ∂Σ, we have +S1(Y ∂Σ, Y ∂Σ) + ⟨ν, ¯N⟩ +⟨µ, ¯N⟩⟨∇f, Y ∂Σ⟩ + ⟨ν, ¯N⟩2 +⟨µ, ¯N⟩2 f +� +h(Y ∂Σ, µ) + h∂B(Y ∂Σ, ¯ν) +� +(A.14) += +−⟨ν, ¯N⟩⟨Y, ¯ν′⟩ + ⟨µ, ¯N⟩h∂B(Y ∂Σ, Y ∂Σ) +Proof. From (A.7) and (A.3), we have +− ⟨ν, ¯N⟩⟨Y, ¯ν′⟩ +(A.15) += ⟨ν, ¯N⟩⟨ ¯∇Y ∂ΣY ∂Σ, ¯ν⟩ + ⟨ν, ¯N⟩ +⟨µ, ¯N⟩⟨Y ∂Σ, ∇∂Σf⟩ + ⟨ν, ¯N⟩2 +⟨µ, ¯N⟩2 f +� +h(Y ∂Σ, µ) + h∂B(Y ∂Σ, ¯ν) +� +Using (2.5), we see +⟨ν, ¯N⟩⟨ ¯∇Y ∂ΣY ∂Σ, ¯ν⟩ = ⟨ ¯∇Y ∂ΣY ∂Σ, −µ + ⟨µ, ¯N⟩ ¯N⟩= S1(Y ∂Σ, Y ∂Σ) − ⟨µ, ¯N⟩h∂B(Y ∂Σ, Y ∂Σ). +The assertion follows. +□ +Lemma A.3. Along ∂Σ, we have +(A.16) +hF (Y Σ, µ) + +1 +⟨µ, ¯N⟩h∂B(Y, (νF )∂Σ) + ⟨µF, ¯N⟩ +⟨µ, ¯N⟩ h∂B(Y, ¯ν) = qFf. +Proof. Using the capillary condition ⟨νF , ¯N⟩ = ω0, (2.8) and the fact that ⟨ ¯∇Y ∂ΣνF , ν⟩ = 0, we +calculate that +⟨µ, ¯N⟩hF (Y ∂Σ, µ) = ⟨µ, ¯N⟩⟨ ¯∇Y ∂ΣνF, µ⟩ = ⟨ ¯∇Y ∂ΣνF , ¯N⟩ = −⟨νF, ¯∇Y ∂Σ ¯N⟩ +(A.17) += −⟨νF , ¯ν⟩h∂B(Y ∂Σ, ¯ν) − h∂B(Y ∂Σ, (νF )∂Σ) += −⟨µF , ¯N⟩h∂B(Y ∂Σ, ¯ν) − h∂B(Y ∂Σ, (νF )∂Σ). +Here in the last equality we used (3.8). +From (A.1), (A.2), (A.3) and (A.17), we obtain that +hF (Y Σ, µ) + +1 +⟨µ, ¯N⟩h∂B(Y, (νF )∂Σ) + ⟨µF , ¯N⟩ +⟨µ, ¯N⟩ h∂B(Y, ¯ν) +(A.18) += hF (Y ∂Σ, µ) + ⟨Y, µ⟩hF (µ, µ) + +1 +⟨µ, ¯N⟩h∂B(Y, (νF )∂Σ) + ⟨µF , ¯N⟩ +⟨µ, ¯N⟩ h∂B(Y, ¯ν) += − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩fhF(µ, µ) + +1 +⟨µ, ¯N⟩ +� +h∂B(Y − Y ∂Σ, (νF )∂Σ) + ⟨µF , ¯N⟩h∂B(Y − Y ∂Σ, ¯ν) +� += − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩fhF(µ, µ) + +1 +⟨µ, ¯N⟩2 +� +h∂B(¯ν, (νF )∂Σ) + ⟨µF , ¯N⟩h∂B(¯ν, ¯ν) +� +f += qFf. + +STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE +15 +□ +Proof of Proposition 3.1. For an admissible variation Y , we calculate the first variation of +anisotropic energy EF as follows +E′ +F (0) = A′ +F (0) + ω0A′ +W(0) = +� +Σ +∂ +∂t +��� +t=0F(ν)dA + +� +Σ +F(ν) ∂ +∂t +��� +t=0dAt + ω0 +� +∂Σ +⟨¯ν, Y ⟩ds += +� +Σ +⟨ ¯∇F(ν), ν′⟩ + F(ν)divΣY dA + ω0 +� +∂Σ +⟨¯ν, Y ⟩ds += +� +Σ +⟨DF(ν), −∇f + SΣ(Y Σ)⟩ + F(ν)(divΣY Σ + fH)dA + ω0 +� +∂Σ +⟨¯ν, Y ⟩ds += +� +Σ +(divΣDF + F(ν)H)fdA + +� +∂Σ +F(ν)⟨Y, µ⟩ − f⟨DF, µ⟩ds + ω0 +� +∂Σ +⟨¯ν, Y ⟩ds += +� +Σ +HFfdA + +� +∂Σ +⟨F(ν)µ − ⟨DF, µ⟩ν + ω0¯ν, Y ⟩ds += +� +Σ +HFfdA + +� +∂Σ +⟨µF + ω0¯ν, Y ⟩ds. +□ +Proof of Proposition 3.5. Let x : Σ → B be an anisotropic capillary CAMC (anisotropic +capillary minimal, respectively) immersion, that is, HF = const (HF = 0, respectively) in Σ +and ⟨νF , ¯N⟩ = ω0 on ∂Σ and x(t, ·) be a volume-preserving (compactly supported, respectively) +admissible variation. It is direct to see that the first variational formula is true for any t ∈ (−ǫ, ǫ), +that is, +E′ +F (t) = +� +Σt +HFfdA + +� +∂Σt +⟨µF + ω0¯ν, Y ⟩ds. +Thus we have +E′′ +F (0) = +� +Σ +H′ +FfdA + HF +�� +Σ +fdA +�′ ++ +� +∂Σ +⟨Y ′, µF + ω0¯ν⟩ + ⟨Y, µ′ +F + ω0¯ν′⟩ds + +� +∂Σ +⟨Y, µF + ω0¯ν⟩ ∂ +∂t +��� +t=0dst. +Observe that in the case of CAMC with volume preserving variation, we have +(A.19) +�� +Σ +fdA +�′ += V′′(0) = 0, +and in the case of anisotropic minimal, HF = 0. Hence in both cases, the term HF +�� +Σ fdA +�′ = 0. +Also, ⟨Y, µF + ω0¯ν⟩ = 0 along ∂Σ since Y |∂Σ ∈ T(∂B). Moreover, we have the evolution +equation (see [33]) +H′ +F = −(divΣ(AF ∇f) + ⟨AF ◦ dν, dν⟩f). +It follows that +E′′ +F (0) = − +� +Σ +(divΣ(AF ∇f) + ⟨AF ◦ dν, dν⟩f)fdA + +� +∂Σ +⟨Y ′, µF + ω0¯ν⟩ + ⟨Y, µ′ +F + ω0¯ν′⟩ds. +(A.20) +So to prove the formula for E′′ +F (0) we only need to compute the boundary term +(A.21) +� +∂Σ +⟨Y ′, µF + ω0¯ν⟩ds + +� +∂Σ +⟨Y, µ′ +F + ω0¯ν′⟩ds. + +16 +JINYU GUO AND CHAO XIA +We now calculate the first term of (A.21). Since µF + ω0¯ν is parallel to ¯N along ∂Σ, we have +⟨Y ′, µF + ω0¯ν⟩ = ⟨ ¯∇Y Y, µF + ω0¯ν⟩ = −⟨µF , ¯N⟩h∂B(Y, Y ). +(A.22) +Next we calculate the second term of (A.21). According to the definition of µF , by using +(2.3) and (2.4), we see that +µ′ +F = (F(ν)µ − ⟨νF, µ⟩ν)′ +(A.23) += F(ν)′µ + F(ν)µ′ − ⟨ ¯∇F(ν), µ⟩′ν − ⟨DF(ν), µ⟩ν′ += ⟨ ¯∇F(ν), ν′⟩µ + F(ν)µ′ − +� ¯∇2F(ν)(ν′, µ) + ⟨ ¯∇F(ν), µ′⟩ +� +ν − ⟨DF(ν), µ⟩ν′ += ⟨DF(ν), ν′⟩µ + F(ν)µ′ − +� +(D2F(ν) + F(ν)Id)(ν′, µ) + ⟨DF(ν) + F(ν)ν, µ′⟩ +� +ν − ⟨DF(ν), µ⟩ν′ += ⟨DF(ν), ν′⟩µ + F(ν)µ′ − +� +D2F(ν)(ν′, µ) + ⟨DF(ν), µ′⟩ +� +ν − ⟨DF(ν), µ⟩ν′. +We remind here that ¯∇F is the Euclidean covariant derivative on the one-homogenous extension +of F, and DF is the covariant derivative of F with respect to Sn. +It follows that +⟨Y, µ′ +F + ω0¯ν′⟩ = −D2F(ν)(ν′, µ)f + F(ν)⟨Y, µ′⟩ + ω0⟨Y, ¯ν′⟩ ++ ⟨DF(ν), ν′⟩⟨Y, µ⟩ − ⟨DF(ν), µ′⟩f − ⟨DF(ν), µ⟩⟨Y, ν′⟩ +:= I + II + III + IV + V + V I. +(A.24) +We now tackle the above terms one by one. Using (A.5), we have +I = −D2F(ν)(ν′, µ)f +(A.25) += −⟨(D2F(ν) + F(ν)Id)µ, ν′⟩f + F(ν)⟨µ, ν′⟩f += −⟨AF(ν) · µ, −∇f + SΣ(Y Σ)⟩f + F(ν)⟨µ, ν′⟩f += ⟨AF (ν)∇f, µ⟩f − hF (Y Σ, µ)f − F(ν)⟨µ′, ν⟩f. +Utilizing (A.6), (A.14) and (A.3), we get +⟨Y, µ′⟩ = ⟨µ′, ν⟩f + S1(Y ∂Σ, Y ∂Σ) + ⟨ν, ¯N⟩ +⟨µ, ¯N⟩⟨∇f, Y ∂Σ⟩ +(A.26) ++ ⟨ν, ¯N⟩2 +⟨µ, ¯N⟩2 fh(Y ∂Σ, µ) + +� +1 + ⟨ν, ¯N⟩2 +⟨µ, ¯N⟩2 +� +h∂B(Y ∂Σ, ¯ν)f += ⟨µ′, ν⟩f − ⟨ν, ¯N⟩⟨Y, ¯ν′⟩ + ⟨µ, ¯N⟩h∂B(Y ∂Σ, Y ∂Σ) + h∂B(Y ∂Σ, ¯ν)f += ⟨µ′, ν⟩f − ⟨ν, ¯N⟩⟨Y, ¯ν′⟩ + ⟨µ, ¯N⟩h∂B(Y ∂Σ, Y ). +Note that ¯ν = +1 +⟨µ, ¯ +N⟩ν − ⟨ν, ¯ +N⟩ +⟨µ, ¯ +N⟩ ¯N. It follows that +S2(Y ∂Σ, Y ∂Σ) = +1 +⟨µ, ¯N⟩h(Y ∂Σ, Y ∂Σ) − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩h∂B(Y ∂Σ, Y ∂Σ). +(A.27) + +STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE +17 +Applying (A.7), (A.27) and (A.3), we have +⟨Y, ¯ν′⟩ = S2(Y ∂Σ, Y ∂Σ) − +1 +⟨µ, ¯N⟩⟨∇f, Y ∂Σ⟩ − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩2 f +� +h(Y ∂Σ, µ) + h∂B(Y ∂Σ, ¯ν) +� +(A.28) += +1 +⟨µ, ¯N⟩h(Y ∂Σ, Y ∂Σ) − +1 +⟨µ, ¯N⟩⟨∇f, Y ∂Σ⟩ +− ⟨ν, ¯N⟩ +⟨µ, ¯N⟩2 fh(Y ∂Σ, µ) − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩h∂B(Y ∂Σ, Y ). +Combining (A.26) with (A.28), by using the capillary condition (3.9), we obtain that +II + III = F(ν)⟨Y, µ′⟩ + ω0⟨Y, ¯ν′⟩ +(A.29) += F(ν)⟨µ′, ν⟩f + (−F(ν)⟨ν, ¯N⟩ + ω0)⟨Y, ¯ν′⟩ + F(ν)⟨µ, ¯ +N⟩h∂B(Y ∂Σ, Y ) += F(ν)⟨µ′, ν⟩f + ⟨µ, ¯N⟩⟨νF , µ⟩⟨Y, ¯ν′⟩ + F(ν)⟨µ, ¯N⟩h∂B(Y ∂Σ, Y ) += F(ν)⟨µ′, ν⟩f + ⟨µF, ¯N⟩h∂B(Y ∂Σ, Y ) ++ ⟨νF, µ⟩ +� +h(Y ∂Σ, Y ∂Σ) − ⟨∇f, Y ∂Σ⟩ − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩fh(Y ∂Σ, µ) +� +. +Now we claim that +IV + V + V I +(A.30) += +⟨DF(ν), ν′⟩⟨Y, µ⟩ − ⟨DF(ν), µ′⟩f − ⟨DF(ν), µ⟩⟨Y, ν′⟩ += +−⟨νF, µ⟩ +� +h(Y ∂Σ, Y ∂Σ) − ⟨∇f, Y ∂Σ⟩ − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩fh(Y ∂Σ, µ) +� +− +f +⟨µ, ¯N⟩h∂B(Y, (νF )∂Σ), +In fact, from (A.5), (A.2) and (A.1) we have +IV = ⟨DF(ν), ν′⟩⟨Y, µ⟩ = − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩f⟨DF(ν), SΣ(Y Σ) − ∇f⟩ +(A.31) += − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩f +� +⟨DF(ν), +� +α +h(Y Σ, τα)τα − ∇∂Σf⟩ + ⟨DF(ν), µ⟩(h(Y Σ, µ) − ∇µf) +� +, +and +V I = −⟨DF(ν), µ⟩⟨Y, ν′⟩ = −⟨DF(ν), µ⟩⟨Y Σ, −∇f + SΣ(Y Σ)⟩ +(A.32) += −⟨DF(ν), µ⟩ +� +−⟨∇f, Y ∂Σ⟩ + ⟨ν, ¯N⟩ +⟨µ, ¯N⟩f∇µf + h(Y ∂Σ, Y ∂Σ) − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩fh(Y ∂Σ, µ) − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩fh(Y Σ, µ) +� +. +Since µ = +1 +⟨µ, ¯ +N⟩ ¯N − ⟨ν, ¯ +N⟩ +⟨µ, ¯ +N⟩ν, we see +S1(Y ∂Σ) = +1 +⟨µ, ¯N⟩ +� +α +h∂B(Y ∂Σ, τα)τα − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩ +� +α +h(Y ∂Σ, τα)τα. +(A.33) + +18 +JINYU GUO AND CHAO XIA +Using (A.6), (A.33) and (A.3), we deduce +V = −⟨DF(ν), µ′⟩f +(A.34) += − +� +DF(ν), +1 +⟨µ, ¯N⟩ +� +α +h∂B(Y ∂Σ, τα)τα − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩ +� +α +h(Y ∂Σ, τα)τα + ⟨ν, ¯N⟩ +⟨µ, ¯N⟩∇∂Σf +� +f +− +� +DF(ν), ⟨ν, ¯N⟩2 +⟨µ, ¯N⟩2 f +� +α +h(µ, τα)τα + +f +⟨µ, ¯N⟩2 +� +α +h∂B(¯ν, τα)τα +� +f += − +� +DF(ν), ⟨ν, ¯N⟩ +⟨µ, ¯N⟩∇∂Σf − ⟨ν, ¯N⟩ +⟨µ, ¯N⟩ +� +α +h(Y Σ, τα)τα + +1 +⟨µ, ¯N⟩ +� +α +h∂B(Y, τα)τα +� +f. +Now the above claim (A.30) follows by combining (A.31), (A.32) and (A.34). +Putting I-V I into (A.24), we get +⟨Y, µ′ +F + ω0¯ν′⟩ +(A.35) += f +� +⟨AF ∇f, µ⟩ − hF (Y Σ, µ) − +1 +⟨µ, ¯N⟩h∂B(Y, (νF )∂Σ) +� ++ ⟨µF , ¯N⟩h∂B(Y ∂Σ, Y ). +Combining (A.35) and (A.22), using (A.3) and (A.16), we have +⟨Y, µ′ +F + ω0¯ν′⟩ + ⟨Y ′, µF + ω0¯ν⟩ +(A.36) += f +� +⟨AF ∇f, µ⟩ − hF (Y Σ, µ) − +1 +⟨µ, ¯N⟩h∂B(Y, (νF )∂Σ) − ⟨µF , ¯N⟩ +⟨µ, ¯N⟩ h∂B(Y, ¯ν) +� += f (⟨AF ∇f, µ⟩ − qF f) . +The proof is completed by (A.20) and (A.36). +□ +Acknowledgements. 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Zhang, Uniqueness for volume-constraint local energy-minimizing sets in a half-space or a ball, +arXiv: 2106.14780v3, 2021. +Department of Mathematics, Tsinghua University, Beijing, 100084, China +Email address: guojinyu@mail.tsinghua.edu.cn +School of Mathematical Sciences, Xiamen University, 361005, Xiamen, P.R. China +Email address: chaoxia@xmu.edu.cn + diff --git a/otE1T4oBgHgl3EQfOwPh/content/tmp_files/load_file.txt b/otE1T4oBgHgl3EQfOwPh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..58ab5668c4fef93d13ff03e9933acbdd8a60f95a --- /dev/null +++ b/otE1T4oBgHgl3EQfOwPh/content/tmp_files/load_file.txt @@ -0,0 +1,885 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf,len=884 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='03020v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='DG] 8 Jan 2023 STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE JINYU GUO AND CHAO XIA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In this paper, we study stability problem of anisotropic capillary hypersurfaces in an Euclidean half-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We prove that any compact immersed anisotropic capillary constant anisotropic mean curvature hypersurface in the half-space is weakly stable if and only if it is a truncated Wulff shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' On the other hand, we prove a Bernstein-type theorem for stable anisotropic capillary minimal surfaces in the three dimensional half-space under Euclidean area growth assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Introduction Capillary phenomena appear in the study of the equilibrium shape of liquid drops and crystals in a given solid container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The mathematical model has been established through the work of Young, Laplace, Gauss and others, as a variational problem on minimizing a free energy func- tional under a volume constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' For more detailed description on the isotropic and anisotropic capillary phenomena, we refer to [18] and [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' A capillary hypersurface in a domain B of a Riemannian manifold is an immersed constant mean curvature (CMC) hypersurface in B which intersects ∂B at a constant contact angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Cap- illary hypersurfaces are the stationary points of an energy functional under volume preserving variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' As a special and important case, free boundary CMC hypersurfaces are the station- ary points of an area functional under volume preserving variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' A capillary hypersurface is called weakly stable if it is local energy-minimizing under volume preserving variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The stability of free boundary CMC or capillary hypersurfaces was initiated by Ros-Vergasta [42] and Ros-Souam [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The classification problem for stable compact free boundary CMC or cap- illary hypersurfaces in an Euclidean ball and in an Euclidean half-space Rn+1 + have been studied intensively, see for instance [38, 39, 4, 1, 46, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The classification has been completed eventu- ally by Wang and the second-named author [50] for the Euclidean ball case and Souam [47] for the Euclidean half-space case respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' This generalizes the classical result of Barbosa-do Carmo [5] on the classfication of stable closed CMC hypersurfaces in Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' See also recent work [22, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' A modern formulation of Gauss’ model of capillary phenomena includes a possibly anisotropic surface tension density, which we are interested in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let F : Sn → R+ be a positive smooth function such that the matrix (D2F + FId) is positive definite, where D2F is the Hessian of F and Id denotes the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' F determines a unique strictly convex hypersurface WF in Rn+1, which is called the Wulff shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' For a closed hypersurface Σ immersed in Rn+1, the anisotropic area functional is given by AF(Σ) = � Σ F(ν)dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' anisotropic capillary hypersurfaces, Wulff shape, stability, Bernstein’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' JG is supported by Shuimu Tsinghua Scholar Program (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2022SM046) and China Postdoctoral Science Foun- dation (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2022M720079).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' CX is supported by the NSFC (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='11871406, 12271449).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' 1 2 JINYU GUO AND CHAO XIA The well-known Wulff theorem (see for example [19, 49]) says that the Wulff shape (up to trans- lation and homothety) is the global minimizer to the anisotropic area functional under fixed volume constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' From variational point of view, the stationary points for the anisotropic area functional under volume-preserving variations are closed hypersurfaces with constant anisotropic mean curvature (CAMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Palmer [40] (see also Winklmann [52]) proved that Wulff shape (up to translation and homothety) is the only stable CAMC hypersurfaces, which is the anisotropic counterpart of Barbosa-do Carmo’s [5] result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' For more rigidity results on closed CAMC hyper- surfaces and related problems, we refer to [20, 24, 25, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' For a compact, orientable hypersurface Σ immersed in some container B ⊂ Rn+1 with bound- ary ∂Σ, which intersects ∂B transversely, the anisotropic energy functional is given by EF (Σ) = AF (Σ) + ω0AW(Σ), where ω0 ∈ R is a real number, AW(Σ) is so-called wetting area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We remark that throughout this paper, Σ will be always orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The global minimizer of EF under fixed volume constraint has been characterized by Winter- bottom [54] (see also [31]) as a truncated Wulff shape (it is also called a Winterbottom shape), which can be viewed as the capillary counterpart of Wulff shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' For anisotropic free energy functionals involving a gravitational potential energy term, the existence, the regularity and boundary regularity for global or local minimizers have been studied by De Giorgi [13], Almgren [2], Taylor [48] and De Philippis-Maggi [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' For the symmetry and uniqueness of global minimizers, we refer to the work of Baer [3] for a class of F with certain symmetry and the work of Gonzalez [21] in the isotropic case via a symmetrization technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' From variational point of view, the stationary points of EF under fixed volume constraint are the anisotropic capillary CAMC hypersurfaces, which are of CAMC and satisfy an anisotropic capillary condition (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='9) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In a series of papers [31, 32, 33], Koiso-Palmer studied the anisotropic capillary hypersurfaces in a slab (the domain bounded by two parallel hyperplane) and their stabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In Koiso-Palmer’s paper, the second variation of EF is derived for a class of anisotropy satisfying certain symmetric condition in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In this paper, we first compute the second variation of EF for any anisotropic F in any dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Hence we give a clear characterization for stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' An anisotropic capillary CAMC immersion x : Σ → B ⊂ Rn+1 is weakly stable if and only if − � Σ (divΣ(AF ∇f) + ⟨AF ◦ dν, dν⟩f)f dA + � ∂Σ (⟨AF ∇f, µ⟩ − qFf) f ds ≥ 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1) for any f ∈ C∞(Σ) satisfying � Σ f dA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Here qF is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='14) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' For some notations involved in the above stability inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1), we refer to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Recently, Koiso [28, 27] studied the stability problem of anisotropic capillary CAMC hypersur- faces in a wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In particular, she proved that a compact stable immersed anisotropic capillary CAMC hypersurface Σ in Rn+1 + with boundary ∂Σ must be a truncated Wulff shape, provided ∂Σ ⊂ Rn is embedded for n = 2 and is convex for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' This extends the result of Choe-Koiso [8] in the isotropic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' As we mentioned, in the isotropic case, Souam [47] classified stable capillary hypersurfaces in Rn+1 + without any additional assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' It is natural to ask whether the embeddedness condition for n = 2 and the convexity condition for n ≥ 3 in Koiso’s result [28, 27] can be removed for the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In this paper we will give an affirmative answer and our main result is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' A compact, immersed anisotropic capillary CAMC hypersurface in Rn+1 + is weakly stable if and only if it is a truncated Wulff shape, up to translation and homothety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' As a special case, we have a classification for stable anisotropic free boundary CAMC hyper- surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' A compact, immersed anisotropic free boundary CAMC hypersurface in Rn+1 + is weakly stable if and only if it is a truncated Wulff shape, up to translation and homothety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1 is based on the stability inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1) and the following Minkowski- type formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2) � Σ � n(F(ν) + ω0⟨EF n+1, ν⟩) − HF⟨x, ν⟩ � dA = 0, where EF n+1 ∈ Rn+1 is a constant vector given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2) has been proved by Jia, Wang, Zhang and the second-named author [26, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3], which was used to prove an Alexandrov-type theorem for embedded anisotropic capillary hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' It is an standard approach to apply Minkowski-type formula involving no boundary terms to handle the stability for free boundary or capillary problems, see for example [1, 22, 47, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In the second part of this paper, we are interested in (strongly) stable anisotropic capillary minimal surfaces in the half-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The second variational formula gives the following charac- terization of strong stability for anisotropic capillary minimal hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' An anisotropic capillary minimal immersion x : Σ → B ⊂ Rn+1 is (strongly) stable if and only if − � Σ (divΣ(AF ∇f) + ⟨AF ◦ dν, dν⟩f)f dA + � ∂Σ (⟨AF ∇f, µ⟩ − qFf) f ds ≥ 0, for any f ∈ C∞ c (Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' A classical Bernstein theorem, proved Fischer-Colbrie-Schoen [17], do Carmo-Peng [12] and Pogorelov [41] independently, says that the only complete stable minimal surfaces in R3 are flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Quite recently, Chodosh-Li [9] resolved a well-known conjecture of Schoen [7, Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='12] that the only complete stable minimal hypersurfaces in R4 are flat, see also Catino-Mastrolia- Roncoroni [6] for another proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Schoen-Simon-Yau [44] have shown that any complete sta- ble minimal hypersurfaces in Rn+1 with n + 1 ≤ 6 with Euclidean area growth must be flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Bernstein-type theorem for the anisotropic case in R3 has been proved by White [51] under the Euclidean area growth assumption, by Lin [36] when F is C2-close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Under similar assumptions, Bernstein-type theorem for the anisotropic case has been studied by Simon [45], Winklmann [53], Chodosh-Li [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' It is still open question whether these extra assumptions could be removed, even in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We refer to Chodosh-Li’s paper [10] on recent progress for stable anisotropic minimal hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Initiated by the min-max construction for capillary minimal surfaces, Bernstein-type theorem for capillary minimal surfaces in R3 + has recently attracted much attentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In particular, Li- Zhou-Zhu [34] and De Masi-De Philippis [16], independently, proved that any properly immersed stable capillary minimal surfaces in R3 + with quadratic area growth must be a half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' By us- ing Fischer-Colbrie-Schoen’s technique, Hong-Saturnino [23] proved the Bernstein-type theorem without Euclidean area growth assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' It is natural to consider the case of anisotropic capillary minimal surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Here we prove the following Bernstein-type theorem for stable anisotropic capillary minimal surfaces in R3 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' 4 JINYU GUO AND CHAO XIA Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2 (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let Σ be an immersed anisotropic capillary minimal surface in R3 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Assume that Σ has Euclidean area growth, that is, there exists some C > 0 such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3) Area (Σ ∩ Br(0)) < Cr2 for any r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Then Σ is stable if and only if Σ is a half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In case F ≡ 1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2 reduces to [34, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2] or [16, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The remaining part of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In Section 2 we review some def- initions and notations about anisotropic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In Section 3 we calculate the first and the second variation formula of anisotropic energy functional in a general domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In Section 4 we present some useful geometric formulas for anisotropic capillary hypersurfaces in Rn+1 + and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In Section 5 we discuss the stability of noncompact anisotropic capillary minimal surface in R3 + and prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Notations and Preliminaries Let F : Sn → R+ be a positive smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Denote by DF and D2F the gradient and Hessian of F on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Then we require the matrix (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1) AF := (D2F + FId)|x > 0 for any x ∈ Sn, where Id denotes the identity on TxSn and “ > ” means the matrix is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We define the map Φ : Sn −→ Rn+1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2) x �→ F(x)x + DF(x) whose image WF = Φ(Sn) is a smooth strictly convex hypersurface in Rn+1 called the Wulff shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We may regard F as a convex function on Rn+1 by one-homogenous extension of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Precisely, set F(x) = |x|F( x |x|) when x ̸= 0 and F(0) = 0, the new F : Rn+1 → R is a one-homogenous function on Rn+1 which is C2 ∈ (Rn+1 \\ {0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We use ¯∇ to denote the Euclidean corvariant derivative of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' It is standard to see that for x ∈ Sn and V, W ∈ TxSn, we have ¯∇F(x) = DF(x) + F(x)x = Φ(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3) ¯∇2F(x)(V, W) = (D2F + FId)(x)(V, W) = AF (x)(V, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='4) Let B be a closed region in an (n+1)-dimensional Euclidean space Rn+1 with smooth boundary ∂B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let x : Σ → B be an isometric immersion from a n-dimensional smooth manifold Σ such that ∂Σ ⊂ ∂B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We denote by ¯∇, ¯∆ and ¯∇2 the gradient, the Laplacian and the Hessian on Rn+1 respectively, while by ∇, ∆ and ∇2 the gradient, the Laplacian and the Hessian on Σ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let T(∂Σ) and N(∂Σ) be the tangent bundle and the normal bundle of ∂Σ as a co-dimensional two submainfolds in Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We will use the following terminology for four normal vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We choose one of the unit normal vector field along x and denote it by ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We denote by ¯N the unit outward normal to ∂B in B and µ be the unit outward normal to ∂Σ in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let ¯ν be the unit normal to ∂Σ in ∂B such that the bases {ν, µ} and {¯ν, ¯N} have the same orientation in N(∂Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Hence, in N(∂Σ), the following relations hold: µ = −⟨ν, ¯N⟩¯ν + ⟨µ, ¯N⟩ ¯N, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5) ν = ⟨µ, ¯N⟩¯ν + ⟨ν, ¯N⟩ ¯N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='6) STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE 5 Equivalently, ¯ν = −⟨ν, ¯N⟩µ + ⟨µ, ¯N⟩ν, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='7) ¯N = ⟨µ, ¯N⟩µ + ⟨ν, ¯N⟩ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='8) We always assume that Σ interesects ∂B transversally, so that ⟨µ, ¯N⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' For a vector field Y on Rn+1, we denote Y Σ and Y ∂Σ to be the tangential projection of Y on TΣ and on T(∂Σ) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Denote by h and H the second fundamental form and the mean curvature of the immersion x respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Precisely, h(X, Y ) = ⟨ ¯∇Xν, Y ⟩ for X, Y ∈ TΣ and H = trg(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Denote by h∂B the second fundamental form of ∂B in B, that is, h∂B(X, Y ) = ⟨ ¯∇X ¯N, Y ⟩ for X, Y ∈ T(∂B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let νF be the anisotropic normal of Σ given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='9) νF = Φ(ν) := DF(ν) + F(ν)ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Hence DF(ν) = νF − ⟨νF , ν⟩ν = νΣ F ∈ TΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The anisotropic principal curvatures � κF i �n i=1 of Σ are given by the eigenvalues of the anisotropic Weingarten map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='10) dνF = AF (ν) ◦ dν : TΣ → TΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The eigenvalues are real since (AF) is positive definite and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The anisotropic second fundamental form and anisotropic mean curvature are denoted respectively by hF (X, Y ) = ⟨ ¯∇XνF, Y ⟩ = (AF ◦ h)(X, Y ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='11) HF = trg(dνF ) = trg(AF (ν) ◦ dν) = divΣ(DF(ν)) + HF(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='12) When F ≡ 1, we see AF = IdSn and hence hF and HF are the usual second fundamental form h and mean curvature H respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The first and second variational formula Let x : Σ → B be an isometric immersion such that x(∂Σ) = x(Σ) ∩ ∂B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' By a compactly supported admissible variation of x, we mean a differentiable map x : (−ǫ, ǫ)×Σ → B such that x(t, ·) : Σ → B is an immersion satisfying x(t, intΣ) ⊂ intB, x(t, ∂Σ) ⊂ ∂B and the support of ∂ ∂tx(t, ·) is compact for every t ∈ (−ǫ, ǫ) and x(0, ·) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The anisotropic area functional AF : (−ǫ, ǫ) → R and the oriented volume functional V : (−ǫ, ǫ) → R are given by AF(t) = � ˜Σ F(ν)dAt, V(t) = � [0,t]×˜Σ x(t, ·)∗dV, where ˜Σ ⊆ Σ is the support of the variation ∂ ∂tx(t, ·), dAt is the area element of Σ with respect to the metric induced by x(t, ·) and dV is the volume element in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' When Σ is compact, then ˜Σ = Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' An admissible variation is said to be volume-preserving if V(t) = V(0) = 0 for each t ∈ (−ǫ, ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The wetting area functional AW(t) : (−ǫ, ǫ) → R are defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1) AW(t) = � [0,t]×(∂Σ∩˜Σ) x(t, ·)∗dA∂B, where dA∂B is the area element of ∂B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' 6 JINYU GUO AND CHAO XIA Fix a real number ω0 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The anisotropic energy functional EF(t) : (−ǫ, ǫ) → R is defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2) EF (t) = AF(t) + ω0AW(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' It is well known that the first variation formulas of V and AW for a compactly supported admissible variation with variation vector field Y = ∂ ∂tx(t, ·)|t=0 ∈ C∞ c (Σ) such that Y |∂Σ ∈ T(∂B) are given by V′(0) = � Σ ⟨Y, ν⟩dA, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3) A′ W(0) = � ∂Σ ⟨Y, µ⟩ds, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='4) where dA and ds are the area element of Σ and ∂Σ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Next we derive the first variational formula for EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let x(·, t) be a compactly supported admissible variation with variational vector field Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5) E′ F (0) = � Σ HF⟨Y, ν⟩dA + � ∂Σ ⟨Y, µF + ω0¯ν⟩ds, where µF := ⟨νF , ν⟩µ − ⟨νF , µ⟩ν = F(ν)µ − ⟨νF , µ⟩ν ∈ N(∂Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='6) The first variational formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5) is known to experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' For completeness, we will give a proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' An interesting property for µF is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Along ∂Σ, we have ⟨µF , ¯ν⟩ = −⟨νF, ¯N⟩, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='7) ⟨µF, ¯N⟩ = ⟨νF , ¯ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' It follows directly by the definition of µF, νF and also the relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Stationary points of EF under compactly supported volume preserving admis- sible variation are hypersurfaces with constant anisotropic mean curvature (CAMC) satisfying the anisotropic capillary boundary condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='9) ⟨νF , ¯N⟩ = ω0 along ∂Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Similarly, stationary points of EF under any compactly supported admissible variation are hy- persurfaces with zero anisotropic mean curvature and satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' From the first variation formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3), it is clear x has constant anisotropic mean curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Next we show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' From the first variation formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5) and admissible condition, we have ⟨Y, µF + ω0¯ν⟩ = 0 for any Y ∈ T(∂B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='10) Note that µF + ω0¯ν ∈ N(∂Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Since any Y ∈ T(∂B) can be split as Y = ⟨Y, ¯ν⟩¯ν + Y ∂Σ, where Y ∂Σ ∈ T(∂Σ), we see that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='10) is equivalent that ⟨¯ν, µF ⟩ = −ω0 along ∂Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='11) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='7), we see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='11) is equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' □ STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE 7 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' An immersion x : Σ → B is said to be anisotropic capillary CAMC if it has CAMC and satisfies boundary condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In particular, Σ is called anisotropic free boundary CAMC hypersurface if ω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' An immersion x : Σ → B is said to be anisotropic capillary minimal if HF = 0 and ⟨νF , ¯N⟩ = ω0 along Σ ∩ ∂B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' For a volume-preserving admissible variation Y , by splitting Y = Y Σ + fν, we see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='12) � Σ f dA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Conversely, we have the following fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' ([1, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1]) Let x : Σ → B be a compact immersed hypersurface with ∂Σ ⊆ ∂B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Then for a given f ∈ C∞(Σ) satisfying � Σ f dA = 0, there exists a volume-preserving admissible variation of x with variational vector field having fν as its normal part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Next we give the second variational formula for the energy functional EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' This formula is new in the anisotropic capillary case, even for n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let x : Σ → B be an immersed anisotropic capillary CAMC (anisotropic capillary minimal, respectively) hypersurface and x(·, t) be a volume-preserving (compactly sup- ported, respectively) admissible variation with variational vector field Y having fν as its normal part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Then E′′ F(0) = − � Σ (divΣ(AF ∇f) + ⟨AF ◦ dν, dν⟩f)f dA + � ∂Σ (⟨AF ∇f, µ⟩ − qFf) f ds, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='13) where qF := 1 ⟨µ, ¯N⟩2 � ⟨µF , ¯N⟩h∂B(¯ν, ¯ν) + h∂B(¯ν, (νF )∂Σ) � − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩hF (µ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='14) We postpone the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5 to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' An anisotropic capillary CAMC immersion x : Σ → B is called weakly stable if E′′ F(0) ≥ 0 for any volume-preserving admissible variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' An anisotropic capillary minimal immersion x : Σ → B is called (strongly) stable if E′′ F (0) ≥ 0 for any compactly supported admissible variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' As a consequence of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='4, we get the following Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' An anisotropic capillary CAMC immersion x : Σ → B is weakly stable if and only if − � Σ (divΣ(AF ∇f) + ⟨AF ◦ dν, dν⟩f)f dA + � ∂Σ (⟨AF ∇f, µ⟩ − qFf) f ds ≥ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='15) for any f satisfying � Σ f dA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' An anisotropic capillary minimal immersion x : Σ → B is (strongly) stable if and only if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='15) is satisfied for any f ∈ C∞ c (Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' 8 JINYU GUO AND CHAO XIA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Rigidity for stable anisotropic capillary hypersurfaces in Rn+1 + In this and next section, we consider the anisotropic capillary hypersurfaces in Rn+1 + , that is the domain is B = Rn+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In this setting, h∂B ≡ 0 and ¯N = −En+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Abuse of notation, we use µn+1 and νn+1 to denote ⟨µ, En+1⟩ and ⟨ν, En+1⟩ on Σ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In this case, qF in the stability inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='15) is given by qF = − νn+1 µn+1 hF (µ, µ) = 1 F(ν) � ω0 µn+1 + ⟨νF , µ⟩ � hF (µ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1) The following proposition is a fundamental fact for anisotropic capillary hypersurfaces in Rn+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let x : Σ → Rn+1 + be an anisotropic capillary immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Then µ is an anisotropic principal direction of ∂Σ in Σ, that is, hF (e, µ) = 0 for any e ∈ T(∂Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' For any e ∈ T(∂Σ), by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='8), we have 0 = ∇e⟨νF , En+1⟩ = ⟨ ¯∇eνF , En+1⟩ = ⟨ ¯∇eνF, µn+1µ + νn+1ν⟩ = µn+1⟨ ¯∇eνF , µ⟩ + νn+1⟨ ¯∇eνF, ν⟩ = µn+1hF (e, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Since µn+1 ̸= 0 along ∂Σ, we get the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' □ In [26], the following anisotropic Minkowski-type formula has been proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' ([26, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3]) Let x : Σ → Rn+1 + be a compact anisotropic capillary immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2) � Σ � n(F(ν) + ω0⟨EF n+1, ν⟩) − HF ⟨x, ν⟩ � dA = 0 where EF n+1 is a constant vector field defined by EF n+1 = � Φ(En+1) F (En+1) if ω0 < 0, − Φ(−En+1) F (−En+1) if ω0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3) Note that EF n+1 satisfies ⟨EF n+1, En+1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' From boundary condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='9) and the anisotropic Cauchy-Schwarz inequality (see for example [25, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='4]), we know that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='4) ω0 ∈ (−F(En+1), F(−En+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' It was proved in [26, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='4] that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5) F(z) + ω0 � EF n+1, z � > 0, for any z ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Since Sn is compact, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='6) 0 < C1 ≤ F(z) + ω0 � EF n+1, z � ≤ C2, for any z ∈ Sn, where C1 and C2 are constant only depending on ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Next we derive the equations for several geometric quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Denote the F-Jacobi operator JF := divΣ(AF ∇·) + ⟨AF ◦ dν, dν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The following identities holds on Σ: JF F(ν) = ⟨∇HF, DF|ν⟩ + tr(h2 F ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='7) JF ⟨EF n+1, ν⟩ = ⟨EF n+1, ∇HF⟩, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='8) JF ⟨x, ν⟩ = ⟨x, ∇HF ⟩ + HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='9) STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The above formulas have been shown in [37] (also see [7, 52]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' For the convenience of reader, we give a direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' For a fixed p ∈ Σ, let {ei}n i=1 at p be the local orthonormal basis and ∇e1ej|p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In the following we calculate at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We have divΣ(AF ∇(F(ν))) = (AijFk ◦ νhkj),i = (Aij ◦ ν),i(Fk ◦ ν)hkj + Aij((Fk ◦ ν)hkj),i = AijphpiFkhkj + Aij(F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='kshsihkj + Fkhkji) = (Aijphpihkj + Aijhkji)Fk + AijhsihkjF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='ks = (Aijhij),kFk + Aijhsihkj(Ask − Fδks) = ⟨∇HF, DF|ν⟩ + tr(h2 F) − tr(AFh2)F(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' divΣ(AF ∇⟨EF n+1, ν⟩) = (Aij∇j⟨EF n+1, ν⟩),i = (Aij⟨EF n+1, ¯∇jν⟩),i = Aijkhki⟨EF n+1, ¯∇jν⟩ + Aij⟨EF n+1, ¯∇i ¯∇jν⟩ = (Aijkhkihjp + Aijhjpi)⟨EF n+1, ep⟩ − Aijhjkhki⟨EF n+1, ν⟩ = (Aijhij),p⟨EF n+1, ep⟩ − tr(AFh2)⟨EF n+1, ν⟩ = ⟨EF n+1, ∇HF⟩ − tr(AFh2)⟨EF n+1, ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' divΣ(AF ∇⟨x, ν⟩) = (Aij∇j⟨x, ν⟩),i = (Aij ◦ ν⟨x, ¯∇jν⟩),i = Aijkhki⟨x, ¯∇jν⟩ + Aij(hij + ⟨x, ¯∇i ¯∇jν⟩) = (Aijkhkihjp + Aijhjpi)⟨x, ep⟩ + HF − Aijhjkhki⟨x, ν⟩ = (Aijhij),p⟨x, ep⟩ + HF − tr(AF h2)⟨x, ν⟩ = ⟨x, ∇HF ⟩ + HF − tr(AF h2)⟨x, ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In the above computation we have used Codazzi equation hijk = hikj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' □ Next we verify the boundary equations for the geometric quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let x : Σ → Rn+1 + be an anisotropic capillary immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Then along ∂Σ, we have ⟨AF ∇⟨x, ν⟩, µ⟩ = qF⟨x, ν⟩, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='10) ⟨AF ∇[F(ν) + ω0⟨EF n+1, ν⟩], µ⟩ = qF(F(ν) + ω0⟨EF n+1, ν⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' In this proof we always take value along ∂Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' From Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='8), we have ⟨AF ∇⟨x, ν⟩, µ⟩ = ⟨AF ◦ dν(x), µ⟩ = hF (µ, µ)⟨x, µ⟩ = − νn+1 µn+1 hF (µ, µ)⟨x, ν⟩ = qF⟨x, ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Here we have used the fact xn+1 = 0 on ∂Rn+1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Similarly, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='8), we have ⟨AF ∇[F(ν) + ω0⟨EF n+1, ν⟩], µ⟩ = ⟨AF ◦ dν( ¯∇F(ν) + ω0EF n+1), µ⟩ = hF (µ, µ)⟨ ¯∇F(ν) + ω0EF n+1, µ⟩ = hF (µ, µ) � ¯∇F(ν) + ω0EF n+1, 1 µn+1 En+1 − νn+1 µn+1 ν � = qF(F(ν) + ω0⟨EF n+1, ν⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' 10 JINYU GUO AND CHAO XIA where in the last equality we use boundary condition ⟨νF , En+1⟩ = ⟨ ¯∇F(ν), En+1⟩ = −ω0 and the fact ⟨EF n+1, En+1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' □ Initiated by Minkowski-type formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2), we define (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='12) ϕ := n(F(ν) + ω0⟨EF n+1, ν⟩) − HF ⟨x, ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let x : Σ → Rn+1 + be a compact anisotropic capillary CAMC immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Then there hold: JF ϕ = ntr(h2 F ) − H2 F, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='13) ⟨AF ∇ϕ, µ⟩ = qFϕ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='14) � Σ ϕ dA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='13) follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='14) from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='15) is the Minkowski- type formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' A compact, immersed anisotropic capillary CAMC hypersurface in Rn+1 + is weakly stable if and only if it is a truncated Wulff shape, up to translation and homothety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We first notice that a truncated Wulff shape in Rn+1 + is energy-minimizing ([54, 31]), hence it is weakly stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Next, we let x : Σ → Rn+1 + be a compact weakly stable anisotropic capillary CAMC immersion and show it is a truncated Wulff shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='15), we know that ϕ is an admissible test function in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Therefore, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='14), we have 0 ≤ − � Σ ϕJF ϕ + � ∂Σ ϕ(⟨AF ∇ϕ, µ⟩ − qF ϕ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='16) = − � Σ [n(F(ν) + ω0⟨EF n+1, ν⟩) − HF⟨x, ν⟩]JF ϕ = − � Σ n(F(ν) + ω0⟨EF n+1, ν⟩)JF ϕ + HF � Σ ⟨x, ν⟩JF ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We compute the last term of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='16) by Green’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='9), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='15), we get � Σ ⟨x, ν⟩JF ϕ = � Σ ϕJF ⟨x, ν⟩ + � ∂Σ ⟨x, ν⟩⟨AF ∇ϕ, µ⟩ − ϕ⟨AF ∇⟨x, ν⟩, µ⟩ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='17) = HF � Σ ϕ + � ∂Σ ⟨x, ν⟩(qF ϕ) − ϕ(qF ⟨x, ν⟩), = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Hence, inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='17) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='16), we see � Σ (F(ν) + ω0⟨EF n+1, ν⟩)(ntr(h2 F ) − H2 F )dA ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='18) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5) and the fact that ntr(h2 F ) ≥ H2 F , we see the above inequality is in fact an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' It follows that ntr(h2 F ) = H2 F and in turn Σ is anisotropic umbilical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' By [40], Σ is a part of the Wulff shape (up to translation and homothety).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' □ STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Bernstein-type theorem for anisotropic capillary minimal surfaces In this section we prove the following Bernstein-type theorem for properly immersed, anisotropic capillary minimal surfaces in R3 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let Σ be an immersed anisotropic capillary minimal surface in R3 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Assume that Σ has Euclidean area growth, that is, there exists some C > 0 such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1) Area (Σ ∩ Br(0)) < Cr2 for any r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Then Σ is stable if and only if Σ is a half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' If Σ is a half-plane, then qF = 0 and it is clear that for f ∈ C∞ c (Σ), E′′ F (0) = � Σ AF (∇f, ∇f)dA ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Hence it is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Next let Σ be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let ψ := F(ν) + ω0⟨EF n+1, ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='7)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='11) we see that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2) � JF ψ = trh2 F in Σ, ⟨AF ∇ψ, µ⟩ = qFψ on ∂Σ, For f ∈ C∞ c (Σ), we put ψf into stability inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='15): (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3) 0 ≤ E′′ F (0) = − � Σ ψfJF (ψf)dA + � ∂Σ ψf[⟨AF ∇(ψf), µ⟩ − qFψf]ds, we now calculate the first term of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3) by integration by parts, − � Σ ψfJF(ψf)dA (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='4) = − � Σ ψf[fJFψ + 2⟨AF ∇ψ, ∇f⟩ + ψdiv(AF ∇f)]dA = − � Σ ψf 2trh2 F + 1 2⟨AF ∇ψ2, ∇f 2⟩ + ψ2fdiv(AF ∇f)dA = − � Σ ψf 2trh2 F + ψ2fdiv(AF ∇f)dA + 1 2 � Σ ψ2div(AF ∇f 2)dA − 1 2 � ∂Σ ψ2⟨AF ∇f 2, µ⟩ds = − � Σ ψf 2trh2 F − ψ2⟨AF ∇f, ∇f⟩dA − � ∂Σ ψ2f⟨AF∇f, µ⟩ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Next we compute the boundary term of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3), � ∂Σ ψf[⟨AF ∇(ψf), µ⟩ − qFψf]ds = � ∂Σ ψf[f(⟨AF ∇ψ, µ⟩ − qF ψ) + ψ⟨AF ∇f, µ⟩]ds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5) = � ∂Σ ψ2f⟨AF∇f, µ⟩ds Inserting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='4)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3), we get (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='6) � Σ ψf 2trh2 F dA ≤ � Σ ψ2⟨AF ∇f, ∇f⟩dA From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='6) we obtain that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='7) � Σ f 2trh2 FdA ≤ C2 2C−1 1 � Σ ⟨AF ∇f, ∇f⟩dA ≤ C2 2C−1 1 Λ � Σ |∇f|2dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' where Λ is maximal eigenvalue of AF on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' 12 JINYU GUO AND CHAO XIA Using the area growth condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3) and choosing a standard logarithmic cutoff function (see for example [11, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='37]) for f in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='7), we deduce that trh2 F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We conclude that Σ is a half-plane in R3 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proof of the first and second variational formulas The appendix is devoted to prove the first and second variational formulas, that is Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let Y ∈ T(∂B) such that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1) Y = Y Σ + fν = Y ∂Σ + ⟨Y, µ⟩µ + fν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We have from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='8) that 0 = ⟨Y, ¯N⟩ = ⟨µ, ¯N⟩⟨Y, µ⟩ + ⟨ν, ¯N⟩⟨Y, ν⟩ on ∂Σ, Therefore, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2) ⟨Y, µ⟩ = − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩f on ∂Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' On the other hand, from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='7), we see Y can be also expressed as follows (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3) Y = Y ∂Σ − f ⟨µ, ¯N⟩(⟨ν, ¯N⟩µ − ⟨µ, ¯N⟩ν) = Y ∂Σ + f ⟨µ, ¯N⟩ ¯ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' It follows that, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='4) ⟨Y, ¯ν⟩ = 1 ⟨µ, ¯N⟩f on ∂Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We use a prime to denote the time derivative at t = 0 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let ∇∂Σ denote the gradient on ∂Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let SΣ, S∂B denote the shape operator of Σ in B with respect to ν and ∂B in B with respect to ¯N respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let S1 and S2 denote the shape operator of ∂Σ in Σ with respect to µ, and of ∂Σ in ∂B with respect to ¯ν respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Then we have ν′ = SΣ(Y Σ) − ∇f, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5) µ′ = (−h(Y Σ, µ) + ∇µf)ν + S1(Y ∂Σ) + ⟨ν, ¯N⟩ ⟨µ, ¯N⟩ ∇∂Σf (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='6) + ⟨ν, ¯N⟩2 ⟨µ, ¯N⟩2 f � SΣ(µ) − h(µ, µ)µ � + 1 ⟨µ, ¯N⟩2 f � S∂B(¯ν) − h∂B(¯ν, ¯ν)¯ν � , ¯ν′ = S2(Y ∂Σ) − 1 ⟨µ, ¯N⟩∇∂Σf − h∂B(Y, ¯ν) ¯N (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='7) − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩2 f � SΣ(µ) − h(µ, µ)µ + S∂B(¯ν) − h∂B(¯ν, ¯ν)¯ν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let {ei}n i=1 be an orthonormal basis of TpΣ for some p ∈ Σ and denote ei(t) = (x(t, ·))∗(ei).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Using the fact ⟨ei(t), ν(t)⟩ = 0 and [ei(t), Y (t)] = 0, we have ν′ = n � i=1 ⟨ν′, ei⟩ei = − n � i=1 ⟨ν, e′ i⟩ei = − n � i=1 ⟨ν, ¯∇eiY ⟩ei = − n � i=1 ⟨ν, ¯∇ei(Y Σ + fν)⟩ei = n � i=1 ⟨SΣ(Y Σ), ei⟩ei − n � i=1 df(ei)ei = SΣ(Y Σ) − ∇f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' This is (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' As a consequence of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5) we get (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='8) ⟨µ′, ν⟩ = −⟨µ, ν′⟩ = −h(Y Σ, µ) + ∇µf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let now {τα}n−1 α=1 be an orthonormal basis of Tp(∂Σ) and denote τα(t) = (x(t, ·))∗(τα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2) and the fact [τα(t), Y (t)] = 0, we have ⟨µ′, τα⟩ = −⟨µ, τ ′ α⟩ = −⟨µ, ¯∇ταY ⟩ = − � µ, ¯∇τα � Y ∂Σ − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩fµ + fν �� (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='9) = −⟨µ, ¯∇ταY ∂Σ⟩ + d � ⟨ν, ¯N⟩ ⟨µ, ¯N⟩f � (τα) − f⟨µ, ¯∇ταν⟩ = −⟨µ, ¯∇ταY ∂Σ⟩ + � ⟨µ, ¯N⟩d⟨ν, ¯ N⟩(τα) − ⟨ν, ¯N⟩d⟨µ, ¯ N⟩(τα) � f ⟨µ, ¯N⟩2 + ⟨ν, ¯N⟩ ⟨µ, ¯N⟩df(τα) − fh(µ, τα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='7), we see ⟨µ, ¯N⟩d⟨ν, ¯N⟩(τα) − ⟨ν, ¯N⟩d⟨µ, ¯N⟩(τα) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='10) = ⟨µ, ¯N⟩⟨ ¯∇ταν, ¯N⟩ − ⟨ν, ¯N⟩⟨ ¯∇ταµ, ¯N⟩ + d ¯N(¯ν, τα) = ⟨µ, ¯N⟩2⟨ ¯∇ταν, µ⟩ − ⟨ν, ¯N⟩2⟨ ¯∇ταµ, ν⟩ + h∂B(τα, ¯ν) = h(µ, τα) + h∂B(¯ν, τα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Replacing (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='10) in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='9), we get ⟨µ′, τα⟩ = −⟨µ, ¯∇ταY ∂Σ⟩ + ⟨ν, ¯N⟩ ⟨µ, ¯N⟩df(τα) + ⟨ν, ¯N⟩2 ⟨µ, ¯N⟩2 h(µ, τα)f + 1 ⟨µ, ¯N⟩2 h∂B(¯ν, τα)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='11) It follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='8) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='11) that µ′ = ⟨µ′, µ⟩µ + ⟨µ′, ν⟩ν + n−1 � α=1 ⟨µ′, τα⟩τα (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='12) = (−h(Y Σ, µ) + ∇µf)ν + S1(Y ∂Σ) + ⟨ν, ¯N⟩ ⟨µ, ¯N⟩ ∇∂Σf + ⟨ν, ¯N⟩2 ⟨µ, ¯N⟩2 f � SΣ(µ) − h(µ, µ)µ � + 1 ⟨µ, ¯N⟩2 f � S∂B(¯ν) − h∂B(¯ν, ¯ν)¯ν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' This is (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' 14 JINYU GUO AND CHAO XIA Lastly, using [τα(t), Y (t)] = 0 again and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3), we have ⟨¯ν′, τα⟩ = −⟨¯ν, τ ′ α⟩ = −⟨¯ν, ¯∇ταY ⟩ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='13) = −⟨¯ν, ¯∇ταY ∂Σ⟩ − d � f ⟨µ, ¯N⟩ � (τα) = ⟨S2(Y ∂Σ), τα⟩ − 1 ⟨µ, ¯N⟩df(τα) − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩2 f � h(µ, τα) + h∂B(¯ν, τα) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Here the last equality we used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Now (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='7) follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='13) and the fact ⟨¯ν′, ¯N⟩ = −h∂B(Y, ¯ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Along ∂Σ, we have S1(Y ∂Σ, Y ∂Σ) + ⟨ν, ¯N⟩ ⟨µ, ¯N⟩⟨∇f, Y ∂Σ⟩ + ⟨ν, ¯N⟩2 ⟨µ, ¯N⟩2 f � h(Y ∂Σ, µ) + h∂B(Y ∂Σ, ¯ν) � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='14) = −⟨ν, ¯N⟩⟨Y, ¯ν′⟩ + ⟨µ, ¯N⟩h∂B(Y ∂Σ, Y ∂Σ) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='7) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3), we have − ⟨ν, ¯N⟩⟨Y, ¯ν′⟩ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='15) = ⟨ν, ¯N⟩⟨ ¯∇Y ∂ΣY ∂Σ, ¯ν⟩ + ⟨ν, ¯N⟩ ⟨µ, ¯N⟩⟨Y ∂Σ, ∇∂Σf⟩ + ⟨ν, ¯N⟩2 ⟨µ, ¯N⟩2 f � h(Y ∂Σ, µ) + h∂B(Y ∂Σ, ¯ν) � Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5), we see ⟨ν, ¯N⟩⟨ ¯∇Y ∂ΣY ∂Σ, ¯ν⟩ = ⟨ ¯∇Y ∂ΣY ∂Σ, −µ + ⟨µ, ¯N⟩ ¯N⟩= S1(Y ∂Σ, Y ∂Σ) − ⟨µ, ¯N⟩h∂B(Y ∂Σ, Y ∂Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' □ Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Along ∂Σ, we have (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='16) hF (Y Σ, µ) + 1 ⟨µ, ¯N⟩h∂B(Y, (νF )∂Σ) + ⟨µF, ¯N⟩ ⟨µ, ¯N⟩ h∂B(Y, ¯ν) = qFf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Using the capillary condition ⟨νF , ¯N⟩ = ω0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='8) and the fact that ⟨ ¯∇Y ∂ΣνF , ν⟩ = 0, we calculate that ⟨µ, ¯N⟩hF (Y ∂Σ, µ) = ⟨µ, ¯N⟩⟨ ¯∇Y ∂ΣνF, µ⟩ = ⟨ ¯∇Y ∂ΣνF , ¯N⟩ = −⟨νF, ¯∇Y ∂Σ ¯N⟩ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='17) = −⟨νF , ¯ν⟩h∂B(Y ∂Σ, ¯ν) − h∂B(Y ∂Σ, (νF )∂Σ) = −⟨µF , ¯N⟩h∂B(Y ∂Σ, ¯ν) − h∂B(Y ∂Σ, (νF )∂Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Here in the last equality we used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='17), we obtain that hF (Y Σ, µ) + 1 ⟨µ, ¯N⟩h∂B(Y, (νF )∂Σ) + ⟨µF , ¯N⟩ ⟨µ, ¯N⟩ h∂B(Y, ¯ν) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='18) = hF (Y ∂Σ, µ) + ⟨Y, µ⟩hF (µ, µ) + 1 ⟨µ, ¯N⟩h∂B(Y, (νF )∂Σ) + ⟨µF , ¯N⟩ ⟨µ, ¯N⟩ h∂B(Y, ¯ν) = − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩fhF(µ, µ) + 1 ⟨µ, ¯N⟩ � h∂B(Y − Y ∂Σ, (νF )∂Σ) + ⟨µF , ¯N⟩h∂B(Y − Y ∂Σ, ¯ν) � = − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩fhF(µ, µ) + 1 ⟨µ, ¯N⟩2 � h∂B(¯ν, (νF )∂Σ) + ⟨µF , ¯N⟩h∂B(¯ν, ¯ν) � f = qFf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE 15 □ Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' For an admissible variation Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' we calculate the first variation of anisotropic energy EF as follows E′ F (0) = A′ F (0) + ω0A′ W(0) = � Σ ∂ ∂t ��� t=0F(ν)dA + � Σ F(ν) ∂ ∂t ��� t=0dAt + ω0 � ∂Σ ⟨¯ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Y ⟩ds = � Σ ⟨ ¯∇F(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' ν′⟩ + F(ν)divΣY dA + ω0 � ∂Σ ⟨¯ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Y ⟩ds = � Σ ⟨DF(ν),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' −∇f + SΣ(Y Σ)⟩ + F(ν)(divΣY Σ + fH)dA + ω0 � ∂Σ ⟨¯ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Y ⟩ds = � Σ (divΣDF + F(ν)H)fdA + � ∂Σ F(ν)⟨Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' µ⟩ − f⟨DF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' µ⟩ds + ω0 � ∂Σ ⟨¯ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Y ⟩ds = � Σ HFfdA + � ∂Σ ⟨F(ν)µ − ⟨DF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' µ⟩ν + ω0¯ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Y ⟩ds = � Σ HFfdA + � ∂Σ ⟨µF + ω0¯ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Y ⟩ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' □ Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Let x : Σ → B be an anisotropic capillary CAMC (anisotropic capillary minimal, respectively) immersion, that is, HF = const (HF = 0, respectively) in Σ and ⟨νF , ¯N⟩ = ω0 on ∂Σ and x(t, ·) be a volume-preserving (compactly supported, respectively) admissible variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' It is direct to see that the first variational formula is true for any t ∈ (−ǫ, ǫ), that is, E′ F (t) = � Σt HFfdA + � ∂Σt ⟨µF + ω0¯ν, Y ⟩ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Thus we have E′′ F (0) = � Σ H′ FfdA + HF �� Σ fdA �′ + � ∂Σ ⟨Y ′, µF + ω0¯ν⟩ + ⟨Y, µ′ F + ω0¯ν′⟩ds + � ∂Σ ⟨Y, µF + ω0¯ν⟩ ∂ ∂t ��� t=0dst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Observe that in the case of CAMC with volume preserving variation, we have (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='19) �� Σ fdA �′ = V′′(0) = 0, and in the case of anisotropic minimal, HF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Hence in both cases, the term HF �� Σ fdA �′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Also, ⟨Y, µF + ω0¯ν⟩ = 0 along ∂Σ since Y |∂Σ ∈ T(∂B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Moreover, we have the evolution equation (see [33]) H′ F = −(divΣ(AF ∇f) + ⟨AF ◦ dν, dν⟩f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' It follows that E′′ F (0) = − � Σ (divΣ(AF ∇f) + ⟨AF ◦ dν, dν⟩f)fdA + � ∂Σ ⟨Y ′, µF + ω0¯ν⟩ + ⟨Y, µ′ F + ω0¯ν′⟩ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='20) So to prove the formula for E′′ F (0) we only need to compute the boundary term (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='21) � ∂Σ ⟨Y ′, µF + ω0¯ν⟩ds + � ∂Σ ⟨Y, µ′ F + ω0¯ν′⟩ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' 16 JINYU GUO AND CHAO XIA We now calculate the first term of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Since µF + ω0¯ν is parallel to ¯N along ∂Σ, we have ⟨Y ′, µF + ω0¯ν⟩ = ⟨ ¯∇Y Y, µF + ω0¯ν⟩ = −⟨µF , ¯N⟩h∂B(Y, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='22) Next we calculate the second term of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' According to the definition of µF , by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='4), we see that µ′ F = (F(ν)µ − ⟨νF, µ⟩ν)′ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='23) = F(ν)′µ + F(ν)µ′ − ⟨ ¯∇F(ν), µ⟩′ν − ⟨DF(ν), µ⟩ν′ = ⟨ ¯∇F(ν), ν′⟩µ + F(ν)µ′ − � ¯∇2F(ν)(ν′, µ) + ⟨ ¯∇F(ν), µ′⟩ � ν − ⟨DF(ν), µ⟩ν′ = ⟨DF(ν), ν′⟩µ + F(ν)µ′ − � (D2F(ν) + F(ν)Id)(ν′, µ) + ⟨DF(ν) + F(ν)ν, µ′⟩ � ν − ⟨DF(ν), µ⟩ν′ = ⟨DF(ν), ν′⟩µ + F(ν)µ′ − � D2F(ν)(ν′, µ) + ⟨DF(ν), µ′⟩ � ν − ⟨DF(ν), µ⟩ν′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' We remind here that ¯∇F is the Euclidean covariant derivative on the one-homogenous extension of F, and DF is the covariant derivative of F with respect to Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' It follows that ⟨Y, µ′ F + ω0¯ν′⟩ = −D2F(ν)(ν′, µ)f + F(ν)⟨Y, µ′⟩ + ω0⟨Y, ¯ν′⟩ + ⟨DF(ν), ν′⟩⟨Y, µ⟩ − ⟨DF(ν), µ′⟩f − ⟨DF(ν), µ⟩⟨Y, ν′⟩ := I + II + III + IV + V + V I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='24) We now tackle the above terms one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5), we have I = −D2F(ν)(ν′, µ)f (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='25) = −⟨(D2F(ν) + F(ν)Id)µ, ν′⟩f + F(ν)⟨µ, ν′⟩f = −⟨AF(ν) · µ, −∇f + SΣ(Y Σ)⟩f + F(ν)⟨µ, ν′⟩f = ⟨AF (ν)∇f, µ⟩f − hF (Y Σ, µ)f − F(ν)⟨µ′, ν⟩f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Utilizing (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='6), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='14) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3), we get ⟨Y, µ′⟩ = ⟨µ′, ν⟩f + S1(Y ∂Σ, Y ∂Σ) + ⟨ν, ¯N⟩ ⟨µ, ¯N⟩⟨∇f, Y ∂Σ⟩ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='26) + ⟨ν, ¯N⟩2 ⟨µ, ¯N⟩2 fh(Y ∂Σ, µ) + � 1 + ⟨ν, ¯N⟩2 ⟨µ, ¯N⟩2 � h∂B(Y ∂Σ, ¯ν)f = ⟨µ′, ν⟩f − ⟨ν, ¯N⟩⟨Y, ¯ν′⟩ + ⟨µ, ¯N⟩h∂B(Y ∂Σ, Y ∂Σ) + h∂B(Y ∂Σ, ¯ν)f = ⟨µ′, ν⟩f − ⟨ν, ¯N⟩⟨Y, ¯ν′⟩ + ⟨µ, ¯N⟩h∂B(Y ∂Σ, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Note that ¯ν = 1 ⟨µ, ¯ N⟩ν − ⟨ν, ¯ N⟩ ⟨µ, ¯ N⟩ ¯N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' It follows that S2(Y ∂Σ, Y ∂Σ) = 1 ⟨µ, ¯N⟩h(Y ∂Σ, Y ∂Σ) − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩h∂B(Y ∂Σ, Y ∂Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='27) STABLE ANISOTROPIC CAPILLARY HYPERSURFACES IN THE HALF-SPACE 17 Applying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='7), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='27) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3), we have ⟨Y, ¯ν′⟩ = S2(Y ∂Σ, Y ∂Σ) − 1 ⟨µ, ¯N⟩⟨∇f, Y ∂Σ⟩ − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩2 f � h(Y ∂Σ, µ) + h∂B(Y ∂Σ, ¯ν) � (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='28) = 1 ⟨µ, ¯N⟩h(Y ∂Σ, Y ∂Σ) − 1 ⟨µ, ¯N⟩⟨∇f, Y ∂Σ⟩ − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩2 fh(Y ∂Σ, µ) − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩h∂B(Y ∂Σ, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='26) with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='28), by using the capillary condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='9), we obtain that II + III = F(ν)⟨Y, µ′⟩ + ω0⟨Y, ¯ν′⟩ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='29) = F(ν)⟨µ′, ν⟩f + (−F(ν)⟨ν, ¯N⟩ + ω0)⟨Y, ¯ν′⟩ + F(ν)⟨µ, ¯ N⟩h∂B(Y ∂Σ, Y ) = F(ν)⟨µ′, ν⟩f + ⟨µ, ¯N⟩⟨νF , µ⟩⟨Y, ¯ν′⟩ + F(ν)⟨µ, ¯N⟩h∂B(Y ∂Σ, Y ) = F(ν)⟨µ′, ν⟩f + ⟨µF, ¯N⟩h∂B(Y ∂Σ, Y ) + ⟨νF, µ⟩ � h(Y ∂Σ, Y ∂Σ) − ⟨∇f, Y ∂Σ⟩ − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩fh(Y ∂Σ, µ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Now we claim that IV + V + V I (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='30) = ⟨DF(ν), ν′⟩⟨Y, µ⟩ − ⟨DF(ν), µ′⟩f − ⟨DF(ν), µ⟩⟨Y, ν′⟩ = −⟨νF, µ⟩ � h(Y ∂Σ, Y ∂Σ) − ⟨∇f, Y ∂Σ⟩ − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩fh(Y ∂Σ, µ) � − f ⟨µ, ¯N⟩h∂B(Y, (νF )∂Σ), In fact, from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='5), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='2) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='1) we have IV = ⟨DF(ν), ν′⟩⟨Y, µ⟩ = − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩f⟨DF(ν), SΣ(Y Σ) − ∇f⟩ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='31) = − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩f � ⟨DF(ν), � α h(Y Σ, τα)τα − ∇∂Σf⟩ + ⟨DF(ν), µ⟩(h(Y Σ, µ) − ∇µf) � , and V I = −⟨DF(ν), µ⟩⟨Y, ν′⟩ = −⟨DF(ν), µ⟩⟨Y Σ, −∇f + SΣ(Y Σ)⟩ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='32) = −⟨DF(ν), µ⟩ � −⟨∇f, Y ∂Σ⟩ + ⟨ν, ¯N⟩ ⟨µ, ¯N⟩f∇µf + h(Y ∂Σ, Y ∂Σ) − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩fh(Y ∂Σ, µ) − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩fh(Y Σ, µ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Since µ = 1 ⟨µ, ¯ N⟩ ¯N − ⟨ν, ¯ N⟩ ⟨µ, ¯ N⟩ν, we see S1(Y ∂Σ) = 1 ⟨µ, ¯N⟩ � α h∂B(Y ∂Σ, τα)τα − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩ � α h(Y ∂Σ, τα)τα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='33) 18 JINYU GUO AND CHAO XIA Using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='6), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='33) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3), we deduce V = −⟨DF(ν), µ′⟩f (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='34) = − � DF(ν), 1 ⟨µ, ¯N⟩ � α h∂B(Y ∂Σ, τα)τα − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩ � α h(Y ∂Σ, τα)τα + ⟨ν, ¯N⟩ ⟨µ, ¯N⟩∇∂Σf � f − � DF(ν), ⟨ν, ¯N⟩2 ⟨µ, ¯N⟩2 f � α h(µ, τα)τα + f ⟨µ, ¯N⟩2 � α h∂B(¯ν, τα)τα � f = − � DF(ν), ⟨ν, ¯N⟩ ⟨µ, ¯N⟩∇∂Σf − ⟨ν, ¯N⟩ ⟨µ, ¯N⟩ � α h(Y Σ, τα)τα + 1 ⟨µ, ¯N⟩ � α h∂B(Y, τα)τα � f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Now the above claim (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='30) follows by combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='31), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='32) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Putting I-V I into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='24), we get ⟨Y, µ′ F + ω0¯ν′⟩ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='35) = f � ⟨AF ∇f, µ⟩ − hF (Y Σ, µ) − 1 ⟨µ, ¯N⟩h∂B(Y, (νF )∂Σ) � + ⟨µF , ¯N⟩h∂B(Y ∂Σ, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='35) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='22), using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='3) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='16), we have ⟨Y, µ′ F + ω0¯ν′⟩ + ⟨Y ′, µF + ω0¯ν⟩ (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='36) = f � ⟨AF ∇f, µ⟩ − hF (Y Σ, µ) − 1 ⟨µ, ¯N⟩h∂B(Y, (νF )∂Σ) − ⟨µF , ¯N⟩ ⟨µ, ¯N⟩ h∂B(Y, ¯ν) � = f (⟨AF ∇f, µ⟩ − qF f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The proof is completed by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='20) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' □ Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The authors would like to thank Professor Haizhong Li and Professor Guofang Wang for their constant support and their interest on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' The first author would also like to thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Han Hong for interesting discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Ainouz and R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' Department of Mathematics, Tsinghua University, Beijing, 100084, China Email address: guojinyu@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='cn School of Mathematical Sciences, Xiamen University, 361005, Xiamen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content=' China Email address: chaoxia@xmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE1T4oBgHgl3EQfOwPh/content/2301.03020v1.pdf'} diff --git a/pdE1T4oBgHgl3EQfiQRF/content/2301.03249v1.pdf b/pdE1T4oBgHgl3EQfiQRF/content/2301.03249v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3a9e5603f52607cf7277a37f28c948524aa9d16c --- /dev/null +++ b/pdE1T4oBgHgl3EQfiQRF/content/2301.03249v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:86b24f552408b6e32874404741c5157ee1afb205e079998443e7eb1d3c0e8a26 +size 465478 diff --git a/pdE1T4oBgHgl3EQfiQRF/vector_store/index.pkl b/pdE1T4oBgHgl3EQfiQRF/vector_store/index.pkl new file mode 100644 index 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“hearing” the shapes of drums and bells +By Guo-Wei Wei +January 13, 2023 +Abstract +Mark Kac asked a famous question in 1966, “Can one hear the shape of a drum?”, a spectral geometry +problem that has intrigued mathematicians for the last six decades and is important to many other fields, +such as architectural acoustics, audio forensics, pattern recognition, radiology, and imaging science. A +related question is how to hear the shape of a drum. We show that the answer was given in the set of 65 +Zenghouyi chime bells dated back to 475-433 B.C. in China. The set of chime bells gradually varies their +sizes and weights to enable melodies, intervals, and temperaments. The same design principle was used +in many other musical instruments, such as xylophones, pan flutes, pianos, etc. We reveal that there +is a fascinating connection between the progression pattern of many musical instruments and filtration +(or spectral sequence) in topological data analysis (TDA). We argue that filtration-induced evolutionary +de Rham-Hodge theory provides a new mathematical foundation for musical instruments. Its discrete +counterpart, persistent Laplacians and many other persistent topological Laplacians, including persistent +sheaf Laplacians and persistent path Laplacians are briefly discussed. +(a) +(b) +(c) +(d) +Figure 1: The progression patterns in musical instruments. +a. The Zonghouyi chime bells. b. An xylophone. c. A +pan flute. d. Strings of a grand piano. Image courtesy of +Wikipedia. +“Can one hear the shape of a drum?”, a famous ques- +tion posed by Mark Kac in 1966 [10], has intrigued math- +ematicians for generations. In other words, if you hear +the sound from a drum, i.e., the set of overtones pro- +duced by the drum, can you infer its shape? +Mathe- +matically, the essence of the spectral geometry question +is whether the shape can be uniquely determined from +the eigenvalues of the Laplacian operator defined on the +shape. There are many examples of isospectral mani- +folds which are not isometric in two and higher dimen- +sional settings [6, 15, 17]. However, the problem is not +closed yet — one can discern the shapes of certain ge- +ometric types from their sounds [9, 16]. The question +has a far-reaching impact on many fields beyond math- +ematics, such as architectural acoustics, audio forensics, +pattern recognition, radiology, imaging science, and mu- +sical science [3,4]. +In musical composition, it is a common practice to +use a drum set of varying sizes, instead of a single piece +of drum, to facilitate a tonic harmonic progression, which is a foundation of harmony in modern music. The +human brain is trained from the drum set to distinguish the overtones of individual drum. The comparative +training/learning from a set of drums is the basis to hear the shape of a drum from a drum set. +Interestingly, to create tonal harmony, the ancient Chinese built a set of 65 Zenghouyi chime bells dated +back to 475-433 B.C. in the Warring States Period in China (Figure 1a). The shape of each chime bell was +designed to produce a distinct sound. The set of 65 chime bells gradually varies in their sizes and weights, +1 +arXiv:2301.05025v1 [math.HO] 11 Jan 2023 + +-FOR.TUMI-U.K.ranging from 153.4 centimeters (60.4 in) to 20.4 centimeters (8.0 in) in height and from 203.6 kilograms (449 +lb) to 2.4 kilograms (5.3 lb) in weight. The set of Zenghouyi chime bells covers from C2 to D7 tonal range +and can play all twelve half tones in the middle area of the tonal range [21], enabling melodies, intervals, and +temperaments. The same design principle is used in many other musical instruments, such as xylophones +(Figure 1b), pan flutes (Figure 1c), and pianos (Figure 1d). Due to the close proximity among chime bells, +resonance among them may occur when they are struck, giving rise to prolonged harmony. Made of bronze, +one of the most precious metals available at the time, chime bells were used in various rituals, ceremonies, +and entertainment. +There is a fascinating and apparent connection between the progression pattern of musical instruments +(Figure 1) and mathematical filtration (Figure 2a), a spectral sequence technique [13] widely used in homo- +logical algebra and topological data analysis (TDA). As a new branch of mathematics, TDA uses topological +and geometric concepts to understand and extract topological patterns and structures in data. The main +workhorse of TDA is persistent homology [5,22], a branch of algebraic topology that creates a mathematical +microscopy of a point cloud by filtration. Through the comparative analysis of the topological invariant +changes induced by the filtration, persistent homology delineates the shape of data [12]. Paired with ad- +vanced machine learning and deep learning algorithms, persistent homology has had tremendous success in +data science. It has been established as one of the most powerful tools in simplifying the geometric com- +plexity and reducing the high dimensionality of biomolecular interactions [11], revolutionizing drug discovery +and the forecasting of emerging viral variants. +a +(b) +(c) +(a) +Figure 2: Illustration of filtration and persistent Laplacian +spectra. a. Filtration of a point cloud. b-c. The persistent +Laplacian analysis of the point cloud. Here, βα +j and λα +j (j = +0, 1) are persistent Betti-j numbers and the first nonzero +eigenvalues of the jth persistent Laplacian, respectively. +Note that λα +j capture the homotopic shape evolution of +data (see the frequency changes after α = 1.5) that is not +reflected in βα +j . +The similarity between λα +0 and λα +1 is a +coincidence. Image courtesy of Dr. Jian Jiang. +However, persistent homology is not directly appli- +cable to the tonal analysis of chime bells. First, the set +of chime bells may be regarded as the result of a set of +evolving chime bell manifolds, rather than that of the fil- +tration of a point cloud. Additionally, there is no change +in topological invariants associated the homotopic shape +evolution of the set of chime bells. Finally, persistent ho- +mology cannot present a frequency or tonal analysis of +chime bells or many other musical instruments. +Recently, an evolutionary de Rham-Hodge method +has been proposed as a multiscale generalization of the +classical de Rham-Hodge theory, a landmark of the 20th +Century’s mathematics [1]. +It provides a multiscale +geometric and topological analysis of filtration-induced +manifolds, on which a family of evolutionary Hodge +Laplacians can be defined on the set of chime bells to +characterize their tonal evolution. In association with +a family of evolutionary de Rham complexes, evolution- +ary Hodge Laplacians reveal the full set of topological +invariants, such as Betti numbers, in their kernel or null +space dimensions. +The point-cloud counterpart of the evolutionary de +Rham-Hodge method on manifolds is called persistent spectral graph [18] (also known as persistent Laplacians +[14]) on simplicial complexes. Like the evolutionary de Rham-Hodge method, persistent Laplacians not only +return the full set of topological invariants in their harmonic spectra as persistent homology does but also +capture the homotopic shape evolution of data during the filtration in their first non-harmonic spectra +(Figure 2b and 2c), for which persistent homology cannot describe. +A generalization of persistent Laplacians was made through the sheaf theory [8] and the resulting per- +2 + +aα = 0.7 +α = 1.4 +a = 1.81 +a = 2.06 +6.0 +2 +3.5 +1.0 +0 +0.75 1.00 1.25 1.50 +1.75 2.00 +2.25 +2.50 +radius a +1 +2 +0 +O +0.75 1.00 1.25 1.50 1.75 +52.00 +2.25 +2.50 +radius asistent sheaf Laplacians [20] allow the embedding of heterogeneous characters in topological invariants, +e.g., encoding non-geometric information in a geometry-based simplicial complex. Another generalization +is persistent path Laplacians [19], built from the path complex and path homology [2, 7]. Persistent path +Laplacians are designed for directed graphs and directed networks. These new persistent topological Lapla- +cians not only lay a mathematical foundation for the tonal analysis in musical science but also significantly +extend the applicable domain and power of TDA. +References +[1] J. Chen, R. Zhao, Y. Tong, and G.-W. Wei. +Evolutionary de Rham-Hodge method. +Discrete and +Continuous Dynamical Systems. Series B, 26(7):3785, 2021. +[2] S. Chowdhury and F. M´emoli. Persistent path homology of directed networks. In Proceedings of the +Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1152–1169. SIAM, 2018. +[3] L. Cosmo, M. Panine, A. Rampini, M. Ovsjanikov, M. M. Bronstein, and E. Rodol`a. Isospectralization, +or how to hear shape, style, and correspondence. +In Proceedings of the IEEE/CVF Conference on +Computer Vision and Pattern Recognition, pages 7529–7538, 2019. +[4] I. Dokmani´c, R. Parhizkar, A. Walther, Y. M. Lu, and M. Vetterli. Acoustic echoes reveal room shape. +Proceedings of the National Academy of Sciences, 110(30):12186–12191, 2013. +[5] H. Edelsbrunner, J. Harer, et al. Persistent homology-A survey. Contemporary Mathematics, 453:257– +282, 2008. +[6] C. Gordon, D. Webb, and S. Wolpert. Isospectral plane domains and surfaces via riemannian orbifolds. +Inventiones Mathematicae, 110(1):1–22, 1992. +[7] A. Grigor’yan, Y. Lin, Y. Muranov, and S.-T. Yau. Homologies of path complexes and digraphs. arXiv +preprint arXiv:1207.2834, 2012. +[8] J. Hansen and R. Ghrist. Toward a spectral theory of cellular sheaves. Journal of Applied and Compu- +tational Topology, 3(4):315–358, 2019. +[9] H. Hezari, Z. Lu, and J. Rowlett. The dirichlet isospectral problem for trapezoids. Journal of Mathe- +matical Physics, 62(5):051511, 2021. +[10] M. Kac. Can one hear the shape of a drum? The American Mathematical Monthly, 73(4P2):1–23, 1966. +[11] J. Liu, K.-L. Xia, J. Wu, S. S.-T. Yau, and G.-W. Wei. Biomolecular topology: Modelling and analysis. +Acta Mathematica Sinica, English Series, 38(10):1901–1938, 2022. +[12] P. Y. Lum, G. Singh, A. Lehman, T. Ishkanov, M. Vejdemo-Johansson, M. Alagappan, J. Carlsson, and +G. Carlsson. Extracting insights from the shape of complex data using topology. Scientific Reports, +3(1):1–8, 2013. +[13] J. McCleary. A user’s guide to spectral sequences. Number 58. Cambridge University Press, 2001. +[14] F. M´emoli, Z. Wan, and Y. Wang. Persistent laplacians: Properties, algorithms and implications. SIAM +Journal on Mathematics of Data Science, 4(2):858–884, 2022. +[15] J. Milnor. +Eigenvalues of the Laplace operator on certain manifolds. +Proceedings of the National +Academy of Sciences, 51(4):542–542, 1964. +[16] A. W. Reid. Isospectrality and commensurability of arithmetic hyperbolic 2-and 3-manifolds. Duke +Mathematical Journal, 65(2):215–228, 1992. +3 + +[17] M.-F. Vign´eras. +Vari´et´es riemanniennes isospectrales et non isom´etriques. +Annals of Mathematics, +112(1):21–32, 1980. +[18] R. Wang, D. D. Nguyen, and G.-W. Wei. Persistent spectral graph. International Journal for Numerical +Methods in Biomedical Engineering, 36(9):e3376, 2020. +[19] R. Wang and G.-W. Wei. Persistent path Laplacian. Foundations of Data Sciences, 5(1):26–55, 2023. +[20] X. Wei and G.-W. Wei. Persistent sheaf Laplacians. arXiv preprint arXiv:2112.10906, 2021. +[21] Y. Yan, K. Chai, H. Liang, and L. Kong. Physics involvement in ancient chinese chime bells. In AIP +Conference Proceedings, volume 1517, pages 43–48. American Institute of Physics, 2013. +[22] A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete Comput. Geom., 33:249–274, +2005. +Guo-Wei Wei is an MSU Foundation Professor at Michigan State University. His research concerns the +mathematical foundations of biological science and data science. +4 + diff --git a/sdE4T4oBgHgl3EQfVgyf/content/tmp_files/load_file.txt b/sdE4T4oBgHgl3EQfVgyf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2fe4c98cc95eedec1fc00a6da9d1be8cdbffa615 --- /dev/null +++ b/sdE4T4oBgHgl3EQfVgyf/content/tmp_files/load_file.txt @@ -0,0 +1,267 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf,len=266 +page_content='Topological data analysis “hearing” the shapes of drums and bells By Guo-Wei Wei January 13, 2023 Abstract Mark Kac asked a famous question in 1966, “Can one hear the shape of a drum?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=', a spectral geometry problem that has intrigued mathematicians for the last six decades and is important to many other fields, such as architectural acoustics, audio forensics, pattern recognition, radiology, and imaging science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' A related question is how to hear the shape of a drum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' We show that the answer was given in the set of 65 Zenghouyi chime bells dated back to 475-433 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' in China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' The set of chime bells gradually varies their sizes and weights to enable melodies, intervals, and temperaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' The same design principle was used in many other musical instruments, such as xylophones, pan flutes, pianos, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' We reveal that there is a fascinating connection between the progression pattern of many musical instruments and filtration (or spectral sequence) in topological data analysis (TDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' We argue that filtration-induced evolutionary de Rham-Hodge theory provides a new mathematical foundation for musical instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Its discrete counterpart, persistent Laplacians and many other persistent topological Laplacians, including persistent sheaf Laplacians and persistent path Laplacians are briefly discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 1: The progression patterns in musical instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' The Zonghouyi chime bells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' An xylophone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' A pan flute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Strings of a grand piano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Image courtesy of Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' “Can one hear the shape of a drum?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=', a famous ques- tion posed by Mark Kac in 1966 [10], has intrigued math- ematicians for generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' In other words, if you hear the sound from a drum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=', the set of overtones pro- duced by the drum, can you infer its shape?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Mathe- matically, the essence of the spectral geometry question is whether the shape can be uniquely determined from the eigenvalues of the Laplacian operator defined on the shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' There are many examples of isospectral mani- folds which are not isometric in two and higher dimen- sional settings [6, 15, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' However, the problem is not closed yet — one can discern the shapes of certain ge- ometric types from their sounds [9, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' The question has a far-reaching impact on many fields beyond math- ematics, such as architectural acoustics, audio forensics, pattern recognition, radiology, imaging science, and mu- sical science [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' In musical composition, it is a common practice to use a drum set of varying sizes, instead of a single piece of drum, to facilitate a tonic harmonic progression, which is a foundation of harmony in modern music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' The human brain is trained from the drum set to distinguish the overtones of individual drum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' The comparative training/learning from a set of drums is the basis to hear the shape of a drum from a drum set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Interestingly, to create tonal harmony, the ancient Chinese built a set of 65 Zenghouyi chime bells dated back to 475-433 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' in the Warring States Period in China (Figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' The shape of each chime bell was designed to produce a distinct sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' The set of 65 chime bells gradually varies in their sizes and weights, 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='05025v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='HO] 11 Jan 2023 FOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='TUMI-U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='ranging from 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='4 centimeters (60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='4 in) to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='4 centimeters (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='0 in) in height and from 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='6 kilograms (449 lb) to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='4 kilograms (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='3 lb) in weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' The set of Zenghouyi chime bells covers from C2 to D7 tonal range and can play all twelve half tones in the middle area of the tonal range [21], enabling melodies, intervals, and temperaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' The same design principle is used in many other musical instruments, such as xylophones (Figure 1b), pan flutes (Figure 1c), and pianos (Figure 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Due to the close proximity among chime bells, resonance among them may occur when they are struck, giving rise to prolonged harmony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Made of bronze, one of the most precious metals available at the time, chime bells were used in various rituals, ceremonies, and entertainment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' There is a fascinating and apparent connection between the progression pattern of musical instruments (Figure 1) and mathematical filtration (Figure 2a), a spectral sequence technique [13] widely used in homo- logical algebra and topological data analysis (TDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' As a new branch of mathematics, TDA uses topological and geometric concepts to understand and extract topological patterns and structures in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' The main workhorse of TDA is persistent homology [5,22], a branch of algebraic topology that creates a mathematical microscopy of a point cloud by filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Through the comparative analysis of the topological invariant changes induced by the filtration, persistent homology delineates the shape of data [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Paired with ad- vanced machine learning and deep learning algorithms, persistent homology has had tremendous success in data science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' It has been established as one of the most powerful tools in simplifying the geometric com- plexity and reducing the high dimensionality of biomolecular interactions [11], revolutionizing drug discovery and the forecasting of emerging viral variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' a (b) (c) (a) Figure 2: Illustration of filtration and persistent Laplacian spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Filtration of a point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' b-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' The persistent Laplacian analysis of the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Here, βα j and λα j (j = 0, 1) are persistent Betti-j numbers and the first nonzero eigenvalues of the jth persistent Laplacian, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Note that λα j capture the homotopic shape evolution of data (see the frequency changes after α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='5) that is not reflected in βα j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' The similarity between λα 0 and λα 1 is a coincidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Image courtesy of Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Jian Jiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' However, persistent homology is not directly appli- cable to the tonal analysis of chime bells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' First, the set of chime bells may be regarded as the result of a set of evolving chime bell manifolds, rather than that of the fil- tration of a point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Additionally, there is no change in topological invariants associated the homotopic shape evolution of the set of chime bells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Finally, persistent ho- mology cannot present a frequency or tonal analysis of chime bells or many other musical instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Recently, an evolutionary de Rham-Hodge method has been proposed as a multiscale generalization of the classical de Rham-Hodge theory, a landmark of the 20th Century’s mathematics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' It provides a multiscale geometric and topological analysis of filtration-induced manifolds, on which a family of evolutionary Hodge Laplacians can be defined on the set of chime bells to characterize their tonal evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' In association with a family of evolutionary de Rham complexes, evolution- ary Hodge Laplacians reveal the full set of topological invariants, such as Betti numbers, in their kernel or null space dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' The point-cloud counterpart of the evolutionary de Rham-Hodge method on manifolds is called persistent spectral graph [18] (also known as persistent Laplacians [14]) on simplicial complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Like the evolutionary de Rham-Hodge method, persistent Laplacians not only return the full set of topological invariants in their harmonic spectra as persistent homology does but also capture the homotopic shape evolution of data during the filtration in their first non-harmonic spectra (Figure 2b and 2c), for which persistent homology cannot describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' A generalization of persistent Laplacians was made through the sheaf theory [8] and the resulting per- 2 aα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='7 α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='4 a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='81 a = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='0 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='50 radius a 1 2 0 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='75 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='50 radius asistent sheaf Laplacians [20] allow the embedding of heterogeneous characters in topological invariants, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=', encoding non-geometric information in a geometry-based simplicial complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Another generalization is persistent path Laplacians [19], built from the path complex and path homology [2, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Persistent path Laplacians are designed for directed graphs and directed networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' These new persistent topological Lapla- cians not only lay a mathematical foundation for the tonal analysis in musical science but also significantly extend the applicable domain and power of TDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Chen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Tong, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Evolutionary de Rham-Hodge method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Discrete and Continuous Dynamical Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Series B, 26(7):3785, 2021.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' American Institute of Physics, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Zomorodian and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Carlsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Computing persistent homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Discrete Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=', 33:249–274, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' Guo-Wei Wei is an MSU Foundation Professor at Michigan State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' His research concerns the mathematical foundations of biological science and data science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} +page_content=' 4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE4T4oBgHgl3EQfVgyf/content/2301.05025v1.pdf'} diff --git 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b/uNAyT4oBgHgl3EQf0fm1/content/tmp_files/2301.00720v1.pdf.txt @@ -0,0 +1,1698 @@ +Quantum Circuit Resizing +Movahhed Sadeghi∗ +The Pennsylvania State University, +Department of +Computer Science and Engineering +Email: mus883@psu.edu +∗corresponding Author +Soheil Khadirsharbiyani +The Pennsylvania State University, +Department of +Computer Science and Engineering +Email: szk921@psu.edu +Mahmut Taylan Kandemir +The Pennsylvania State University, +Department of +Computer Science and Engineering +Email: mtk2@psu.edu +Abstract—Existing quantum systems provide very limited phys- +ical qubit counts, trying to execute a quantum algorithm/circuit +on them that have a higher number of logical qubits than +physically available lead to a compile-time error. Given that it +is unrealistic to expect existing quantum systems to provide, in +near future, sufficient number of qubits that can accommodate +large circuit, there is a pressing need to explore strategies that +can somehow execute large circuits on small systems. +In this paper, first, we perform an analysis to identify the +qubits that are most suitable for circuit resizing. Our results +reveal that, in most quantum programs, there exist qubits that +can be reused mid-program to serially/sequentially execute the +circuit employing fewer qubits. Motivated by this observation, we +design, implement and evaluate a compiler-based approach that +i) identifies the qubits that can be most beneficial for serial circuit +execution; ii) selects those qubits to reuse at each step of execution +for size minimization of the circuit; and iii) minimizes Middle +Measurement (MM) delays due to impractical implementation of +shots to improve the circuit reliability. Furthermore, since our +approach intends to execute the circuits sequentially, the crosstalk +errors can also be optimized as a result of the reduced number +of concurrent gates. +The experimental results indicate that our proposed approach +can (i) execute large circuits that initially cannot fit into small +circuits, on small quantum hardware, and (ii) can significantly +improve the PST of the results by 2.1X when both original and +our serialized programs can fit into the target quantum hardware. +I. INTRODUCTION +Quantum computers are introduced to enhance the com- +putational capacity for complex problems such as machine +learning [8] and chemistry simulation [23]. Over the last +decade, several vendors unveiled their quantum hardware in +an effort to benefit from quantum phenomena to execute a +quantum program on real systems. Currently, vendors like +Google, IBM, and Intel support up to 72, 127, and 49 qubits +(quantum bits) [19], [24], [25], respectively. However, due to +errors introduced in the real implementation of qubits, such +as coherence and gate errors, existing quantum computers +are highly unreliable. To solve this issue, Quantum Error +Correction Codes (QEC) algorithms [11], [18], [20], [33] have +been introduced, but they typically require 10-100 additional +qubits to create a single fault-tolerant qubit. +Due to this enormous extra qubit requirements brought by +QEC, which is impractical considering the size of the current +systems, Noisy Intermediate Scale Quantum Computers (NISQ) +[18] have been developed to execute small-to-medium circuits +on current hardware while aiming to minimize the error rate by +different techniques. Several works targeting NISQ machines +have been introduced to minimize different errors, aiming +to improve the reliability of the system. While some works +such as [32] try to execute a given quantum program on the +most reliable set of qubits available in the target hardware, +others [28], [32], [48], [51] aim to minimize gate errors by +optimizing the number of SWAP operations (SWAP operations +are used to transfer the content of one qubit to another when +the qubits involved in a 2-qubit operation do not have a direct +physical connection between them). Despite these efforts, the +problem of executing a large quantum circuit with lots of +logical qubits on existing small quantum systems with only a +few physical qubits seems to be one of the biggest challenges +in quantum computing. +Till recently, there had been no mechanism to solve this +problem successfully. However, quantum vendors have recently +introduced the Middle Reset (MR) and Middle Measurement +(MM) gates, which can be utilized to resize quantum circuits +during the execution. By utilizing MR/MM, theoretically, +famous quantum algorithms like Bernstein-Vazirani [7] with +any qubit count can be executed only using 2 qubits [3]. +Additionally, when running on 2 qubits, no SWAP operations +is needed since the 2 physical qubits to which the program is +assigned usually have a connection between them, eliminating +the gate error due to SWAP operations in the process. Although +this approach can be highly effective in improving the system’s +reliability when the qubits we want to reset are correctly +identified, to our knowledge, there are no prior works in this +area that exploit the potential of these gates in an automated +fashion. +In this paper, first, we perform an analysis to identify the +qubits that are most suitable for circuit resizing. Our results +reveal that, in most quantum programs, there exist qubits that +can be reused mid-program to serially execute the circuit +employing fewer qubits. Motivated by this observation, we +design, implement and evaluate a compiler-based approach +that i) identifies the qubits that can be most beneficial for +serial circuit execution; ii) selects those qubits to reuse at +each step of execution for size minimization of the circuit1; +1When it leads to no confusion, we will use the terms ”serializability”, +”sequential/serial execution”, ”circuit reduction”, and ”circuit resizing”, inter- +changeably. +1 +arXiv:2301.00720v1 [cs.ET] 30 Dec 2022 + +and iii) minimizes Middle Measurement (MM) delays due to +impractical implementation of shots2 to improve the circuit +reliability. Furthermore, since our approach intends to execute +the circuits sequentially, the crosstalk errors can also be +optimized as a result of the reduced number of concurrent +gates. To summarize, in this paper, we make the following +main contributions: +• We observe that there is only one constraint that need to be +satisfied in a quantum circuit for the circuit to be ”serially +executable”. More detail about this constraint is given in +Section IV. +• Based on this observation, we present an algorithm that +selects qubits in a way that the serial execution opportunity +is maximized; hence, the size of the resulting circuit is +minimal. +• We present a proof demonstrating that our proposed ap- +proach really minimizes the number of qubits needed to +execute a quantum program (i.e., it completely serializes +it). Consequently, we avoid system size compilation error +whenever it is possible to do so. +• We observe that the current concept of shot does not fully +work with our proposal and leads to coherence errors in +the results. To solve it, we present a new concept/feature +in quantum systems called iteration, which leads to highly +reliable results. +• We present experimental evidence showing the effectiveness +of our proposed approach. The experimental results indicate +that our proposed approach can (i) execute large circuits +that initially cannot fit into small circuits, on small quantum +hardware, and (ii) can significantly improve the PST of +the results by 2.1x when both original and our serialized +programs can fit into the target quantum hardware. +The remainder of this paper is organized as follows. In Sec- +tion II, we give a background on quantum computing covering +quantum computing basics, currently available quantum hard- +ware, and MM/MR gates. In Section II-D, we go over the prior +works relevant to this study and explain their shortcomings. +In Section III, we motivate our work by characterizing the +problem, and in Section IV, we explain the technical details +of our proposed approach to circuit minimization/serialization. +In Section V-A, we present our evaluation setup and describe +the workloads as well as our evaluation methodology. The +performance and sensitivity results are presented, respectively, +Sections V-B and V-C. In Section VI, we give a summary +of our major conclusions, and finally, discuss future research +directions in Section VII. +II. BACKGROUND AND RELATED WORK +In this section, we give an overview of quantum computing +and discuss representative quantum systems that are currently +available and their salient features. +2Numerous executions, known as ”shots”, are often carried out in order to +obtain the chance of getting the right answer from quantum hardware. +A. Quantum Computing Basics +Quantum computation is based on qubits, as opposed to +classical computing, which is based on bits. Compared to a +classical bit which represents a value of 0 or 1, a qubit is +represented as a vector that holds a ”state” between 0 and 1, +which is defined as follows: |ϕ⟩ = α|0⟩+β|1⟩. This leads to +an exponential growth in state space in terms of the number +of qubits [46]. For example, having two qubits gives us a +state space of |ϕ⟩ = α00|00⟩ + α01|01⟩ + α10|10⟩ + α11|11⟩. +Consequently, algorithms whose state/search space grows +exponentially can potentially benefit from quantum computing +by operating on qubits and linearizing their state space [46]. +A quantum gate is the basic building block of a quantum +program. It typically operates on a small number of qubits. +Example quantum gates include SWAP gate, NOT-gate square +root, Controlled-NOT gate (C-NOT) and other controlled gates. +It is to be emphasized that a quantum gate that operates on +multiple qubits can execute only in the presence of a direct +link between the involved qubits.3 In reality, qubits are prone +to a variety of errors, such as coherence error, gate error, and +crosstalk. Qubits can only keep their state for a finite amount +of time, which leads to coherence error. This error increases +exponentially over time by a factor of +t +T1 or +t +T2, where T1 is +the time it takes to transition from a state of |1⟩ to |0⟩ and T2 is +the time it takes to transition from a middle state to a |0⟩ state +for a qubit, according to [38]. Gate error happens because of +the operations executing on qubits, and crosstalk happens due +to the interaction between different qubits during concurrently +running operations. Additionally, during the measurement, there +is another type of error, called readout error, that affects the +reliability of the outputs. Note that T1, T2, gate error, and +readout error are the system’s characteristics and are different +across different systems. +To solve the reliability concerns, quantum systems require +quantum error correction (QEC) to ensure the correctness of +the results. However, currently, QEC codes need the addition +of numerous ”extra” qubits to ensure the reliability of a single +qubit, which is impractical given the limited scale of present +systems. This brings about the NISQ [43] era, which permits the +execution of small-to-medium size circuits on quantum systems +without any QEC techniques. In NISQ (Noisy Intermediate- +Scale Quantum) devices, the connections between qubits are +limited, for reliability purposes, to avoid the requirement of +QEC [48]. Instead, NISQ machines rely heavily on SWAP +operations [36] – gate operations that swap the state of two +linked/neighboring qubits4, to perform an operation between +two qubits that are not adjacent. More specifically, when +an operation is anticipated between two qubits that are not +neighbors, one of the qubits would swap across links until it +becomes a neighbor of the other qubit [38]. Following that, +3We distinguish between ”logical qubits” and ”physical qubits”; the former +is an abstract qubit in a quantum program or quantum circuit, whereas the +latter represents a physical device (which can be implemented in various +ways/technologies) that acts as a two-state quantum system. +4Neighboring qubits are qubits that are connected to each other via a direct +physical link. +2 + +Fig. 1: (a) Architecture of the system, (b) Sample circuit with +no SWAP operation, and (c) The same circuit with the addition +of a SWAP operation. +the original gate operation between them can finally take place. +Fig. 1 shows an example of how SWAP operations are added +to make a circuit executable. Fig. 1-a shows the architecture of +the system. Based on this architecture, for the circuit shown in +Fig. 1-b, there is no direct connection between Q2 and Q3 for +the red CNOT to be executed. Therefore, we switch the content +of Q1 and Q2 by using the SWAP operation so that we can +run the CNOT operation mentioned above. After adding the +SWAP, the CNOT operation is performed between Q1 and Q3, +generating the final result. To sustain reliability, as the scale +of quantum systems grows, the number of links across qubits +reduces. Current IBMQ systems, for example, include one to +three links per qubit, with the bulk of qubits connected to only +two links [1]. This in turn causes an increase in the number +of swap gates required for a circuit to execute on them. +Current NISQ systems operate in a QAOA (Quantum +Approximate Optimization Algorithm) [16] fashion. QAOA +is a paradigm that combines classical computers and NISQ +systems [16]. More specifically, a quantum program compiles +into either a quantum circuit or a batch of quantum circuits. +At the end of the execution of each circuit, the resulting qubits +(output) are measured and their measured values are stored in +the ”classical computer memory”. Subsequently, the qubits are +reset to a state of |0⟩ for the next circuit to execute. Note that +due to the ”probabilistic nature” of quantum computing, the +results of various executions may differ. As a result, several +runs, also known as ”shots,” are often performed to determine +the probability of having the correct answer. +B. Different Types of Dependency in QC +Qiskit provides a circuit directed acyclic graph DAG for +users, in which roots and leaves represent qubits and other +nodes represent gate operations. Additionally, the edges rep- +resent the qubits used by each gates operations as shown +in the example of Fig. 6. The use cases of DAG include +(but are not limited to) providing order of gate operations, +dependencies between qubits, parallel operations at each stage, +Fig. 2: Two equivalent circuits obtain from ignoring false +dependencies, However they both can be minimized to 2 qubits +and critical depth of the circuit. Programmer uses for DAG +include computation of circuit execution speed, locating parallel +CNOT operations to avoid cross-talk, locating the circuit’s +critical CNOT path for different optimizations, etc. While +DAG provides the dependencies between operations, it does +not differentiate between false and true dependencies. False +dependencies exist in DAG due to the sequence of operations +but do not influence the subsequent qubit/operation, meaning +that changing the sequence does not affect the outcome. +However, true dependencies can affect the final result of the +system if the dependency is not satisfied (order of operations +change). In Fig. 6, for example, the CNOT operation between +q0 and q5 has no influence on the value of q1 following its +CNOT operation with q5, but in the DAG, they are dependent +due to the sequence of the program. Fig. 2 depicts an example +in which two circuits are equivalent and can transform to +each other by taking advantage of false dependencies. False +dependencies can provide the opportunity to change the order of +the operations and transform circuits [29]. Attaining these false +dependencies is easy during the early stages of compilation +but becomes impractical to detect during the execution of the +operations. +C. New Features: MM and MR +Starting in 2020, IBM Quantum systems (IBMQ) started +to gradually include Middle Measurement (MM) and Middle +Reset (MR) gates into their quantum systems [3], aiming to +provide qubit reuse during the course of program execution. To +our knowledge, all IBMQ quantum systems currently support +these features. In the early days of these gates’ debut, IBMQ +provided an instance of the Bernstein-Vazirani [7], [15] circuit +(a 5-qubit version is shown in Fig. 6) to demonstrate the +possibility of serial execution of quantum circuits for improved +reliability [3]. The presented results demonstrate that, by +employing the MM and MR gates, some quantum circuits +(but not all) can be executed ”serially/sequentially” using a +smaller number of qubits. An example is illustrated in Fig. 5. +Still, to our knowledge, there is no automated approach to +downsize/serialize a given quantum circuit by taking advantage +of MM/MR. As a result, currently, if the circuit’s number of +qubits exceeds the target system’s physical qubit count, the +compiler generates an ”error” and does not execute the circuit +on the target system, whereas, with a proper/careful use of MM +and MR, depending on the circuit at hand, the compiler can still +be able to execute the circuit successfully. This paper proposes +a compiler-based ”circuit resizing/serialization” approach that +automates the disciplined use of MM/MR so that a given +3 + +H +Q0 +Q1 +Q2 +Q3 +Q4 +XH +H +SWAP +D +XHH +Hquantum circuit can execute on fewer qubits – in a serial +fashion – without any compile-time error. +D. Related Work +In this part, we discuss quantum compilation methods, quan- +tum programming languages, and most recent accomplishments +in this area of research. While quantum computing is still in its +infancy (in terms of both hardware or software), its potential +advantages over so-called classical computing for particular +algorithms, e.g., in the context of drug discovery, machine +learning and prime factorization, are very promising [23], +[42], [46]. Quantum systems are currently being heavily +researched, and the major efforts focus on the areas of +compiler support [10], [29], [30], [39], [45], operating system +support [44], and programming languages [22]. +In particular, several quantum programming languages, +including Q# [47], OpenQASM 3.0 [12], Silq [9], Scaffold [22], +Scaffcc [22], and QCOR [35], [37], have been developed over +last couple of decades. It is to be noted however that, at present +time, these languages are very close to the low-level assembly +code and depend partly on user-supplied gate insertions. For +example, Qiskit (IBM’s open-source software development kit +for working with quantum computers at the level of circuits, +pulses, and algorithms [4]) employs such a strategy. +Compilation of quantum programs consists of no more +than three main stages: i) matrix-to-gates conversion, ii) +IR optimizations, and iii) logical-to-physical qubit mapping +and circuit execution. In this context, a matrix represents a +function/system which is applied on sample inputs (a vector of +inputs) to generate a final output (an output vector). In the first +step, the matrix is translated into 1-qubit and 2-qubits gates +using an algorithm such as Fowler [17], [18]. If, on the other +hand, the programmer encodes the quantum circuit directly +(i.e., if he/she inputs the gates instead of the matrix), then the +preceding phase – Fowler gate production – can be omitted. +The Fowler [17] method is probably the most well-known +(first-stage) compilation technique, which takes a target matrix +of the Hilbert space as input and searches for gate matrices at +each step in order to approach the target matrix within a certain +proximity, called the Fowler Distance, which is calculated as +follows: +dist(U,Ul) = +� +2 − tr |U . U† +l | +2 +. +Note that Fowler Distance is calculated over multiple iteration +and at each step, this distance is calculated until we reach +the desired threshold. Unfortunately, the complexity of this +approach can be exponential; hence, it is preferable for a +quantum programmer to input the circuit using quantum +gates or input a circuit-matrix hybrid, which consists of +different circuit parts; each has either a gate representation or a +matrix representation. Therefore, the complexity of the Fowler +algorithm can be reduced, thereby improving its efficiency. +Current compilers try to optimize this procedure by reducing +the exponent base by restricting the collection of existing gates. +Booth et al. [10] offer further optimizations by i) using a +bidirectional search and ii) modifying the basis to a modified +Pauli basis to facilitate the Fowler distance computations. +The second stage of a quantum compiler performs IR +(intermediate representation) level optimizations. Note that +the IR abstraction layers employed by compilers targeting +quantum programs range from source-code level abstractions +to system-level abstractions. In modern quantum compilers, +LLVM [26] is the standard, which is close to machine-level +IR. Recent studies have implemented a variety of LLVM-based +optimizations aimed at various domains [39]. For example, +Paulihedral [29] is one of the most recent works; it proposes +retaining a gate matrix IR abstraction until the final levels +for more straightforward circuit optimizations, such as depth +reduction, gate cancellation optimization and swap reduction, +by ignoring false dependencies (i.e, dependencies that are in +the DAG representation of the circuit due to the order of +operations, while the order is not actually important) between +layers of gate production and re-ordering the gate operations. +This approach fits well to enhance our work since discovering +false dependencies at the final stage of compilation (where our +approach is embedded) would be complex. It can also be used +to arrange gate layers according to the volume of their true +dependencies in order to maximize serialization opportunities. +The last stage of the compilation of a quantum program +involves quantum circuit-level optimizations including the +mapping of logical qubits to physical qubits, gate cancellation, +swap reduction, etc. Optimizations at this stage focus mostly +on boosting the reliability of the output. Works such as [15], +[32], [40] concentrate on the important problem of mapping +logical qubits (qubits in the quantum program) to physical +qubits (qubits that are physically implemented in the target +quantum machine) to improve reliability. On the other hand, +Murali et.al. [38] study gate scheduling to boost reliability. +Other efforts focus on optimizing the number of shots [5], +dynamic decoupling optimizations for reliability [13], ancillary +reuse [41], [41], and system selection optimizations [44]. Our +approach proposed in this paper operates at this last stage +compilation of a quantum program. Note however that we +differ from the related work in that our approach i) focuses +on minimizing the number of qubits to prevent ”system +size compilation errors”; to offer the possibility of getting +output from a large quantum circuit considering system size +limitations; and ii) increasing the circuit’s output reliability via +serial execution. +III. MOTIVATION AND PROBLEM DEFINITION +This section describes the three main factors that have +motivated us to design a compiler-based strategy for minimizing +the size of a given quantum circuit. We have identified two +major impediments to raising the qubit count on existing NISQ +systems. Additionally, we noticed that some of the well-known +quantum circuits, such as Bernstein-Vazirani [3], [7], [15], can +be converted into more ”serialized” circuits by using the MM +and MR gates in the IBM QAOA systems. +4 + +A. Size Limitation in NISQ Systems +The continuous demand for more processing power requires +the development of quantum computers with large number +of physical qubits. While the addition of a single qubit may +result in an exponential rise in the compute capability of a +quantum system, the absence of a physical ”quantum memory” +places the whole processing weight on qubits. While QAOA +systems combine quantum computing with a classical memory, +the quantum states still remain ”unmemorable”. +With all these demands for more qubits on quantum +computers, the reliability of physical qubits and their connecting +links remain as the main issue. Furthermore, as the system +grows in size, the reliability degradation becomes even more +difficult to avoid. This results in a decrease in the number of +links between qubits to eliminate crosstalk noise [6], and as a +result, leads to more scattered/distributed qubits, lowering the +qubit processing speeds. Nonetheless, owing to the exponential +expansion of processing power, this cost of processing speed +may not be the primary concern. Still the decrease in the +number of links in larger NISQ systems, needs the inclusion +of multiple extra SWAP operations to execute the quantum +circuit. +B. Reliability Concerns with Larger Systems +The majority of qubits in current IBMQ system architectures, +for example, have two links, whereas a few have one or three +links. A large quantum circuit will have various ”entangled”5 +operations, necessitating the use of numerous SWAP gates +to complete the execution. This not only results in extra +gate errors but also in excessive crosstalk and coherence +faults; the latter is owing to possible differences in the +completion times of different qubits [31]. Thus, the recently- +proposed logical-to-physical qubit mapping techniques, e.g., +NASSC [31] and Qcloud [32], try to reduce such SWAP +costs. In particular, recently, quantum computing research has +primarily focused on mapping [32], scheduling [38], [40], +[49], and even architectural-level strategies [50], for reducing +crosstalk errors. On the other hand, to minimize coherence +faults, dynamic decoupling strategies have been developed, the +most recent of which is ADAPT [13]. +While these strategies can help us reduce various errors +in quantum circuits, errors cannot be completely eliminated, +owing to the physical properties of qubits and quantum gates. +For example, executing a 12-qubit circuit like Bernstein- +Vazirani, which can be categorized as one of the relatively +simple ”medium-size” quantum circuits, on a sample IBMQ +device results in entirely unreliable outputs (a fidelity of +0.007 is reported in [3]). However, by employing the circuit +minimization technique, this fidelity can increase up to 0.31 +(400x faster compared to the prior case [3]). Consequently, there +is a strong motivation for exploring strategies that minimize the +qubit requirements of a given quantum circuit in the absence +of a quantum memory, and to our knowledge, not only there +5Entanglement is a technique used by quantum computers to create a +”correlated state” across multiple qubits, where changing one affects the other. +Fig. 3: Two unresizable circuits. +is no published algorithm that can do this minimization for +general circuits, but also there is no solution for the MM delay +problem, which will be detailed later in the paper. +C. How to Resize the Circuit via the MM and MR Gates? +The first step in building a circuit for the current NISQ +machines is defining the required number of qubits and classical +registers for measurement. The programmer encounters an error +if the number of qubits in the circuit to be implemented exceeds +the system size. For instance, defining a Bernstein-Vazirani +circuit with a size of 10 logical qubits and attempting to execute +it on a system with 7 physical qubits results in a compilation +error. In reality, since current quantum systems support both +the MM and MR gates, qubits can be reused to incarnate (and +execute) additional qubits, and thus this error can be resolved. +As stated in [3], the Bernstein-Vazirani circuit of any size +can be executed, in principle, sequentially using no more than +2 qubits via the MM and MR gates. While executing this +circuit sequentially contradicts the algorithm’s stated goal of +demonstrating the parallelism intrinsic in quantum computing, +it makes sense on a NISQ system. This is because, as previously +mentioned, the NISQ systems depend on SWAP gate operations, +which results in the circuit running sequentially. While this +potential has been highlighted before [3], to our knowledge, +no works have been offered to use it and prevent this error. +Although, as illustrated in Fig. 3a, not only a fully-entangled +circuit –like Fig. 3a– cannot be reduced to a smaller circuit, +but also it requires extra ancilla qubits to generate this large +gate from the physical available gates. These circuits are +primarily constructed using compiler techniques based entirely +on quantum theory and make assumptions that are more +compatible with a theoretical quantum computer. In comparison, +current compilers like Qiskit [4] and SCAFFC [22] are more +considerate of the circuit design constraints imposed by existing +hardware. For instance, Paulihedral [29], a state-of-the-art +compiler-based approach, generates quantum circuits using +only unitary and CNOT gates, which have a higher chance +of producing resizable circuits. These circuits can achieve +significant size reductions when certain constraints are met, +which will be elaborated later in the design section. A circuit +is ”resizable” if it contains at least one qubit that may +attain its final state without the need of all other qubits. +Fig. 4 depicts an example quantum circuit in which q0 and q1 +complete their tasks in this fashion. For q0 to finish its task, +we only need q0 and q1, and for q1 to finish its task only q0, +q1 and q2 are required. Hence, the constraint stated above is +satisfied, and this circuit is resizable. +5 + +Z +H +4 +H +H +H +H +4 +H +H +Y +4 +H +4 +4 +H +4Fig. 4: A resizable circuit. +Our goal in this paper is to minimize the size of a given +quantum circuit by running it in a serial/sequential manner by +employing the MM and MR gates, and also to optimize output +reliability for circuits executing in architectures with limited +number of links per qubit. Thus, our main novelties include +i) giving a polynomial time algorithm to minimize the size of +a given quantum circuit; ii) providing an implementation of +this algorithm and proving that it really minimizes the input +circuit; and iii) avoiding the reliability issues due to delays of +MM gates, which happen through multiple shots. +IV. OVERALL DESIGN +The first step to shrink the size of the circuit is to identify +the criteria which makes a circuit serially executable. Based +on our observations, the only criteria is: +A circuit is ”serializable” if and only if there +exists a qubit that can complete its final gate +operation without the activation of other qubits +on the circuit. +Fig. 5 shows a serializable circuit and its serial execution +with the minimum necessary qubits, whereas Fig. 3 depicts +two circuits that do not meet our serializability requirement. +As an example, in Fig. 3A, none of the qubits can complete +its operation without activating the remaining qubits. On the +other hand, in the circuit shown in Fig 5, q0 can complete its +task with the assistance of only q1. Therefore, the first circuit +should be executed in parallel, while the second circuit can be +serialized (i.e., its size can be reduced). +Fig. 5: A sample cat state n4 circuit and its serial execution. +A. Requirements for Circuit Size Reduction (Sequential Execu- +tion) +In this section we carefully explain the reasons why the +circuits shown in Fig. 3A and 3B are not resizable (cannot +run serially). First, let us discuss the use-cases for which the +MM/MR gates are beneficial. The MM gate enables users to +measure a qubit after it has completed its final action, while +the MR gate is utilized to reset it to a |0⟩ state. Together, these +two operations enable any result qubits to be measured and +reset so that it can be used by other qubits. For garbage qubits6, +on the other hand, only an MR gate is needed (for them to be +resued). +Now let us discuss the reason behind why Fig. 3A and +Fig. 3B are not resizable. For Fig. 3A, we can see that none of +the qubits this circuit possesses can perform its task/operation in +isolation from the other qubits owing to the gate that entangles +them all (i.e., complete entanglement). Therefore, resetting any +of these qubits is not beneficial for shrinking the size of the +circuit. Fig. 3B shows a more complex circuit to demonstrate +the criteria that should be satisfied in order to serially execute +a circuit as much as possible. In this example, Q0 is used in +OP1 and OP4, needing Q1 for its completion. However, OP2 +and OP3 should also be executed before we can reuse Q0 (OP2 +and OP2 should finish before OP4), meaning that Q0 needs +all the qubits to be available when it finishes its last operation. +Q1 has operations with all the other qubits, eliminating the +possibility that it can be reused for serial execution. Q2 has +operations with Q1 and Q3 (OP3 and OP2), still, it needs OP1 +to be executed beforehand, meaning it needs all the qubits to +be available before its final operation. Q3 is similar to Q2 since +it does not have any operation with Q1, still needing OP1 to be +executed first. Therefore, we cannot reuse any of these qubits, +eliminating the possibility that they can be used as a choice for +resizing the circuit. Therefore, even if MM and/or MR gates are +employed on the qubits, it does not provide any benefits since +no qubits exist to be executed on the reset qubit. Thereby, it +denotes this circuit is not resizable and cannot benefit from our +scheme. Based on these examples, we introduce the concept +of activation, meaning that a qubit can finish its task without +directly or indirectly using all other qubits. Note that not having +an operation with other qubits does not satisfy the activation +requirement since there are cases where qubits do not have an +operation with each other but still need each other to complete +their operations first, such as Fig. 3B. It is important to note +that the order of operations is extracted from DAG, which +may contain false dependencies. In our scheme, we do not +differentiate between false and true dependencies since we +do not want to change the structure of the circuit. Still, it is +possible to feed the algorithm with the true dependencies and +reorder the operations if possible, but our approach mainly +focuses on changing the execution and not the circuit. +The most beneficial qubits for resizing/serializing a given +quantum circuit are those qubits that can complete their tasks +with the minimum number of activation from other qubits. By +prioritizing the qubits based on the number of activation from +other qubits, in our algorithm, we reduce the circuit size as +much as possible. Therefore, in our algorithm, we introduce an +additional constraint aiming at maximizing the improvement +6Garbage qubits are qubits that are employed to help to generate the output +of the program but are not themselves output of the circuit. +6 + +R +RH +εb zbb ob +H +10) +
    S[qm] +qubits permanently. Further, adding qm even right before +qi will lead to S[qi] = S[qi] − S[qm] and release qm for qi +for use, thereby increasing the size of the circuit at most +by S[qm] + S[qi] − S[qm] − 1, leading to a smaller circuit, +which is clearly a contradiction. +• The qi and qm dependency lists have no intersection. In +this case, they are likely to be two separate circuits (C1 +and C2) we are trying to resize (in the example Bernstein- +Vazirani circuit above, based on the dependency lists, for +our purposes, we can consider [q0,q1,q5],[q2],[q3],[q4] as +separate circuits), with C1 being the circuit containing qi +and C2 being the one that has qm. Since they are like two +separate circuits, it is clear that any minimization needs them +to work in a serial fashion, meaning that either we start with +C1 and then load C2 or vice-versa. In either case, irrespective +of the selection order of qm or qi, the final minimum-sized +circuit will be of size Max{C1,C2}, leading to the two cases +with an equal number of physical qubits used and will be a +contradiction. +• The qi and qm dependency lists intersect, but the qm +dependency list is not a proper subset of the qi dependency +list. In this scenario, we have one of the following two +cases: +• qm is an element of the qi dependency list, which is similar +to the first case with qm dependency list is a subset of qi +dependency list. +• qm is not an element of the qi dependency list. In this +case, qm adds fewer qubits than qi but guarantees at least +one reset qubit, which provides one available qubit for +qi. It is worth to note that, due to the intersection of qm +and qi dependency lists, some elements of qi dependency +list would be already added to the target circuit. Now, by +adding the remaining elements of qi, a target circuit with +the size of qi dependency list is created, given that qm is +already in the target circuit. Hence, selecting qm leads to a +final circuit with same or less number of qubits compared +to selecting qi first, which is contradicts our assumption. +Therefore, by using contradiction, we prove that our algo- +rithm can minimize the size requirement if changing the order +of operations is not attainable. +D. Complexity Analysis of the Proposed Algorithm +In this section, we study the timing complexity of our +proposal. First, the algorithm must extract dependencies from +the DAG of the circuit. As indicated in Section II, the roots +and leaves of the DAG represent qubits, the remaining nodes +represent gates, and the edges capture the qubits of the +corresponding gate operations. Assuming a circuit with n qubits +and m total gate operations, we need n qubits to check from +leaf to root and at most m operations to check for each qubit; +hence, this phase of our algorithm takes O(nm) complete. +The subsequent step of the algorithm consists of circuit +resizing based on the dependency lists of the qubits obtain in +the previous phase and saved as a list of lists named l-list. The +outside loop is a while-loop on all n dependency lists of qubits, +and inside we sort this l-list based on the dependency list size +(each dependency list is an element of l-list), which accounts +for O(nlogn) and may be optimized to On if implemented by +locating the minimum size element of l-list at each iteration. +Note that each iteration includes the addition of a maximum +of m gates; therefore, this phase of the algorithm takes at +most O(mn2 logn). Overall, the complexity of the algorithm +is: O(nm)+O(mn2 logn) = O(mn2 logn). +Based on the results, we want to emphasize that our approach +to optimizing (serialize) a given circuit has a polynomial +complexity, which is good. For example, trying to resize a +circuit of size 1000, with one million gates to a minimum +100 qubits takes less than 10 seconds on a 2016 Intel core i7 +6950X system. +E. Iterations vs Shots +. While our algorithm seems promising, as demonstrated in +Section IV-D, there is an underlying issue in current quantum +hardware that needs to be carefully addressed. In current +systems, the quantum circuit is executed over multiple shots +to attain the probability of success in achieving the correct +distribution results [16]. However, we observed that when a +circuit containing an MM gate is executed using the same +strategy, instead of running the circuit until the final operation, +the current systems [1] execute the circuit until the MM gate +for multiple shots, attaining the results and continuing with the +execution of the remainder of the circuit after that. This will +cause the other qubits to be stalled for the number of shots +multiplied by the duration of the executed operations on the +measured qubit, which can be substantial in practice. Based +on the numbers attained from a sample system of the new +generation of ibmq, ibmq kolkata [1], the readout latency is +675µS and the best available T1/T2 for the qubits is 214µS +(which is the best value available for this system). If the +circuit is stalled for 1000 shots, it means that the other qubits +should wait for a minimum of 1000×675µS = 675mS, which +is significantly higher than T1/T2 (more than 3000x), causing +the other qubits to become |0⟩ state in the process. None of +9 + +PST report on ibmq-lima +Circuit Name +# of Qubits in Normal +Execution +# of Qubits in Sequen- +tial Execution +PST +Total Gate Count +CNOT Gate Count +bv n14 [12] +14 +2 +51.2% +121 +13 +bv n19 [12] +19 +2 +42.2% +166 +18 +wstate n27 [12] +27 +3 +35.4% +593 +124 +ghz state n23 [12] +23 +2 +52.2% +70 +22 +swap test n25 [12] +25 +3 +67.6% +482 +174 +cat state n22 [12] +22 +2 +53.2% +66 +21 +rd53 139 [34] +8 +5 +60.9% +251 +140 +AVG PST(%) +51.8% +AVG gate count +214 +AVG CNOT count +53.1 +TABLE I: PST results for large benchmark circuits on a 5-qubit ibmq-lima machine (our worst-case scenario). +the current Dynamic Decoupling [13] techniques can solve an +idle period of this magnitude. +To solve this issue, instead of running a circuit over 1000 +shots, we execute the circuit using 1 shot and run this circuit +in a for-loop for 1000 iterations. Doing so can solve the issue +mentioned above while attaining the final results. It is important +to note that, by using this technique, we are not introducing +any new significant change in the current systems; rather, we +provide a simple method for solving an issue related to current +quantum hardware. More specifically, we are encouraging the +vendors to add the concept of iterations, which can eliminate +the need for unnecessary waiting in the queue. This can be +done by tracking the existing job ID and executing the whole +circuit instead of a portion of it if the user specifies iteration +numbers instead of shots count in the algorithm. +V. EXPERIMENTAL EVALUATION +In this section, first, we discuss the methodology we adopted +in our experiments. We then evaluate our proposal using +two different scenarios. Firstly, by using a small quantum +hardware, we optimize (serialize) the circuits that cannot fit +into this hardware using our technique, to show the scalability +of our approach. The goal behind experimenting with this +scenario is to demonstrate that our approach can be used to +execute quantum circuits on quantum hardware with lower +qubit capacity. Note that, by default (without our approach) +such circuits would not execute on the target (small) quantum +hardware. Secondly, using a larger quantum hardware, we +evaluate and compare our proposal with original circuits when +they can be executed on the quantum hardware. This scenario +aims at giving a PST comparison of our proposal against the +normal (parallel) and at revealing the improvements we provide +due to the factors such as gate reduction. +A. Methodology +We evaluate and present our results using two IBMQ +systems, namely, ibmq lima and ibmq kolkata [1]. ibmq lima +is a 5-qubit system that has an architecture in a T-like +Fig.. For simulating ibmq lima, we execute the experiment +using FakelimaV2(), the most recent simulator for the latter +system supplied by Qiskit [2], [4]. Compared to the earlier +version of Fakelima(), FakelimaV2() supports more coherence +error-related optimizations like instructions duration, pulse +scheduling, etc [2]. For simulated ibmq kolkata, we evaluated +our results using FakekolkataV2(), which is the fake-backend +for the ibmq Kolkata hardware. Note that ibmq kolkata is a +new generation IBMQ system with 27 qubits, each having 1 +to 3 links. +While our presented results are based on simulations, we +want to emphasize that our comparison is accurate since we +eliminate most of the crosstalk by using serialization. It is +because crosstalk occurs when multiple qubits are operated +concurrently by different operations (mostly CNOTs) and since +most of our quantum circuits can be executed on 2-3 qubits, +we are not facing crosstalk in any of the results presented. For +our baseline (parallel execution), on the other hand, there may +be some crosstalk cases that are ignored; consequently, the +baseline results we report can be overestimation in terms of +PST. Therefore, our benefits can be expected to be even higher +in real quantum hardware. +Our serialized execution results are reported using 1000 +iterations, each containing 1 shot/iteration, as discussed in +Section IV-E. For our baseline results, on the other hand, +all the results are reported using 1000 shots per workload. +The mapping policy is set to the default mapping that +qiskit/qiskit.transpile employs. We report and compare the +result by using gate count and PST. Note that gate count is +an important metric since it shows the effects of the total gate +error. PST is calculated using the following formula (Eval is +the correct expected value for the results): +PST = ∑Eval +i +PST[i] +Count[Eval] +B. Results and Discussion +Table I shows our results on a 5-qubit system for benchmarks +as large as 27 qubits. While these (original) circuits clearly +cannot run on 5 qubits in a parallel (normal) fashion, we are +able to execute them and obtain reliable outputs by using the +proposed strategy. Our results indicate that, on average, we +are shrinking the size of the circuits tested by 8.87x while +achieving an average PST of 51.7%, which is significant based +on the size of the workloads. +For our experiments, the results are reported on a 5-qubit +system, which is the minimum qubit size for the commercially +available quantum systems. We use this system to show that +i) while prior works cannot execute these algorithms on small +hardware, our approach can execute them and achieve results +with high reliability, and (ii) there exist some large quantum +circuits that benefit from our proposal when targeting even +10 + +PST report on ibmq-Kolkata +Circuit Name +Parallel +Execution PST +Serial Execution +PST +Parallel +Execution +Gate +Count +Serial Execution +Gate Count +Parallel +Execution CNOT +Count +Serial Execution +CNOT Count +bv n14 [12] +26.9% +77.8% +285 +121 +187 +13 +bv n19 [12] +21.1% +68.8% +559 +166 +426 +18 +wstate n27 [12] +13.7% +56.1% +1016 +593 +571 +124 +ghz state n23 [12] +20.5% +69.8% +307 +70 +280 +22 +swap test n25 [12] +43.9% +53.8% +768 +482 +484 +174 +cat state n22 [12] +31.3% +73% +185 +66 +159 +21 +rd53 139 [34] +64.5% +67.6% +245 +222 +137 +111 +Average +31.7% +66.7% +480.7 +245.7 +301 +69 +AVG PST Gain(%) +210.4% (˜2.1 X) +AVG gate reduction(%) +48.9% (˜0.5 X) +TABLE II: Comparison of our sequential execution and baseline execution on an ibmq kolkata (27-qubit system) simulator. +the smallest quantum hardware available. We predict that, by +increasing the qubit size and/or using a newer generation of +quantum hardware, our proposal can achieve a better PST for +a larger set of workloads. +For Bernstein-Vazirani, we report two results with 14 and +19 qubits. As shown in the Table I, the PST decreases from +51.2% to 42.2% for 14 qubits to 19 qubits on ibmq lima. +While, theoretically, any Bernstein-Vazirani circuit with a given +number of qubits can be executed on two qubits [3] by using the +MM and MR gates, fitting larger Bernstein-Vazirani circuits +with any number of qubits into 2 qubits does not always +generate a reliable output since it leads to significant coherence +error. Therefore, we predict there will be a cap to serialization +based on the coherence error of the underlying hardware and +the circuit depth. +C. Sensitivity Analysis on larger systems8 +In this part, we compare the results of our serialized +execution to baseline (parallel) results on a newer generation +system. Table II shows the PST and gate count results with +the ibmq kolkata system. We believe that sequential execution, +when applicable, is superior to parallel execution for three +major reasons: +• Low average number of links on current systems leads +to excessive SWAP operations in parallel execution. Our +technique can significantly improves this problem. +• The new generation of quantum systems is trending toward +improving the measurement gate (readout error), which leads +to the minimization of our proposal’s overhead. +• The new generation of the system also has better T1/T2. +Therefore, this leads to coherence error decreasing exponen- +tially, which is the main concern for sequential execution. +Therefore, we argue that the current quantum systems are +very well suited for sequential execution; in fact, their low +average link count is not a good fit for parallel execution, +which is the state of the art. Our experimental results indicate +that the proposed scheme achieves a 2.1x PST improvement +while reducing the number of gates by 48.9%, compared to +running the original circuit on the same system. This is because, +by serializing the execution, we are reducing the number of +SWAPs needed to migrate the qubits over the links. Thus, by +8Benchmarks are obtained from QASM [12] and Revlib [34]. +decreasing the gate error rate, we are able to achieve highly +reliable results. +Compared to the results reported in Table I, we observe a +significant PST improvement with the ibmq kolkata system. +The reason behind this is that the newer generation IBMQ +systems have better system characteristics such as readout +latency and T1/T2. Additionally, since the newer generation of +quantum hardware leads to lower readout latency, we expect +the effectiveness of our approach to be even higher in future +systems. +VI. CONCLUSION +In this paper, we presented an approach for sequentially +executing quantum circuits and resizing them into the smallest +qubit count necessary to fit them into a small-sized system +using the MM/MR gates. We demonstrated the correctness +of our proposed method and provided a complexity analysis +demonstrating that it operates in O(mn2 logn) time. We also +showed its scalability by choosing the smallest system with +the largest impact on coherence error, which is the primary +concern in sequential execution, and reported the appropriate +level of reliability for a large fraction of the (large) benchmark +circuits offered by QASM [27]. +We suggested the notion of iterations number over the prior +concept of shots number in order to reduce coherence error and +deliver reliable results on a small worst-case system for large +benchmark circuits, as opposed to no results at all (compilation +size error). Sequential execution can possibly compensate for +quantum devices’ lack of memory to a certain degree by +resizing the circuits and allowing small chunks of circuit to +execute on processing qubits. We also showed, via simulations, +that, on a modern NISQ system with 27 qubits, owing to +the reduction in the number of links in the current NISQ [1] +designs, our proposed sequential execution can boost reliability +by a factor of two (avg. 2.1x PST improvement) and reduce +gate counts by almost half by minimizing the SWAP operations. +VII. FUTURE WORK +Our approach acts on the basis of dependencies extracted +from a DAG to decrease the circuit size via serial/sequential +execution while preserving the circuit’s structure. As a future +direction, one can explore an approach to obtain true depen- +dencies in polynomial time and feed them as input to our +algorithm for additional size reductions. However, we plan +to explore a potentially better strategy, which is to construct +11 + +circuits in early stages (e.g., Pualihedral IR [29]) such that the +DAG dependencies do not impact the size reduction, as shown +in Fig. 6 (BV-n circuit). In this way, the circuit is constructed +in an ideal fashion from the viewpoint of serial execution, and +we can employ our algorithm while preserving the circuit’s +structure and changing just its execution method. +Future quantum hardware can enhance qubits connectivity +(links), owing to anticipated improvements in fidelity and/or +reliability. We suggest developing techniques for combination of +sequential execution and current parallel execution based on the +architecture of the systems in order to increase the circuit’s reli- +ability and reduce its gate count. We also recommend mapping +policies improvements and the development of the dynamic +decoupling (DD) strategies for sequential execution of circuits +to further boost reliability, similar to the techniques that have +been established for the existing method of execution [32], [40], +[48]. Additionally, based on the observations in Section IV-E, +we also encourage hardware designers to incorporate support +for the concept of iterations in addition to the shots. +VIII. DISCLAIMER +We are aware of two works concurrent with our paper. These +two works have recently appeared in arXiv, and aim to reduce +qubit requirements via qubit reuse. One of these works [14] +(dated October 14th, 2022) was applied on top of ion trap +machines with all-to-all connections in an effort to reduce +resource usage. For smaller circuits, a SAT-based technique +is used to determine optimal utilization, whereas a heuristic +method is employed for larger circuits. However, this study did +not consider one of the main advantages of our strategy, which +is to minimize the number of SWAP operations. In addition, our +compiler-based approach can outperform the dynamic algorithm +in [14] through our greedy strategy, which has been proven to +be optimal in this paper. The other work [21] (dated November +3rd, 2022) explores a similar approach to fit a program in the +selected hardware. Our paper differs from that work in that, +we prove that our algorithm really minimizes the circuit in a +polynomial amount of time, whereas the mentioned work does +not, and We want to emphasize these two arXiv papers are +concurrent works to ours, and our proposal had been submitted +to (and rejected in) an earlier conference deadline (August 1st, +2022), predating both of these arXiv submissions. 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Zhang, “A +depth-aware swap insertion scheme for the qubit mapping problem,” +arXiv preprint arXiv:2002.07289, 2020. +14 + diff --git a/uNAyT4oBgHgl3EQf0fm1/content/tmp_files/load_file.txt b/uNAyT4oBgHgl3EQf0fm1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a40334a7d62de533fa217431c4bf8c862e58f287 --- /dev/null +++ b/uNAyT4oBgHgl3EQf0fm1/content/tmp_files/load_file.txt @@ -0,0 +1,1403 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf,len=1402 +page_content='Quantum Circuit Resizing Movahhed Sadeghi∗ The Pennsylvania State University, Department of Computer Science and Engineering Email: mus883@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='edu ∗corresponding Author Soheil Khadirsharbiyani The Pennsylvania State University, Department of Computer Science and Engineering Email: szk921@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='edu Mahmut Taylan Kandemir The Pennsylvania State University, Department of Computer Science and Engineering Email: mtk2@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='edu Abstract—Existing quantum systems provide very limited phys- ical qubit counts, trying to execute a quantum algorithm/circuit on them that have a higher number of logical qubits than physically available lead to a compile-time error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Given that it is unrealistic to expect existing quantum systems to provide, in near future, sufficient number of qubits that can accommodate large circuit, there is a pressing need to explore strategies that can somehow execute large circuits on small systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In this paper, first, we perform an analysis to identify the qubits that are most suitable for circuit resizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Our results reveal that, in most quantum programs, there exist qubits that can be reused mid-program to serially/sequentially execute the circuit employing fewer qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Motivated by this observation, we design, implement and evaluate a compiler-based approach that i) identifies the qubits that can be most beneficial for serial circuit execution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' ii) selects those qubits to reuse at each step of execution for size minimization of the circuit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' and iii) minimizes Middle Measurement (MM) delays due to impractical implementation of shots to improve the circuit reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Furthermore, since our approach intends to execute the circuits sequentially, the crosstalk errors can also be optimized as a result of the reduced number of concurrent gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' The experimental results indicate that our proposed approach can (i) execute large circuits that initially cannot fit into small circuits, on small quantum hardware, and (ii) can significantly improve the PST of the results by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='1X when both original and our serialized programs can fit into the target quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' INTRODUCTION Quantum computers are introduced to enhance the com- putational capacity for complex problems such as machine learning [8] and chemistry simulation [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Over the last decade, several vendors unveiled their quantum hardware in an effort to benefit from quantum phenomena to execute a quantum program on real systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Currently, vendors like Google, IBM, and Intel support up to 72, 127, and 49 qubits (quantum bits) [19], [24], [25], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' However, due to errors introduced in the real implementation of qubits, such as coherence and gate errors, existing quantum computers are highly unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' To solve this issue, Quantum Error Correction Codes (QEC) algorithms [11], [18], [20], [33] have been introduced, but they typically require 10-100 additional qubits to create a single fault-tolerant qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Due to this enormous extra qubit requirements brought by QEC, which is impractical considering the size of the current systems, Noisy Intermediate Scale Quantum Computers (NISQ) [18] have been developed to execute small-to-medium circuits on current hardware while aiming to minimize the error rate by different techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Several works targeting NISQ machines have been introduced to minimize different errors, aiming to improve the reliability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' While some works such as [32] try to execute a given quantum program on the most reliable set of qubits available in the target hardware, others [28], [32], [48], [51] aim to minimize gate errors by optimizing the number of SWAP operations (SWAP operations are used to transfer the content of one qubit to another when the qubits involved in a 2-qubit operation do not have a direct physical connection between them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Despite these efforts, the problem of executing a large quantum circuit with lots of logical qubits on existing small quantum systems with only a few physical qubits seems to be one of the biggest challenges in quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Till recently, there had been no mechanism to solve this problem successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' However, quantum vendors have recently introduced the Middle Reset (MR) and Middle Measurement (MM) gates, which can be utilized to resize quantum circuits during the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' By utilizing MR/MM, theoretically, famous quantum algorithms like Bernstein-Vazirani [7] with any qubit count can be executed only using 2 qubits [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Additionally, when running on 2 qubits, no SWAP operations is needed since the 2 physical qubits to which the program is assigned usually have a connection between them, eliminating the gate error due to SWAP operations in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Although this approach can be highly effective in improving the system’s reliability when the qubits we want to reset are correctly identified, to our knowledge, there are no prior works in this area that exploit the potential of these gates in an automated fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In this paper, first, we perform an analysis to identify the qubits that are most suitable for circuit resizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Our results reveal that, in most quantum programs, there exist qubits that can be reused mid-program to serially execute the circuit employing fewer qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Motivated by this observation, we design, implement and evaluate a compiler-based approach that i) identifies the qubits that can be most beneficial for serial circuit execution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' ii) selects those qubits to reuse at each step of execution for size minimization of the circuit1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 1When it leads to no confusion, we will use the terms ”serializability”, ”sequential/serial execution”, ”circuit reduction”, and ”circuit resizing”, inter- changeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='00720v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='ET] 30 Dec 2022 and iii) minimizes Middle Measurement (MM) delays due to impractical implementation of shots2 to improve the circuit reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Furthermore, since our approach intends to execute the circuits sequentially, the crosstalk errors can also be optimized as a result of the reduced number of concurrent gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' To summarize, in this paper, we make the following main contributions: We observe that there is only one constraint that need to be satisfied in a quantum circuit for the circuit to be ”serially executable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' More detail about this constraint is given in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Based on this observation, we present an algorithm that selects qubits in a way that the serial execution opportunity is maximized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' hence, the size of the resulting circuit is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' We present a proof demonstrating that our proposed ap- proach really minimizes the number of qubits needed to execute a quantum program (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=', it completely serializes it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Consequently, we avoid system size compilation error whenever it is possible to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' We observe that the current concept of shot does not fully work with our proposal and leads to coherence errors in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' To solve it, we present a new concept/feature in quantum systems called iteration, which leads to highly reliable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' We present experimental evidence showing the effectiveness of our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' The experimental results indicate that our proposed approach can (i) execute large circuits that initially cannot fit into small circuits, on small quantum hardware, and (ii) can significantly improve the PST of the results by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='1x when both original and our serialized programs can fit into the target quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In Sec- tion II, we give a background on quantum computing covering quantum computing basics, currently available quantum hard- ware, and MM/MR gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In Section II-D, we go over the prior works relevant to this study and explain their shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In Section III, we motivate our work by characterizing the problem, and in Section IV, we explain the technical details of our proposed approach to circuit minimization/serialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In Section V-A, we present our evaluation setup and describe the workloads as well as our evaluation methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' The performance and sensitivity results are presented, respectively, Sections V-B and V-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In Section VI, we give a summary of our major conclusions, and finally, discuss future research directions in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' BACKGROUND AND RELATED WORK In this section, we give an overview of quantum computing and discuss representative quantum systems that are currently available and their salient features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 2Numerous executions, known as ”shots”, are often carried out in order to obtain the chance of getting the right answer from quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Quantum Computing Basics Quantum computation is based on qubits, as opposed to classical computing, which is based on bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Compared to a classical bit which represents a value of 0 or 1, a qubit is represented as a vector that holds a ”state” between 0 and 1, which is defined as follows: |ϕ⟩ = α|0⟩+β|1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' This leads to an exponential growth in state space in terms of the number of qubits [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' For example, having two qubits gives us a state space of |ϕ⟩ = α00|00⟩ + α01|01⟩ + α10|10⟩ + α11|11⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Consequently, algorithms whose state/search space grows exponentially can potentially benefit from quantum computing by operating on qubits and linearizing their state space [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' A quantum gate is the basic building block of a quantum program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' It typically operates on a small number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Example quantum gates include SWAP gate, NOT-gate square root, Controlled-NOT gate (C-NOT) and other controlled gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' It is to be emphasized that a quantum gate that operates on multiple qubits can execute only in the presence of a direct link between the involved qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='3 In reality, qubits are prone to a variety of errors, such as coherence error, gate error, and crosstalk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Qubits can only keep their state for a finite amount of time, which leads to coherence error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' This error increases exponentially over time by a factor of t T1 or t T2, where T1 is the time it takes to transition from a state of |1⟩ to |0⟩ and T2 is the time it takes to transition from a middle state to a |0⟩ state for a qubit, according to [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Gate error happens because of the operations executing on qubits, and crosstalk happens due to the interaction between different qubits during concurrently running operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Additionally, during the measurement, there is another type of error, called readout error, that affects the reliability of the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Note that T1, T2, gate error, and readout error are the system’s characteristics and are different across different systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' To solve the reliability concerns, quantum systems require quantum error correction (QEC) to ensure the correctness of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' However, currently, QEC codes need the addition of numerous ”extra” qubits to ensure the reliability of a single qubit, which is impractical given the limited scale of present systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' This brings about the NISQ [43] era, which permits the execution of small-to-medium size circuits on quantum systems without any QEC techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In NISQ (Noisy Intermediate- Scale Quantum) devices, the connections between qubits are limited, for reliability purposes, to avoid the requirement of QEC [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Instead, NISQ machines rely heavily on SWAP operations [36] – gate operations that swap the state of two linked/neighboring qubits4, to perform an operation between two qubits that are not adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' More specifically, when an operation is anticipated between two qubits that are not neighbors, one of the qubits would swap across links until it becomes a neighbor of the other qubit [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Following that, 3We distinguish between ”logical qubits” and ”physical qubits”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' the former is an abstract qubit in a quantum program or quantum circuit, whereas the latter represents a physical device (which can be implemented in various ways/technologies) that acts as a two-state quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 4Neighboring qubits are qubits that are connected to each other via a direct physical link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 1: (a) Architecture of the system, (b) Sample circuit with no SWAP operation, and (c) The same circuit with the addition of a SWAP operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' the original gate operation between them can finally take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 1 shows an example of how SWAP operations are added to make a circuit executable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 1-a shows the architecture of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Based on this architecture, for the circuit shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 1-b, there is no direct connection between Q2 and Q3 for the red CNOT to be executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Therefore, we switch the content of Q1 and Q2 by using the SWAP operation so that we can run the CNOT operation mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' After adding the SWAP, the CNOT operation is performed between Q1 and Q3, generating the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' To sustain reliability, as the scale of quantum systems grows, the number of links across qubits reduces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Current IBMQ systems, for example, include one to three links per qubit, with the bulk of qubits connected to only two links [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' This in turn causes an increase in the number of swap gates required for a circuit to execute on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Current NISQ systems operate in a QAOA (Quantum Approximate Optimization Algorithm) [16] fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' QAOA is a paradigm that combines classical computers and NISQ systems [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' More specifically, a quantum program compiles into either a quantum circuit or a batch of quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' At the end of the execution of each circuit, the resulting qubits (output) are measured and their measured values are stored in the ”classical computer memory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Subsequently, the qubits are reset to a state of |0⟩ for the next circuit to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Note that due to the ”probabilistic nature” of quantum computing, the results of various executions may differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' As a result, several runs, also known as ”shots,” are often performed to determine the probability of having the correct answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Different Types of Dependency in QC Qiskit provides a circuit directed acyclic graph DAG for users, in which roots and leaves represent qubits and other nodes represent gate operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Additionally, the edges rep- resent the qubits used by each gates operations as shown in the example of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' The use cases of DAG include (but are not limited to) providing order of gate operations, dependencies between qubits, parallel operations at each stage, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 2: Two equivalent circuits obtain from ignoring false dependencies, However they both can be minimized to 2 qubits and critical depth of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Programmer uses for DAG include computation of circuit execution speed, locating parallel CNOT operations to avoid cross-talk, locating the circuit’s critical CNOT path for different optimizations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' While DAG provides the dependencies between operations, it does not differentiate between false and true dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' False dependencies exist in DAG due to the sequence of operations but do not influence the subsequent qubit/operation, meaning that changing the sequence does not affect the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' However, true dependencies can affect the final result of the system if the dependency is not satisfied (order of operations change).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 6, for example, the CNOT operation between q0 and q5 has no influence on the value of q1 following its CNOT operation with q5, but in the DAG, they are dependent due to the sequence of the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 2 depicts an example in which two circuits are equivalent and can transform to each other by taking advantage of false dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' False dependencies can provide the opportunity to change the order of the operations and transform circuits [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Attaining these false dependencies is easy during the early stages of compilation but becomes impractical to detect during the execution of the operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' New Features: MM and MR Starting in 2020, IBM Quantum systems (IBMQ) started to gradually include Middle Measurement (MM) and Middle Reset (MR) gates into their quantum systems [3], aiming to provide qubit reuse during the course of program execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' To our knowledge, all IBMQ quantum systems currently support these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In the early days of these gates’ debut, IBMQ provided an instance of the Bernstein-Vazirani [7], [15] circuit (a 5-qubit version is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 6) to demonstrate the possibility of serial execution of quantum circuits for improved reliability [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' The presented results demonstrate that, by employing the MM and MR gates, some quantum circuits (but not all) can be executed ”serially/sequentially” using a smaller number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' An example is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Still, to our knowledge, there is no automated approach to downsize/serialize a given quantum circuit by taking advantage of MM/MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' As a result, currently, if the circuit’s number of qubits exceeds the target system’s physical qubit count, the compiler generates an ”error” and does not execute the circuit on the target system, whereas, with a proper/careful use of MM and MR, depending on the circuit at hand, the compiler can still be able to execute the circuit successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' This paper proposes a compiler-based ”circuit resizing/serialization” approach that automates the disciplined use of MM/MR so that a given 3 H Q0 Q1 Q2 Q3 Q4 XH H SWAP D XHH Hquantum circuit can execute on fewer qubits – in a serial fashion – without any compile-time error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Related Work In this part, we discuss quantum compilation methods, quan- tum programming languages, and most recent accomplishments in this area of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' While quantum computing is still in its infancy (in terms of both hardware or software), its potential advantages over so-called classical computing for particular algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=', in the context of drug discovery, machine learning and prime factorization, are very promising [23], [42], [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Quantum systems are currently being heavily researched, and the major efforts focus on the areas of compiler support [10], [29], [30], [39], [45], operating system support [44], and programming languages [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In particular, several quantum programming languages, including Q# [47], OpenQASM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='0 [12], Silq [9], Scaffold [22], Scaffcc [22], and QCOR [35], [37], have been developed over last couple of decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' It is to be noted however that, at present time, these languages are very close to the low-level assembly code and depend partly on user-supplied gate insertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' For example, Qiskit (IBM’s open-source software development kit for working with quantum computers at the level of circuits, pulses, and algorithms [4]) employs such a strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Compilation of quantum programs consists of no more than three main stages: i) matrix-to-gates conversion, ii) IR optimizations, and iii) logical-to-physical qubit mapping and circuit execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In this context, a matrix represents a function/system which is applied on sample inputs (a vector of inputs) to generate a final output (an output vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In the first step, the matrix is translated into 1-qubit and 2-qubits gates using an algorithm such as Fowler [17], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' If, on the other hand, the programmer encodes the quantum circuit directly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=', if he/she inputs the gates instead of the matrix), then the preceding phase – Fowler gate production – can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' The Fowler [17] method is probably the most well-known (first-stage) compilation technique, which takes a target matrix of the Hilbert space as input and searches for gate matrices at each step in order to approach the target matrix within a certain proximity, called the Fowler Distance, which is calculated as follows: dist(U,Ul) = � 2 − tr |U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' U† l | 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Note that Fowler Distance is calculated over multiple iteration and at each step, this distance is calculated until we reach the desired threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Unfortunately, the complexity of this approach can be exponential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' hence, it is preferable for a quantum programmer to input the circuit using quantum gates or input a circuit-matrix hybrid, which consists of different circuit parts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' each has either a gate representation or a matrix representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Therefore, the complexity of the Fowler algorithm can be reduced, thereby improving its efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Current compilers try to optimize this procedure by reducing the exponent base by restricting the collection of existing gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Booth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' [10] offer further optimizations by i) using a bidirectional search and ii) modifying the basis to a modified Pauli basis to facilitate the Fowler distance computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' The second stage of a quantum compiler performs IR (intermediate representation) level optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Note that the IR abstraction layers employed by compilers targeting quantum programs range from source-code level abstractions to system-level abstractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In modern quantum compilers, LLVM [26] is the standard, which is close to machine-level IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Recent studies have implemented a variety of LLVM-based optimizations aimed at various domains [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' For example, Paulihedral [29] is one of the most recent works;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' it proposes retaining a gate matrix IR abstraction until the final levels for more straightforward circuit optimizations, such as depth reduction, gate cancellation optimization and swap reduction, by ignoring false dependencies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='e, dependencies that are in the DAG representation of the circuit due to the order of operations, while the order is not actually important) between layers of gate production and re-ordering the gate operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' This approach fits well to enhance our work since discovering false dependencies at the final stage of compilation (where our approach is embedded) would be complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' It can also be used to arrange gate layers according to the volume of their true dependencies in order to maximize serialization opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' The last stage of the compilation of a quantum program involves quantum circuit-level optimizations including the mapping of logical qubits to physical qubits, gate cancellation, swap reduction, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Optimizations at this stage focus mostly on boosting the reliability of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Works such as [15], [32], [40] concentrate on the important problem of mapping logical qubits (qubits in the quantum program) to physical qubits (qubits that are physically implemented in the target quantum machine) to improve reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' On the other hand, Murali et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' [38] study gate scheduling to boost reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Other efforts focus on optimizing the number of shots [5], dynamic decoupling optimizations for reliability [13], ancillary reuse [41], [41], and system selection optimizations [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Our approach proposed in this paper operates at this last stage compilation of a quantum program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Note however that we differ from the related work in that our approach i) focuses on minimizing the number of qubits to prevent ”system size compilation errors”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' to offer the possibility of getting output from a large quantum circuit considering system size limitations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' and ii) increasing the circuit’s output reliability via serial execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' MOTIVATION AND PROBLEM DEFINITION This section describes the three main factors that have motivated us to design a compiler-based strategy for minimizing the size of a given quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' We have identified two major impediments to raising the qubit count on existing NISQ systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Additionally, we noticed that some of the well-known quantum circuits, such as Bernstein-Vazirani [3], [7], [15], can be converted into more ”serialized” circuits by using the MM and MR gates in the IBM QAOA systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Size Limitation in NISQ Systems The continuous demand for more processing power requires the development of quantum computers with large number of physical qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' While the addition of a single qubit may result in an exponential rise in the compute capability of a quantum system, the absence of a physical ”quantum memory” places the whole processing weight on qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' While QAOA systems combine quantum computing with a classical memory, the quantum states still remain ”unmemorable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' With all these demands for more qubits on quantum computers, the reliability of physical qubits and their connecting links remain as the main issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Furthermore, as the system grows in size, the reliability degradation becomes even more difficult to avoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' This results in a decrease in the number of links between qubits to eliminate crosstalk noise [6], and as a result, leads to more scattered/distributed qubits, lowering the qubit processing speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Nonetheless, owing to the exponential expansion of processing power, this cost of processing speed may not be the primary concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Still the decrease in the number of links in larger NISQ systems, needs the inclusion of multiple extra SWAP operations to execute the quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Reliability Concerns with Larger Systems The majority of qubits in current IBMQ system architectures, for example, have two links, whereas a few have one or three links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' A large quantum circuit will have various ”entangled”5 operations, necessitating the use of numerous SWAP gates to complete the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' This not only results in extra gate errors but also in excessive crosstalk and coherence faults;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' the latter is owing to possible differences in the completion times of different qubits [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Thus, the recently- proposed logical-to-physical qubit mapping techniques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=', NASSC [31] and Qcloud [32], try to reduce such SWAP costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In particular, recently, quantum computing research has primarily focused on mapping [32], scheduling [38], [40], [49], and even architectural-level strategies [50], for reducing crosstalk errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' On the other hand, to minimize coherence faults, dynamic decoupling strategies have been developed, the most recent of which is ADAPT [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' While these strategies can help us reduce various errors in quantum circuits, errors cannot be completely eliminated, owing to the physical properties of qubits and quantum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' For example, executing a 12-qubit circuit like Bernstein- Vazirani, which can be categorized as one of the relatively simple ”medium-size” quantum circuits, on a sample IBMQ device results in entirely unreliable outputs (a fidelity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='007 is reported in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' However, by employing the circuit minimization technique, this fidelity can increase up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='31 (400x faster compared to the prior case [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Consequently, there is a strong motivation for exploring strategies that minimize the qubit requirements of a given quantum circuit in the absence of a quantum memory, and to our knowledge, not only there 5Entanglement is a technique used by quantum computers to create a ”correlated state” across multiple qubits, where changing one affects the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 3: Two unresizable circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' is no published algorithm that can do this minimization for general circuits, but also there is no solution for the MM delay problem, which will be detailed later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' How to Resize the Circuit via the MM and MR Gates?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' The first step in building a circuit for the current NISQ machines is defining the required number of qubits and classical registers for measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' The programmer encounters an error if the number of qubits in the circuit to be implemented exceeds the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' For instance, defining a Bernstein-Vazirani circuit with a size of 10 logical qubits and attempting to execute it on a system with 7 physical qubits results in a compilation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In reality, since current quantum systems support both the MM and MR gates, qubits can be reused to incarnate (and execute) additional qubits, and thus this error can be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' As stated in [3], the Bernstein-Vazirani circuit of any size can be executed, in principle, sequentially using no more than 2 qubits via the MM and MR gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' While executing this circuit sequentially contradicts the algorithm’s stated goal of demonstrating the parallelism intrinsic in quantum computing, it makes sense on a NISQ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' This is because, as previously mentioned, the NISQ systems depend on SWAP gate operations, which results in the circuit running sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' While this potential has been highlighted before [3], to our knowledge, no works have been offered to use it and prevent this error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Although, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 3a, not only a fully-entangled circuit –like Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 3a– cannot be reduced to a smaller circuit, but also it requires extra ancilla qubits to generate this large gate from the physical available gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' These circuits are primarily constructed using compiler techniques based entirely on quantum theory and make assumptions that are more compatible with a theoretical quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In comparison, current compilers like Qiskit [4] and SCAFFC [22] are more considerate of the circuit design constraints imposed by existing hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' For instance, Paulihedral [29], a state-of-the-art compiler-based approach, generates quantum circuits using only unitary and CNOT gates, which have a higher chance of producing resizable circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' These circuits can achieve significant size reductions when certain constraints are met, which will be elaborated later in the design section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' A circuit is ”resizable” if it contains at least one qubit that may attain its final state without the need of all other qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 4 depicts an example quantum circuit in which q0 and q1 complete their tasks in this fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' For q0 to finish its task, we only need q0 and q1, and for q1 to finish its task only q0, q1 and q2 are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Hence, the constraint stated above is satisfied, and this circuit is resizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 5 Z H 4 H H H H 4 H H Y 4 H 4 4 H 4Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 4: A resizable circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Our goal in this paper is to minimize the size of a given quantum circuit by running it in a serial/sequential manner by employing the MM and MR gates, and also to optimize output reliability for circuits executing in architectures with limited number of links per qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Thus, our main novelties include i) giving a polynomial time algorithm to minimize the size of a given quantum circuit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' ii) providing an implementation of this algorithm and proving that it really minimizes the input circuit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' and iii) avoiding the reliability issues due to delays of MM gates, which happen through multiple shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' OVERALL DESIGN The first step to shrink the size of the circuit is to identify the criteria which makes a circuit serially executable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Based on our observations, the only criteria is: A circuit is ”serializable” if and only if there exists a qubit that can complete its final gate operation without the activation of other qubits on the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 5 shows a serializable circuit and its serial execution with the minimum necessary qubits, whereas Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 3 depicts two circuits that do not meet our serializability requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' As an example, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 3A, none of the qubits can complete its operation without activating the remaining qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' On the other hand, in the circuit shown in Fig 5, q0 can complete its task with the assistance of only q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Therefore, the first circuit should be executed in parallel, while the second circuit can be serialized (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=', its size can be reduced).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 5: A sample cat state n4 circuit and its serial execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Requirements for Circuit Size Reduction (Sequential Execu- tion) In this section we carefully explain the reasons why the circuits shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 3A and 3B are not resizable (cannot run serially).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' First, let us discuss the use-cases for which the MM/MR gates are beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' The MM gate enables users to measure a qubit after it has completed its final action, while the MR gate is utilized to reset it to a |0⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Together, these two operations enable any result qubits to be measured and reset so that it can be used by other qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' For garbage qubits6, on the other hand, only an MR gate is needed (for them to be resued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Now let us discuss the reason behind why Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 3A and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 3B are not resizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' For Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 3A, we can see that none of the qubits this circuit possesses can perform its task/operation in isolation from the other qubits owing to the gate that entangles them all (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=', complete entanglement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Therefore, resetting any of these qubits is not beneficial for shrinking the size of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 3B shows a more complex circuit to demonstrate the criteria that should be satisfied in order to serially execute a circuit as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In this example, Q0 is used in OP1 and OP4, needing Q1 for its completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' However, OP2 and OP3 should also be executed before we can reuse Q0 (OP2 and OP2 should finish before OP4), meaning that Q0 needs all the qubits to be available when it finishes its last operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Q1 has operations with all the other qubits, eliminating the possibility that it can be reused for serial execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Q2 has operations with Q1 and Q3 (OP3 and OP2), still, it needs OP1 to be executed beforehand, meaning it needs all the qubits to be available before its final operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Q3 is similar to Q2 since it does not have any operation with Q1, still needing OP1 to be executed first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Therefore, we cannot reuse any of these qubits, eliminating the possibility that they can be used as a choice for resizing the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Therefore, even if MM and/or MR gates are employed on the qubits, it does not provide any benefits since no qubits exist to be executed on the reset qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Thereby, it denotes this circuit is not resizable and cannot benefit from our scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Based on these examples, we introduce the concept of activation, meaning that a qubit can finish its task without directly or indirectly using all other qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Note that not having an operation with other qubits does not satisfy the activation requirement since there are cases where qubits do not have an operation with each other but still need each other to complete their operations first, such as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' It is important to note that the order of operations is extracted from DAG, which may contain false dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' In our scheme, we do not differentiate between false and true dependencies since we do not want to change the structure of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Still, it is possible to feed the algorithm with the true dependencies and reorder the operations if possible, but our approach mainly focuses on changing the execution and not the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' The most beneficial qubits for resizing/serializing a given quantum circuit are those qubits that can complete their tasks with the minimum number of activation from other qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' By prioritizing the qubits based on the number of activation from other qubits, in our algorithm, we reduce the circuit size as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' Therefore, in our algorithm, we introduce an additional constraint aiming at maximizing the improvement 6Garbage qubits are qubits that are employed to help to generate the output of the program but are not themselves output of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNAyT4oBgHgl3EQf0fm1/content/2301.00720v1.pdf'} +page_content=' 6 R RH εb zbb ob H 10)
      0, set +C′ +t = tNZ/2dMt−NZ/2, +C′′ +t = t−NZ/2 � +dM�∗ tNZ/2, +Ct = 1 +2 (C′ +t + C′′ +t ) , +Dt = 1 +2 (C′′ +t − C′ +t) , +6 + +then C′′ +t is the adjoint of C′ +t with respect to hE . Ct is a superconnection and Dt is +an odd element of Ω(S, End(E)), and +C2 +t = −D2 +t . +For X ∈ TZ, let X∗ ∈ T ∗Z correspond to X by the metric gTZ. +Set c(X) = +X∗ ∧ −iX. Then +Ct = +√ +t +2 DZ + ∇E,u − +1 +2 +√ +tc(T). +Let f(a) = a exp(a2), and ϕ : Ωeven (S) → Ωeven (S) as follows, +ϕω = (2πi)−kω, +for ω ∈ Ω2k(S). +For any t > 0, the operator Dt is a fiberwise-elliptic differential operator. Then +f (Dt) is a fiberwise trace class operator. For t > 0, put +f ∧ � +C′ +t, hE� +:= ϕ Trs +�N Z +2 f ′ (Dt) +� +. +Put +χ(Z, F) := �dim Z +i=0 (−1)i rank Hi(Z, F), +χ′(Z, F) := �dim Z +i=0 (−1)ii rank Hi (Z, F) . +Definition 2.1. The analytic torsion form T +� +T HM, gTZ, hF � +is a form on S which +is given by +T +� +T HM, gTZ, hF � += − +� +∞ +0 +� +f ∧ � +C′ +t, hE� +− χ′(Z, F) +2 +− χ(Z, F) dim Z − 2χ′(Z, F) +4 +f ′ +�i +√ +t +2 +�� dt +t . +It follows from [3, Theorem 3.21] that T +� +T HM, gTZ, hF � +is well defined. The +degree 0 part of T +� +T HM, gTZ, hF � +is nothing but the fiberwise analytic torsions. +This is why T +� +T HM, gTZ, hF � +is referred to as analytic torsion forms. +2.2 +Analytic torsion forms for manifolds with boundary +Let N ⊆ M be a hypersurface transversal to Z. We suppose that π|N : N → S is +a fibration of fiber Y := N ∩ Z. Suppose that N cuts M into two pieces M1 and +M2, and fiberwisely, cut Z into two pieces Z1 and Z2. Let πi : Mi → S be the +restriction of π to Mi. Then πi is a fibration of Zi (i = 1, 2). Let Fi, T HMi and hFi +be the restriction of F, T HM and hF to Mi respectively (i = 1, 2). First, identify a +neighborhood U1 of ∂M1 with (−2, −1]×N, and identify ∂M1 with {−1}×N; Then +identify a neighborhood U2 of ∂M2 with [1, 2) × N, and identify ∂M2 with {1}× N. +Let (s, z) be the coordinate of Ui w.r.t. +the identification above, pi : Ui → S, +(s, z) → π|N(z) be the canonical submersion (i = 1, 2). +7 + +Let Ω(Zi, F) denotes the space of F|Zi-valued smooth differential forms. +Ωabs (Z1, F) = +� +ω ∈ Ω (Z1, F) : i ∂ +∂s ω +��� +∂Z1 = i ∂ +∂s +� +dZ1ω +���� +∂Zj += 0 +� +, +Ωrel (Z2, F) = +� +ω ∈ Ω (Z2, F) : de ∧ ω|∂Zj = ds ∧ +� +dZ2,∗ω +��� +∂Z2 = 0 +� +. +We write Ei = Ωbd(Zi; F) for short if the choice of abs/rel is clear. +Let DZi be the restriction of DZ on Zi acting on Ωbd(Zi; Fi). Similarly, we have +dMi, ∇Ei, Ct,i, Dt,i e.t.c. +For t > 0, put +f ∧ � +C′ +t,i, hEi� +:= ϕ Trs +�N Z +2 f ′ (Dt,i) +� +, +where hEi = (·, ·)L2(Zi) is the metric induced by gTZi and hFi. +Put +χbd(Zi, F) := �dim Z +i=0 (−1)i rank Hi +bd(Zi, Fi), +χ′(Zi, Fi) := �dim Z +i=0 (−1)ii rank Hi +bd (Zi, Fi) . +Definition 2.2. The analytic torsion form Ti +� +T HMi, gTZi, hFi� +is a form on S +which is given by +Ti +� +T HMi, gTZi, hFi� += − +� +∞ +0 +� +f ∧ � +C′ +t,′, hEi� +− χ′(Zi, Fi) +2 +− χ(Zi, Fi) dim Z − 2χ′(Zi, Fi) +4 +f ′ +�i +√ +t +2 +�� dt +t . +It follows from [26, Theorem 2.17] that Ti +� +T HMi, gTZi, hFi� +is well defined. +2.3 +Analytic torsion form for Witten deformations +Let N ⊆ M be a hypersurface transversal to Z. We suppose that π|N : N → S is +a fibration of fiber Y := N ∩ Z. Suppose that N cuts M into two pieces M1 and +M2, and fiberwisely, cut Z into two pieces Z1 and Z2. Let πi : Mi → S be the +restriction of π to Mi. Then πi is a fibration of Zi (i = 1, 2). Let Fi, T HMi and hFi +be the restriction of F, T HM and hF to Mi respectively (i = 1, 2). First, identify a +neighborhood U1 of ∂M1 with (−2, −1]×N, and identify ∂M1 with {−1}×N; Then +identify a neighborhood U2 of ∂M2 with [1, 2) × N, and identify ∂M2 with {1}× N. +Let (s, z) be the coordinate of Ui w.r.t. +the identification above, pi : Ui → S, +(s, z) → π|N(z) be the canonical submersion (i = 1, 2). +Let +¯ +M = M1 ∪ [−1, 1] × N ∪ M2, and ¯F, h ¯F , gT ¯Z and T H ¯ +M be the natural +extensions of F, hF , gTZ and T HM to +¯ +M. +We still have a natural extension of +fiberation ¯ +M → S. Similar, we have notation ¯Z for the fiber Z. +Let pT be a family of odd smooth functions on [−2, 2], such that +(a) pT |[1,2] ≡ T/2, +8 + +(b) pT |[1/16,1](s) = −Tρ(eT 2(1 − s))(s − 1)2/2 + T/2 , where ρ ∈ C∞ +c ([0, ∞)), such +that 0 ≤ ρ ≤ 1, ρ[0,1/2] ≡ 0, ρ[3/4,∞] ≡ 1, |ρ′| ≤ δ1 and |ρ′′| ≤ δ2 for some +universal constant δ1 and δ2, +(c) C1T ≤ |p′ +T |(s) ≤ 2C1T, |p′′ +T | ≤ C2T for some universal constants C1 and C2 +whenever s ∈ [0, 1/16]. +Then one can see that C3T||s| − 1| ≤ |p′ +T |(s) ≤ 2C3T||s| − 1| and |p′′ +T |(s) ≤ C4T +whenever ||s| − 1| ≤ e−T 2 for some universal constant C3 and C4. +We could think pT as a function on +¯ +M. Still denotes pT to be its fiberwise +restriction. Let d ¯Z +T := d ¯Z + dpT ∧, d +¯Z,∗ +T +be the adjoint of d ¯Z +T . Then D ¯Z +T := d ¯Z +T + +d +¯Z,∗ +T , ∆T := (D ¯Z +T )2. Similarly, we have notation d ¯ +M +T , ∇ ¯E, Ct,T , Dt,T e.t.c. +For t > 0, put +f ∧ � +C′ +t,T , h +¯E� +:= ϕ Trs +�N Z +2 f ′ (Dt,T ) +� +, +where ¯E = Ω( ¯Z, ¯F), and h ¯E = (·, ·)L2( ¯Z) is the metric on Ω( ¯Z, ¯F) induced by gT ¯Z +and h ¯F . Let H( ¯Z; ¯F)(T) be the bundle on S, whose fiber at θ ∈ S is the cohomology +H(Zθ; Fθ)(T) with respect to d ¯Z +T . +Definition 2.3. The analytic torsion form T +� +T H ¯ +M, gT ¯Z, h ¯F � +(T) is a form on S +which is given by +T +� +T H ¯ +M, gT ¯Z, h +¯F � +(T) += − +� +∞ +0 +� +f ∧ � +C′ +t,T , h +¯E� +− χ′( ¯Z, ¯F) +2 +− χ( ¯Z, ¯F) dim Z − 2χ′( ¯Z, ¯F) +4 +f ′ +�i +√ +t +2 +�� dt +t . +It follows from [3, Theorem 3.21] and discussions in §2.3.1 that T +� +T HM, gTZ, hF � +is well defined for a fixed T > 0. +Lastly, for the sake of convenience, (·, ·)L2 (resp. ∥ · ∥L2 := +� +(·, ·)L2) will be +adopted to represent (·, ·)L2(Z) (resp. ∥ · ∥L2(Z) := +� +(·, ·)L2(Z)) , (·, ·)L2( ¯Z) (resp. +∥ · ∥L2( ¯Z) := +� +(·, ·)L2( ¯Z)) or (·, ·)L2(Zi)(resp. ∥ · ∥L2(Zi) := +� +(·, ·)L2(Zi)) (i = 1, 2), +when the context is clear. +2.3.1 +Witten Laplacian v.s. Weighted Laplacian +Instead of deforming the de Rham differential d ¯Z, we could also deform the metric +h ¯F : let h ¯F +T := e−2pT h ¯F . Similarly, g ¯Z and h ¯F +T induce an L2-norm h ¯E +T = (·, ·)L2( ¯Z),T +on ¯E = Ω( ¯Z; ¯F). +Then the formal adjoint δ +¯Z,∗ +T +of d ¯Z w.r.t. the (·, ·)L2,T is then given by epT d +¯Z,∗ +T e−pT . +Then ˜D ¯Z +T := d ¯Z + δ +¯Z,∗ +T +, ˜∆T := ( ˜D ¯Z +T )2. Similarly, we have notation ˜d ¯ +M +T , ˜Ct,T , ˜Dt,T +e.t.c. +9 + +The Weighted Laplacian ˜∆T := d ¯Zδ +¯Z,∗ +T ++δ +¯Z,∗ +T +d ¯Z, one can see that ˜∆T = epT ∆T e−pT . +Let lk(T) be the k-th eigenvalue of ˜∆T , then lk(T) = λk(T). Moreover, if u is an +eigenform of ∆T w.r.t. eigenvalue λ, then epT u is an eigenform of ˜∆T w.r.t. eigen- +value λ. +As a result, f ∧ � +C′ +t,T , h ¯E� += f ∧ � +˜C′ +t,T , h ¯E +T +� +. +2.3.2 +Absolute/Relative boundary conditions for weighted Lapla- +cian +Let ¯ +M1 := M1 ∪ [−1, 0] × M = M− +0 , +¯ +M2 := M2 ∪ [0, 1] × N = M+ +0 , and ¯Z1 := +Z1 ∪ [−1, 0] × Y = Z− +0 , ¯Z2 := Z2 ∪ [0, 1] × Y = Z+ +0 . Let ¯Fi be the restriction of ¯F +on ¯ +Mi (i = 1, 2). Set +Ωabs( ¯Z1; ¯F1) := +� +ω ∈ Ω( ¯Z1; ¯F1) : i ∂ +∂s ω = 0, i ∂ +∂s d +¯Z1ω = 0 on {0} × Y +� +, +Ωrel( ¯Z2; ¯F2)T := +� +ω ∈ Ω( ¯Z2; ¯F2) : ds ∧ ω = 0, ds ∧ d +¯Z2,∗ +T +ω = 0 on {0} × Y +� +. +Let ˜∆T,i be the restriction of ˜∆T acting on Ωbd( ¯Zi; ¯Fi). Then by Hodge theory, +ker( ˜∆T,i) ∼= Hbd( ¯Zi; ¯Fi). Similarly, we have notation ˜d ¯ +M +T,i, ˜Ct,T,i, ˜Dt,T,i e.t.c. +Definition 2.4. The analytic torsion form Ti +� +T H ¯ +Mi, gT ¯Zi, h +¯Fi +T +� +is a form on S +which is given by +Ti +� +T H ¯ +Mi, gT ¯Zi, h +¯Fi +T +� +(T) += − +� ∞ +0 +� +f ∧ � +˜C′ +t,T ′, h +¯Ei +T +� +− χ′(Zi, Fi) +2 +− χ(Zi, Fi) dim Z − 2χ′(Zi, Fi) +4 +f ′ +�i +√ +t +2 +�� dt +t . +Lastly, g ¯Zi and h +¯Fi +T induce an L2-norm (·, ·)L2( ¯Zi),T on Ωbd( ¯Zi; ¯Fi). +For the sake of convenience, (·, ·)L2,T (resp. +∥ · ∥L2,T := +� +(·, ·)L2,T ) will be +adopted to represent (·, ·)L2( ¯Z),T (resp. ∥ · ∥L2( ¯Z),T := +� +(·, ·)L2( ¯Z),T ) or (·, ·)L2( ¯Zi),T +(resp. ∥ · ∥L2( ¯Zi),T := +� +(·, ·)L2( ¯Zi),T ) (i = 1, 2), when the context is clear. +2.4 +Analytic torsion form for complex of finite dimen- +sional vector bundles +Let X be a closed manifold. Let +(E, ν) : 0 → E0 +ν→ E1 +ν→ · · · ν→ En → 0. +be a flat complex of complex vector bundles on X. That is ∇E = ⊕n +i=0∇Ei is a flat +connection on E = ⊕n +i=0Ei and ν is a flat chain map, meaning by +� +∇E�2 = 0, ν2 = 0, ∇Eν = 0. +10 + +Then ν + ∇E is a flat superconnection of total degree 1 . By [3, §2(a)], the coho- +mology H(E, v) of the complex is a vector bundle on X, and let ∇H(E,v) be the flat +connection on H(E, v) induced by ∇E. Let +d(E) = �n +i=0(−1)iirankEi, +d(H(E, v)) = �n +i=0(−1)iirankHi(E, v). +Let hE = ⊕hEi be a metric on E = ⊕Ei. Let ν∗ and ∇E,∗ be the formal adjoint +of ν and ∇E with respect to hE. Let N be the number operator on E, i.e. N acts +by multiplication by i on Ei. Set +f +� +∇E, hE� += �n +i=0(−1)if +� +∇Ei, hEi� +f +� +∇H(E,v), hH(E,v)� += �n +i=0(−1)if +� +∇Hi(E,v), hH(E,v)� +For t > 0, let +Dt = 1 +2 +√ +t (v∗ − v) + 1 +2(∇E,∗ − ∇E). +The analytic torsion form for the complex of finite dimensional vector bundles is +defined as +Definition 2.5. +Tf +� +ν + ∇E, hE� += − +� ∞ +0 +� +ϕ Trs +�1 +2Nf ′ (Dt) +� +− 1 +2d(H(E, v)) − 1 +2 (d(E) − d(H(E, v)) f ′ +�i +√ +t +2 +�� dt +t . +3 +Intermidiate Results +In this section, we will state and prove some intermediate results to prove Theorem +1.1. For each θ ∈ S, denote D ¯Z +T (θ) (resp. DZi(θ) and ˜D +¯Zi +T (θ)) to be the restriction +of D ¯Z +T (resp. DZi and ˜D ¯Z +T ) on ¯Zθ (resp. Zi,θ and ¯Zi,θ), ∆T(θ) := (D ¯Z +T (θ))2 (resp. +∆i(θ) := (DZ +i (θ))2 and ˜∆T,i(θ)). Here Zi,θ := π−1 +i +(θ) (i = 1, 2). +Let λk(T, θ) be the k-th eigenvalue of ∆T (θ), λk(θ) be the k-th eigenvalue of +∆1(θ) ⊕ ∆2(θ) acting on Ωabs(Zθ,1; F1,θ) ⊕ Ωrel(Zθ,2; F2,θ), and ˜λk(T, θ) be the k-th +eigenvalue of ˜∆T,1(θ) ⊕ ˜∆T,2(θ). +Moreover, to avoid heavy notation, we will denote ¯Z, ¯F, DZi, ∆T et cetera for +¯Zθ, ¯Fθ, DZi +θ , ∆T (θ) et cetera, if there is no need to specify the base point θ. +Theorem 3.1. limT→∞ λk(T, θ) = limT→∞ ˜λk(T, θ) = λk(θ) uniformly in S. That +is, for example, for every ǫ > 0, there exists Tk(ǫ) > 0 that doesn’t depend on θ, +such that whenever T ≥ Tk, |λk(T, θ) − λk(θ)| < ǫ. +By Hodge theory, there exists k0 ∈ Z, such that whenever k < k0 λk(θ) ≡ 0, +λk0(θ) ̸= 0. Let δ := 1 +2 infθ∈S λk0(θ) > 0. Then by Theorem 3.1, when T is large +enough, all eigenvalues of ∆T inside [0, δ] converge to 0 as T → ∞. +11 + +Let Ωsm( ¯Z, ¯F)(T) be the vector bundle over S, such that for all θ ∈ S, Ωsm( ¯Zθ, ¯Fθ)(T) +is the space generated by eigenforms of ∆T for eigenvalues inside [0, δ], and Pδ(T) +be the orthogonal projection w.r.t. Ωsm( ¯Z, ¯F)(T). +Let ∇δ,T := Pδ∇ ¯E and Dδ,T +t +:= ∇δ,T − ∇δ,T,∗ − 1 +2 +√ +t(d ¯Z +T − d +¯Z,∗ +T ). For t > 0, put +f ∧ +la +� +C′ +t,T , hE� +:= ϕ Trs +�N Z +2 f ′ (Dt,T ) +� +−ϕ Trs +�N Z +2 Pδ(T)f ′ � +Pδ(T)Dδ,T +t +Pδ(T) +� +Pδ(T) +� +, +f ∧ +sm +� +C′ +t,T , hE� +:= ϕ Trs +�N Z +2 Pδ(T)f ′ � +Pδ(T)Dδ,T +t +Pδ(T) +� +Pδ(T) +� +. +Then proceeding as in the proof of [3, Theorem 2.13] (simply replace Pker(V ) +λ +in [3, +(2.46)] by Pδ(T)(λ − +√ +t(d ¯Z +T − d +¯Z,∗ +T ))−1 e.t.c.), +Theorem 3.2. For some θ-independent constant C(T), C′(T) > 0, such that for +t ≥ 1 +��f ∧ +la +� +C′ +t,T , hE��� ≤ C(T) +√ +t . +For t ∈ (0, 1], +����f ∧ +la +� +C′ +t,T , hE� +− χ( ¯Z, ¯F) dim(Z) − 2d(Ωsm( ¯ +M, ¯F)(T)) +4 +���� ≤ C′(T)t. +Here we put a metric gTS on S, and for α ∈ Ω(S), |α| := +� +gTS(α, α). +Remark 3.3. The key challenge in this article is the possible T-dependence of the +constants C(T) and C′(T). Similar issues can be addressed by introducing a two- +parameter deformation and taking the adiabatic limit of analytic torsion forms, as +is done in [20]. However, we use a different approach in this paper. +We figure +out that when t ∈ [1, ∞), +���f ∧ +la +� +C′ +t,T , hE���� could be bounded by a (T, θ)-independent +measurable function G(t), such that t−1G(t) is in L1([1, ∞)); when t ∈ [0, 1], +the positive degree component of f ∧ +la +� +C′ +t,T , hE� +could be bounded by C′t for some +(T, θ)-independent C′, while the degree 0 component of f ∧ +la +� +C′ +t,T , hE� +is related +to the analytic torsion. +To deal with the degree 0 component, rather than the +adiabatic limit used in [21], a coupling technique is introduced in [25, §7.1]. To- +gether with Theorem 3.1 and dominated convergence theorem, one can understand +Tla +� +T H ¯ +M, gT ¯Z, h ¯F � +(T) as T → ∞. +By Theorem 3.2, we could set +T L +la +� +T H ¯ +M, gT ¯Z, h +¯F � +(T) += − +� ∞ +1 +� +f ∧ +la +� +C′ +t,T , h +¯E� +− χ( ¯Z, ¯F) dim(Z) − 2d(Ωsm( ¯ +M, ¯F)(T)) +4 +f ′ +�i +√ +t +2 +�� dt +t , +12 + +T S +la +� +T H ¯ +M, gT ¯Z, h +¯F � +(T) += − +� 1 +0 +� +f ∧ +la +� +C′ +t,T , h +¯E� +− χ( ¯Z, ¯F) dim(Z) − 2d(Ωsm( ¯ +M, ¯F)(T)) +4 +f ′ +�i +√ +t +2 +�� dt +t . +Similarly, let Pi be the be the orthogonal projection from Ω(Zi, Fi) to ker(∆i) +i = 1, 2, and ˜Pi(T) be the be the orthogonal projection from Ω( ¯Zi, ¯Fi) to ker( ˜∆T,i) +i = 1, 2. +Let ∇Hi := Pi∇Ei and DHi +t +:= ∇Hi − ∇Hi,∗ + 1 +2 +√ +t(dZi − dZi,∗). +For t > 0, put +f ∧ +la +� +C′ +t,i, hE� +:= ϕ Trs +�N Z +2 f ′ (Dt,i) +� +− ϕ Trs +�N Z +2 Pif ′ � +PiDHi +t Pi +� +Pi +� += ϕ Trs +�N Z +2 f ′ (Dt,i) +� +− χ′(Zi, Fi) +2 +(By [3, Theorem 3.15]). +Similarly, +f ∧ +la +� +˜C′ +t,T,i, hE +T +� +:= ϕ Trs +�N Z +2 f ′ � +˜Dt,T,i +�� +− χ′(Zi, Fi) +2 +. +Set +T L +la,i +� +T HMi, gTZi, hFi� += − +� ∞ +1 +� +f ∧ +la +� +C′ +t,i, hEi� +− χ(Zi, Fi) dim(Z) − 2χ′(Zi, Fi) +4 +f ′ +�i +√ +t +2 +�� dt +t , +T S +la,i +� +T HMi, gTZi, hFi� += − +� 1 +0 +� +f ∧ +la +� +C′ +t,i, hEi� +− χ(Zi, Fi) dim(Z) − 2χ′(Zi, Fi) +4 +f ′ +�i +√ +t +2 +�� dt +t . +Similarly, one has T L +la,i +� +T H ¯ +Mi, gT ¯Zi, h +¯Fi +T +� +(T) and T S +la,i +� +T H ¯ +Mi, gT ¯Zi, h +¯Fi +T +� +(T). +Theorem 3.4. +lim +T→∞ f ∧ +la +� +C′ +t,T , hE� += lim +T→∞ +2 +� +i=1 +f ∧ +la +� +˜C′ +t,T,i, h +¯E +T +� += +2 +� +i=1 +f ∧ +la +� +C′ +t,i, hE� +. +More precisely, for example, for every ǫ > 0, there exists T0(t, ǫ) > 0, such that +whenever T ≥ T0(t, ǫ), +∥f ∧ +la +� +C′ +t,T , hE� +− +2 +� +i=1 +f ∧ +la +� +C′ +t,i, hE� +∥L∞ < ǫ. +Here we put a metric gTS on S, and for α ∈ Ω(S), +∥α∥L∞ := sup +θ∈S +|α|(θ), +where |α| := +� +gTS(α, α). +13 + +Theorem 3.5. +lim +T→∞ T L +la +� +T H ¯ +M, gT ¯Z, h +¯F � +(T) = lim +T→∞ +2 +� +i=1 +T L +la,i +� +T H ¯ +Mi, gT ¯Zi, h +¯Fi +T +� +(T) += +2 +� +i=1 +T L +la,i +� +T HMi, gTZi, hFi� +. +The limit is taken w.r.t. the topology described in Theorem 3.4. +Theorem 3.6. +T S +la +� +T H ¯ +M, gT ¯Z, h +¯F � +(T) = +2 +� +i=1 +T S +la,i +� +T HMi, gTZi, hFi� +−(T−log(2))χ(Y )rank(F)/2+o(1), +T S +la,i +� +T H ¯ +Mi, gT ¯Zi, h +¯Fi +T +� +(T) = T S +la,i +� +T HMi, gTZi, hFi� +−T log(2)χ(Y )rank(F)/4+o(1). +as T → ∞. +As a result, +T S +la +� +T H ¯ +M, gT ¯Z, h +¯F � +(T) = +2 +� +i=1 +T S +la,i +� +T H ¯ +Mi, gT ¯Zi, h +¯Fi +T +� +(T)+log(2)χ(Y )rank(F)/2+o(1). +The limit is taken w.r.t. the topology described in Theorem 3.4. +Next, we have the following Mayer-Vietoris exact sequence (c.f. [5, (0.16)]) of +vector bundles over S: +MV : · · · +∂k−1 +→ Hk +rel +� ¯Z2; ¯F +� ek +→ Hk � ¯Z; ¯F +� rk +→ Hk +abs +� ¯Z1; ¯F +� ∂k +→ · · · . +(3) +Let ˜Ωsm( ¯Z, ¯F)(T) be the space generated by eigenforms of ˜∆T for eigenvalues +inside [0, δ], then by our discussion above in §2.3.1, +˜Ωsm( ¯Z, ¯F)(T) = epT Ωsm( ¯Z, ¯F)(T). +And ˜Pδ(T) := epT Pδ(T)e−pT : L2Ω( ¯Z, ¯F)(T) → ˜Ωsm( ¯Z, ¯F)(T) is the orthogonal +projection. Let H( ¯Z; ¯F)(T) := ker( ˜∆T ), and H( ¯Zi; ¯Fi)(T) := ker( ˜∆T,i). We also +have the following Mayer-Vietoris exact sequence induced by Hodge theory and (3) +MV(T) : · · · +∂k−1,T +→ +Hk � ¯Z2; ¯F2 +� +(T) +ek,T +→ Hk � ¯Z; ¯F +� +(T) +rk,T +→ Hk � ¯Z1; ¯F1 +� +(T) +∂k,T +→ · · · +(4) +with metric and flat connections induced by Hodge theory. Let T (T) be the analytic +torsion form for this complex. +For any L2 differential form w on Zi (or ¯Zi), let E(w) be the extension of w, s.t. +outside Zi (or ¯Zi), E(w) = 0. We will not distinguish w and E(w) if the context is +clear. +14 + +Next, when T is large enough, we have a sequence of morphism of vector bundles +0 → Hk( ¯Z2, ¯F2)(T) +˜ek,T +→ ˜Ωk +sm( ¯Z, ¯F)(T) +˜rk,T +→ Hk( ¯Z1, ¯F1)(T) → 0. +(5) +Here ˜ek,T is given by u �→ ˜Pδ(T)E(u) for all u ∈ ker( ˜∆T,1). And ˜rk,T is given +by u �→ ˜P1(T)(u| ¯Z1) for all u ∈ ˜Ωk +sm( ¯Z, ¯F)(T). Recall that ˜Pi(T) is the orthogonal +projection w.r.t. ker( ˜∆T,i) (i = 1, 2). +Let η ∈ C∞ +c ([0, 1]), such that η|[0,1/4] ≡ 0, η|[1/2,1] ≡ 1. +For u = (u1, u2) ∈ +ker (∆1) ⊕ ker (∆2), let QT : Ωabs (Z1; F1) ⊕ Ωrel (Z2; F2) → Ω( ¯Z, ¯F) be +QT (u)(x) := + + + +ui(x), if x ∈ Zi, +η(−s)u(−1, y)e−pT (s)−T/2, if x = (s, y) ∈ [−1, 0] × Y, +η(s)u(1, y)epT (s)−T/2, if x = (s, y) ∈ [0, 1] × Y. +And let ˜QT = epT QT . One can see that +Proposition 3.7. For u ∈ ker (∆1) ⊕ ker (∆2), +∥QT u − u∥2 +L2 ≤ +C +√ +T ∥u∥2 +L2, +��Pδ(T)QT (u) − u +��2 +L2 ≤ +C∥u∥2 +L2 +√ +T +. +for some constant C that is independent of T. +As a result, +��� ˜QT u − epT u +��� +2 +L2,T ≤ +C +√ +T ∥epT u∥2 +L2,T, +��� ˜Pδ(T) ˜QT (u) − epT u +��� +2 +L2,T ≤ +C∥epT u∥2 +L2,T +√ +T +. +Recall that ∥ · ∥L2,T is the norm induced by gT ¯Z and h ¯F +T := e−2pT h ¯F . +As a result, when T is large enough, ˜Pδ(T) ˜QT (u) spans ˜Ωsm( ¯Z, ¯F)(T) for u ∈ +ker (∆1) ⊕ ker (∆2). +Proof. For u ∈ ker (∆1) ⊕ ker (∆2), set uT = Pδ(T)QT (u), vT = QT (u) − uT . First, +by trace theorem and Garding’s inequality, +� +Y +��ui +� +(−1)i, y +���2 dvolY ≤ C +� +Zi +|ui|2 + |∇ui|2 dvol ≤ C′ +� +Zi +|ui|2 dvol. +(6) +for some constants C, C′ that doesn’t depends on T. By (6) and a straightforward +computation, one can see that +∥QT u − u∥2 +L2 ≤ C +√ +T +∥u∥2 +L2 +(7) +and +���D +¯Z +T QT u +��� +2 +L2 ≤ C +√ +T +∥u∥2 +L2. +(8) +15 + +Moreover, +δ ∥vT ∥2 +L2 ≤ +���D +¯Z +T vT +��� +2 +L2 ≤ +���D +¯Z +T QT u +��� +2 +L2 . +(9) +(8) and (9) then imply that +∥vT ∥2 +L2 ≤ +C +δ +√ +T +∥u∥L2, +i.e., +���Pδ(T)QT (u) − QT (u) +��� +2 +L2 ≤ C∥u∥2 +L2 +δ +√ +T +. +(10) +The proposition then follows form (7) and (10). +Notice that if u ∈ Ωbd(Zi, Fi), then QT u ∈ Ωbd( ¯Zi, ¯Fi). By Hodge theory and +Theorem 3.1, when T is big enough, all eigenvalues of ˜∆T,i inside [0, δ] must be 0. +Let ˜Pi(T) be the orthogonal projection from L2Ω( ¯Zi; ¯Fi)(T) to ker( ˜∆T,i). Similarly, +one has +Proposition 3.8. For u ∈ ker (∆i), +��� ˜QT u − epT u +��� +2 +L2,T ≤ +C +√ +T ∥epT u∥2 +L2,T, +��� ˜Pi(T) ˜QT (u) − epT u +��� +2 +L2,T ≤ +C∥epT u∥2 +L2,T +√ +T +. +As a result, when T is large enough, +˜Pi(T) ˜QT (u) spans H( ¯Zi, ¯Fi)(T) for u ∈ +ker (∆i). +Proposition 3.9. ˜ek,T and ˜r∗ +k,T are almost isometric embeddings as T → ∞, where +˜r∗ +k,T is the adjoint of ˜rk,T. +That is, for example, for any u ∈ Hk( ¯Z2; ¯F2)(T), +limT→∞ +∥˜ek,T u∥L2,T +∥uT ∥L2,T += 1. +Proof. +• ˜ek,T is almost isometric. +For any u ∈ Hk( ¯Z1, ¯F1)(T), there exists uT ∈ ker(∆1) ∩ Ωk +rel(Z1, F1) such that +u = ˜Pi(T) ˜QT (uT ), then by Proposition 3.8, +∥u∥2 +L2,T ≥ (1 − C +√ +T +)∥epT uT ∥2 +L2,T . +(11) +While +∥˜ek,T u∥2 +L2,T = ∥ ˜Pδ(T)u∥2 +L2,T +≤ ∥ ˜Pδ(T)QT uT ∥2 +L2,T + C∥epT uT ∥2 +L2 +√ +T +(By Proposition 3.8 and the fact that ∥ ˜Pδ(T)∥ = 1) +≤ ∥epT uT ∥2 +L2,T (1 + C′ +√ +T +) (By Proposition 3.7). +(12) +16 + +It follows from (11) and (12) that +lim sup +T→∞ +∥˜ek,T u∥2 +L2,T +∥u∥L2,T += 1. +Similarly, +lim inf +T→∞ +∥˜ek,Tu∥2 +L2,T +∥u∥L2,T += 1. +• ˜r∗ +k,T is almost isometric. +For u ∈ Hk( ¯Z2, ¯F2)(T), we first show that ˜r∗ +k,Tu = ˜Pδ(T)E(u) : Notice that for any +v ∈ ˜Ωsm( ¯Z, ¯F)(T), +(˜rk,Tv, u)L2( ¯Z2),T = (v, E(u))L2( ¯Z),T = (v, ˜Pδ(T)E(u))L2( ¯Z),T . +Following the same steps as above, one can show that ˜r∗ +k,T is almost isometric. +Theorem 3.10. With maps ˜ek,T and ˜rk,T given above, the sequence (5) is exact. +Proof. Let ⟨·, ·⟩T be the pointwise inner product on each fiber that is induced by +gT ¯Z and h ¯F +T . +• In follows from Proposition 3.9 that ˜ek,T and ˜r∗ +k,T are injective when T is large. +• E(ker( ˜∆T,1)) ⊂ ker(δ +¯Z,∗ +T +) and E(ker( ˜∆T,2)) ⊂ ker(d ¯Z): +Let u ∈ ker( ˜∆T,2). First, since u satisfies relative boundary conditions, integra- +tion by parts shows that for any β ∈ Ω( ¯Z, ¯F), +� +¯Z⟨E(u), δ +¯Z,∗ +T +β⟩T dvol = 0. Thus, +E(u) ∈ ker(d ¯Z). Similarly, E(ker( ˜∆T,1)) ⊂ ker(δ +¯Z,∗ +T +). +• im ˜ek,T = ker ˜rk,T : +For the dimension reason, it suffices to show that im˜ek,T ⊂ ker ˜rk,T. That is, it +suffices to show im˜ek,T ⊥ im˜r∗ +k,T . First, for any ui ∈ ker( ˜∆T,i), i = 1, 2, it’s clear +that +(E(u1), E(u2))L2,T = 0. +(13) +Since E(u1) ∈ ker(δ +¯Z,∗ +T +), E(u2) ∈ ker(d ¯Z), one can see that (1 − ˜Pδ(T))u1 ∈ +im δ +¯Z,∗ +T +, (1 − ˜Pδ(T))u1 ∈ im d ¯Z, which means +((1 − ˜Pδ(T))E(u1), (1 − ˜Pδ(T))E(u2))L2,T = 0. +(14) +By (13) and (14), +(˜ek,T u2, ˜r∗ +k,Tu1)L2,T = 0. +17 + +Moreover, we have the following complexes of finite dimensional vector bundles +0 → H0( ¯Zi, ¯Fi)(T) +0→ H1( ¯Zi, ¯Fi)(T) +0→ · · · +0→ Hdim Z( ¯Zi, ¯Fi)(T) → 0 +(15) +0 → ˜Ω0 +sm( ¯Z, ¯F)(T) d ¯ +Z +→ ˜Ω1 +sm( ¯Z, ¯F)(T) d ¯ +Z +→ · · · d ¯ +Z +→ ˜Ωdim Z +sm +( ¯Z, ¯F)(T) → 0. +(16) +Integration by parts as in the proof of Theorem 3.10, one can show easily that +Proposition 3.11. d ¯Z ◦ ˜ek,T = 0, ˜rk,T ◦ d ¯Z = 0. +Hence, by Theorem 3.10 and Proposition 3.11, we get the following long exact +sequence again +MV(T) : · · · +∂k−1,T +→ +Hk � ¯Z2; ¯F2 +� +(T) +ek,T +→ Hk � ¯Z; ¯F +� +(T) +rk,T +→ Hk � ¯Z1; ¯F1 +� +(T) +∂k,T +→ · · · +(17) +with metric and connection induced by Hodge theory. +Let +Tsm(T H ¯ +M, gT ¯Z, h +¯F )(T) +:= − +� ∞ +0 +� +f ∧ +sm(C′ +t,T , h +¯E) − χ′(Z, F) +2 +� ++ χ′(Z, F) − d(Ωsm( ¯ +M, ¯F)(T)) +2 +f ′(i +√ +t +2 )dt +t . +The following Theorem will be proved in §6. +Theorem 3.12. limT→∞ Tsm(T H ¯ +M, gT ¯Z, h ¯F )(T) − T (T) = 0. +It follows from anomaly formulas (c.f. [3, Theorem 2.24 and Theorem 3.24] and +[27, Theorem 1.5]) that +Theorem 3.13. In QS/QS +0 , +log T (T H ¯ +M, gT ¯Z, h +¯F )(T) − +2 +� +i=1 +log Ti(T H ¯ +Mi, gT ¯Zi, h +¯Fi +T )(T) − log T (T) += log T (T HM, gTZ, hF ) − +2 +� +i=1 +log Ti(T HMi, gTZi, hFi) − log T . +Proof of Theorem 1.1. It follows from Theorem 3.5, 3.6 and 3.12 that +log T (T H ¯ +M, gT ¯Z, h +¯F )(T) − +2 +� +i=1 +log Ti(T H ¯ +Mi, gT ¯Zi, h +¯Fi +T )(T) − log T (T) += log(2)χ(Y )rank(F)/2 + o(1). +Thus, by Theorem 3.13, Theorem 1.1 follows. +18 + +4 +Convergence of Eigenvalues +First, it’s straightforward to check that +Lemma 4.1. Let F → X be a flat complex vector bundle over a compact smooth +manifold X, f be a smooth function on X. Let hF +l , gTX +l +, ∇F +l be smooth families of +metrics and connections over F → X, l ∈ [0, 1]. Let dl : Ω∗(X; F) → Ω∗+1(X; F) be +the covariant derivative w.r.t. ∇F +l , and d∗ +l be the adjoint of dF +l . Let df,l := dF +l +df∧, +and d∗ +f,l be the adjoint of df,l. Then for any u ∈ Ω, ǫ > 0, there exists δ > 0 that +doesn’t depend on u and f, such that whenever |l1 − l2| < δ, one has +� +X |df,l1u|2 +l1 + |d∗ +f,l1u|2 +l1 +� +X |u|2 +l1 +≤ (1 + ǫ) +� +X |df,l2u|2 +l2 + |d∗ +f,l2u|2 +l2 +� +X |u|2 +l2 +. +Here | · |l is the metric on Λ∗(TZ) ⊗ F induced by hF +l and gTX +l +. +First, one observes that λk(T, θ) has uniform upper bounds: +Lemma 4.2. Fix k ∈ Z+. There exists an increasing sequence {Λk}∞ +k=1 of constants, +such that λk(T, θ) ≤ Λk. +Proof. For a fixed θ ∈ S, it follows from [25, Lemma 4.1] that there exists {Λk(θ)}, +such that λk(T, θ) ≤ Λk. It follows from Lemma 4.1 that when θ′ is close to θ, +λk(T, θ′) ≤ 2λT,θ. The existence of Λk then follows from the compactness of S. +Corollary 4.3. λk(T, θ) is a family of equicontinuous function on S. That is, for +any θ ∈ S, ǫ > 0, there exists a neighborhood U of θ that doesn’t depends on T, +whenever θ′ ∈ S, |λk(T, θ) − λk(T, θ′)| < ǫ. +Proof. By Lemma 4.1 and Lemma 4.2, when θ′ is closed to θ, |λk(T, θ)−λk(T, θ′)| ≤ +ǫλk(T, θ) ≤ ǫΛk. +Next, it follows from [25, Lemma 4.2] and compactness of S that, +Lemma 4.4. Let u ∈ Ω( ¯Z; ¯F), such that +� +¯Z |u|2 = 1 and +� +¯Z |D ¯Z +T u|2 ≤ λ. Then for +s ∈ [−2, −1 + +� +2 +T ] ∪ [1 − +� +2 +T , 2] +� +Y +|u|2(s, y) ≤ C(1 + λ) +if T is large enough. Here the constant C is independent of T and θ, | · | is the +metric on Λ∗( ¯Z) ⊗ ¯F| ¯Z induced by h ¯F and gT ¯Z. +Proof of Theorem 3.1. Fix θ ∈ S, [25, Theorem 3.1] implies that limT→∞ λk(T, θ) = +λk(θ). Since S is compact, the uniformness follows from Corollary 4.3 and continuity +of λk(θ). +Similarly, one can show limT→∞ ˜λk(T, θ) = λk(θ). +19 + +Recall that for u ∈ Ωabs(Z1, F1) ⊕ Ωrel(Z2, F2), +QT (u)(x) := + + + + + +ui(x), if x ∈ Zi, +η(−s)u(−1, y)e−pT (s)−T/2, if x = (s, y) ∈ [−1, 0] × Y , +η(s)u(1, y)epT (s)−T/2, if x = (s, y) ∈ [0, 1] × Y . +It follows from Lemma 4.4 and the construction of QT that +Lemma 4.5. Let u ∈ Ωbd(Zi, Fi) such that ∥dZ + dZ,∗u∥L2 ≤ λ∥u∥L2. Then +∥QT (u) − E(u)∥2 +L2 ≤ C(1 + λ) +√ +T +∥u∥2 +L2. +Here the constant C is independent of T and θ. +It follows from [25, Lemma 5.1 and Lemma 5.2] and the compactness of S that +Lemma 4.6. There exists constants c1, c2, c3, c4 and c5 independent of T and θ, +such that λk(T, θ) ≥ uk(T). Here {uk(T)}∞ +k=1 is the collection of 4 copies of {vl(T)+ +c4m2/(dim M−1)}∞ +l=1,m=1 and 2 copy of {c5l2/ dim M}, listed in the increasing order and +counted with multiplicity. Moreover, {vk(T)}∞ +k=1 is the collection of {T max{c1l − +c2, 0}}∞ +l=1 and {c3l2}∞ +l=1, listed in the increasing order and counted with multiplicity. +4.1 +Convergence of f ∧ +la +� +C′ +t,T, h ¯E� +and the large time con- +tributions +Let VT = d ¯Z +T − d +¯Z,∗ +T +and ¯Ft := Dt,T − +√ +t +2 (d ¯Z +T − d +¯Z,∗ +T ), then all eigenvalues of VT are +pure imaginary. Moreover, ¯Ft is nilpotent. Similarly, one has Ft,i and Ft. Moreover, +¯Ft|Zi = Ft,i. +(18) +We define ¯F j +t inductively: set ¯F 0 +t = ¯Ft, and ¯F j+1 +t += [VT , ¯F j +t ]. Since T H ¯ +M, gT ¯Z +and h ¯F are product-type, by a straightforward computation, +Lemma 4.7. There exists (T, θ)-independent Cj > 0, s.t. ∥ ¯F j +t ∥ ≤ Cj. +It’s trivial that +Lemma 4.8. If τ is pure imagenary and |Re(λ)| = 1, then +|λ − τ|−1 ≤ C |λ| +|τ| +or +|λ − τ|−1 ≤ 1 +for some universal constant C. +20 + +Lemma 4.9. Let u be a normal eigenform w.r.t. an eigenvalue µ for VT (Moreover, +assume that |µ| ≥ 1), then for any j ∈ Z+, t > 0, there exits (T, θ)-independent +Cj > 0, such that +��� +�� +f ′(Dt,T ) − f ′( +√ +tVT ) +� +u, u +� +L2 +��� ≤ +Cj +√ +t|µ|j . +Proof. Let γ be the oriented contour given by {z ∈ C : |Re(z)| = 1}. Proceeding as +in the proof of [3, Theorem 2.13], +f ′ (Dt,T ) − f ′ �√ +tVT +� += +dim S +� +l=1 +� +γ +f ′(λ) +� +(λ − +√ +tVT )−1 ¯Ft +�l +(λ − +√ +tVT )−1dλ. +(19) +First, notice that +����� +�� +γ +f ′(λ)(λ − +√ +tVT )−1 ¯Ft(λ − +√ +tVT )−1udλ, u +� +L2 +����� += +����� +�� +γ +f ′(λ) ¯Ft(λ − +√ +tVT )−1udλ, (¯λ + +√ +tVT )−1u +� +L2 +����� += +����� +�� +γ +f ′(λ)(λ − +√ +tµ)−2 ¯Ftudλ, u +� +L2 +����� = 0. +(20) +Recall that ¯F 0 +t := ¯Ft, and ¯F j+1 +t +:= [VT , ¯F j +t ]. Through a simple calculation, +[ ¯F j +t , (λ − +√ +tVT )−1] = (λ − +√ +tVT )−1√ +t ¯F j+1 +t +(λ − +√ +tVT )−1. +(21) +Consequently, by (21), Lemma 4.8 and Lemma 4.7, if λ ∈ γ, +�� +(λ − +√ +tVT )−1 ¯Ft +�2 +(λ − +√ +tVT )−1u, u +� += +�j−1 +� +k=1 +(λ − +√ +tVT )−(k+1)√ +t +k ¯F k−1 +t +¯Ft(λ − +√ +tVT )−1 ++(λ − +√ +tVT )−j√ +t +j ¯F j−1 +t +((λ − +√ +tVT )−1) ¯Ft(λ − +√ +tVT )−1u, u +� +≤ +�j−1 +� +k=1 +(λ − +√ +tVT )−(k+1)√ +t +k ¯F k−1 +t +¯Ft(λ − +√ +tVT )−1u, u +� ++ C +���(λ − +√ +tµ)−(j+1)√ +t +j��� +≤ +�j−1 +� +k=1 +(λ − +√ +tVT )−(k+1)√ +t +k ¯F k−1 +t +¯Ft(λ − +√ +tVT )−1u, u +� ++ C|λ|j+1|µ|−(j+1)√ +t +−1; +(22) +21 + +similarly, +�� +(λ − +√ +tVT )−1 ¯Ft +�2 +(λ − +√ +tVT )−1u, u +� +≥ +�j−1 +� +k=1 +(λ − +√ +tVT )−(k+1)√ +t +k ¯F k−1 +t +¯Ft(λ − +√ +tVT )−1u, u +� +− C|λ|j+1|µ|−(j+1)√ +t +−1. +(23) +By (22) and (23), proceeding as in (19), one can see that +����� +�� +γ +f ′(λ) +� +(λ − +√ +tVT )−1 ¯Ft +�2 +(λ − +√ +tVT )−1udλ, u +� +L2 +����� ≤ +Cj +√ +t|µ|j . +(24) +Similarly, one can show +����� +�� +γ +f ′(λ) +� +(λ − +√ +tVT )−1 ¯Ft +�l +(λ − +√ +tVT )−1udλ, u +� +L2 +����� ≤ +Cj,l +√ +t|µ|j . +(25) +We also have +Lemma 4.10. Let u be an eigenform of ∆i w.r.t. eigenvalue µ, then for |Re(λ)| = 1, +∥(λ − Dt,T )−1QT u − E((λ − Dt,i)−1u)∥2 +L2 ≤ C(µ2 + µ + 1)(1 + λ)∥u∥2 +L2 +√ +T +, +and +∥(λ − Dt,T )−1E(u) − E((λ − Dt,i)−1u)∥2 +L2 ≤ C(µ2 + µ + 1)(1 + λ)∥u∥2 +L2 +√ +T +. +Proof. Integration by parts as in the proof of Theorem 3.10 shows that for u ∈ +Ωbd(Zi, Fi), (i = 1, 2) +Dt,T QT u|Zi = Dt,iu, +(26) +and by the construction of QT +∥Dt,T QT u − E(Dt,iu)∥2 +L2 ≤ C(1 + µ2)∥u∥2 +√ +T +. +(27) +Notice that if |Re(λ)| = 1, ∥(λ − Dt,i)−1∥ ≤ 1, by (26) and Lemma 4.5, +∥(λ − Dt,T )QT (λ − Dt,i)−1u − QT(u)∥L2 ≤ C(1 + λ)(1 + µ2)∥u∥2 +√ +T +. +(28) +22 + +Since we also have ∥(λ − Dt,T )−1∥ ≤ 1 for |Re(λ)| = 1, (28) implies that +∥QT (λ − Dt,i)−1u − (λ − Dt,T )−1QT (u)∥L2 ≤ C(1 + λ)(1 + µ2)∥u∥2 +√ +T +. +(29) +By Lemma 4.5 and (29), the lemma follows. +Proof of Theorem 3.4. Let {uk}∞ +k=1 be normal eigenforms for ∆T such that {uk} +forms an orthonormal basis. +Fix ǫ > 0. By Lemma 4.9, there exists k0(ǫ, t) > 0, such that +� +k≥k0 +��� +�� +f ′(Dt,T ) − f ′( +√ +tVT ) +� +uk, uk +� +L2 +��� < ǫ. +(30) +By Lemma 4.6, we may assume that +� +k≥k0 +��� +� +f ′( +√ +tVT )uk, uk +� +L2 +��� ≤ C +� +k≥k0 +(1 + λ2 +k(T, θ))e−tλk(T,θ) < ǫ. +(31) +By (30) and (31), +� +k≥k0 +��� +f ′(Dt,T )uk, uk +� +L2 +�� < 2ǫ. +(32) +Let {vk}∞ +k=1 be normal eigenforms for ∆1⊕∆2 such that {vk} forms an orthonor- +mal basis. +Similarly, one may assume that +� +k≥k0 +2 +� +i=1 +��� +f ′(Dt,i)vk, vk +� +L2 +�� < 2ǫ. +(33) +Let {vk}k0 +k=1 be orthonormal eigenforms with respect to eigenvalues {λk}k0 +k=1. +Let Ek0( ¯Z, ¯F) be the space generated by eigenforms with respect to eigenvalues +λ1(T, θ), ...λk0(T, θ). Set Pk0(T) be the orthogonal projection from L2Ω( ¯Z, ¯F) to +Ek0( ¯Z, ¯F). Proceeding as in the proof of Propositon 3.8, one can see that Ek0( ¯Z, ¯F) +is generated by {Pk0(T)QT vk}k0 +k=1 if T is large. +Moreover, +∥Pk0(T)QT vk − QT vk∥2 +L2 ≤ C(λk0(T, θ) + 1) +√ +T +. +(34) +∥vk − QT vk∥2 +L2 ≤ C(λk0(T, θ) + 1) +√ +T +. +(35) +Let {uk(T)} be the Gram-Schmidt Orthogonalization of {Pk0(T)QT uk}k0 +k=1. Then +Lemma 4.5 implies that +∥uk(T) − uk∥2 +L2 ≤ C(λk0(T, θ) + 1) +√ +T +. +(36) +23 + +∥Pk0(T)QT uk − uk(T)∥2 +L2 ≤ C(λk0(T, θ) + 1) +√ +T +. +(37) +Procceding as in the proof of [3, Theorem 2.13], one can show that there exists +(T, θ)-independent C > 0, such that +���ϕf ′ (Dt,T ) − Pδ(T)ϕf ′ � +Pδ(T)Dδ,T +t +Pδ(T) +� +Pδ(T) +��� ≤ C +√ +t. +(38) +(Comparing with Theorem 3.2, we are looking at operator norm, instead of trace.) +Let f ′ +la(Dt,T ) := ϕN ¯ +Z +2 +� +f ′ (Dt,T ) − Pδ(T)f ′� +Pδ(T)Dδ,T +t +Pδ(T) +� +Pδ(T) +� +. +Hence, by (34), (35), (36), (37), (38) and Lemma 4.10, there exists T(ǫ, t) > 0, +such that when T > T(ǫ, t) +∥ +k0 +� +k=1 +� +f ′ +la(Dt,T )uk(T), uk(T) +� +L2 − A(t)∥L∞ +≤ ∥ +k0 +� +k=1 +� +f ′ +la(Dt,T )QT vk, QT vk +� +L2 − A(t)∥L∞ + ǫ +≤ 2ǫ. +(39) +Here for simplicity, set +A(t) := +k0 +� +k=1 +2 +� +i=1 +� +ϕN +2 +� +f ′ (Dt,i) − Pif ′(PiDHi +t Pi)Pi +� +vk, vk +� +L2 . +By (32), (33) and (39), +lim +T→∞ f ∧ +la +� +C′ +t,T , h +¯E� += +2 +� +i=1 +f ∧ +la +� +C′ +t,i, hE� +. +Similarly, one can show +lim +T→∞ +2 +� +i=1 +f ∧ +la +� +˜C′ +t,T,i, h +¯E +T +� += +2 +� +i=1 +f ∧ +la +� +C′ +t,i, hE� +. +Proof of Theorem 3.5. By Lemma 4.6, Lemma 4.9 and Theorem 3.1, proceeding as +in the proof of [25, Theorem 3.2], one can see that there exists a measurable function +G(t) on [1, ∞), s.t. G(t)/t is L1([1, ∞)-integrable (G is independent of T and θ). +Moreover, +|f ∧ +la +� +C′ +t,T , h +¯E� +| ≤ G(t). +24 + +By Theorem 3.4 and the dominate convergence theorem, +lim +T→∞ T L +la +� +T H ¯ +M, gT ¯Z, h +¯F � +(T) = +2 +� +i=1 +T L +la,i +� +T HMi, gTZi, hFi� +. +Similarly, +lim +T→∞ +2 +� +i=1 +T L +la,i +� +T H ¯ +Mi, gT ¯Zi, h +¯Fi +T +� +(T) = +2 +� +i=1 +T L +la,i +� +T HMi, gTZi, hFi� +. +5 +The Small Time Contributions +5.1 +Several Hodge Laplacians +To show the gluing formula for f ∧, we introduce several Hodge Laplacians. +Let ∆R +B,1 be the Hodge Laplacian on [−2, −1] with absolute boundary condi- +tions. It’s easy to see that ker(∆R +B,1) is one-dimensional and generated by constant +functions. Thus, +Trs((1 + 2∆B,1)e−t∆R +B,1) = lim +t→∞ Trs((1 + 2∆B,1)e−t∆R +B,1) = 1. +(40) +Let ∆R +B,2 be the Hodge Laplacian on [1, 2] with relative boundary conditions. +Similarly, +Trs((1 + 2∆B,2)e−t∆R +B,2) = −1. +(41) +Let ¯∆B be the Hodge laplacian on Ω([−2, 2]) satisfying the absolute boundary +condition on −2, and relative boundary condition on 2. +We can also regards pT as a smooth function in (−2, 2), and let ∆R +T be the +Witten Laplacian on (−2, 2) with respect to pT , with absolute boundary condition +on −2, and relative boundary condition on 2. +5.2 +Gluing formulas for f ∧ � +C′ +t,i, hE� +and f ∧ � +C′ +t,T, h ¯E� +Let Dt,Y := Dt|Y . Let ηi(i = 1, 2) be a smooth function on (−∞, ∞) satisfying +1. 0 ≤ ηi ≤ 1; +2. η1 ≡ 1 in (−∞, −3/2),η1 ≡ 0 in (−5/4, ∞); +3. η2 ≡ 1 in (3/2, ∞),η2 ≡ 0 in (−∞, 5/4). +25 + +We can think ηi as a function on Zi(i = 1, 2). +Let ˜f(a) = (1+2a)ea, then ˜f(a2) = f ′(a). Proceeding as in [25, §6] or [2, §13(b)], +since T HM, gTZ and hF are porduct-type near N, for some C, c > 0, +��� +2 +� +i=1 +ϕ Trs +� +N Zf ′ (Dt,i) +� +− +2 +� +i=1 +ϕ Trs +� +N Zηif ′ (Dt,i) +� +− +2 +� +i=1 +ϕ Trs +� +N Zf ′ (Dt,Y ) ⊗ ˜f(t∆R +B,i) +� ++ +2 +� +i=1 +ϕ Trs +� +N Zηif ′ (Dt,Y ) ⊗ ˜f(t ¯∆B) +� ��� +L∞ ≤ C exp(−c/t). +(42) +Next, notice that [−2, 2] × Y , the number operator can be decomposed as N Z = +N Y + N R canonically (Here N Y and N R are the number operator on Y and R +components respectively). By (40), (41) and [3, Theorem 3.15], +2 +� +i=1 +ϕ Trs +� +N Zf ′ (Dt,Y ) ⊗ ˜f(t∆R +B,i) +� += +2 +� +i=1 +ϕ Trs +� +N Y f ′ (Dt,Y ) ⊗ ˜f(t∆R +B,i) +� ++ +2 +� +i=1 +ϕ Trs +� +f ′ (Dt,Y ) ⊗ N R ˜f(t∆R +B,i) +� +. += +2 +� +i=1 +χ(Y )rank(F) Trs(N R ˜f(t∆R +B,i)). +(43) +Similarly, for some (T, θ)-independent C, c > 0, +���ϕ Trs +� +N +¯Zf ′ (Dt,T ) +� +− +2 +� +i=1 +ϕ Trs +� +N Zηif ′ (Dt,i) +� +− ϕ Trs +� +N Zf ′ (Dt,Y ) ⊗ ˜f(t∆R +T ) +� ++ +2 +� +i=1 +ϕ Trs +� +N Zηif ′ (Dt,Y ) ⊗ ˜f(t ¯∆B) +� ��� +L∞ +≤ C exp(−c/t). +(44) +Moreover, Trs(e−t∆R +T ) = limt→∞ Trs(e−t∆R +T ) = dim(ker(∆R +T )0) − dim(ker(∆R +T )1). +Here ker(∆R +T )i denotes the space of harmonic i-forms(i = 0, 1). Since pT is odd, one +can see easily that if u(s) ∈ ker(∆R +T )0, then u(−s)ds ∈ ker(∆R +T )1. +As a result, Trs((1 + 2∆T )e−t∆R +T ) = 0. Proceeding as before, +ϕ Trs +� +N Zf ′ (Dt,Y ) ⊗ ˜f(t∆R +T ) +� += χ(Y )rank(F) Trs(N R ˜f(t∆R +T )). +(45) +26 + +5.3 +Proof of Theorem 3.6 +For a differential form w, let w0 denote its degree 0 component, and w+ := w − w0. +Proceeding as in [3, Proposition 2.18], for some (T, θ)-independent C, +����ϕ Trs +�N Z +2 Pδ(T)f ′ � +Pδ(T)Dδ,T +t +Pδ(T) +� +Pδ(T) +� +− +2 +� +i=1 +ϕ Trs +�N Z +2 Pif ′ � +PiDHi +t Pi +� +Pi +������ +L∞ +≤ Ct. +(46) +It follows from (42), (43), (44), (45), (46), Theorem 3.4 and dominated convergence +theorem that +� +T S +la +� +T H ¯ +M, gT ¯Z, h +¯F � +(T) +�+ += +2 +� +i=1 +� +T S +la,i +� +T HMi, gTZi, hFi��+ + o(1). +It follows from [25, Theorem 3.3] that +� +T S +la +� +T H ¯ +M, gT ¯Z, h +¯F � +(T) +�0 += +2 +� +i=1 +� +T S +la,i +� +T HMi, gTZi, hFi��0 − (T − log(2))χ(Y )rank(F)/2 + o(1). +Similarly, one can show that +� +T S +la,i +� +T H ¯ +Mi, gT ¯Zi, h +¯Fi +T +� +(T) +�+ += +� +T S +la,i +� +T HMi, gTZi, hFi��+ + o(1), +and +� +T S +la +� +T H ¯ +M, gT ¯Z, h +¯F � +(T) +�0 += +2 +� +i=1 +� +T S +la,i +� +T HMi, gTZi, hFi��0 − Tχ(Y )rank(F)/4 + o(1). +6 +Proof of Theorem 3.12 +From now on, we assume that T > 0 is large enough. +6.1 +Some Estimate of harmonic forms on the tube +Let w ∈ H( ¯ +Mi, ¯Fi)(T) such that ∥w∥L2,T = 1. Notice that ˜∆T = epT ∆T e−pT . By +Agmon estimate (c.f. [7, Theorem 1.1]), +27 + +Proposition 6.1. On [−1/2, 0] × Y or [0, 1/2] × Y, |e−pT w| ≤ Ce−cT for some +constant C > 0, c ∈ (0, 1/2). +Proposition 6.2. +∥ ˜Pδ(T)E(w) − E(w)∥L2,T ≤ Ce−cT +for some C > 0, c ∈ (0, 1/2). +Proof. We may as well assume that w ∈ H( ¯Z1, ¯F1)(T). Let η ∈ C∞ +c (R), s.t. η(s) = 1 +if s ∈ (−∞, −1/8) and η|[−1/16,0] ≡ 0, we can treat η as a smooth function on ¯Z. +By Proposition 6.1, +∥w − ηw∥L2,T ≤ Ce−cT +(47) +and +| ˜DT ηw|L2,T = |η′w|L2,T ≤ C′e−cT . +(48) +Proceeding as in the proof of Proposition 3.7, the proposition follows. +Proceeding as in the proof of Lemma 4.4, one has +Lemma 6.3. When |s − (−1)i| ≤ +2 +√ +T , +� +Y |e−pT w|2(s, y)dvolY ≤ C. +Lemma 6.4. For w ∈ H( ¯Z1, ¯F1)(T), +| +� −1/16 +−1 +� +Y +(s + 1)2|e−pT w|2dvolY ds| ≤ +C +T 3/2 . +For w ∈ H( ¯ +M2, ¯F2)(T), +| +� 1 +1/16 +� +Y +(s − 1)2|e−pT w|2dvolY ds| ≤ +C +T 3/2 . +Proof. WLOG, assume w ∈ H( ¯Z1, ¯F1)(T). Since ∆T e−pT w = 0, one can see that +� +¯Z1 +|(d + d∗)e−pT w|2 + p′′ +T |e−pT w|2 + |p′ +T |2|e−pT w|2dvol = 0. +(49) +By Lemma 6.3 and (49), +| +� −1/16 +−1+ +2 +√ +T +� +Y +T 2(s + 1)2|e−pT w|2dvolY ds| ≤ C +� −1+ +2 +√ +T +−1 +T|e−pT w|2dvol ≤ C +√ +T. +That is, +| +� −1/16 +−1+ +2 +√ +T +� +Y +(s + 1)2|e−pT w|2dvolY ds| ≤ +C +T 3/2 . +(50) +It follows from Lemma 6.3 that +| +� −1+ +2 +√ +T +−1 +� +Y +(s + 1)2|e−pT w|2dvolY ds| ≤ +C +T 3/2 . +(51) +The lemma then follows from (50) and (51). +28 + +6.2 +Estimate of small eigenvalues +Lemma 6.5. When T is big enough, ∥ ∂ +∂T Pδ(T)∥ ≤ C for some (T, θ)-independent +C > 0. Moreover, there exists a uniformly bounded operator UT , such that +∂ +∂T Pδ(T) = [ ¯D +¯Z +T , UT ]. +(52) +Here ¯D ¯Z +T := d ¯Z +T −d +¯Z,∗ +T . Similar statements hold if we replace Pδ(T) by ˜Pδ(T), ˜Pi(T) +e.t.c. +Proof. Let γ be the circle of radius +� +3δ/2 with center 0, and oriented positively. +Then +Pδ(T) = +� +γ +(λ − D +¯Z +T )−1dλ. +As a result, +∂ +∂T Pδ(T) = +� +γ +(λ − D +¯Z +T )−1 ∂ +∂T D +¯Z +T (λ − D +¯Z +T )−1dλ. +Now, one can check easily that when T is large enough, ∥(λ−D ¯Z +T )−1∥ ≤ +� +( +� +3/2 − 1)δ +�−1 +, +and +| ∂ +∂T +∂ +∂spT(s)| ≤ C. +Thus, ∥ ∂ +∂T Pδ(T)∥ ≤ C. +Notice that +∂ +∂T D ¯Z +T = [ ¯D ¯Z +T , ∂ +∂T pT ], | ∂ +∂T pT | ≤ C and ¯D ¯Z +T commutes with D ¯Z +T , one +has (52) for +UT = +� +γ +(λ − D +¯Z +T )−1 ∂ +∂T pT(λ − D +¯Z +T )−1dλ. +When T is large enough, +˜Ωk +sm( ¯Z, ¯F)(T) = ˜ek,TH( ¯Z2, ¯F2)(T) ⊕ ˜r∗ +k,TH( ¯Z1, ¯F1)(T). +(53) +Let ˜e−1 +k,T be the inverse of ˜ek,T |˜ek,T H( ¯Z1, ¯F1)(T), and ˜r−1 +k,T be the inverse of ˜rk,T |˜r∗ +k,T H( ¯Z2, ¯F2)(T). +Next, we will put a family of metric gT on Habs( ¯Z1, ¯F1) ⊕ Hrel( ¯Z2, ¯F2) when T +is large enough. +First, we define a map RT : Hk +abs( ¯Z1, ¯F1)⊕Hk +rel( ¯Z2, ¯F2) → ˜Ωsm( ¯ +M, ¯F) as follows. +For [u] ∈ Hk +abs( ¯Z1, ¯F1) represented by u ∈ Ωk +abs( ¯Z1, ¯F1), set RT ([u]) := ˜ek,T ˜P1(u) = +˜Pδ(T)E( ˜P1(T)u). For [v] ∈ Hk +rel( ¯Z2, ¯F2)0 represented by v ∈ Ωk +rel( ¯Z2, ¯F2), set RT ([v]) := +˜r−1 +k,T ˜P2(v). +Then set gT (w, w) := (RT w, RT w)L2,T for w ∈ Hk +abs( ¯Z1, ¯F1) ⊕ Hk +rel( ¯Z2, ¯F2)0. +Lemma 6.6. There exists (T, θ)-independent C > 0, such that 1− +C +T 3/2 ≤ ∥g−1 +T +∂ +∂T gT ∥ ≤ +1 + +C +T 3/2 . Here the operator norm is taken with respect to gT . +29 + +Proof. In the following, some ˜P2(T)u should be understood as E( ˜P2(T)u). First, +it’s clear that for a family of projection operators P(T), +P(T) ∂ +∂T P(T)P(T) = 0. +(54) +For any [u], [v] ∈ Hk +rel( ¯Z2, ¯F2), +∂ +∂T gT ([u], [v]) = ∂ +∂T (RT [u], RT [v])L2,T = ∂ +∂T ( ˜Pδ(T) ˜P2(T)u, ˜Pδ(T) ˜P2(T)v)L2,T += ( ∂ +∂T +˜Pδ(T) ˜P2(T)u, ˜Pδ(T) ˜P2(T)v)L2,T + ( ˜Pδ(T) ˜P2(T)u, ∂ +∂T +˜Pδ(T) ˜P2(T)v)L2,T ++ ( ˜Pδ(T) ∂ +∂T +˜P2(T)u, ˜Pδ(T) ˜P2(T)v)L2,T + ( ˜Pδ(T) ˜P2(T)u, ˜Pδ(T) ∂ +∂T +˜P2(T)v)L2,T +− 2( ∂ +∂T pT ˜Pδ(T) ˜P2(T)u, ˜Pδ(T) ˜P2(T)v)L2,T. +(55) +Let η ∈ C∞ +c (R), s.t. η(s) = 1 if s ∈ (1/8, ∞) and η|[0,1/16] ≡ 0, we can treat η as +a smooth function on ¯ +M. +By Proposition 6.2, (47), (48) and Lemma 6.5, +( ∂ +∂T +˜Pδ(T) ˜P2(T)u, ˜Pδ(T) ˜P2(T)v)L2,T +≤ e−cT ∥ ˜P2(T)u∥L2,T∥ ˜P2(T)v∥L2,T + ( ∂ +∂T +˜Pδ(T)η ˜P2(T)u, η ˜P2(T)v)L2,T +≤ e−cT ∥ ˜P2(T)u∥L2,T∥ ˜P2(T)v∥L2,T + ([ ¯D +¯Z +T , UT ]η ˜P2(T)u, η ˜P2(T)v)L2,T +≤ e−cT ∥ ˜P2(T)u∥L2,T∥ ˜P2(T)v∥L2,T ++ (UT ¯D +¯Z +T η ˜P2(T)u, η ˜P2(T)v)L2,T + (η ˜P2(T)u, UT ¯D +¯Z +T η ˜P2(T)v)L2,T +≤ Ce−cT ∥ ˜P2(T)u∥L2,T ∥ ˜P2(T)v∥L2,T. +(56) +By Hodge theory, one can see that if T ′ ≥ T, then ˜P2(T ′) ˜P2(T) = ˜P2(T ′), hence +∂ +∂T +˜P2(T) = ∂ +∂T +˜P2(T) ˜P2(T). +(57) +By (54), (57), Proposition 6.1 and Lemma 6.5, +( ˜Pδ(T) ∂ +∂T +˜P2(T)u, ˜Pδ(T) ˜P2(T)v)L2,T +≤ e−cT∥ ˜P2(T)u∥L2,T ∥ ˜P2(T)v∥L2,T + ( ∂ +∂T +˜P2(T) ˜P2(T)u, ˜P2(T)v)L2,T += e−cT∥ ˜P2(T)u∥L2,T ∥ ˜P2(T)v∥L2,T + ( ˜P2(T) ∂ +∂T +˜P2(T) ˜P2(T)u, v)L2,T += e−cT∥ ˜P2(T)u∥L2,T ∥ ˜P2(T)v∥L2,T. +(58) +By Proposition 6.2, Lemma 6.3 and Lemma 6.4 +30 + +2( ∂ +∂T pT ˜Pδ(T) ˜P2(T)u, ˜Pδ(T) ˜P2(T)v)L2,T +≤ e−cT ∥ ˜P2(T)u∥L2,T ∥ ˜P2(T)v∥L2,T + 2( ∂ +∂T pT ˜P2(T)u, ˜P2(T)v)L2,T +≤ +C +T 3/2 ∥ ˜P2(T)u∥L2,T ∥ ˜P2(T)v∥L2,T − ( ˜P2(T)u, ˜P2(T)v)L2,T . +(59) +It follows (55), (56), (58), (59) that for any [u], [v] ∈ Hk +rel( ¯Z2, ¯F2), +∂ +∂T gT ([u], [v]) ≤ +C +T 3/2 ∥ ˜P2(T)u∥L2,T ∥ ˜P2(T)v∥L2,T + ( ˜P2(T)u, ˜P2(T)v)L2,T . +(60) +Similarly, for any [u], [v] ∈ Hk +rel( ¯Z2, ¯F2), +∂ +∂T gT ([u], [v]) ≥ − C +T 3/2 ∥ ˜P2(T)u∥L2,T∥ ˜P2(T)v∥L2,T + ( ˜P2(T)u, ˜P2(T)v)L2,T . (61) +By Lemma 6.2, proceeding as in Proposition 3.9, one can see that +1 − Ce−cT ≤ ∥˜r∗ +k,T∥ ≤ 1 + Ce−cT . +Similarly, for any [u], [v] ∈ Hk +abs( ¯Z1, ¯F1), +∂ +∂T gT ([u], [v]) ≤ +C +T 3/2 ∥ ˜P1(T)u∥L2,T ∥ ˜P1(T)v∥L2,T − ( ˜P1(T)u, ˜P1(T)v)L2,T , +(62) +∂ +∂T gT ([u], [v]) ≥ − C +T 3/2 ∥ ˜P1(T)u∥L2,T∥ ˜P1(T)v∥L2,T − ( ˜P1(T)u, ˜P1(T)v)L2,T ; (63) +for any [u] ∈ Hk +rel( ¯Z1, ¯F1), [v] ∈ Hk +abs( ¯Z2, ¯F2), +| ∂ +∂T gT ([u], [v])| ≤ +C +T 3/2 ∥ ˜P1(T)u∥L2,T ∥ ˜P2(T)v∥L2,T. +(64) +Lastly, by Proposition 6.1, for any [u], [v] ∈ Hk +rel( ¯Z1, ¯F1) ⊕ Hk +abs( ¯Z2, ¯F2), +|gT ([u], [v]) − ( ˜Pi(T)u, ˜Pi(T)v)L2,T | ≤ e−cT ∥ ˜Pi(T)u∥L2,T∥ ˜Pi(T)v∥L2,T . +(65) +It follows (62), (63), (64) and (65) that +1 − +C +T 3/2 ≤ ∥g−1 +T +∂ +∂T gT ∥ ≤ 1 + +C +T 3/2 . +Next, we define a differential ˜∂ : Hk−1 +abs ( ¯Z1, ¯F1)⊕Hk−1 +rel ( ¯Z2, ¯F2)0 → Hk +abs( ¯Z1, ¯F1)⊕ +Hk +rel( ¯Z2, ¯F2)0, ([u], [v]) �→ (0, ∂k[u]). Recall that ∂k is the map in Mayer-Vietoris se- +quence (3). Then one can check easily that RT ˜∂R−1 +T += d ¯Z|˜Ωsm( ¯ +M, ¯F) and RT ˜∂∗ +T R−1 +T += +δ +¯Z,∗ +T +|˜Ωsm( ¯ +M, ¯F), where ˜∂∗ +T is the dual of ˜∂ w.r.t. gT . +It follows from the statement in Step 1 in the proof of [21, Theorem 3.8] and +Lemma 6.6 that +Corollary 6.7. When T is large enough, all nonzero eigenvalues of ∆T inside [0, δ] +are actually inside [c2 +1e−2T , c2 +2e−2T ] for some c2 > c1 > 0. +31 + +6.3 +Comparison of connections +H( ¯Zi, ¯Fi)(T) has a flat connection ∇Hi,T := ˜Pi(T)∇ ¯Ei(c.f. [3, Proposition 2.6]). +Recall that if T is large enough, +˜Ωk +sm( ¯Z, ¯F)(T) = ˜ek,THk( ¯Z2, ¯F2)(T) ⊕ ˜r∗ +k,THk( ¯Z1, ¯F1)(T). +(66) +Recall that ˜e−1 +k,T is the inverse of ˜ek,T|˜ek,T H( ¯Z1, ¯F1)(T), and ˜r−1 +k,T is the inverse of +˜rk,T|˜r∗ +k,T H( ¯Z2, ¯F2)(T), then ˜Ωsm( ¯Z, ¯F)(T) has a flat connection +∇H,T := ˜ek,T ∇H1,T ˜e−1 +k,T ⊕ ˜r−1 +k,T∇H2,T ˜rk,T. +Moreover, let ∇δ,T := ˜Pδ(T)∇ ¯E, then ∇δ,T is another connection on ˜Ωsm( ¯Z, ¯F)(T). +In this subsection, we are going to compare ∇H,T and ∇δ,T . +First, one has +Lemma 6.8. When T is big enough, ∥[∇ ¯E, ˜Pδ(T)]∥ ≤ C for some (T, θ)-independent +C > 0. Moreover, there exists a uniformly bounded operator valued 1-form AT , such +that +[∇ +¯E, ˜Pδ(T)] = [d +¯Z, AT ]. +(67) +Similar statements hold if we replace ˜Pδ(T) by ˜Pi(T) (i = 1, 2). +Proof. Doing functional calculus as in Lemma 6.5 for ˜D ¯Z +T , one has ∥[∇E, Pδ(T)]∥ ≤ +C. Since [d ¯Z, ∇ ¯E] = 0, one can see that +[∇ +¯E, ˜∆T ] = [d +¯Z, [∇ +¯E, δ +¯Z,∗ +T +]]. +Since gT ¯Z, T H ¯ +M and h ¯F +T are product type near N, one can see that [∇ ¯E, δ +¯Z,∗ +T +] is +uniformly bounded. Doing functional calculus for ˜∆T as in Lemma 6.5, the lemma +follows. +Let Kδ +T = ∇δ,T − ∇δ,T,∗, KH +T = ∇H,T − ∇H,T,∗ , KT = Kδ +T − KH +T . +Lemma 6.9. When T is large enough, ∥KT ∥ ≤ C exp(−ρT) for some (T, θ)- +independent C > 0, ρ ∈ (0, 1). +Proof. For u ∈ ˜ek,THk( ¯Z2, ¯F2)(T), there exists v ∈ Hk( ¯Z2, ¯F2)(T), such that u = +˜ek,Tv. Then +∇δ,T u = ˜Pδ(T)∇ +¯E ˜Pδ(T)E(v) = ˜Pδ(T)∇ +¯EE(v) + ˜Pδ(T)[ ˜Pδ(T), ∇ +¯E]E(v). (68) +Recall that E(v) is an extension of v, s.t. outside ¯Z1, E(v) = 0. Integration by parts +as in the proof of Theorem 3.10 shows that E(v) is d ¯Z-closed. Hence, by Lemma +6.8 and Corollary 6.7, +32 + +∥ ˜Pδ(T)[ ˜Pδ(T), ∇ +¯E]E(v)∥ = ∥Pδ(T)d +¯ZAT E(v)∥ += ∥d +¯ZPδ(T)AT E(v)∥ ≤ Ce−T ∥v∥. +(69) +Similarly, +∇H,T u = ˜Pδ(T)E( ˜P2(T)∇ +¯E2v) = ˜Pδ(T)∇ +¯EE(v) + ˜Pδ(T)E([ ˜P2(T), ∇ +¯E2]v). +(70) +Moreover, Integration by parts as in the proof of Theorem 3.10 shows that for +w ∈ Ωrel( ¯Z2, ¯F2)(T), +d +¯ZE(w) = E(d +¯Z2w). +(71) +Thus, by (71), Lemma 6.8 and Corollary 6.7, +∥ ˜Pδ(T)E([ ˜P2(T), ∇ +¯E2]v)∥ = ∥Pδ(T)d +¯ZE(AT,2v)∥ += ∥d +¯ZPδ(T)E(AT,2v)∥ ≤ Ce−T ∥v∥. +(72) +It follows from (68), (69), (70) and (72) and Proposition 6.2 that +∥(∇δ,T − ∇H,T)u∥ ≤ Ce−T ∥u∥. +(73) +While for any u1, u2 ∈ ˜ek,T Hk( ¯Z2, ¯F2)(T), +(KT u1, u2)L2,T = ((∇δ,T − ∇H,T )u1, u2)L2,T + (u1, (∇δ,T − ∇H,T )u2)L2,T . +(74) +Hence, by (73) and (74) +|(KT u1, u2)L2,T | ≤ Ce−T ∥u1∥L2,T ∥u2∥L2,T. +(75) +Similarly, one can show that restricted on ˜r∗ +k,TH( ¯Z1, ¯F1)(T), ∥∇δ,T,∗ − ∇H,T,∗∥ ≤ +Ce−T . Similarly, for u1, u2 ∈ ˜r∗ +k,TH( ¯Z1, ¯F1)(T), +|(KT u1, u2)L2,T | ≤ Ce−T ∥u1∥L2,T ∥u2∥L2,T. +(76) +Since T H ¯ +M, gT ¯Z and h ¯F are product-type near N, +∥∇ +¯E − ∇ +¯E,∗∥ ≤ C. +(77) +Suppose vi ∈ H( ¯Zi, ¯Fi)(T), i = 1, 2, and u1 = ˜r∗ +k,Tv1, u2 = ˜ek,Tv2. Let η ∈ C∞ +c (R), +s.t. η(s) = 1 if s ∈ (−∞, −1/8) and η|[−1/16,0] ≡ 0. Then we could think η as a +function on ¯Z. +33 + +By Proposition 6.1, 6.2 and (77) , there exists ρ ∈ (0, 1), s.t. +((∇δ,T − ∇δ,T,∗)u1, u2)L2,T = ((∇δ,T − ∇δ,T,∗) ˜Pδ(T)E(v1), ˜Pδ(T)E(v2))L2,T += ((∇ +¯E − ∇ +¯E,∗) ˜Pδ(T)E(v1), ˜Pδ(T)E(v2))L2,T +≤ Ce−ρT ∥u1∥L2,T ∥u2∥L2,T + ((∇ +¯E − ∇ +¯E,∗)ηE(v1), E(v2))L2,T += Ce−ρT ∥u1∥L2,T ∥u2∥L2,T + (η(∇ +¯E − ∇ +¯E,∗)E(v1), E(v2))L2,T += Ce−ρT ∥u1∥L2,T ∥u2∥L2,T. +(78) +Since for any ui ∈ H( ¯Zi, ¯Fi)(T), i = 1, 2, +((∇H,T − ∇H,T,∗)u1, u2)L2,T = 0. +(79) +By (78) and (79), for any ui ∈ H( ¯Zi, ¯Fi)(T), i = 1, 2, +|(KT u1, u2)L2,T | ≤ Ce−ρT ∥u1∥L2,T ∥u2∥L2,T. +(80) +The lemma then follows from (75), (76) and (80). +Let Dδ,T +t += ∇δ,T −∇δ,T,∗+ +√ +t(d ¯Z −δ +¯Z,∗ +T +), DH,T +t += ∇H,T −∇H,T,∗+ +√ +t(d ¯Z −δ +¯Z,∗ +T +). +It follows from Corollary 6.7 and Lemma 6.9 that +Corollary 6.10. For t ≥ 1, +| Trs +� +Nf ′(Dδ,T +t +) − Nf ′(DH,T +t +) +� +| ≤ Ce(1−ρ)T +√ +t +for some (T, θ)-independent C > 0, where ρ is the constant in Lemma 6.9. +For t > 0, +| Trs +� +Nf ′(Dδ,T +t +) − Nf ′(DH,T +t +) +� +| ≤ C′e−2T t +for some (T, θ)-independent C′ > 0. +Proof. Let Kδ +T = ∇δ,T − ∇δ,T,∗, KH +T = ∇H,T − ∇H,T,∗ , KT = Kδ +T − KH +T . +By Lemma 6.9, +∥KT ∥ ≤ Ce−ρT . +(81) +Let γ be the oriented contour given by {z ∈ C : |Re(z)| = 1}. Then by Corollary +6.7, when T is large (c.f. [3, Theorem 2.13]), for “•”=“H” or “δ” +f ′(D•,T +t +) = +� +γ +f ′(λ)(λ − D•,T +t +)−1dλ. +For λ ∈ γ, let V = d ¯Z − δ +¯Z,∗ +T +, then +� +λ − D•,T +t +�−1 += +� +1 − +� +λ − +√ +tV +�−1 +K• +T +�−1 � +λ − +√ +tV +�−1 +. +(82) +34 + +By Corollary 6.7 and Lemma 4.8, for λ ∈ γ +∥ +� +λ − +√ +tV +�−1 +− Pker V +λ +∥ ≤ CeT |λ| +√ +t +; +(83) +or +∥ +� +λ − +√ +tV +�−1 +− Pker V +λ +∥ ≤ 2. +(84) +Also +� +1 − +� +λ − +√ +tV +�−1 +K• +T +�−1 += +dim S +� +i=0 +�� +λ − +√ +tV +�−1 +K• +T +�i +, +(85) +Proceeding as in the proof of [3, Theorem 2.13], by (81), (82), (83), (84) and (85) +| Trs +� +N +� +f ′(Dδ,T +t +) − f ′(Pker V Kδ +T Pker V ) − f ′(DH,T +t +) + f ′(Pker V KH +T Pker V ) +�� +| +≤ Ce(1−ρ)T +√ +t +. +(86) +While by [3, Proposition 1.3], +Trs(Nf ′(Pker V Kδ +T Pker V )) = Trs(Nf ′(Pker V KH +T Pker V )) = Trs(Nf ′(0)). +Hence, when t ∈ [1, ∞), +| Trs(Nf ′(Dδ,T +t +)) − Trs(Nf ′(DH,T +t +))| ≤ Ce1−ρT +√ +t +. +Although ∇δ,T is not flat, the argument in [3, Proposition 1.3] still works. Hence, +we have +Trs +� +Nf ′(Dδ,T +0 ) − Nf ′(DH,T +0 +) +� += 0. +(87) +Moreover, by a straightforward computation, +∂ +∂t Trs +� +Nf ′(Dδ,T +t +) − Nf ′(DH,T +t +) +� += Trs +� +(d +¯Zδ +¯Z,∗ +T +− δ +¯Z,∗ +T +d +¯Z) +� ˜f ′((Dδ,T +t +)2) − ˜f ′((DH,T +t +)2) +�� +. +(88) +Recall that ˜f(a) = (1 + 2a)ea, hence ˜f(a2) = f ′(a). +By Corollary 6.7, +∥(d +¯Zδ +¯Z,∗ +T +− δ +¯Z,∗ +T +d +¯Z)( ˜f ′((Dδ,T +t +)2) − ˜f ′((DH,T +t +)2))∥ ≤ Ce−2T . +(89) +By (87), (88) and (89), +| Trs +� +Nf ′(Dδ,T +t +) − Nf ′(DH,T +t +) +� +| ≤ C′e−2T t. +35 + +Let +Tf +� +d +¯Z + ∇δ,T� += − +� ∞ +0 +� +ϕ Trs +�1 +2Nf ′ � +Dδ,T +t +�� +− 1 +2χ′(Z, F) − d(Ωsm( ¯ +M, ¯F)(T)) − χ′(Z, F) +2 +f ′ +�i +√ +t +2 +�� dt +t , +and +Tf +� +d +¯Z + ∇H,T� += − +� ∞ +0 +� +ϕ Trs +�1 +2Nf ′ � +DH,T +t +�� +− 1 +2χ′(Z, F) − d(Ωsm( ¯ +M, ¯F)(T)) − χ′(Z, F) +2 +f ′ +�i +√ +t +2 +�� dt +t . +Corollary 6.11. limT→∞ +� +Tf +� +d ¯Z + ∇δ,T� +− Tf +� +d ¯Z + ∇H,T �� += 0. That is, +lim +T→∞ +� +Tsm(T H ¯ +M, gT ¯Z, h +¯F )(T) − Tf +� +d +¯Z + ∇H,T�� += 0. +Proof. Fix ρ′ ∈ (0, ρ), where ρ is the constant in Lemma 6.9. By the second in- +equality in Corollary 6.10, +� e2T −2ρ′T +0 +| Trs +�1 +2Nf ′ � +DH,T +t +�� +− Trs +�1 +2Nf ′ � +Dδ,T +t +�� +|dt +t ≤ Ce−2ρ′T . +(90) +By the first inequality in Corollary 6.10, +� ∞ +e2T −2ρ′T | Trs +�1 +2Nf ′ � +DH,T +t +�� +− Trs +�1 +2Nf ′ � +Dδ,T +t +�� +|dt +t ≤ Ce(ρ′−ρ)T . +(91) +Proof of Theorem 3.12. +Since ∇H2,T ˜∂k,T = ˜∂k,T∇H1,T and ˜∂k,T = ˜e−1 +k,T d ¯Z˜r−1 +k,T, one has [∇H,T , d ¯Z] = 0. +It follows from Proposition 3.9, [26, Lemma 2.2] and [9, Theorem 7.37] that in +QS/QS +0, +lim +T→∞ Tf +� +d +¯Z + ∇H,T� +− T (T) = 0. +The theorem then follows from Corollary 6.11. +References +[1] J. M. Bismut and S. Goette. Families torsion and Morse functions. Ast´erisque, +275, 2001. +[2] J.-M. Bismut and G. Lebeau. Complex immersions and Quillen metrics. 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In Abhandlungen aus dem +Mathematischen Seminar der Universit¨at Hamburg, volume 11, pages 102–109. +Springer, 1935. +[24] J. B. Wagoner. Diffeomorphisms, K2, and analytic torsion. In Algebraic and ge- +ometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., +1976), Part, volume 1, pages 23–33, 1976. +[25] J. Yan. Witten deformation for non-Morse functions and gluing formula for +analytic torsions. arXiv:2301.01990. +[26] J. Zhu. On the gluing formula of real analytic torsion forms. International +Mathematics Research Notices, 2015(16):6793–6841, 2015. +[27] J. Zhu. Gluing formula of real analytic torsion forms and adiabatic limit. Israel +Journal of Mathematics, 215(1):181–254, 2016. +38 + diff --git a/ydE0T4oBgHgl3EQftQH5/content/tmp_files/load_file.txt b/ydE0T4oBgHgl3EQftQH5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..72443a984e752e6039e002d363dea692462f74af --- /dev/null +++ b/ydE0T4oBgHgl3EQftQH5/content/tmp_files/load_file.txt @@ -0,0 +1,1404 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf,len=1403 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='02591v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='DG] 6 Jan 2023 A new proof of gluing formula for analytic torsion forms Junrong Yan ∗ January 9, 2023 Abstract By extending the author’s prior work [25] to the family case, this paper presents a new proof of the gluing formula for the analytic torsion forms, considerably sim- plifying the proof given by Puchol-Zhang-Zhu [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Contents 1 Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 Overview .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4 Organization .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 5 2 Preliminary 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 Definition of Bismut-Lott analytic torsion forms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2 Analytic torsion forms for manifolds with boundary .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3 Analytic torsion form for Witten deformations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 Witten Laplacian v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Weighted Laplacian .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2 Absolute/Relative boundary conditions for weighted Laplacian 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4 Analytic torsion form for complex of finite dimensional vector bundles 10 3 Intermidiate Results 11 4 Convergence of Eigenvalues 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 Convergence of f ∧ la � C′ t,T , h ¯E� and the large time contributions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 20 ∗Beijing International Center for Mathematical Research, Peking University, Beijing, China 100871, j yan@bicmr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Supported by Boya Postdoctoral Fellowship at Peking University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 1 5 The Small Time Contributions 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 Several Hodge Laplacians .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2 Gluing formulas for f ∧ � C′ t,i, hE� and f ∧ � C′ t,T , h ¯E� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 27 6 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='12 27 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 Some Estimate of harmonic forms on the tube .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 27 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2 Estimate of small eigenvalues .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 29 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3 Comparison of connections .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 32 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 Overview Let (M, g) be a closed Riemannian manifold associated with a flat complex vec- tor bundle F → M with a Hermitian metric hF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The corresponding Ray-Singer analytic torsion [22] is the determinant of the Hodge-Laplacian on F-valued differ- ential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The Ray-Singer analytic has a well-known topological counterpart, the Reidemeister torsion (R-torsion)[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' According to the well-known Cheeger- M¨uller/Bismut-Zhang theorem, the two torsions are related [6, 17, 18, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It was conjectured that R-torsion and Ray-Singer torsion can be extended to invariants of a C∞ fibration π : M → S of closed fiber Z, associated with a flat complex vector bundle F → M [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Bismut and Lott [3] then construct analytic torsion forms (BL-torsion), which are even forms on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Igusa, motivated by the work of Bismut and Lott, developed the Igusa-Klein torsion, a higher topological torsion (IK-torsion) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' As an application of IK-torsion, Goette, Igusa, and Williams [11, 10] uncover fiber bundles’ exotic smooth structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then it becomes a natural and significant question to investigate the connection between these higher torsion invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Under the assumption that there exists a fiberwise Morse function [1, 9], Bismut and Goette established a higher version of the Cheeger-M¨uller/Bismut- Zhang theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Lastly, an interesting relation between the Bismut-Freed connection and analytic torsion forms was observed by Dai and Zhang [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Higher torsion invariants were axiomatized by Igusa [13], and Igusa showed that IK-torsion complies with his axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' His axiomatization consists of two axioms: the axiom of additivity and the axiom of transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' And any higher torsion invariant that satisfies Igusa’s axioms is simply a linear combination of IK-torsion and the higher Miller-Morita-Mumford class [16, 19, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' BL-torsion is proven to satisfy the transfer axiom thanks to the work of Ma [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Recently, Puchol-Zhang-Zhu [20] established the additivity of BL-torsion (gluing formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' In this paper, a different, considerably less complicated proof of the gluing formula is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let’s assume that N ⊂ M is a hypersurface that fiberwisely divides Z into two parts, Z1 and Z2, and that it also divides M into two pieces, M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' In this 2 paper, we prove a gluing formula using the Witten deformation for non-Morse func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We choose a family of smooth functions pT where limT→∞ pT contains critical loci M1 and M2 with ”Morse indices” 0 and 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Assuming T varies from 0 to ∞, we may adopt the philosophy of Witten deformation to see the relationship between the analytic torsion forms on M and analytic torsion forms on the two components above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By exploring Witten deformation for non-Morse functions, this paper tries to give light on how to establish the family version of Cheeger-M¨uller theorem for general flat bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Acknowledgment: The author is appreciative of Professor Xianzhe Dai’s consis- tently stimulating conversation and encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The author also appreciates the insightful discussion with Martin Puchol and Yeping Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2 Main results Let M → S be a fibration of closed fiber Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let TZ ⊂ TM be the vertical tangent bundle of the fiber bundle with metric gTZ, and T ∗Z be its dual bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let F → M be a flat complex vector bundle with Hermitian metric hF , and ∇F be a flat connection on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Fix a splitting of TM TM = T HM ⊕ TZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let QS be the space of closed even forms, QS 0 be the space of exact forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Based on the data (T HM, gTZ, hF ), one can define analytic torsion forms T (T HM, gTZ, hF ) ∈ QS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' See Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 for the definition of T (T HM, gTZ, hF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let N ⊆ M be a hypersurface transversal to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We suppose that π|N : N → S is a fibration of fiber Y := N ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Suppose that N cuts M into two pieces M1 and M2, and fiberwisely, cut Z into two pieces Z1 and Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let πi : Mi → S be the restriction of π to Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then πi is a fibration of Zi (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let Fi, T HMi and hFi be the restriction of F, T HM and hF to Mi respectively (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We put the relative boundary condition on the boundary of Z1 and the absolute boundary condition on the boundary of Z2 (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The analytic torsion form for fibration Mi with boundary equipped with absolute/relative boundary condition could be defined, denoted by Ti(T HMi, gTZi, hFi) ∈ QS (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' See §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For θ ∈ S, let Zθ := π−1θ, Fθ := F|Zθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let H (Z, F) be the Z-graded vector bundle over S whose fiber over θ ∈ S is the cohomology H (Zθ, Fθ) of the sheaf of locally flat sections of F on Zθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We have a Mayer-Vietoris exact sequence of flat complex vector bundles over S, · · → Hp rel (Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' F2) → Hp (Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' F) → Hp abs (Z1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' F1) → · · · (1) 3 with metric given by Hodge theory and connection induced by ∇E and ∇Ei (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' [3, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let T ∈ QS be the torsion form for the exact sequence (1)(c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We assume that gTZ, hF and T HM are product-type near N, then Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' In QS/QS 0 , T (T HM, gTZ, hF ) − T1(T HM1, gTZ1, hF1) − T2(T HM2, gTZ2, hF2) − T = 1 2χ(Y )rank(F) log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For any closed oriented submanifold S′ ⊆ S, the map � S′ : QS → R descends to a linear function on QS/QS 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It follows from Stokes’ formula and the de Rham theorem that these linear functions separate the elements of QS/QS,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1, it is, therefore, sufficient to show the case in which S is a closed manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3 Main ideas Let N ⊂ M be a hypersurface cutting M into two pieces M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let U be a neighbor of N, such that U ∩ M1 is diffeomorphic to (−2, −1] × N and identify ∂M1 with {−1} × N, U ∩ M2 is diffeomorphic to [1, 2) × N and identify ∂M2 with {1} × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Moreover, assume that on U, T HM, gTZ and hF are product-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We then glue M1, M2 and [−1, 1]×Y naturally, we get a new fiberation ¯ M → S, which is isomorphic to the original fiberation M → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let pT be a smooth function on ¯ M, such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' pT |M1 ≡ −T/2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' pT |M2 ≡ T/2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' pT |[−1,0]×Y (s, y) ≈ T(s + 1)2/2 − T/2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' pT |[−1,0]×Y (s, y) ≈ −T(s − 1)2/2 + T/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For simplicity, we solely describe the situation where S = {pt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' In this case, the analytic torsion form reduces to the logarithm of analytic torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let dF T := dF + dpT ∧, dF,∗ T be the formal adjoint of dF T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Set DT := dF T + dF,∗ T , ∆T := D2 T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let λk be the k-th eigenvalue (counted with multiplicities) of ∆1 ⊕ ∆2 (acting on Ωrel(M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' F1)⊕Ωabs(M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' F2)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let {λk(T)} be the k-th eigenvalue (counted with multiplicities) of ∆T (acting on Ω( ¯ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' One has a nice observation that lim T→∞ λk(T) = λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (2) 4 We temporarily assume that dim Hk(M) = dim Hk(M1) + dim Hk(M2, ∂M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let T (gT ¯ M, hF , ∇F )(T) be the analytic torsion with respect to ∆F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then based on (2), naively, one should expect that lim T→∞ T (gT ¯ M, hF , ∇F )(T) = T (gTM1, hF1, ∇F1) + T (gTM2, hF2, ∇F2), and lim T→0 T (gT ¯ M, hF , ∇F )(T) = T (gT ¯ M, hF , ∇F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' As a result, the relationship between analytic torsion on M and analytic torsion on the two pieces above could be seen, proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Now, we no longer assume that dim Hk(M) = dim Hk(M1)+dim Hk(M2, ∂M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Note that for large T, the bundle Ωsm(M, F)(T) generated by the eigenforms of ∆T with respect to small eigenvalues is of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The second key ingredient in our proof is the relationship between the analytic torsion form for the exact sequence (1) and Ωsm(M, F)(T) as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' See §6 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4 Organization In §2, we will give a brief review of analytic torsion forms and establish the basic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' In §3, we state and prove several intermediate results, then prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' While Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='12 will be proved in subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' In §4, we investigate the behavior of eigenvalues as T → ∞ and prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' In §5, we establish Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='12 is proved in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 2 Preliminary Let π : M → S be a fibration of C∞ manifolds with closed fibre Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let TZ ⊂ TM be the vertical tangent bundle of the fiber bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let F be a flat complex vector bundle on M with Hermitian metric hF and a flat connection ∇F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For θ ∈ S, Zθ := π−1(θ) and Fθ := F|Zθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' From now on, we assume that S is closed, and T HM, gTZ and hF are product- type near N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' That is, there exists a neighborhood U of N, such that U = (−1, 1)×N, and let (s, z) be its coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then T HU = {0} × T HN for some splitting TN = T HN ⊕ TY , gTZ|U = ds ⊗ ds + gTY for some metric gTY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let hF Y := hF |{0}×Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For any v ∈ F(s,y), let Pγ ∈ End(F(s,z), F(0,z)) be the parallel transport associated with ∇F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' the path γ(t) = (st, z), t ∈ [0, 1], then we require that hF (v, v) = hF Y (Pγv, Pγv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Thus, ∇F ∂ ∂s hF = 0 in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' If T is sufficiently large, all constants appearing in this paper are at least inde- pendent of T and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The notations C and c, et cetera, denote constants that may vary based on context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 Definition of Bismut-Lott analytic torsion forms Let T HM be a sub-bundle of TM such that TM = T HM ⊕ TZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let P TZ denote the projection from TM to TZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' If U ∈ TS, let U H be the lift of U in T HW, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' π∗U H = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For a Hilbert bundle H → X, Γ(X, H) denotes the space of smooth sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Now let E = Ω(Z, F), which is the bundle over S, such that for θ ∈ S, Ω(Z, F)|Zθ = Ω(Zθ, Fθ), that is, E = ⊕dim Z i=0 Ei is the smooth infinite-dimensional Z-graded vector bundle over S, whose fiber at θ ∈ S is Γ (Zθ, (Λ (T ∗Z) ⊗ F) |Zθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then Γ � S, Ωi(Z, F) � = Γ � M, Λi (T ∗Z) ⊗ F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For s ∈ Γ(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' E) and U ∈ Γ(S, TS), the Lie differential LUH acts on Γ(S, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Set ∇E Us = LUHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then ∇E is a connection on E preserving the Z-grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' If U1, U2 are vector fields on S, put T (U1, U2) = −P TZ � U H 1 , U H 2 � ∈ Γ(M, TZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We denote iT ∈ Ω2 � S, Hom � E•, E•−1�� be the 2 -form on S which, to vector fields U1, U2 on S, assigns the operation of interior multiplication by T (U1, U2) on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let dZ : Ω∗(Z, F) → Ω∗+1(Z, F) be exterior differentiation along fibers induced by ∇F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We consider dZ to be an element of Γ � S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Hom � E•, E•+1�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By [3, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4], we have dM = dZ + ∇E + iT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Here dM : Ω∗(M, F) → Ω∗+1(M, F) is induced by ∇F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' So dM is a flat superconnec- tion of total degree 1 on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (dM)2 = 0 implies that � dZ�2 = 0, � ∇E, dZ� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let gTZ be a metric on TZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let hE be the metric on E induced by hF and gTZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Sometime we will also denote hE by (·, ·)L2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let ∇E,∗, dZ,∗, dM,∗ be the formal adjoint of ∇E, dZ, dM with respect to hE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Set DZ = dZ + dZ∗, ∇E,u = 1 2 � ∇E + � ∇E�∗� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let N Z be the number operator of E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' acts by multiplication by k on the space Γ � W, Λk (T ∗Z) ⊗ F � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For t > 0, set C′ t = tNZ/2dMt−NZ/2, C′′ t = t−NZ/2 � dM�∗ tNZ/2, Ct = 1 2 (C′ t + C′′ t ) , Dt = 1 2 (C′′ t − C′ t) , 6 then C′′ t is the adjoint of C′ t with respect to hE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Ct is a superconnection and Dt is an odd element of Ω(S, End(E)), and C2 t = −D2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For X ∈ TZ, let X∗ ∈ T ∗Z correspond to X by the metric gTZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Set c(X) = X∗ ∧ −iX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then Ct = √ t 2 DZ + ∇E,u − 1 2 √ tc(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let f(a) = a exp(a2), and ϕ : Ωeven (S) → Ωeven (S) as follows, ϕω = (2πi)−kω, for ω ∈ Ω2k(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For any t > 0, the operator Dt is a fiberwise-elliptic differential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then f (Dt) is a fiberwise trace class operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For t > 0, put f ∧ � C′ t, hE� := ϕ Trs �N Z 2 f ′ (Dt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Put χ(Z, F) := �dim Z i=0 (−1)i rank Hi(Z, F), χ′(Z, F) := �dim Z i=0 (−1)ii rank Hi (Z, F) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The analytic torsion form T � T HM, gTZ, hF � is a form on S which is given by T � T HM, gTZ, hF � = − � +∞ 0 � f ∧ � C′ t, hE� − χ′(Z, F) 2 − χ(Z, F) dim Z − 2χ′(Z, F) 4 f ′ �i √ t 2 �� dt t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It follows from [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='21] that T � T HM, gTZ, hF � is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The degree 0 part of T � T HM, gTZ, hF � is nothing but the fiberwise analytic torsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' This is why T � T HM, gTZ, hF � is referred to as analytic torsion forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2 Analytic torsion forms for manifolds with boundary Let N ⊆ M be a hypersurface transversal to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We suppose that π|N : N → S is a fibration of fiber Y := N ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Suppose that N cuts M into two pieces M1 and M2, and fiberwisely, cut Z into two pieces Z1 and Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let πi : Mi → S be the restriction of π to Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then πi is a fibration of Zi (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let Fi, T HMi and hFi be the restriction of F, T HM and hF to Mi respectively (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' First, identify a neighborhood U1 of ∂M1 with (−2, −1]×N, and identify ∂M1 with {−1}×N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then identify a neighborhood U2 of ∂M2 with [1, 2) × N, and identify ∂M2 with {1}× N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let (s, z) be the coordinate of Ui w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' the identification above, pi : Ui → S, (s, z) → π|N(z) be the canonical submersion (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 7 Let Ω(Zi, F) denotes the space of F|Zi-valued smooth differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Ωabs (Z1, F) = � ω ∈ Ω (Z1, F) : i ∂ ∂s ω ��� ∂Z1 = i ∂ ∂s � dZ1ω ���� ∂Zj = 0 � , Ωrel (Z2, F) = � ω ∈ Ω (Z2, F) : de ∧ ω|∂Zj = ds ∧ � dZ2,∗ω ��� ∂Z2 = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We write Ei = Ωbd(Zi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' F) for short if the choice of abs/rel is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let DZi be the restriction of DZ on Zi acting on Ωbd(Zi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, we have dMi, ∇Ei, Ct,i, Dt,i e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For t > 0, put f ∧ � C′ t,i, hEi� := ϕ Trs �N Z 2 f ′ (Dt,i) � , where hEi = (·, ·)L2(Zi) is the metric induced by gTZi and hFi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Put χbd(Zi, F) := �dim Z i=0 (−1)i rank Hi bd(Zi, Fi), χ′(Zi, Fi) := �dim Z i=0 (−1)ii rank Hi bd (Zi, Fi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The analytic torsion form Ti � T HMi, gTZi, hFi� is a form on S which is given by Ti � T HMi, gTZi, hFi� = − � +∞ 0 � f ∧ � C′ t,′, hEi� − χ′(Zi, Fi) 2 − χ(Zi, Fi) dim Z − 2χ′(Zi, Fi) 4 f ′ �i √ t 2 �� dt t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It follows from [26, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='17] that Ti � T HMi, gTZi, hFi� is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3 Analytic torsion form for Witten deformations Let N ⊆ M be a hypersurface transversal to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We suppose that π|N : N → S is a fibration of fiber Y := N ∩ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Suppose that N cuts M into two pieces M1 and M2, and fiberwisely, cut Z into two pieces Z1 and Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let πi : Mi → S be the restriction of π to Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then πi is a fibration of Zi (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let Fi, T HMi and hFi be the restriction of F, T HM and hF to Mi respectively (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' First, identify a neighborhood U1 of ∂M1 with (−2, −1]×N, and identify ∂M1 with {−1}×N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then identify a neighborhood U2 of ∂M2 with [1, 2) × N, and identify ∂M2 with {1}× N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let (s, z) be the coordinate of Ui w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' the identification above, pi : Ui → S, (s, z) → π|N(z) be the canonical submersion (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let ¯ M = M1 ∪ [−1, 1] × N ∪ M2, and ¯F, h ¯F , gT ¯Z and T H ¯ M be the natural extensions of F, hF , gTZ and T HM to ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We still have a natural extension of fiberation ¯ M → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similar, we have notation ¯Z for the fiber Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let pT be a family of odd smooth functions on [−2, 2], such that (a) pT |[1,2] ≡ T/2, 8 (b) pT |[1/16,1](s) = −Tρ(eT 2(1 − s))(s − 1)2/2 + T/2 , where ρ ∈ C∞ c ([0, ∞)), such that 0 ≤ ρ ≤ 1, ρ[0,1/2] ≡ 0, ρ[3/4,∞] ≡ 1, |ρ′| ≤ δ1 and |ρ′′| ≤ δ2 for some universal constant δ1 and δ2, (c) C1T ≤ |p′ T |(s) ≤ 2C1T, |p′′ T | ≤ C2T for some universal constants C1 and C2 whenever s ∈ [0, 1/16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then one can see that C3T||s| − 1| ≤ |p′ T |(s) ≤ 2C3T||s| − 1| and |p′′ T |(s) ≤ C4T whenever ||s| − 1| ≤ e−T 2 for some universal constant C3 and C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We could think pT as a function on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Still denotes pT to be its fiberwise restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let d ¯Z T := d ¯Z + dpT ∧, d ¯Z,∗ T be the adjoint of d ¯Z T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then D ¯Z T := d ¯Z T + d ¯Z,∗ T , ∆T := (D ¯Z T )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, we have notation d ¯ M T , ∇ ¯E, Ct,T , Dt,T e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For t > 0, put f ∧ � C′ t,T , h ¯E� := ϕ Trs �N Z 2 f ′ (Dt,T ) � , where ¯E = Ω( ¯Z, ¯F), and h ¯E = (·, ·)L2( ¯Z) is the metric on Ω( ¯Z, ¯F) induced by gT ¯Z and h ¯F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let H( ¯Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F)(T) be the bundle on S, whose fiber at θ ∈ S is the cohomology H(Zθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Fθ)(T) with respect to d ¯Z T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The analytic torsion form T � T H ¯ M, gT ¯Z, h ¯F � (T) is a form on S which is given by T � T H ¯ M, gT ¯Z, h ¯F � (T) = − � +∞ 0 � f ∧ � C′ t,T , h ¯E� − χ′( ¯Z, ¯F) 2 − χ( ¯Z, ¯F) dim Z − 2χ′( ¯Z, ¯F) 4 f ′ �i √ t 2 �� dt t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It follows from [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='21] and discussions in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 that T � T HM, gTZ, hF � is well defined for a fixed T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Lastly, for the sake of convenience, (·, ·)L2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ∥ · ∥L2 := � (·, ·)L2) will be adopted to represent (·, ·)L2(Z) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ∥ · ∥L2(Z) := � (·, ·)L2(Z)) , (·, ·)L2( ¯Z) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ∥ · ∥L2( ¯Z) := � (·, ·)L2( ¯Z)) or (·, ·)L2(Zi)(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ∥ · ∥L2(Zi) := � (·, ·)L2(Zi)) (i = 1, 2), when the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 Witten Laplacian v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Weighted Laplacian Instead of deforming the de Rham differential d ¯Z, we could also deform the metric h ¯F : let h ¯F T := e−2pT h ¯F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, g ¯Z and h ¯F T induce an L2-norm h ¯E T = (·, ·)L2( ¯Z),T on ¯E = Ω( ¯Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then the formal adjoint δ ¯Z,∗ T of d ¯Z w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' the (·, ·)L2,T is then given by epT d ¯Z,∗ T e−pT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then ˜D ¯Z T := d ¯Z + δ ¯Z,∗ T , ˜∆T := ( ˜D ¯Z T )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, we have notation ˜d ¯ M T , ˜Ct,T , ˜Dt,T e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 9 The Weighted Laplacian ˜∆T := d ¯Zδ ¯Z,∗ T +δ ¯Z,∗ T d ¯Z, one can see that ˜∆T = epT ∆T e−pT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let lk(T) be the k-th eigenvalue of ˜∆T , then lk(T) = λk(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Moreover, if u is an eigenform of ∆T w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' eigenvalue λ, then epT u is an eigenform of ˜∆T w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' eigen- value λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' As a result, f ∧ � C′ t,T , h ¯E� = f ∧ � ˜C′ t,T , h ¯E T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2 Absolute/Relative boundary conditions for weighted Lapla- cian Let ¯ M1 := M1 ∪ [−1, 0] × M = M− 0 , ¯ M2 := M2 ∪ [0, 1] × N = M+ 0 , and ¯Z1 := Z1 ∪ [−1, 0] × Y = Z− 0 , ¯Z2 := Z2 ∪ [0, 1] × Y = Z+ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let ¯Fi be the restriction of ¯F on ¯ Mi (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Set Ωabs( ¯Z1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F1) := � ω ∈ Ω( ¯Z1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F1) : i ∂ ∂s ω = 0, i ∂ ∂s d ¯Z1ω = 0 on {0} × Y � , Ωrel( ¯Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F2)T := � ω ∈ Ω( ¯Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F2) : ds ∧ ω = 0, ds ∧ d ¯Z2,∗ T ω = 0 on {0} × Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let ˜∆T,i be the restriction of ˜∆T acting on Ωbd( ¯Zi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯Fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then by Hodge theory, ker( ˜∆T,i) ∼= Hbd( ¯Zi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯Fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, we have notation ˜d ¯ M T,i, ˜Ct,T,i, ˜Dt,T,i e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The analytic torsion form Ti � T H ¯ Mi, gT ¯Zi, h ¯Fi T � is a form on S which is given by Ti � T H ¯ Mi, gT ¯Zi, h ¯Fi T � (T) = − � ∞ 0 � f ∧ � ˜C′ t,T ′, h ¯Ei T � − χ′(Zi, Fi) 2 − χ(Zi, Fi) dim Z − 2χ′(Zi, Fi) 4 f ′ �i √ t 2 �� dt t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Lastly, g ¯Zi and h ¯Fi T induce an L2-norm (·, ·)L2( ¯Zi),T on Ωbd( ¯Zi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯Fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For the sake of convenience, (·, ·)L2,T (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ∥ · ∥L2,T := � (·, ·)L2,T ) will be adopted to represent (·, ·)L2( ¯Z),T (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ∥ · ∥L2( ¯Z),T := � (·, ·)L2( ¯Z),T ) or (·, ·)L2( ¯Zi),T (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ∥ · ∥L2( ¯Zi),T := � (·, ·)L2( ¯Zi),T ) (i = 1, 2), when the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4 Analytic torsion form for complex of finite dimen- sional vector bundles Let X be a closed manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let (E, ν) : 0 → E0 ν→ E1 ν→ · · · ν→ En → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' be a flat complex of complex vector bundles on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' That is ∇E = ⊕n i=0∇Ei is a flat connection on E = ⊕n i=0Ei and ν is a flat chain map, meaning by � ∇E�2 = 0, ν2 = 0, ∇Eν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 10 Then ν + ∇E is a flat superconnection of total degree 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By [3, §2(a)], the coho- mology H(E, v) of the complex is a vector bundle on X, and let ∇H(E,v) be the flat connection on H(E, v) induced by ∇E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let d(E) = �n i=0(−1)iirankEi, d(H(E, v)) = �n i=0(−1)iirankHi(E, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let hE = ⊕hEi be a metric on E = ⊕Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let ν∗ and ∇E,∗ be the formal adjoint of ν and ∇E with respect to hE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let N be the number operator on E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' N acts by multiplication by i on Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Set f � ∇E, hE� = �n i=0(−1)if � ∇Ei, hEi� f � ∇H(E,v), hH(E,v)� = �n i=0(−1)if � ∇Hi(E,v), hH(E,v)� For t > 0, let Dt = 1 2 √ t (v∗ − v) + 1 2(∇E,∗ − ∇E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The analytic torsion form for the complex of finite dimensional vector bundles is defined as Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Tf � ν + ∇E, hE� = − � ∞ 0 � ϕ Trs �1 2Nf ′ (Dt) � − 1 2d(H(E, v)) − 1 2 (d(E) − d(H(E, v)) f ′ �i √ t 2 �� dt t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 3 Intermidiate Results In this section, we will state and prove some intermediate results to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For each θ ∈ S, denote D ¯Z T (θ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' DZi(θ) and ˜D ¯Zi T (θ)) to be the restriction of D ¯Z T (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' DZi and ˜D ¯Z T ) on ¯Zθ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Zi,θ and ¯Zi,θ), ∆T(θ) := (D ¯Z T (θ))2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ∆i(θ) := (DZ i (θ))2 and ˜∆T,i(θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Here Zi,θ := π−1 i (θ) (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let λk(T, θ) be the k-th eigenvalue of ∆T (θ), λk(θ) be the k-th eigenvalue of ∆1(θ) ⊕ ∆2(θ) acting on Ωabs(Zθ,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' F1,θ) ⊕ Ωrel(Zθ,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' F2,θ), and ˜λk(T, θ) be the k-th eigenvalue of ˜∆T,1(θ) ⊕ ˜∆T,2(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Moreover, to avoid heavy notation, we will denote ¯Z, ¯F, DZi, ∆T et cetera for ¯Zθ, ¯Fθ, DZi θ , ∆T (θ) et cetera, if there is no need to specify the base point θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' limT→∞ λk(T, θ) = limT→∞ ˜λk(T, θ) = λk(θ) uniformly in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' That is, for example, for every ǫ > 0, there exists Tk(ǫ) > 0 that doesn’t depend on θ, such that whenever T ≥ Tk, |λk(T, θ) − λk(θ)| < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By Hodge theory, there exists k0 ∈ Z, such that whenever k < k0 λk(θ) ≡ 0, λk0(θ) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let δ := 1 2 infθ∈S λk0(θ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1, when T is large enough, all eigenvalues of ∆T inside [0, δ] converge to 0 as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 11 Let Ωsm( ¯Z, ¯F)(T) be the vector bundle over S, such that for all θ ∈ S, Ωsm( ¯Zθ, ¯Fθ)(T) is the space generated by eigenforms of ∆T for eigenvalues inside [0, δ], and Pδ(T) be the orthogonal projection w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Ωsm( ¯Z, ¯F)(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let ∇δ,T := Pδ∇ ¯E and Dδ,T t := ∇δ,T − ∇δ,T,∗ − 1 2 √ t(d ¯Z T − d ¯Z,∗ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For t > 0, put f ∧ la � C′ t,T , hE� := ϕ Trs �N Z 2 f ′ (Dt,T ) � −ϕ Trs �N Z 2 Pδ(T)f ′ � Pδ(T)Dδ,T t Pδ(T) � Pδ(T) � , f ∧ sm � C′ t,T , hE� := ϕ Trs �N Z 2 Pδ(T)f ′ � Pδ(T)Dδ,T t Pδ(T) � Pδ(T) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then proceeding as in the proof of [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='13] (simply replace Pker(V ) λ in [3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='46)] by Pδ(T)(λ − √ t(d ¯Z T − d ¯Z,∗ T ))−1 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ), Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For some θ-independent constant C(T), C′(T) > 0, such that for t ≥ 1 ��f ∧ la � C′ t,T , hE��� ≤ C(T) √ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For t ∈ (0, 1], ����f ∧ la � C′ t,T , hE� − χ( ¯Z, ¯F) dim(Z) − 2d(Ωsm( ¯ M, ¯F)(T)) 4 ���� ≤ C′(T)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Here we put a metric gTS on S, and for α ∈ Ω(S), |α| := � gTS(α, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The key challenge in this article is the possible T-dependence of the constants C(T) and C′(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similar issues can be addressed by introducing a two- parameter deformation and taking the adiabatic limit of analytic torsion forms, as is done in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' However, we use a different approach in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We figure out that when t ∈ [1, ∞), ���f ∧ la � C′ t,T , hE���� could be bounded by a (T, θ)-independent measurable function G(t), such that t−1G(t) is in L1([1, ∞));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' when t ∈ [0, 1], the positive degree component of f ∧ la � C′ t,T , hE� could be bounded by C′t for some (T, θ)-independent C′, while the degree 0 component of f ∧ la � C′ t,T , hE� is related to the analytic torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' To deal with the degree 0 component, rather than the adiabatic limit used in [21], a coupling technique is introduced in [25, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' To- gether with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 and dominated convergence theorem, one can understand Tla � T H ¯ M, gT ¯Z, h ¯F � (T) as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2, we could set T L la � T H ¯ M, gT ¯Z, h ¯F � (T) = − � ∞ 1 � f ∧ la � C′ t,T , h ¯E� − χ( ¯Z, ¯F) dim(Z) − 2d(Ωsm( ¯ M, ¯F)(T)) 4 f ′ �i √ t 2 �� dt t , 12 T S la � T H ¯ M, gT ¯Z, h ¯F � (T) = − � 1 0 � f ∧ la � C′ t,T , h ¯E� − χ( ¯Z, ¯F) dim(Z) − 2d(Ωsm( ¯ M, ¯F)(T)) 4 f ′ �i √ t 2 �� dt t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, let Pi be the be the orthogonal projection from Ω(Zi, Fi) to ker(∆i) i = 1, 2, and ˜Pi(T) be the be the orthogonal projection from Ω( ¯Zi, ¯Fi) to ker( ˜∆T,i) i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let ∇Hi := Pi∇Ei and DHi t := ∇Hi − ∇Hi,∗ + 1 2 √ t(dZi − dZi,∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For t > 0, put f ∧ la � C′ t,i, hE� := ϕ Trs �N Z 2 f ′ (Dt,i) � − ϕ Trs �N Z 2 Pif ′ � PiDHi t Pi � Pi � = ϕ Trs �N Z 2 f ′ (Dt,i) � − χ′(Zi, Fi) 2 (By [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, f ∧ la � ˜C′ t,T,i, hE T � := ϕ Trs �N Z 2 f ′ � ˜Dt,T,i �� − χ′(Zi, Fi) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Set T L la,i � T HMi, gTZi, hFi� = − � ∞ 1 � f ∧ la � C′ t,i, hEi� − χ(Zi, Fi) dim(Z) − 2χ′(Zi, Fi) 4 f ′ �i √ t 2 �� dt t , T S la,i � T HMi, gTZi, hFi� = − � 1 0 � f ∧ la � C′ t,i, hEi� − χ(Zi, Fi) dim(Z) − 2χ′(Zi, Fi) 4 f ′ �i √ t 2 �� dt t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, one has T L la,i � T H ¯ Mi, gT ¯Zi, h ¯Fi T � (T) and T S la,i � T H ¯ Mi, gT ¯Zi, h ¯Fi T � (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' lim T→∞ f ∧ la � C′ t,T , hE� = lim T→∞ 2 � i=1 f ∧ la � ˜C′ t,T,i, h ¯E T � = 2 � i=1 f ∧ la � C′ t,i, hE� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' More precisely, for example, for every ǫ > 0, there exists T0(t, ǫ) > 0, such that whenever T ≥ T0(t, ǫ), ∥f ∧ la � C′ t,T , hE� − 2 � i=1 f ∧ la � C′ t,i, hE� ∥L∞ < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Here we put a metric gTS on S, and for α ∈ Ω(S), ∥α∥L∞ := sup θ∈S |α|(θ), where |α| := � gTS(α, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 13 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' lim T→∞ T L la � T H ¯ M, gT ¯Z, h ¯F � (T) = lim T→∞ 2 � i=1 T L la,i � T H ¯ Mi, gT ¯Zi, h ¯Fi T � (T) = 2 � i=1 T L la,i � T HMi, gTZi, hFi� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The limit is taken w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' the topology described in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' T S la � T H ¯ M, gT ¯Z, h ¯F � (T) = 2 � i=1 T S la,i � T HMi, gTZi, hFi� −(T−log(2))χ(Y )rank(F)/2+o(1), T S la,i � T H ¯ Mi, gT ¯Zi, h ¯Fi T � (T) = T S la,i � T HMi, gTZi, hFi� −T log(2)χ(Y )rank(F)/4+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' As a result, T S la � T H ¯ M, gT ¯Z, h ¯F � (T) = 2 � i=1 T S la,i � T H ¯ Mi, gT ¯Zi, h ¯Fi T � (T)+log(2)χ(Y )rank(F)/2+o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The limit is taken w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' the topology described in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Next, we have the following Mayer-Vietoris exact sequence (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' [5, (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='16)]) of vector bundles over S: MV : · · · ∂k−1 → Hk rel � ¯Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F � ek → Hk � ¯Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F � rk → Hk abs � ¯Z1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F � ∂k → · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (3) Let ˜Ωsm( ¯Z, ¯F)(T) be the space generated by eigenforms of ˜∆T for eigenvalues inside [0, δ], then by our discussion above in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1, ˜Ωsm( ¯Z, ¯F)(T) = epT Ωsm( ¯Z, ¯F)(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' And ˜Pδ(T) := epT Pδ(T)e−pT : L2Ω( ¯Z, ¯F)(T) → ˜Ωsm( ¯Z, ¯F)(T) is the orthogonal projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let H( ¯Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F)(T) := ker( ˜∆T ), and H( ¯Zi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯Fi)(T) := ker( ˜∆T,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We also have the following Mayer-Vietoris exact sequence induced by Hodge theory and (3) MV(T) : · · · ∂k−1,T → Hk � ¯Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F2 � (T) ek,T → Hk � ¯Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F � (T) rk,T → Hk � ¯Z1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F1 � (T) ∂k,T → · · · (4) with metric and flat connections induced by Hodge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let T (T) be the analytic torsion form for this complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For any L2 differential form w on Zi (or ¯Zi), let E(w) be the extension of w, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' outside Zi (or ¯Zi), E(w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We will not distinguish w and E(w) if the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 14 Next, when T is large enough, we have a sequence of morphism of vector bundles 0 → Hk( ¯Z2, ¯F2)(T) ˜ek,T → ˜Ωk sm( ¯Z, ¯F)(T) ˜rk,T → Hk( ¯Z1, ¯F1)(T) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (5) Here ˜ek,T is given by u �→ ˜Pδ(T)E(u) for all u ∈ ker( ˜∆T,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' And ˜rk,T is given by u �→ ˜P1(T)(u| ¯Z1) for all u ∈ ˜Ωk sm( ¯Z, ¯F)(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Recall that ˜Pi(T) is the orthogonal projection w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ker( ˜∆T,i) (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let η ∈ C∞ c ([0, 1]), such that η|[0,1/4] ≡ 0, η|[1/2,1] ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For u = (u1, u2) ∈ ker (∆1) ⊕ ker (∆2), let QT : Ωabs (Z1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' F1) ⊕ Ωrel (Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' F2) → Ω( ¯Z, ¯F) be QT (u)(x) := \uf8f1 \uf8f2 \uf8f3 ui(x), if x ∈ Zi, η(−s)u(−1, y)e−pT (s)−T/2, if x = (s, y) ∈ [−1, 0] × Y, η(s)u(1, y)epT (s)−T/2, if x = (s, y) ∈ [0, 1] × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' And let ˜QT = epT QT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' One can see that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For u ∈ ker (∆1) ⊕ ker (∆2), ∥QT u − u∥2 L2 ≤ C √ T ∥u∥2 L2, ��Pδ(T)QT (u) − u ��2 L2 ≤ C∥u∥2 L2 √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' for some constant C that is independent of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' As a result, ��� ˜QT u − epT u ��� 2 L2,T ≤ C √ T ∥epT u∥2 L2,T, ��� ˜Pδ(T) ˜QT (u) − epT u ��� 2 L2,T ≤ C∥epT u∥2 L2,T √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Recall that ∥ · ∥L2,T is the norm induced by gT ¯Z and h ¯F T := e−2pT h ¯F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' As a result, when T is large enough, ˜Pδ(T) ˜QT (u) spans ˜Ωsm( ¯Z, ¯F)(T) for u ∈ ker (∆1) ⊕ ker (∆2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For u ∈ ker (∆1) ⊕ ker (∆2), set uT = Pδ(T)QT (u), vT = QT (u) − uT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' First, by trace theorem and Garding’s inequality, � Y ��ui � (−1)i, y ���2 dvolY ≤ C � Zi |ui|2 + |∇ui|2 dvol ≤ C′ � Zi |ui|2 dvol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (6) for some constants C, C′ that doesn’t depends on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By (6) and a straightforward computation, one can see that ∥QT u − u∥2 L2 ≤ C √ T ∥u∥2 L2 (7) and ���D ¯Z T QT u ��� 2 L2 ≤ C √ T ∥u∥2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (8) 15 Moreover, δ ∥vT ∥2 L2 ≤ ���D ¯Z T vT ��� 2 L2 ≤ ���D ¯Z T QT u ��� 2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (9) (8) and (9) then imply that ∥vT ∥2 L2 ≤ C δ √ T ∥u∥L2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=', ���Pδ(T)QT (u) − QT (u) ��� 2 L2 ≤ C∥u∥2 L2 δ √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (10) The proposition then follows form (7) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Notice that if u ∈ Ωbd(Zi, Fi), then QT u ∈ Ωbd( ¯Zi, ¯Fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By Hodge theory and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1, when T is big enough, all eigenvalues of ˜∆T,i inside [0, δ] must be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let ˜Pi(T) be the orthogonal projection from L2Ω( ¯Zi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯Fi)(T) to ker( ˜∆T,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, one has Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For u ∈ ker (∆i), ��� ˜QT u − epT u ��� 2 L2,T ≤ C √ T ∥epT u∥2 L2,T, ��� ˜Pi(T) ˜QT (u) − epT u ��� 2 L2,T ≤ C∥epT u∥2 L2,T √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' As a result, when T is large enough, ˜Pi(T) ˜QT (u) spans H( ¯Zi, ¯Fi)(T) for u ∈ ker (∆i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ˜ek,T and ˜r∗ k,T are almost isometric embeddings as T → ∞, where ˜r∗ k,T is the adjoint of ˜rk,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' That is, for example, for any u ∈ Hk( ¯Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F2)(T), limT→∞ ∥˜ek,T u∥L2,T ∥uT ∥L2,T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ˜ek,T is almost isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For any u ∈ Hk( ¯Z1, ¯F1)(T), there exists uT ∈ ker(∆1) ∩ Ωk rel(Z1, F1) such that u = ˜Pi(T) ˜QT (uT ), then by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='8, ∥u∥2 L2,T ≥ (1 − C √ T )∥epT uT ∥2 L2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (11) While ∥˜ek,T u∥2 L2,T = ∥ ˜Pδ(T)u∥2 L2,T ≤ ∥ ˜Pδ(T)QT uT ∥2 L2,T + C∥epT uT ∥2 L2 √ T (By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='8 and the fact that ∥ ˜Pδ(T)∥ = 1) ≤ ∥epT uT ∥2 L2,T (1 + C′ √ T ) (By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (12) 16 It follows from (11) and (12) that lim sup T→∞ ∥˜ek,T u∥2 L2,T ∥u∥L2,T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, lim inf T→∞ ∥˜ek,Tu∥2 L2,T ∥u∥L2,T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ˜r∗ k,T is almost isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For u ∈ Hk( ¯Z2, ¯F2)(T), we first show that ˜r∗ k,Tu = ˜Pδ(T)E(u) : Notice that for any v ∈ ˜Ωsm( ¯Z, ¯F)(T), (˜rk,Tv, u)L2( ¯Z2),T = (v, E(u))L2( ¯Z),T = (v, ˜Pδ(T)E(u))L2( ¯Z),T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Following the same steps as above, one can show that ˜r∗ k,T is almost isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' With maps ˜ek,T and ˜rk,T given above, the sequence (5) is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let ⟨·, ·⟩T be the pointwise inner product on each fiber that is induced by gT ¯Z and h ¯F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' In follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='9 that ˜ek,T and ˜r∗ k,T are injective when T is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' E(ker( ˜∆T,1)) ⊂ ker(δ ¯Z,∗ T ) and E(ker( ˜∆T,2)) ⊂ ker(d ¯Z): Let u ∈ ker( ˜∆T,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' First, since u satisfies relative boundary conditions, integra- tion by parts shows that for any β ∈ Ω( ¯Z, ¯F), � ¯Z⟨E(u), δ ¯Z,∗ T β⟩T dvol = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Thus, E(u) ∈ ker(d ¯Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, E(ker( ˜∆T,1)) ⊂ ker(δ ¯Z,∗ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' im ˜ek,T = ker ˜rk,T : For the dimension reason, it suffices to show that im˜ek,T ⊂ ker ˜rk,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' That is, it suffices to show im˜ek,T ⊥ im˜r∗ k,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' First, for any ui ∈ ker( ˜∆T,i), i = 1, 2, it’s clear that (E(u1), E(u2))L2,T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (13) Since E(u1) ∈ ker(δ ¯Z,∗ T ), E(u2) ∈ ker(d ¯Z), one can see that (1 − ˜Pδ(T))u1 ∈ im δ ¯Z,∗ T , (1 − ˜Pδ(T))u1 ∈ im d ¯Z, which means ((1 − ˜Pδ(T))E(u1), (1 − ˜Pδ(T))E(u2))L2,T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (14) By (13) and (14), (˜ek,T u2, ˜r∗ k,Tu1)L2,T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 17 Moreover, we have the following complexes of finite dimensional vector bundles 0 → H0( ¯Zi, ¯Fi)(T) 0→ H1( ¯Zi, ¯Fi)(T) 0→ · · · 0→ Hdim Z( ¯Zi, ¯Fi)(T) → 0 (15) 0 → ˜Ω0 sm( ¯Z, ¯F)(T) d ¯ Z → ˜Ω1 sm( ¯Z, ¯F)(T) d ¯ Z → · · · d ¯ Z → ˜Ωdim Z sm ( ¯Z, ¯F)(T) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (16) Integration by parts as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='10, one can show easily that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' d ¯Z ◦ ˜ek,T = 0, ˜rk,T ◦ d ¯Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Hence, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='10 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='11, we get the following long exact sequence again MV(T) : · · · ∂k−1,T → Hk � ¯Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F2 � (T) ek,T → Hk � ¯Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F � (T) rk,T → Hk � ¯Z1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F1 � (T) ∂k,T → · · · (17) with metric and connection induced by Hodge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let Tsm(T H ¯ M, gT ¯Z, h ¯F )(T) := − � ∞ 0 � f ∧ sm(C′ t,T , h ¯E) − χ′(Z, F) 2 � + χ′(Z, F) − d(Ωsm( ¯ M, ¯F)(T)) 2 f ′(i √ t 2 )dt t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The following Theorem will be proved in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' limT→∞ Tsm(T H ¯ M, gT ¯Z, h ¯F )(T) − T (T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It follows from anomaly formulas (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='24 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='24] and [27, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5]) that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' In QS/QS 0 , log T (T H ¯ M, gT ¯Z, h ¯F )(T) − 2 � i=1 log Ti(T H ¯ Mi, gT ¯Zi, h ¯Fi T )(T) − log T (T) = log T (T HM, gTZ, hF ) − 2 � i=1 log Ti(T HMi, gTZi, hFi) − log T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='6 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='12 that log T (T H ¯ M, gT ¯Z, h ¯F )(T) − 2 � i=1 log Ti(T H ¯ Mi, gT ¯Zi, h ¯Fi T )(T) − log T (T) = log(2)χ(Y )rank(F)/2 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Thus, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='13, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 18 4 Convergence of Eigenvalues First, it’s straightforward to check that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let F → X be a flat complex vector bundle over a compact smooth manifold X, f be a smooth function on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let hF l , gTX l , ∇F l be smooth families of metrics and connections over F → X, l ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let dl : Ω∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' F) → Ω∗+1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' F) be the covariant derivative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ∇F l , and d∗ l be the adjoint of dF l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let df,l := dF l +df∧, and d∗ f,l be the adjoint of df,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then for any u ∈ Ω, ǫ > 0, there exists δ > 0 that doesn’t depend on u and f, such that whenever |l1 − l2| < δ, one has � X |df,l1u|2 l1 + |d∗ f,l1u|2 l1 � X |u|2 l1 ≤ (1 + ǫ) � X |df,l2u|2 l2 + |d∗ f,l2u|2 l2 � X |u|2 l2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Here | · |l is the metric on Λ∗(TZ) ⊗ F induced by hF l and gTX l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' First, one observes that λk(T, θ) has uniform upper bounds: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Fix k ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' There exists an increasing sequence {Λk}∞ k=1 of constants, such that λk(T, θ) ≤ Λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For a fixed θ ∈ S, it follows from [25, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1] that there exists {Λk(θ)}, such that λk(T, θ) ≤ Λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 that when θ′ is close to θ, λk(T, θ′) ≤ 2λT,θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The existence of Λk then follows from the compactness of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' λk(T, θ) is a family of equicontinuous function on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' That is, for any θ ∈ S, ǫ > 0, there exists a neighborhood U of θ that doesn’t depends on T, whenever θ′ ∈ S, |λk(T, θ) − λk(T, θ′)| < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2, when θ′ is closed to θ, |λk(T, θ)−λk(T, θ′)| ≤ ǫλk(T, θ) ≤ ǫΛk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Next, it follows from [25, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2] and compactness of S that, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let u ∈ Ω( ¯Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ¯F), such that � ¯Z |u|2 = 1 and � ¯Z |D ¯Z T u|2 ≤ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then for s ∈ [−2, −1 + � 2 T ] ∪ [1 − � 2 T , 2] � Y |u|2(s, y) ≤ C(1 + λ) if T is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Here the constant C is independent of T and θ, | · | is the metric on Λ∗( ¯Z) ⊗ ¯F| ¯Z induced by h ¯F and gT ¯Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Fix θ ∈ S, [25, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1] implies that limT→∞ λk(T, θ) = λk(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Since S is compact, the uniformness follows from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3 and continuity of λk(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, one can show limT→∞ ˜λk(T, θ) = λk(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 19 Recall that for u ∈ Ωabs(Z1, F1) ⊕ Ωrel(Z2, F2), QT (u)(x) := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ui(x), if x ∈ Zi, η(−s)u(−1, y)e−pT (s)−T/2, if x = (s, y) ∈ [−1, 0] × Y , η(s)u(1, y)epT (s)−T/2, if x = (s, y) ∈ [0, 1] × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4 and the construction of QT that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let u ∈ Ωbd(Zi, Fi) such that ∥dZ + dZ,∗u∥L2 ≤ λ∥u∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then ∥QT (u) − E(u)∥2 L2 ≤ C(1 + λ) √ T ∥u∥2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Here the constant C is independent of T and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It follows from [25, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2] and the compactness of S that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' There exists constants c1, c2, c3, c4 and c5 independent of T and θ, such that λk(T, θ) ≥ uk(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Here {uk(T)}∞ k=1 is the collection of 4 copies of {vl(T)+ c4m2/(dim M−1)}∞ l=1,m=1 and 2 copy of {c5l2/ dim M}, listed in the increasing order and counted with multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Moreover, {vk(T)}∞ k=1 is the collection of {T max{c1l − c2, 0}}∞ l=1 and {c3l2}∞ l=1, listed in the increasing order and counted with multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 Convergence of f ∧ la � C′ t,T, h ¯E� and the large time con- tributions Let VT = d ¯Z T − d ¯Z,∗ T and ¯Ft := Dt,T − √ t 2 (d ¯Z T − d ¯Z,∗ T ), then all eigenvalues of VT are pure imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Moreover, ¯Ft is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, one has Ft,i and Ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Moreover, ¯Ft|Zi = Ft,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (18) We define ¯F j t inductively: set ¯F 0 t = ¯Ft, and ¯F j+1 t = [VT , ¯F j t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Since T H ¯ M, gT ¯Z and h ¯F are product-type, by a straightforward computation, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' There exists (T, θ)-independent Cj > 0, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ∥ ¯F j t ∥ ≤ Cj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It’s trivial that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' If τ is pure imagenary and |Re(λ)| = 1, then |λ − τ|−1 ≤ C |λ| |τ| or |λ − τ|−1 ≤ 1 for some universal constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 20 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let u be a normal eigenform w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' an eigenvalue µ for VT (Moreover, assume that |µ| ≥ 1), then for any j ∈ Z+, t > 0, there exits (T, θ)-independent Cj > 0, such that ��� �� f ′(Dt,T ) − f ′( √ tVT ) � u, u � L2 ��� ≤ Cj √ t|µ|j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let γ be the oriented contour given by {z ∈ C : |Re(z)| = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proceeding as in the proof of [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='13], f ′ (Dt,T ) − f ′ �√ tVT � = dim S � l=1 � γ f ′(λ) � (λ − √ tVT )−1 ¯Ft �l (λ − √ tVT )−1dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (19) First, notice that ����� �� γ f ′(λ)(λ − √ tVT )−1 ¯Ft(λ − √ tVT )−1udλ, u � L2 ����� = ����� �� γ f ′(λ) ¯Ft(λ − √ tVT )−1udλ, (¯λ + √ tVT )−1u � L2 ����� = ����� �� γ f ′(λ)(λ − √ tµ)−2 ¯Ftudλ, u � L2 ����� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (20) Recall that ¯F 0 t := ¯Ft, and ¯F j+1 t := [VT , ¯F j t ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Through a simple calculation, [ ¯F j t , (λ − √ tVT )−1] = (λ − √ tVT )−1√ t ¯F j+1 t (λ − √ tVT )−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (21) Consequently, by (21), Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='8 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='7, if λ ∈ γ, �� (λ − √ tVT )−1 ¯Ft �2 (λ − √ tVT )−1u, u � = �j−1 � k=1 (λ − √ tVT )−(k+1)√ t k ¯F k−1 t ¯Ft(λ − √ tVT )−1 +(λ − √ tVT )−j√ t j ¯F j−1 t ((λ − √ tVT )−1) ¯Ft(λ − √ tVT )−1u, u � ≤ �j−1 � k=1 (λ − √ tVT )−(k+1)√ t k ¯F k−1 t ¯Ft(λ − √ tVT )−1u, u � + C ���(λ − √ tµ)−(j+1)√ t j��� ≤ �j−1 � k=1 (λ − √ tVT )−(k+1)√ t k ¯F k−1 t ¯Ft(λ − √ tVT )−1u, u � + C|λ|j+1|µ|−(j+1)√ t −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (22) 21 similarly, �� (λ − √ tVT )−1 ¯Ft �2 (λ − √ tVT )−1u, u � ≥ �j−1 � k=1 (λ − √ tVT )−(k+1)√ t k ¯F k−1 t ¯Ft(λ − √ tVT )−1u, u � − C|λ|j+1|µ|−(j+1)√ t −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (23) By (22) and (23), proceeding as in (19), one can see that ����� �� γ f ′(λ) � (λ − √ tVT )−1 ¯Ft �2 (λ − √ tVT )−1udλ, u � L2 ����� ≤ Cj √ t|µ|j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (24) Similarly, one can show ����� �� γ f ′(λ) � (λ − √ tVT )−1 ¯Ft �l (λ − √ tVT )−1udλ, u � L2 ����� ≤ Cj,l √ t|µ|j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (25) We also have Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let u be an eigenform of ∆i w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' eigenvalue µ, then for |Re(λ)| = 1, ∥(λ − Dt,T )−1QT u − E((λ − Dt,i)−1u)∥2 L2 ≤ C(µ2 + µ + 1)(1 + λ)∥u∥2 L2 √ T , and ∥(λ − Dt,T )−1E(u) − E((λ − Dt,i)−1u)∥2 L2 ≤ C(µ2 + µ + 1)(1 + λ)∥u∥2 L2 √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Integration by parts as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='10 shows that for u ∈ Ωbd(Zi, Fi), (i = 1, 2) Dt,T QT u|Zi = Dt,iu, (26) and by the construction of QT ∥Dt,T QT u − E(Dt,iu)∥2 L2 ≤ C(1 + µ2)∥u∥2 √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (27) Notice that if |Re(λ)| = 1, ∥(λ − Dt,i)−1∥ ≤ 1, by (26) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5, ∥(λ − Dt,T )QT (λ − Dt,i)−1u − QT(u)∥L2 ≤ C(1 + λ)(1 + µ2)∥u∥2 √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (28) 22 Since we also have ∥(λ − Dt,T )−1∥ ≤ 1 for |Re(λ)| = 1, (28) implies that ∥QT (λ − Dt,i)−1u − (λ − Dt,T )−1QT (u)∥L2 ≤ C(1 + λ)(1 + µ2)∥u∥2 √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (29) By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5 and (29), the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let {uk}∞ k=1 be normal eigenforms for ∆T such that {uk} forms an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Fix ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='9, there exists k0(ǫ, t) > 0, such that � k≥k0 ��� �� f ′(Dt,T ) − f ′( √ tVT ) � uk, uk � L2 ��� < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (30) By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='6, we may assume that � k≥k0 ��� � f ′( √ tVT )uk, uk � L2 ��� ≤ C � k≥k0 (1 + λ2 k(T, θ))e−tλk(T,θ) < ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (31) By (30) and (31), � k≥k0 ��� f ′(Dt,T )uk, uk � L2 �� < 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (32) Let {vk}∞ k=1 be normal eigenforms for ∆1⊕∆2 such that {vk} forms an orthonor- mal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, one may assume that � k≥k0 2 � i=1 ��� f ′(Dt,i)vk, vk � L2 �� < 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (33) Let {vk}k0 k=1 be orthonormal eigenforms with respect to eigenvalues {λk}k0 k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let Ek0( ¯Z, ¯F) be the space generated by eigenforms with respect to eigenvalues λ1(T, θ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='λk0(T, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Set Pk0(T) be the orthogonal projection from L2Ω( ¯Z, ¯F) to Ek0( ¯Z, ¯F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proceeding as in the proof of Propositon 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='8, one can see that Ek0( ¯Z, ¯F) is generated by {Pk0(T)QT vk}k0 k=1 if T is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Moreover, ∥Pk0(T)QT vk − QT vk∥2 L2 ≤ C(λk0(T, θ) + 1) √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (34) ∥vk − QT vk∥2 L2 ≤ C(λk0(T, θ) + 1) √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (35) Let {uk(T)} be the Gram-Schmidt Orthogonalization of {Pk0(T)QT uk}k0 k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5 implies that ∥uk(T) − uk∥2 L2 ≤ C(λk0(T, θ) + 1) √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (36) 23 ∥Pk0(T)QT uk − uk(T)∥2 L2 ≤ C(λk0(T, θ) + 1) √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (37) Procceding as in the proof of [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='13], one can show that there exists (T, θ)-independent C > 0, such that ���ϕf ′ (Dt,T ) − Pδ(T)ϕf ′ � Pδ(T)Dδ,T t Pδ(T) � Pδ(T) ��� ≤ C √ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (38) (Comparing with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2, we are looking at operator norm, instead of trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=') Let f ′ la(Dt,T ) := ϕN ¯ Z 2 � f ′ (Dt,T ) − Pδ(T)f ′� Pδ(T)Dδ,T t Pδ(T) � Pδ(T) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Hence, by (34), (35), (36), (37), (38) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='10, there exists T(ǫ, t) > 0, such that when T > T(ǫ, t) ∥ k0 � k=1 � f ′ la(Dt,T )uk(T), uk(T) � L2 − A(t)∥L∞ ≤ ∥ k0 � k=1 � f ′ la(Dt,T )QT vk, QT vk � L2 − A(t)∥L∞ + ǫ ≤ 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (39) Here for simplicity, set A(t) := k0 � k=1 2 � i=1 � ϕN 2 � f ′ (Dt,i) − Pif ′(PiDHi t Pi)Pi � vk, vk � L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By (32), (33) and (39), lim T→∞ f ∧ la � C′ t,T , h ¯E� = 2 � i=1 f ∧ la � C′ t,i, hE� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, one can show lim T→∞ 2 � i=1 f ∧ la � ˜C′ t,T,i, h ¯E T � = 2 � i=1 f ∧ la � C′ t,i, hE� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='6, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='9 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1, proceeding as in the proof of [25, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2], one can see that there exists a measurable function G(t) on [1, ∞), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' G(t)/t is L1([1, ∞)-integrable (G is independent of T and θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Moreover, |f ∧ la � C′ t,T , h ¯E� | ≤ G(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 24 By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4 and the dominate convergence theorem, lim T→∞ T L la � T H ¯ M, gT ¯Z, h ¯F � (T) = 2 � i=1 T L la,i � T HMi, gTZi, hFi� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, lim T→∞ 2 � i=1 T L la,i � T H ¯ Mi, gT ¯Zi, h ¯Fi T � (T) = 2 � i=1 T L la,i � T HMi, gTZi, hFi� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 5 The Small Time Contributions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 Several Hodge Laplacians To show the gluing formula for f ∧, we introduce several Hodge Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let ∆R B,1 be the Hodge Laplacian on [−2, −1] with absolute boundary condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It’s easy to see that ker(∆R B,1) is one-dimensional and generated by constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Thus, Trs((1 + 2∆B,1)e−t∆R B,1) = lim t→∞ Trs((1 + 2∆B,1)e−t∆R B,1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (40) Let ∆R B,2 be the Hodge Laplacian on [1, 2] with relative boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, Trs((1 + 2∆B,2)e−t∆R B,2) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (41) Let ¯∆B be the Hodge laplacian on Ω([−2, 2]) satisfying the absolute boundary condition on −2, and relative boundary condition on 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We can also regards pT as a smooth function in (−2, 2), and let ∆R T be the Witten Laplacian on (−2, 2) with respect to pT , with absolute boundary condition on −2, and relative boundary condition on 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2 Gluing formulas for f ∧ � C′ t,i, hE� and f ∧ � C′ t,T, h ¯E� Let Dt,Y := Dt|Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let ηi(i = 1, 2) be a smooth function on (−∞, ∞) satisfying 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 0 ≤ ηi ≤ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' η1 ≡ 1 in (−∞, −3/2),η1 ≡ 0 in (−5/4, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' η2 ≡ 1 in (3/2, ∞),η2 ≡ 0 in (−∞, 5/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 25 We can think ηi as a function on Zi(i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let ˜f(a) = (1+2a)ea, then ˜f(a2) = f ′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proceeding as in [25, §6] or [2, §13(b)], since T HM, gTZ and hF are porduct-type near N, for some C, c > 0, ��� 2 � i=1 ϕ Trs � N Zf ′ (Dt,i) � − 2 � i=1 ϕ Trs � N Zηif ′ (Dt,i) � − 2 � i=1 ϕ Trs � N Zf ′ (Dt,Y ) ⊗ ˜f(t∆R B,i) � + 2 � i=1 ϕ Trs � N Zηif ′ (Dt,Y ) ⊗ ˜f(t ¯∆B) � ��� L∞ ≤ C exp(−c/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (42) Next, notice that [−2, 2] × Y , the number operator can be decomposed as N Z = N Y + N R canonically (Here N Y and N R are the number operator on Y and R components respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By (40), (41) and [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='15], 2 � i=1 ϕ Trs � N Zf ′ (Dt,Y ) ⊗ ˜f(t∆R B,i) � = 2 � i=1 ϕ Trs � N Y f ′ (Dt,Y ) ⊗ ˜f(t∆R B,i) � + 2 � i=1 ϕ Trs � f ′ (Dt,Y ) ⊗ N R ˜f(t∆R B,i) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' = 2 � i=1 χ(Y )rank(F) Trs(N R ˜f(t∆R B,i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (43) Similarly, for some (T, θ)-independent C, c > 0, ���ϕ Trs � N ¯Zf ′ (Dt,T ) � − 2 � i=1 ϕ Trs � N Zηif ′ (Dt,i) � − ϕ Trs � N Zf ′ (Dt,Y ) ⊗ ˜f(t∆R T ) � + 2 � i=1 ϕ Trs � N Zηif ′ (Dt,Y ) ⊗ ˜f(t ¯∆B) � ��� L∞ ≤ C exp(−c/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (44) Moreover, Trs(e−t∆R T ) = limt→∞ Trs(e−t∆R T ) = dim(ker(∆R T )0) − dim(ker(∆R T )1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Here ker(∆R T )i denotes the space of harmonic i-forms(i = 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Since pT is odd, one can see easily that if u(s) ∈ ker(∆R T )0, then u(−s)ds ∈ ker(∆R T )1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' As a result, Trs((1 + 2∆T )e−t∆R T ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proceeding as before, ϕ Trs � N Zf ′ (Dt,Y ) ⊗ ˜f(t∆R T ) � = χ(Y )rank(F) Trs(N R ˜f(t∆R T )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (45) 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='6 For a differential form w, let w0 denote its degree 0 component, and w+ := w − w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proceeding as in [3, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='18], for some (T, θ)-independent C, ����ϕ Trs �N Z 2 Pδ(T)f ′ � Pδ(T)Dδ,T t Pδ(T) � Pδ(T) � − 2 � i=1 ϕ Trs �N Z 2 Pif ′ � PiDHi t Pi � Pi ������ L∞ ≤ Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (46) It follows from (42), (43), (44), (45), (46), Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4 and dominated convergence theorem that � T S la � T H ¯ M, gT ¯Z, h ¯F � (T) �+ = 2 � i=1 � T S la,i � T HMi, gTZi, hFi��+ + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It follows from [25, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3] that � T S la � T H ¯ M, gT ¯Z, h ¯F � (T) �0 = 2 � i=1 � T S la,i � T HMi, gTZi, hFi��0 − (T − log(2))χ(Y )rank(F)/2 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, one can show that � T S la,i � T H ¯ Mi, gT ¯Zi, h ¯Fi T � (T) �+ = � T S la,i � T HMi, gTZi, hFi��+ + o(1), and � T S la � T H ¯ M, gT ¯Z, h ¯F � (T) �0 = 2 � i=1 � T S la,i � T HMi, gTZi, hFi��0 − Tχ(Y )rank(F)/4 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 6 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='12 From now on, we assume that T > 0 is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 Some Estimate of harmonic forms on the tube Let w ∈ H( ¯ Mi, ¯Fi)(T) such that ∥w∥L2,T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Notice that ˜∆T = epT ∆T e−pT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By Agmon estimate (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' [7, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1]), 27 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' On [−1/2, 0] × Y or [0, 1/2] × Y, |e−pT w| ≤ Ce−cT for some constant C > 0, c ∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ∥ ˜Pδ(T)E(w) − E(w)∥L2,T ≤ Ce−cT for some C > 0, c ∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' We may as well assume that w ∈ H( ¯Z1, ¯F1)(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let η ∈ C∞ c (R), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' η(s) = 1 if s ∈ (−∞, −1/8) and η|[−1/16,0] ≡ 0, we can treat η as a smooth function on ¯Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1, ∥w − ηw∥L2,T ≤ Ce−cT (47) and | ˜DT ηw|L2,T = |η′w|L2,T ≤ C′e−cT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (48) Proceeding as in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='7, the proposition follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proceeding as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4, one has Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' When |s − (−1)i| ≤ 2 √ T , � Y |e−pT w|2(s, y)dvolY ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For w ∈ H( ¯Z1, ¯F1)(T), | � −1/16 −1 � Y (s + 1)2|e−pT w|2dvolY ds| ≤ C T 3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For w ∈ H( ¯ M2, ¯F2)(T), | � 1 1/16 � Y (s − 1)2|e−pT w|2dvolY ds| ≤ C T 3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' WLOG, assume w ∈ H( ¯Z1, ¯F1)(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Since ∆T e−pT w = 0, one can see that � ¯Z1 |(d + d∗)e−pT w|2 + p′′ T |e−pT w|2 + |p′ T |2|e−pT w|2dvol = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (49) By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3 and (49), | � −1/16 −1+ 2 √ T � Y T 2(s + 1)2|e−pT w|2dvolY ds| ≤ C � −1+ 2 √ T −1 T|e−pT w|2dvol ≤ C √ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' That is, | � −1/16 −1+ 2 √ T � Y (s + 1)2|e−pT w|2dvolY ds| ≤ C T 3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (50) It follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3 that | � −1+ 2 √ T −1 � Y (s + 1)2|e−pT w|2dvolY ds| ≤ C T 3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (51) The lemma then follows from (50) and (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 28 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2 Estimate of small eigenvalues Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' When T is big enough, ∥ ∂ ∂T Pδ(T)∥ ≤ C for some (T, θ)-independent C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Moreover, there exists a uniformly bounded operator UT , such that ∂ ∂T Pδ(T) = [ ¯D ¯Z T , UT ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (52) Here ¯D ¯Z T := d ¯Z T −d ¯Z,∗ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similar statements hold if we replace Pδ(T) by ˜Pδ(T), ˜Pi(T) e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let γ be the circle of radius � 3δ/2 with center 0, and oriented positively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then Pδ(T) = � γ (λ − D ¯Z T )−1dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' As a result, ∂ ∂T Pδ(T) = � γ (λ − D ¯Z T )−1 ∂ ∂T D ¯Z T (λ − D ¯Z T )−1dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Now, one can check easily that when T is large enough, ∥(λ−D ¯Z T )−1∥ ≤ � ( � 3/2 − 1)δ �−1 , and | ∂ ∂T ∂ ∂spT(s)| ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Thus, ∥ ∂ ∂T Pδ(T)∥ ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Notice that ∂ ∂T D ¯Z T = [ ¯D ¯Z T , ∂ ∂T pT ], | ∂ ∂T pT | ≤ C and ¯D ¯Z T commutes with D ¯Z T , one has (52) for UT = � γ (λ − D ¯Z T )−1 ∂ ∂T pT(λ − D ¯Z T )−1dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' When T is large enough, ˜Ωk sm( ¯Z, ¯F)(T) = ˜ek,TH( ¯Z2, ¯F2)(T) ⊕ ˜r∗ k,TH( ¯Z1, ¯F1)(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (53) Let ˜e−1 k,T be the inverse of ˜ek,T |˜ek,T H( ¯Z1, ¯F1)(T), and ˜r−1 k,T be the inverse of ˜rk,T |˜r∗ k,T H( ¯Z2, ¯F2)(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Next, we will put a family of metric gT on Habs( ¯Z1, ¯F1) ⊕ Hrel( ¯Z2, ¯F2) when T is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' First, we define a map RT : Hk abs( ¯Z1, ¯F1)⊕Hk rel( ¯Z2, ¯F2) → ˜Ωsm( ¯ M, ¯F) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For [u] ∈ Hk abs( ¯Z1, ¯F1) represented by u ∈ Ωk abs( ¯Z1, ¯F1), set RT ([u]) := ˜ek,T ˜P1(u) = ˜Pδ(T)E( ˜P1(T)u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For [v] ∈ Hk rel( ¯Z2, ¯F2)0 represented by v ∈ Ωk rel( ¯Z2, ¯F2), set RT ([v]) := ˜r−1 k,T ˜P2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then set gT (w, w) := (RT w, RT w)L2,T for w ∈ Hk abs( ¯Z1, ¯F1) ⊕ Hk rel( ¯Z2, ¯F2)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' There exists (T, θ)-independent C > 0, such that 1− C T 3/2 ≤ ∥g−1 T ∂ ∂T gT ∥ ≤ 1 + C T 3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Here the operator norm is taken with respect to gT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' In the following, some ˜P2(T)u should be understood as E( ˜P2(T)u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' First, it’s clear that for a family of projection operators P(T), P(T) ∂ ∂T P(T)P(T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (54) For any [u], [v] ∈ Hk rel( ¯Z2, ¯F2), ∂ ∂T gT ([u], [v]) = ∂ ∂T (RT [u], RT [v])L2,T = ∂ ∂T ( ˜Pδ(T) ˜P2(T)u, ˜Pδ(T) ˜P2(T)v)L2,T = ( ∂ ∂T ˜Pδ(T) ˜P2(T)u, ˜Pδ(T) ˜P2(T)v)L2,T + ( ˜Pδ(T) ˜P2(T)u, ∂ ∂T ˜Pδ(T) ˜P2(T)v)L2,T + ( ˜Pδ(T) ∂ ∂T ˜P2(T)u, ˜Pδ(T) ˜P2(T)v)L2,T + ( ˜Pδ(T) ˜P2(T)u, ˜Pδ(T) ∂ ∂T ˜P2(T)v)L2,T − 2( ∂ ∂T pT ˜Pδ(T) ˜P2(T)u, ˜Pδ(T) ˜P2(T)v)L2,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (55) Let η ∈ C∞ c (R), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' η(s) = 1 if s ∈ (1/8, ∞) and η|[0,1/16] ≡ 0, we can treat η as a smooth function on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2, (47), (48) and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5, ( ∂ ∂T ˜Pδ(T) ˜P2(T)u, ˜Pδ(T) ˜P2(T)v)L2,T ≤ e−cT ∥ ˜P2(T)u∥L2,T∥ ˜P2(T)v∥L2,T + ( ∂ ∂T ˜Pδ(T)η ˜P2(T)u, η ˜P2(T)v)L2,T ≤ e−cT ∥ ˜P2(T)u∥L2,T∥ ˜P2(T)v∥L2,T + ([ ¯D ¯Z T , UT ]η ˜P2(T)u, η ˜P2(T)v)L2,T ≤ e−cT ∥ ˜P2(T)u∥L2,T∥ ˜P2(T)v∥L2,T + (UT ¯D ¯Z T η ˜P2(T)u, η ˜P2(T)v)L2,T + (η ˜P2(T)u, UT ¯D ¯Z T η ˜P2(T)v)L2,T ≤ Ce−cT ∥ ˜P2(T)u∥L2,T ∥ ˜P2(T)v∥L2,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (56) By Hodge theory, one can see that if T ′ ≥ T, then ˜P2(T ′) ˜P2(T) = ˜P2(T ′), hence ∂ ∂T ˜P2(T) = ∂ ∂T ˜P2(T) ˜P2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (57) By (54), (57), Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5, ( ˜Pδ(T) ∂ ∂T ˜P2(T)u, ˜Pδ(T) ˜P2(T)v)L2,T ≤ e−cT∥ ˜P2(T)u∥L2,T ∥ ˜P2(T)v∥L2,T + ( ∂ ∂T ˜P2(T) ˜P2(T)u, ˜P2(T)v)L2,T = e−cT∥ ˜P2(T)u∥L2,T ∥ ˜P2(T)v∥L2,T + ( ˜P2(T) ∂ ∂T ˜P2(T) ˜P2(T)u, v)L2,T = e−cT∥ ˜P2(T)u∥L2,T ∥ ˜P2(T)v∥L2,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (58) By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='4 30 2( ∂ ∂T pT ˜Pδ(T) ˜P2(T)u, ˜Pδ(T) ˜P2(T)v)L2,T ≤ e−cT ∥ ˜P2(T)u∥L2,T ∥ ˜P2(T)v∥L2,T + 2( ∂ ∂T pT ˜P2(T)u, ˜P2(T)v)L2,T ≤ C T 3/2 ∥ ˜P2(T)u∥L2,T ∥ ˜P2(T)v∥L2,T − ( ˜P2(T)u, ˜P2(T)v)L2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (59) It follows (55), (56), (58), (59) that for any [u], [v] ∈ Hk rel( ¯Z2, ¯F2), ∂ ∂T gT ([u], [v]) ≤ C T 3/2 ∥ ˜P2(T)u∥L2,T ∥ ˜P2(T)v∥L2,T + ( ˜P2(T)u, ˜P2(T)v)L2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (60) Similarly, for any [u], [v] ∈ Hk rel( ¯Z2, ¯F2), ∂ ∂T gT ([u], [v]) ≥ − C T 3/2 ∥ ˜P2(T)u∥L2,T∥ ˜P2(T)v∥L2,T + ( ˜P2(T)u, ˜P2(T)v)L2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (61) By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2, proceeding as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='9, one can see that 1 − Ce−cT ≤ ∥˜r∗ k,T∥ ≤ 1 + Ce−cT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, for any [u], [v] ∈ Hk abs( ¯Z1, ¯F1), ∂ ∂T gT ([u], [v]) ≤ C T 3/2 ∥ ˜P1(T)u∥L2,T ∥ ˜P1(T)v∥L2,T − ( ˜P1(T)u, ˜P1(T)v)L2,T , (62) ∂ ∂T gT ([u], [v]) ≥ − C T 3/2 ∥ ˜P1(T)u∥L2,T∥ ˜P1(T)v∥L2,T − ( ˜P1(T)u, ˜P1(T)v)L2,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (63) for any [u] ∈ Hk rel( ¯Z1, ¯F1), [v] ∈ Hk abs( ¯Z2, ¯F2), | ∂ ∂T gT ([u], [v])| ≤ C T 3/2 ∥ ˜P1(T)u∥L2,T ∥ ˜P2(T)v∥L2,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (64) Lastly, by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1, for any [u], [v] ∈ Hk rel( ¯Z1, ¯F1) ⊕ Hk abs( ¯Z2, ¯F2), |gT ([u], [v]) − ( ˜Pi(T)u, ˜Pi(T)v)L2,T | ≤ e−cT ∥ ˜Pi(T)u∥L2,T∥ ˜Pi(T)v∥L2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (65) It follows (62), (63), (64) and (65) that 1 − C T 3/2 ≤ ∥g−1 T ∂ ∂T gT ∥ ≤ 1 + C T 3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Next, we define a differential ˜∂ : Hk−1 abs ( ¯Z1, ¯F1)⊕Hk−1 rel ( ¯Z2, ¯F2)0 → Hk abs( ¯Z1, ¯F1)⊕ Hk rel( ¯Z2, ¯F2)0, ([u], [v]) �→ (0, ∂k[u]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Recall that ∂k is the map in Mayer-Vietoris se- quence (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then one can check easily that RT ˜∂R−1 T = d ¯Z|˜Ωsm( ¯ M, ¯F) and RT ˜∂∗ T R−1 T = δ ¯Z,∗ T |˜Ωsm( ¯ M, ¯F), where ˜∂∗ T is the dual of ˜∂ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' gT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It follows from the statement in Step 1 in the proof of [21, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='8] and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='6 that Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' When T is large enough, all nonzero eigenvalues of ∆T inside [0, δ] are actually inside [c2 1e−2T , c2 2e−2T ] for some c2 > c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3 Comparison of connections H( ¯Zi, ¯Fi)(T) has a flat connection ∇Hi,T := ˜Pi(T)∇ ¯Ei(c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' [3, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Recall that if T is large enough, ˜Ωk sm( ¯Z, ¯F)(T) = ˜ek,THk( ¯Z2, ¯F2)(T) ⊕ ˜r∗ k,THk( ¯Z1, ¯F1)(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (66) Recall that ˜e−1 k,T is the inverse of ˜ek,T|˜ek,T H( ¯Z1, ¯F1)(T), and ˜r−1 k,T is the inverse of ˜rk,T|˜r∗ k,T H( ¯Z2, ¯F2)(T), then ˜Ωsm( ¯Z, ¯F)(T) has a flat connection ∇H,T := ˜ek,T ∇H1,T ˜e−1 k,T ⊕ ˜r−1 k,T∇H2,T ˜rk,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Moreover, let ∇δ,T := ˜Pδ(T)∇ ¯E, then ∇δ,T is another connection on ˜Ωsm( ¯Z, ¯F)(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' In this subsection, we are going to compare ∇H,T and ∇δ,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' First, one has Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' When T is big enough, ∥[∇ ¯E, ˜Pδ(T)]∥ ≤ C for some (T, θ)-independent C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Moreover, there exists a uniformly bounded operator valued 1-form AT , such that [∇ ¯E, ˜Pδ(T)] = [d ¯Z, AT ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (67) Similar statements hold if we replace ˜Pδ(T) by ˜Pi(T) (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Doing functional calculus as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5 for ˜D ¯Z T , one has ∥[∇E, Pδ(T)]∥ ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Since [d ¯Z, ∇ ¯E] = 0, one can see that [∇ ¯E, ˜∆T ] = [d ¯Z, [∇ ¯E, δ ¯Z,∗ T ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Since gT ¯Z, T H ¯ M and h ¯F T are product type near N, one can see that [∇ ¯E, δ ¯Z,∗ T ] is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Doing functional calculus for ˜∆T as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='5, the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let Kδ T = ∇δ,T − ∇δ,T,∗, KH T = ∇H,T − ∇H,T,∗ , KT = Kδ T − KH T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' When T is large enough, ∥KT ∥ ≤ C exp(−ρT) for some (T, θ)- independent C > 0, ρ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For u ∈ ˜ek,THk( ¯Z2, ¯F2)(T), there exists v ∈ Hk( ¯Z2, ¯F2)(T), such that u = ˜ek,Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then ∇δ,T u = ˜Pδ(T)∇ ¯E ˜Pδ(T)E(v) = ˜Pδ(T)∇ ¯EE(v) + ˜Pδ(T)[ ˜Pδ(T), ∇ ¯E]E(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (68) Recall that E(v) is an extension of v, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' outside ¯Z1, E(v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Integration by parts as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='10 shows that E(v) is d ¯Z-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Hence, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='8 and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='7, 32 ∥ ˜Pδ(T)[ ˜Pδ(T), ∇ ¯E]E(v)∥ = ∥Pδ(T)d ¯ZAT E(v)∥ = ∥d ¯ZPδ(T)AT E(v)∥ ≤ Ce−T ∥v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (69) Similarly, ∇H,T u = ˜Pδ(T)E( ˜P2(T)∇ ¯E2v) = ˜Pδ(T)∇ ¯EE(v) + ˜Pδ(T)E([ ˜P2(T), ∇ ¯E2]v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (70) Moreover, Integration by parts as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='10 shows that for w ∈ Ωrel( ¯Z2, ¯F2)(T), d ¯ZE(w) = E(d ¯Z2w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (71) Thus, by (71), Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='8 and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='7, ∥ ˜Pδ(T)E([ ˜P2(T), ∇ ¯E2]v)∥ = ∥Pδ(T)d ¯ZE(AT,2v)∥ = ∥d ¯ZPδ(T)E(AT,2v)∥ ≤ Ce−T ∥v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (72) It follows from (68), (69), (70) and (72) and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2 that ∥(∇δ,T − ∇H,T)u∥ ≤ Ce−T ∥u∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (73) While for any u1, u2 ∈ ˜ek,T Hk( ¯Z2, ¯F2)(T), (KT u1, u2)L2,T = ((∇δ,T − ∇H,T )u1, u2)L2,T + (u1, (∇δ,T − ∇H,T )u2)L2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (74) Hence, by (73) and (74) |(KT u1, u2)L2,T | ≤ Ce−T ∥u1∥L2,T ∥u2∥L2,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (75) Similarly, one can show that restricted on ˜r∗ k,TH( ¯Z1, ¯F1)(T), ∥∇δ,T,∗ − ∇H,T,∗∥ ≤ Ce−T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Similarly, for u1, u2 ∈ ˜r∗ k,TH( ¯Z1, ¯F1)(T), |(KT u1, u2)L2,T | ≤ Ce−T ∥u1∥L2,T ∥u2∥L2,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (76) Since T H ¯ M, gT ¯Z and h ¯F are product-type near N, ∥∇ ¯E − ∇ ¯E,∗∥ ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (77) Suppose vi ∈ H( ¯Zi, ¯Fi)(T), i = 1, 2, and u1 = ˜r∗ k,Tv1, u2 = ˜ek,Tv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let η ∈ C∞ c (R), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' η(s) = 1 if s ∈ (−∞, −1/8) and η|[−1/16,0] ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then we could think η as a function on ¯Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 33 By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='1, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2 and (77) , there exists ρ ∈ (0, 1), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' ((∇δ,T − ∇δ,T,∗)u1, u2)L2,T = ((∇δ,T − ∇δ,T,∗) ˜Pδ(T)E(v1), ˜Pδ(T)E(v2))L2,T = ((∇ ¯E − ∇ ¯E,∗) ˜Pδ(T)E(v1), ˜Pδ(T)E(v2))L2,T ≤ Ce−ρT ∥u1∥L2,T ∥u2∥L2,T + ((∇ ¯E − ∇ ¯E,∗)ηE(v1), E(v2))L2,T = Ce−ρT ∥u1∥L2,T ∥u2∥L2,T + (η(∇ ¯E − ∇ ¯E,∗)E(v1), E(v2))L2,T = Ce−ρT ∥u1∥L2,T ∥u2∥L2,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (78) Since for any ui ∈ H( ¯Zi, ¯Fi)(T), i = 1, 2, ((∇H,T − ∇H,T,∗)u1, u2)L2,T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (79) By (78) and (79), for any ui ∈ H( ¯Zi, ¯Fi)(T), i = 1, 2, |(KT u1, u2)L2,T | ≤ Ce−ρT ∥u1∥L2,T ∥u2∥L2,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (80) The lemma then follows from (75), (76) and (80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let Dδ,T t = ∇δ,T −∇δ,T,∗+ √ t(d ¯Z −δ ¯Z,∗ T ), DH,T t = ∇H,T −∇H,T,∗+ √ t(d ¯Z −δ ¯Z,∗ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It follows from Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='7 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='9 that Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For t ≥ 1, | Trs � Nf ′(Dδ,T t ) − Nf ′(DH,T t ) � | ≤ Ce(1−ρ)T √ t for some (T, θ)-independent C > 0, where ρ is the constant in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For t > 0, | Trs � Nf ′(Dδ,T t ) − Nf ′(DH,T t ) � | ≤ C′e−2T t for some (T, θ)-independent C′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Let Kδ T = ∇δ,T − ∇δ,T,∗, KH T = ∇H,T − ∇H,T,∗ , KT = Kδ T − KH T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='9, ∥KT ∥ ≤ Ce−ρT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (81) Let γ be the oriented contour given by {z ∈ C : |Re(z)| = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Then by Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='7, when T is large (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='13]), for “•”=“H” or “δ” f ′(D•,T t ) = � γ f ′(λ)(λ − D•,T t )−1dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' For λ ∈ γ, let V = d ¯Z − δ ¯Z,∗ T , then � λ − D•,T t �−1 = � 1 − � λ − √ tV �−1 K• T �−1 � λ − √ tV �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (82) 34 By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='7 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='8, for λ ∈ γ ∥ � λ − √ tV �−1 − Pker V λ ∥ ≤ CeT |λ| √ t ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (83) or ∥ � λ − √ tV �−1 − Pker V λ ∥ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (84) Also � 1 − � λ − √ tV �−1 K• T �−1 = dim S � i=0 �� λ − √ tV �−1 K• T �i , (85) Proceeding as in the proof of [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='13], by (81), (82), (83), (84) and (85) | Trs � N � f ′(Dδ,T t ) − f ′(Pker V Kδ T Pker V ) − f ′(DH,T t ) + f ′(Pker V KH T Pker V ) �� | ≤ Ce(1−ρ)T √ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (86) While by [3, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3], Trs(Nf ′(Pker V Kδ T Pker V )) = Trs(Nf ′(Pker V KH T Pker V )) = Trs(Nf ′(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Hence, when t ∈ [1, ∞), | Trs(Nf ′(Dδ,T t )) − Trs(Nf ′(DH,T t ))| ≤ Ce1−ρT √ t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Although ∇δ,T is not flat, the argument in [3, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='3] still works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Hence, we have Trs � Nf ′(Dδ,T 0 ) − Nf ′(DH,T 0 ) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (87) Moreover, by a straightforward computation, ∂ ∂t Trs � Nf ′(Dδ,T t ) − Nf ′(DH,T t ) � = Trs � (d ¯Zδ ¯Z,∗ T − δ ¯Z,∗ T d ¯Z) � ˜f ′((Dδ,T t )2) − ˜f ′((DH,T t )2) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (88) Recall that ˜f(a) = (1 + 2a)ea, hence ˜f(a2) = f ′(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='7, ∥(d ¯Zδ ¯Z,∗ T − δ ¯Z,∗ T d ¯Z)( ˜f ′((Dδ,T t )2) − ˜f ′((DH,T t )2))∥ ≤ Ce−2T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (89) By (87), (88) and (89), | Trs � Nf ′(Dδ,T t ) − Nf ′(DH,T t ) � | ≤ C′e−2T t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 35 Let Tf � d ¯Z + ∇δ,T� = − � ∞ 0 � ϕ Trs �1 2Nf ′ � Dδ,T t �� − 1 2χ′(Z, F) − d(Ωsm( ¯ M, ¯F)(T)) − χ′(Z, F) 2 f ′ �i √ t 2 �� dt t , and Tf � d ¯Z + ∇H,T� = − � ∞ 0 � ϕ Trs �1 2Nf ′ � DH,T t �� − 1 2χ′(Z, F) − d(Ωsm( ¯ M, ¯F)(T)) − χ′(Z, F) 2 f ′ �i √ t 2 �� dt t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' limT→∞ � Tf � d ¯Z + ∇δ,T� − Tf � d ¯Z + ∇H,T �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' That is, lim T→∞ � Tsm(T H ¯ M, gT ¯Z, h ¯F )(T) − Tf � d ¯Z + ∇H,T�� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Fix ρ′ ∈ (0, ρ), where ρ is the constant in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' By the second in- equality in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='10, � e2T −2ρ′T 0 | Trs �1 2Nf ′ � DH,T t �� − Trs �1 2Nf ′ � Dδ,T t �� |dt t ≤ Ce−2ρ′T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (90) By the first inequality in Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='10, � ∞ e2T −2ρ′T | Trs �1 2Nf ′ � DH,T t �� − Trs �1 2Nf ′ � Dδ,T t �� |dt t ≤ Ce(ρ′−ρ)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' (91) Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Since ∇H2,T ˜∂k,T = ˜∂k,T∇H1,T and ˜∂k,T = ˜e−1 k,T d ¯Z˜r−1 k,T, one has [∇H,T , d ¯Z] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' It follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='9, [26, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='2] and [9, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='37] that in QS/QS 0, lim T→∞ Tf � d ¯Z + ∇H,T� − T (T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' The theorem then follows from Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content='11.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Gluing formula of real analytic torsion forms and adiabatic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' Israel Journal of Mathematics, 215(1):181–254, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} +page_content=' 38' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE0T4oBgHgl3EQftQH5/content/2301.02591v1.pdf'} diff --git a/ytE0T4oBgHgl3EQftgFE/content/2301.02592v1.pdf 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Tinguely1‡, I Pusztai2, VA Izzo3, K S¨arkim¨aki4, T F¨ul¨op2, +DT Garnier1, RS Granetz1, M Hoppe5, C Paz-Soldan6, +A Sundstr¨om2, and R Sweeney1 +1 Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, +MA, USA +2 Department of Physics, Chalmers University of Technology, SE-41296 G¨oteborg, Sweden +3 Fiat Lux, San Diego, CA 92101, USA +4 Max Planck Institute for Plasmaphysics, 85748 Garching, Germany +5 Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Swiss Plasma Center (SPC), +CH-1015 Lausanne, Switzerland +6 Department of Applied Physics and Applied Mathematics, Columbia University, NY, +USA +Abstract. +In [V.A. Izzo et al 2022 Nucl. Fusion 62 096029], state-of-the-art modeling of +thermal and current quench (CQ) MHD coupled with a self-consistent evolution of runaway +electron (RE) generation and transport showed that a non-axisymmetric (n = 1) in-vessel +coil could passively prevent RE beam formation during disruptions in SPARC, a compact +high-field tokamak projected to achieve a fusion gain Q > 2 in DT plasmas. +However, +such suppression requires finite transport of REs within magnetic islands and re-healed flux +surfaces; conservatively assuming zero transport in these regions leads to an upper bound +of RE current ∼1 MA compared to ∼8.7 MA of pre-disruption plasma current. +Further +investigation finds that core-localized electrons, within r/a < 0.3 and with kinetic energies +∼0.2−15 MeV, contribute most to the RE plateau formation. Yet only a relatively small +amount of transport, i.e. +a diffusion coefficient ∼18 m2/s, is needed in the core to fully +mitigate these REs. Properly accounting for (i) the CQ electric field’s effect on RE transport +in islands and (ii) the contribution of significant RE currents to disruption MHD may help +achieve this. +Keywords: Runaway electrons, passive mitigation, transport, disruptions, SPARC +1. Introduction +In [1, 2], a novel method was proposed for passive mitigation of relativistic “runaway +electrons” (REs) generated during tokamak plasma disruptions: First, an in-vessel, non- +axisymmetric coil would be passively energized through mutual coupling to the plasma +current during the disruption’s current quench (CQ); then, the resulting magnetic field +‡ Author to whom correspondence should be addressed: rating@mit.edu +arXiv:2301.01435v1 [physics.plasm-ph] 4 Jan 2023 + +2 +perturbation would enhance stochasticity and transport such that the RE loss rate would +dominate the growth rate, thus preventing RE beam formation. +In [3], such a “Runaway Electron Mitigation Coil” (REMC) was proposed for the +SPARC tokamak [4], a high-field (B0 = 12.2 T), compact (R0 = 1.85 m, a = 0.57 m) device +currently under construction in Devens, Massachusetts, USA. The present REMC design +has a predominantly n = 1 structure and is located on the outboard wall; a similar coil is +planned for the DIII-D tokamak, but on the inboard wall [5]. Several aspects of the SPARC +“Primary Reference Discharge” (PRD) make the RE problem challenging: a large plasma +current (Ip = 8.7 MA) can lead to dangerous exponential RE growth; high core temperatures +Te0 ≈ 20 keV can cause enhanced primary and hot-tail generation; DT fuel provides a seed of +non-thermal electrons through tritium beta decay; and high energy gammas from activated +materials could accelerate electrons via Compton scattering. +However, in [6], modeling of the PRD’s worst-case-scenario CQ (∼3 ms) showed complete +prevention of RE beam formation with the REMC – and ∼5−6 MA of RE current without +it. The modeling workflow included four steps: First, the mutual couplings of all toroidally +conducting structures were simulated in COMSOL [7] to evaluate the REMC’s vacuum +electromagnetic fields during the worst-case CQ. Second, these magnetic fields were applied +at the boundary of a nonlinear, 3D NIMROD [8] simulation to assess the plasma response +and total fields. Third, the stochastic magnetic fields were input into the orbit-following +code ASCOT5 to calculate the advective and diffusive transport [9] of energetic electrons. +Finally, these transport coefficients – A, D as functions of energy, pitch, and radius – were +supplied to the hybrid fluid-kinetic code DREAM [10] for self-consistent evolution of the RE +population. Importantly, in both NIMROD and DREAM, the REMC vacuum fields and +transport coefficients, respectively, were evolved as functions of Ip and not time explicitly. +More recently, in [11], both the thermal quench (TQ) and CQ were modeled for the +SPARC PRD and REMC; the results of this study – which bound the maximum expected +RE current – are summarized in Section 2. Section 3 further explores these bounds in RE +phase space, as well as the minimum transport needed to fully prevent RE beam formation. +Finally, results and opportunities for future modeling are discussed in Section 4. +2. REMC efficacy during the thermal and current quenches +The same workflow presented in [6] and summarized in Section 1 was used in [11] to assess +the SPARC REMC’s efficacy for a full PRD mitigated disruption, i.e. including both the TQ +and CQ. Here, the TQ (∼1 ms in duration) was induced by neon radiation, as in a scenario +where massive gas injection was employed. The main results are captured in Figs. 1 and 2. +The pre-disruption safety factor (q) profile is shown at t = 0 in Fig. 1a with q(0) ∼ 1 and +q = 2 around a normalized poloidal flux value of ψN ≈ 0.75. During the TQ, i.e. the first +∼1 ms of the simulation, the plasma current Ip decreases slightly, with the Ip-spike denoting +the start of the CQ. Around that time, the magnetic perturbation amplitudes δB/B first +peak (see Fig. 1b), and strong nonlinear coupling among odd and even toroidal harmonics + +3 +is observed. Poincar´e plots of magnetic field lines, in Fig. 1a, also show high stochasticity +during this period. +(a) +(b) +Figure 1: (a, upper) Poincar´e plots of (mostly) stochastic magnetic field lines from NIMROD within +the simulation boundary (dashed) and SPARC first wall (solid). (a, lower) The safety factor q-profile +evolution vs normalized poloidal flux (ψN) and time, with the plasma current (Ip) time-evolution +overlaid. (b) Amplitudes of n = 1−10 modes in units of δB/B = +� +Wmag(n)/Wmag(n = 0) with +Wmag the magnetic energy Fourier component. Subplot (a) is reproduced from Figure 5 in [11]. +However, from t ≈ 1−1.5 ms, q(0) increases from 1 to 2, and beyond t > 1.5 ms, +the REMC is no longer resonant with the plasma core (refer to Figure 1 in [6] for more +details). Although the predominantly odd externally applied fields continue to grow as a +the coil current continues to increase, these are now largely non-resonant fields that do not +perturb the flux surfaces, and the nonlinearly excited resonant field components, both odd +and even, decay away. Thus, small islands start to reform, re-healing as closed flux surfaces +by t ≈ 1.8 ms. Note that the contribution from REs to the MHD are not included in these +NIMROD simulations, but the back-reaction is expected to be small for low RE currents +early on. This will be discussed further in Section 4. +Figure 2 shows the self-consistent evolution of Ohmic and RE currents from DREAM, +including the advective and diffusive transport calculated by ASCOT5 in DREAM’s fluid +transport model [12]. Note that the time bases of the DREAM and NIMROD simulations +are not exactly the same; instead, the DREAM simulation is initialized with profiles close +to the time of NIMROD’s Ip-spike. +Importantly, the TQ in DREAM is only modeled for +the final ∼0.1 ms of NIMROD’s ∼1 ms TQ because DREAM requires a monotonic variation + +tcq-0.16 ms +tcq+0.14ms +tcq+ 0.44 ms +tcQ+ 0.74 ms +time = 0.90 ms +time = 1.20 ms +time = 1.50 ms +time = 1.80 ms +1.0 +t-tco [ms] -0.56 +-0.06 +0.44 +0.94 +8.0 +7.2 +0.8 + 6.4 +5.6 +0.6 +4.8 +N +0.4 +4.0 +0.2 +1.000 +0.0 +0.5 +1.0 +1.5 +2.0 +0.8 +times [ms]n=1 +10-2 +n=2 +n=3 +6B/B +10-3 +n=4 +n=5 +n=6 +10-4 +n=7 +n=8 +10-5 +n=9 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +n=10 +Time [ms]4 +of the plasma current from which to map transport coefficients. Thesetransport coefficients +evolve with the plasma current until the final Ip-value of the NIMROD simulation; then, they +are held constant in time (see the vertical dashed line in the upper part of Fig. 2).§ +2 +4 +6 +8 +1.05 1.23 1.49 1.71 1.96 2.24 2.58 +I [MA] +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +10-12 +10-10 +10-8 +10-6 +10-4 +10-2 +t[ms] +Figure 2: Time-traces of Ohmic (solid), RE (dot-dashed), and total (dashed) currents from DREAM; +thick/thin RE currents indicate no/transport within re-healed flux surfaces. Times denoted above +correspond to the NIMROD simulation (see Fig. 1). Transport coefficients are fixed in time after +the vertical dashed line (upper), and surface re-healing begins at the vertical dotted line (lower). +Reproduced from Figure 6 in [11]. +Two scenarios for the RE current are depicted in Fig. 2: In the first, the transport +coefficients are applied as calculated throughout the entire plasma domain, i.e. even inside +the re-healed flux surfaces, and the RE current remains negligibly low (∼1 µA). However, in +the second case, transport inside islands and re-healed flux surfaces is set to zero, which is +perhaps overly conservative. (Explicitly, A = D = 0 wherever D < 1000 m2/s.) The result is +a RE plateau with current ∼1 MA. While this value is an improvement upon the ∼5−6 MA +of RE current expected with no REMC [3,6], it is likely the pessimistic upper bound on the +true value. +Here, it is important to note that the initial hot-tail seed can be sensitive to the TQ +cooling time prescribed in DREAM. In the CQ-only modeling effort of [6], a ∼0.5 ms TQ +from 20 keV to 4 eV led to a ∼50 kA seed which was then dissipated by the REMC, with +similar δB/B ∼ 10−2 as that in Fig. 1b. Further reducing the TQ time to 0.1 ms resulted +in a much higher transient RE beam current of ∼1 MA, which was still suppressed by the +REMC. These results are consistent with all test-particle REs losing confinement during the +TQ in NIMROD for the scenario modeled in this paper [11]. The inclusion of TQ transport +in DREAM, as a function of non-monotonic Ip variation, is left for future work. +§ To enforce transport ambipolarity in DREAM, any change of the electron density on a given flux surface +due to transport is compensated by a change in the bulk electron density of similar size but opposite sign. + +5 +3. Investigating transport inside re-healed flux surfaces +This section explores further the ten-order-of-magnitude difference in the predicted RE +current when transport is/not accounted for within NIMROD’s islands and re-healed flux +surfaces. +Radial profiles of the diffusion coefficient (D) are shown in Fig. 3a, also as a +function of normalized electron momentum, at the last NIMROD simulation time; the values +shown are taken at a representative electron pitch p∥/p = 0.8. There are a few important +notes here: (i) the diffusion coefficients span five orders of magnitude from the plasma +core to edge; (ii) though not shown, the advection coefficients are of similar magnitude +(A[m/s] ∼ D[m2/s]); and (iii) both transport coefficients are relatively insensitive to the +electron pitch in the relevant range p∥/p ∈ [0.8, 1] (see Figure 3 in [6]). +log10(D[m2/s]) +(a) ASCOT5 results. +D[m2/s] +5.62 +10 +17.8 +1000 +1 +2 +3 +4 +5 +6 +7 +1 +10-2 +10-4 +10-6 +10-8 +10-10 +t[ms] +IRE [MA] +(b) DREAM results. +Figure 3: (a) Diffusion coefficients, log10(D [m2/s]), from ASCOT5 vs normalized minor radius +(r/a) and electron momentum normalized to the rest mass (p/mec) at the time indicated by the +vertical dashed lines in Fig. 2(upper) and subplot (b). (b) Time-traces of the RE current from +DREAM when diffusion coefficients less than the noted value are set to zero, i.e. D = 0 within the +similarly styled contours in (a). Note the various linear/logarithmic scales. The legend for curves +in subplot (b) applies to contours in subplot (a). +In Fig. 3a, general trends are seen of rapidly decreasing transport with decreasing radius +and relative insensitivity to electron energy. However, there is a clear feature of “very low” +transport (D < 30 m2/s) for electrons localized in the core (r/a ∼ 0.05−0.2) and with +energies <50 MeV (p/mec < 100). Figure 3b shows the time-evolution of RE current when +the transport coefficients are zeroed in different regions of the phase space in Fig. 3a.∥ +The “base case” is D = 0 wherever D < 1000 m2/s, which effectively includes the entire +core, r/a < 0.3, and leads to the previously seen ∼1 MA RE beam. +Yet reducing this +threshold to D < 10 m2/s leads to negligible RE current. Thus, it is primarily the electron +∥ Note that the diffusivity is used for discrimination of the phase space regions, while the advection coefficients +(not shown here) are also filtered in the same regions. + +6 +population within D ∼ 10−18 m2/s, i.e. localized in r/a ∼ 0.05−0.2 and with kinetic energies +∼0.2−15 MeV (p/mec ∼ 1−30), which contributes most to RE plateau formation. +This problem can be looked at from another angle: What is the minimum transport +needed to fully suppress RE plateau formation? More specifically, within the region of phase +space where D < 1000 m2/s in Fig. 3a (that mostly coincides with the re-healed flux surface +region), which constant value of D is sufficient to yield negligible RE current? As seen in +Fig. 4, full RE beam prevention is only achieved somewhere in the range D = 10−18 m2/s. +Therefore, compared to the highly diffusive edge region (D ≈ 103−105 m2/s), a relatively +small amount of core transport is needed. Importantly, note that the advection coefficient +A[m/s] is set to the same value as D[m2/s] in these phase space regions, but almost identical +results are found when setting A = 0, as diffusion dominates in the narrow radial region of +re-healed flux surfaces (as long as A[m/s] ∼ D[m2/s]). +D[m2/s] +3.16 +5.62 +10 +17.8 +1 +2 +3 +4 +5 +6 +7 +1 +10-2 +10-4 +10-6 +10-8 +10-10 +t[ms] +IRE [MA] +Figure 4: Time-traces of the RE current from DREAM when the diffusion coefficient is set to the +listed value in regions of phase space with D < 1000 m2/s in Fig. 3a. The time indicated by the +vertical dashed line is the same as in Figs. 2 and 3b. +4. Discussion and summary +From the previous sections, it is clear that zeroing the transport in the core (r/a < 0.3) in +DREAM is too conservative and pessimistic, resulting in a ∼1 MA RE beam. Even so, it is +important to note that this current is 5-6 times less than that expected for an unmitigated +RE beam, i.e. no REMC, so even this conservative base case could be considered successful. +ASCOT5 simulations evaluate diffusion coefficients spanning D ≈ 1−1000 m2/s in the core, +but encouragingly only D ∼ 18 m2/s is needed in that region to completely suppress a RE +beam. Perhaps this is one reason why many tokamaks struggle to generate RE plateaus +via “natural” disruptions (as in Alcator C-Mod [13,14], for example), and instead resort to +special “recipes,” although lower plasma current and thus lower avalanching certainly also +play a role. + +7 +However, it is not yet known whether this level of transport is achievable in SPARC. +In [11], it was noted that the degree of field line stochastization predicted by NIMROD could +be affected by several approximations, most notably the presence of a close, ideal wall which +tends to limit MHD mode growth. This approximation will be explored further in future +resistive-wall studies, and perhaps this minimum D-value will even decrease. +We can also approach this from another direction: In what ways can the REMC design +be modified to achieve D > 18 m2/s throughout the plasma? Perhaps most straightforward, +the coil could be moved closer to the plasma and farther from the VV. This would (i) improve +the plasma-coil mutual coupling and reduce the coil’s self-inductance, thereby increasing the +induced coil current, (ii) decrease image currents in the conducting wall, and (iii) enhance +the magnetic perturbation amplitude δB/B in the core. Perhaps a design metric could be +the expected diffusion coefficient computed from vacuum fields `a la [15], D ∝ (δB/B)2. The +coil resistance could also be lowered by changing the coil cross-section, length, and material +(resistivity). That said, many other factors constrain the design, like available space, forces +and stresses, heating, and more. +As discussed in [6], both advection and diffusion tend to increase with RE energy, but +there is a roll-over when the energetic electron drift orbits effectively average over large regions +of stochasticity (see Figure 3 in [6] or [16,17] and others for further details). However, for REs +within healed flux surfaces, perhaps large orbits could lead to “excursions” into stochastic +fields, thus enhancing transport. For example, KORC simulations in [16] found that REs +with Larmor radii similar to island widths could escape them. In addition, the same electric +field accelerating REs causes them to drift radially [18], and this was not accounted for in +these ASCOT5 simulations, but will be pursued in the future.¶ +Perhaps most importantly, the effect of the RE population itself on the magnetic field +and MHD has not yet been fully assessed. Figure 5 shows the time-evolution of the q-profile, +its minimum value, and the internal inductance (ℓi) in DREAM for the base case. Although +slightly later in time than in Fig. 1a, the central safety factor q(0) also surpasses q = 2; +however, unlike the NIMROD results, the increasing RE current then reduces q(0) < 2 at +t ≈ 5 ms and q(0) < 1 at t ≈ 5.5 ms. Thus, in theory, the REMC should regain resonance +in the core beyond t > 5 ms, thereby enhancing transport and reducing the RE current, but +this was not captured in the current workflow. A destructive kink instability might also be +expected, as seen in experiment [19,20], for such low q(0) and high ℓi. +Even then it is not clear what overall effect this self-regulation would have on the RE +beam which already has a current ∼1 MA by t ≈ 5 ms (for the base case with no island +transport). Luckily, the Ohmic current has almost completely decayed by then, and the +relatively long L/R time (>10 ms) of the REMC will maintain the coil current and its +perturbative effect. Furthermore, any additional transport within the re-healed flux surfaces +will help lower this quasi-stationary RE current. A fluid RE model that could capture this +¶ See [17] for simulations of passing and trapped REs during an ITER CQ, including the effects of collisions +and the electric field. + +8 +t[ms] +0 +1 +3 +5 +6.9 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +2 +4 +6 +8 +10 +r[m] +q +(a) +li +min(q) +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +5 +t[ms] +li, min(q) +(b) +Figure 5: +DREAM results for the base case in Fig. 3b with D = 0 below D < 1000 m2/s: +(a) evolution of the safety factor q-profile vs minor radius for five times, and (b) time evolution of +the minimum q-value (dot-dashed) and internal inductance ℓi (solid). +effect has been incorporated into the JOREK [21,22] and M3D-C1 [23] MHD codes; a similar +model is being implemented in NIMROD [24] and benchmarked against the existing codes. +Its application to the SPARC REMC will be pursued in future work. +Acknowledgments +Supported by Commonwealth Fusion Systems, Swedish Research Council (Dnr. 2018-03911), +US DOE Award Numbers DE-FC02-04ER54698 and DE-FG02-95ER54309. This work has +been carried out within the framework of the EUROfusion Consortium, funded by the +European Union via the Euratom Research and Training Programme (Grant Agreement No +101052200 – EUROfusion). Views and opinions expressed are however those of the author(s) +only and do not necessarily reflect those of the European Union or the European Commission. +Neither the European Union nor the European Commission can be held responsible for them. +References +[1] Allen H Boozer. Two beneficial non-axisymmetric perturbations to tokamaks. Plasma Phys. Control. +Fusion, 53:84002–84008, 2011. +[2] H. M. Smith, A. H. Boozer, and P. Helander. +Passive runaway electron suppression in tokamak +disruptions. Physics of Plasmas, 20(7), 2013. +[3] R. Sweeney, A.J. Creely, J. Doody, T. F¨ul¨op, D.T. Garnier, R. Granetz, M. Greenwald, L. Hesslow, +J. Irby, V.A. Izzo, R.J. La Haye, N.C. Logan, K. Montes, C. Paz-Soldan, C. Rea, R.A. Tinguely, +O. Vallhagen, and J. Zhu. 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In APS Division of Plasma Physics Meeting Abstracts, volume 2020, pages +PP12–035, 2020. + diff --git a/zdAzT4oBgHgl3EQfefwA/content/tmp_files/load_file.txt b/zdAzT4oBgHgl3EQfefwA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8a31ae81ee9b072876e8693571135f6dbb83b530 --- /dev/null +++ b/zdAzT4oBgHgl3EQfefwA/content/tmp_files/load_file.txt @@ -0,0 +1,541 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf,len=540 +page_content='On the minimum transport required to passively suppress runaway electrons in SPARC disruptions R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Tinguely1‡,' metadata={'source': 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Applied Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Columbia University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' USA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' In [V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Izzo et al 2022 Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Fusion 62 096029], state-of-the-art modeling of thermal and current quench (CQ) MHD coupled with a self-consistent evolution of runaway electron (RE) generation and transport showed that a non-axisymmetric (n = 1) in-vessel coil could passively prevent RE beam formation during disruptions in SPARC, a compact high-field tokamak projected to achieve a fusion gain Q > 2 in DT plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' However, such suppression requires finite transport of REs within magnetic islands and re-healed flux surfaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' conservatively assuming zero transport in these regions leads to an upper bound of RE current ∼1 MA compared to ∼8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='7 MA of pre-disruption plasma current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Further investigation finds that core-localized electrons, within r/a < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='3 and with kinetic energies ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='2−15 MeV, contribute most to the RE plateau formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Yet only a relatively small amount of transport, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' a diffusion coefficient ∼18 m2/s, is needed in the core to fully mitigate these REs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Properly accounting for (i) the CQ electric field’s effect on RE transport in islands and (ii) the contribution of significant RE currents to disruption MHD may help achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Keywords: Runaway electrons, passive mitigation, transport, disruptions, SPARC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Introduction In [1, 2], a novel method was proposed for passive mitigation of relativistic “runaway electrons” (REs) generated during tokamak plasma disruptions: First, an in-vessel, non- axisymmetric coil would be passively energized through mutual coupling to the plasma current during the disruption’s current quench (CQ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' then, the resulting magnetic field ‡ Author to whom correspondence should be addressed: rating@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='edu arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='01435v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='plasm-ph] 4 Jan 2023 2 perturbation would enhance stochasticity and transport such that the RE loss rate would dominate the growth rate, thus preventing RE beam formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' In [3], such a “Runaway Electron Mitigation Coil” (REMC) was proposed for the SPARC tokamak [4], a high-field (B0 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='2 T), compact (R0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='85 m, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='57 m) device currently under construction in Devens, Massachusetts, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' The present REMC design has a predominantly n = 1 structure and is located on the outboard wall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' a similar coil is planned for the DIII-D tokamak, but on the inboard wall [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Several aspects of the SPARC “Primary Reference Discharge” (PRD) make the RE problem challenging: a large plasma current (Ip = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='7 MA) can lead to dangerous exponential RE growth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' high core temperatures Te0 ≈ 20 keV can cause enhanced primary and hot-tail generation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' DT fuel provides a seed of non-thermal electrons through tritium beta decay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' and high energy gammas from activated materials could accelerate electrons via Compton scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' However, in [6], modeling of the PRD’s worst-case-scenario CQ (∼3 ms) showed complete prevention of RE beam formation with the REMC – and ∼5−6 MA of RE current without it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' The modeling workflow included four steps: First, the mutual couplings of all toroidally conducting structures were simulated in COMSOL [7] to evaluate the REMC’s vacuum electromagnetic fields during the worst-case CQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Second, these magnetic fields were applied at the boundary of a nonlinear, 3D NIMROD [8] simulation to assess the plasma response and total fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Third, the stochastic magnetic fields were input into the orbit-following code ASCOT5 to calculate the advective and diffusive transport [9] of energetic electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Finally, these transport coefficients – A, D as functions of energy, pitch, and radius – were supplied to the hybrid fluid-kinetic code DREAM [10] for self-consistent evolution of the RE population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Importantly, in both NIMROD and DREAM, the REMC vacuum fields and transport coefficients, respectively, were evolved as functions of Ip and not time explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' More recently, in [11], both the thermal quench (TQ) and CQ were modeled for the SPARC PRD and REMC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' the results of this study – which bound the maximum expected RE current – are summarized in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Section 3 further explores these bounds in RE phase space, as well as the minimum transport needed to fully prevent RE beam formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Finally, results and opportunities for future modeling are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' REMC efficacy during the thermal and current quenches The same workflow presented in [6] and summarized in Section 1 was used in [11] to assess the SPARC REMC’s efficacy for a full PRD mitigated disruption, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' including both the TQ and CQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Here, the TQ (∼1 ms in duration) was induced by neon radiation, as in a scenario where massive gas injection was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' The main results are captured in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' The pre-disruption safety factor (q) profile is shown at t = 0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 1a with q(0) ∼ 1 and q = 2 around a normalized poloidal flux value of ψN ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' During the TQ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' the first ∼1 ms of the simulation, the plasma current Ip decreases slightly, with the Ip-spike denoting the start of the CQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Around that time, the magnetic perturbation amplitudes δB/B first peak (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 1b), and strong nonlinear coupling among odd and even toroidal harmonics 3 is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Poincar´e plots of magnetic field lines, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 1a, also show high stochasticity during this period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' (a) (b) Figure 1: (a, upper) Poincar´e plots of (mostly) stochastic magnetic field lines from NIMROD within the simulation boundary (dashed) and SPARC first wall (solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' (a, lower) The safety factor q-profile evolution vs normalized poloidal flux (ψN) and time, with the plasma current (Ip) time-evolution overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' (b) Amplitudes of n = 1−10 modes in units of δB/B = � Wmag(n)/Wmag(n = 0) with Wmag the magnetic energy Fourier component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Subplot (a) is reproduced from Figure 5 in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' However, from t ≈ 1−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 ms, q(0) increases from 1 to 2, and beyond t > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 ms, the REMC is no longer resonant with the plasma core (refer to Figure 1 in [6] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Although the predominantly odd externally applied fields continue to grow as a the coil current continues to increase, these are now largely non-resonant fields that do not perturb the flux surfaces, and the nonlinearly excited resonant field components, both odd and even, decay away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Thus, small islands start to reform, re-healing as closed flux surfaces by t ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='8 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Note that the contribution from REs to the MHD are not included in these NIMROD simulations, but the back-reaction is expected to be small for low RE currents early on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' This will be discussed further in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Figure 2 shows the self-consistent evolution of Ohmic and RE currents from DREAM, including the advective and diffusive transport calculated by ASCOT5 in DREAM’s fluid transport model [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Note that the time bases of the DREAM and NIMROD simulations are not exactly the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' instead, the DREAM simulation is initialized with profiles close to the time of NIMROD’s Ip-spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Importantly, the TQ in DREAM is only modeled for the final ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='1 ms of NIMROD’s ∼1 ms TQ because DREAM requires a monotonic variation tcq-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='16 ms tcq+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='14ms tcq+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='44 ms tcQ+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='74 ms time = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='90 ms time = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='20 ms time = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='50 ms time = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='80 ms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='0 t-tco [ms] -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='94 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='8 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='8 times [ms]n=1 10-2 n=2 n=3 6B/B 10-3 n=4 n=5 n=6 10-4 n=7 n=8 10-5 n=9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 n=10 Time [ms]4 of the plasma current from which to map transport coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Thesetransport coefficients evolve with the plasma current until the final Ip-value of the NIMROD simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' then, they are held constant in time (see the vertical dashed line in the upper part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='§ 2 4 6 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='49 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='58 I [MA] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 5 10-12 10-10 10-8 10-6 10-4 10-2 t[ms] Figure 2: Time-traces of Ohmic (solid), RE (dot-dashed), and total (dashed) currents from DREAM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' thick/thin RE currents indicate no/transport within re-healed flux surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Times denoted above correspond to the NIMROD simulation (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Transport coefficients are fixed in time after the vertical dashed line (upper), and surface re-healing begins at the vertical dotted line (lower).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Reproduced from Figure 6 in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Two scenarios for the RE current are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 2: In the first, the transport coefficients are applied as calculated throughout the entire plasma domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' even inside the re-healed flux surfaces, and the RE current remains negligibly low (∼1 µA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' However, in the second case, transport inside islands and re-healed flux surfaces is set to zero, which is perhaps overly conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' (Explicitly, A = D = 0 wherever D < 1000 m2/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=') The result is a RE plateau with current ∼1 MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' While this value is an improvement upon the ∼5−6 MA of RE current expected with no REMC [3,6], it is likely the pessimistic upper bound on the true value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Here, it is important to note that the initial hot-tail seed can be sensitive to the TQ cooling time prescribed in DREAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' In the CQ-only modeling effort of [6], a ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 ms TQ from 20 keV to 4 eV led to a ∼50 kA seed which was then dissipated by the REMC, with similar δB/B ∼ 10−2 as that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Further reducing the TQ time to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='1 ms resulted in a much higher transient RE beam current of ∼1 MA, which was still suppressed by the REMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' These results are consistent with all test-particle REs losing confinement during the TQ in NIMROD for the scenario modeled in this paper [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' The inclusion of TQ transport in DREAM, as a function of non-monotonic Ip variation, is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' § To enforce transport ambipolarity in DREAM, any change of the electron density on a given flux surface due to transport is compensated by a change in the bulk electron density of similar size but opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Investigating transport inside re-healed flux surfaces This section explores further the ten-order-of-magnitude difference in the predicted RE current when transport is/not accounted for within NIMROD’s islands and re-healed flux surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Radial profiles of the diffusion coefficient (D) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 3a, also as a function of normalized electron momentum, at the last NIMROD simulation time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' the values shown are taken at a representative electron pitch p∥/p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' There are a few important notes here: (i) the diffusion coefficients span five orders of magnitude from the plasma core to edge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' (ii) though not shown, the advection coefficients are of similar magnitude (A[m/s] ∼ D[m2/s]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' and (iii) both transport coefficients are relatively insensitive to the electron pitch in the relevant range p∥/p ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='8, 1] (see Figure 3 in [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' log10(D[m2/s]) (a) ASCOT5 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' D[m2/s] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='62 10 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='8 1000 1 2 3 4 5 6 7 1 10-2 10-4 10-6 10-8 10-10 t[ms] IRE [MA] (b) DREAM results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Figure 3: (a) Diffusion coefficients, log10(D [m2/s]), from ASCOT5 vs normalized minor radius (r/a) and electron momentum normalized to the rest mass (p/mec) at the time indicated by the vertical dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 2(upper) and subplot (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' (b) Time-traces of the RE current from DREAM when diffusion coefficients less than the noted value are set to zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' D = 0 within the similarly styled contours in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Note the various linear/logarithmic scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' The legend for curves in subplot (b) applies to contours in subplot (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 3a, general trends are seen of rapidly decreasing transport with decreasing radius and relative insensitivity to electron energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' However, there is a clear feature of “very low” transport (D < 30 m2/s) for electrons localized in the core (r/a ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='05−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='2) and with energies <50 MeV (p/mec < 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Figure 3b shows the time-evolution of RE current when the transport coefficients are zeroed in different regions of the phase space in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='∥ The “base case” is D = 0 wherever D < 1000 m2/s, which effectively includes the entire core, r/a < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='3, and leads to the previously seen ∼1 MA RE beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Yet reducing this threshold to D < 10 m2/s leads to negligible RE current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Thus, it is primarily the electron ∥ Note that the diffusivity is used for discrimination of the phase space regions, while the advection coefficients (not shown here) are also filtered in the same regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 6 population within D ∼ 10−18 m2/s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' localized in r/a ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='05−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='2 and with kinetic energies ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='2−15 MeV (p/mec ∼ 1−30), which contributes most to RE plateau formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' This problem can be looked at from another angle: What is the minimum transport needed to fully suppress RE plateau formation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' More specifically, within the region of phase space where D < 1000 m2/s in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 3a (that mostly coincides with the re-healed flux surface region), which constant value of D is sufficient to yield negligible RE current?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 4, full RE beam prevention is only achieved somewhere in the range D = 10−18 m2/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Therefore, compared to the highly diffusive edge region (D ≈ 103−105 m2/s), a relatively small amount of core transport is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Importantly, note that the advection coefficient A[m/s] is set to the same value as D[m2/s] in these phase space regions, but almost identical results are found when setting A = 0, as diffusion dominates in the narrow radial region of re-healed flux surfaces (as long as A[m/s] ∼ D[m2/s]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' D[m2/s] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='62 10 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='8 1 2 3 4 5 6 7 1 10-2 10-4 10-6 10-8 10-10 t[ms] IRE [MA] Figure 4: Time-traces of the RE current from DREAM when the diffusion coefficient is set to the listed value in regions of phase space with D < 1000 m2/s in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' The time indicated by the vertical dashed line is the same as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 2 and 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Discussion and summary From the previous sections, it is clear that zeroing the transport in the core (r/a < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='3) in DREAM is too conservative and pessimistic, resulting in a ∼1 MA RE beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Even so, it is important to note that this current is 5-6 times less than that expected for an unmitigated RE beam, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' no REMC, so even this conservative base case could be considered successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' ASCOT5 simulations evaluate diffusion coefficients spanning D ≈ 1−1000 m2/s in the core, but encouragingly only D ∼ 18 m2/s is needed in that region to completely suppress a RE beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Perhaps this is one reason why many tokamaks struggle to generate RE plateaus via “natural” disruptions (as in Alcator C-Mod [13,14], for example), and instead resort to special “recipes,” although lower plasma current and thus lower avalanching certainly also play a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 7 However, it is not yet known whether this level of transport is achievable in SPARC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' In [11], it was noted that the degree of field line stochastization predicted by NIMROD could be affected by several approximations, most notably the presence of a close, ideal wall which tends to limit MHD mode growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' This approximation will be explored further in future resistive-wall studies, and perhaps this minimum D-value will even decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' We can also approach this from another direction: In what ways can the REMC design be modified to achieve D > 18 m2/s throughout the plasma?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Perhaps most straightforward, the coil could be moved closer to the plasma and farther from the VV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' This would (i) improve the plasma-coil mutual coupling and reduce the coil’s self-inductance, thereby increasing the induced coil current, (ii) decrease image currents in the conducting wall, and (iii) enhance the magnetic perturbation amplitude δB/B in the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Perhaps a design metric could be the expected diffusion coefficient computed from vacuum fields `a la [15], D ∝ (δB/B)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' The coil resistance could also be lowered by changing the coil cross-section, length, and material (resistivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' That said, many other factors constrain the design, like available space, forces and stresses, heating, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' As discussed in [6], both advection and diffusion tend to increase with RE energy, but there is a roll-over when the energetic electron drift orbits effectively average over large regions of stochasticity (see Figure 3 in [6] or [16,17] and others for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' However, for REs within healed flux surfaces, perhaps large orbits could lead to “excursions” into stochastic fields, thus enhancing transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' For example, KORC simulations in [16] found that REs with Larmor radii similar to island widths could escape them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' In addition, the same electric field accelerating REs causes them to drift radially [18], and this was not accounted for in these ASCOT5 simulations, but will be pursued in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='¶ Perhaps most importantly, the effect of the RE population itself on the magnetic field and MHD has not yet been fully assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Figure 5 shows the time-evolution of the q-profile, its minimum value, and the internal inductance (ℓi) in DREAM for the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Although slightly later in time than in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 1a, the central safety factor q(0) also surpasses q = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' however, unlike the NIMROD results, the increasing RE current then reduces q(0) < 2 at t ≈ 5 ms and q(0) < 1 at t ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Thus, in theory, the REMC should regain resonance in the core beyond t > 5 ms, thereby enhancing transport and reducing the RE current, but this was not captured in the current workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' A destructive kink instability might also be expected, as seen in experiment [19,20], for such low q(0) and high ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Even then it is not clear what overall effect this self-regulation would have on the RE beam which already has a current ∼1 MA by t ≈ 5 ms (for the base case with no island transport).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Luckily, the Ohmic current has almost completely decayed by then, and the relatively long L/R time (>10 ms) of the REMC will maintain the coil current and its perturbative effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Furthermore, any additional transport within the re-healed flux surfaces will help lower this quasi-stationary RE current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' A fluid RE model that could capture this ¶ See [17] for simulations of passing and trapped REs during an ITER CQ, including the effects of collisions and the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 8 t[ms] 0 1 3 5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content='5 0 2 4 6 8 10 r[m] q (a) li min(q) 0 1 2 3 4 5 6 0 1 2 3 4 5 t[ms] li, min(q) (b) Figure 5: DREAM results for the base case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 3b with D = 0 below D < 1000 m2/s: (a) evolution of the safety factor q-profile vs minor radius for five times, and (b) time evolution of the minimum q-value (dot-dashed) and internal inductance ℓi (solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' effect has been incorporated into the JOREK [21,22] and M3D-C1 [23] MHD codes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' a similar model is being implemented in NIMROD [24] and benchmarked against the existing codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Its application to the SPARC REMC will be pursued in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Acknowledgments Supported by Commonwealth Fusion Systems, Swedish Research Council (Dnr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' 2018-03911), US DOE Award Numbers DE-FC02-04ER54698 and DE-FG02-95ER54309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' This work has been carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 – EUROfusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Neither the European Union nor the European Commission can be held responsible for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' References [1] Allen H Boozer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} +page_content=' Two beneficial non-axisymmetric perturbations to tokamaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdAzT4oBgHgl3EQfefwA/content/2301.01435v1.pdf'} 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b/ztAyT4oBgHgl3EQfPPYT/content/tmp_files/2301.00019v1.pdf.txt @@ -0,0 +1,1284 @@ +Tailoring fusion-based error correction for high thresholds to biased fusion failures +Kaavya Sahay, Jahan Claes, and Shruti Puri +Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA +Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA +(Dated: January 3, 2023) +We introduce fault-tolerant (FT) architectures for error correction with the XZZX cluster state +based on performing measurements of two-qubit Pauli operators Z ⊗ Z and X ⊗ X, or fusions, on a +collection of few-body entangled resource states. Our construction is tailored to be effective against +noise that predominantly causes faulty X ⊗ X measurements during fusions. This feature offers +practical advantage in linear optical quantum computing with dual-rail photonic qubits, where failed +fusions only erase X ⊗ X measurement outcomes. By applying our construction to this platform, +we find a record high FT threshold to fusion failures exceeding 25% in the experimentally relevant +regime of non-zero loss rate per photon, considerably simplifying hardware requirements. +Introduction— Fault-tolerant (FT) error correction en- +ables arbitrary suppression of errors as long as the error +rate is below a constant threshold, making scalable quan- +tum computation possible. It is important to take into +consideration the underlying physical operations avail- +able when designing a FT architecture. +For example, +if the entangling operations are inherently probabilistic +or if the noise in these operations destroys the qubits, +then the framework of FT measurement-based error cor- +rection (MBEC) is more natural [1–3]. MBEC is imple- +mented using a cluster state [4–8], which is a many-body +entangled state and may be obtained from a stabilizer +error correcting code using a process called foliation [9– +12]. Outcomes of single-qubit measurements performed +on the cluster state are used to reconstruct the underly- +ing stabilizers and correct errors [9, 13]. +The measured +qubits are removed from the entangled state, allowing +considerable flexibility in how the measurements are real- +ized in hardware. For example, it is possible for the mea- +surement to be destructive and erase the measured qubit. +The most well known cluster state is the Raussendorf- +Harrington-Goyal (RHG) cluster state [9, 13, 14] which +is a foliation of the standard surface code [15]. Recently, +the XZZX cluster state was introduced [16], which is a +foliation of the XZZX surface code [17, 18]. +In the most common MBEC framework, the cluster +state is generated using a set of commuting two-qubit +entangling gates. Alternatively, one could start with a +collection of few-body entangled states and then stitch +them together into a many-body entangled cluster state +using measurements of two-qubit Pauli operators X ⊗ X +and Z ⊗ Z, also called fusions or Bell measurements, +which may be implemented destructively [19]. This ap- +proach has been referred to as fusion-based error cor- +rection (FBEC) [3] and is a more natural choice for ar- +chitectures where high-fidelity fusions are native to the +hardware like discrete variable photonic qubits [19], con- +tinuous variable qubits [20, 21], and Majorana-based +qubits [22]. The FBEC framework has been studied for +error correction with the RHG cluster state [3] and, more +recently, with the foliated floquet color code [23]. +In this paper, we introduce fusion-based architecture +for error correction with the XZZX cluster state. +We +present two constructions, one based on using a collection +of 4-body entangled resource states and the other based +on using a set of 6-body entangled resource states. Im- +portantly, both the constructions offer practical advan- +tage when noise in the fusion circuit is biased so that the +Z ⊗Z measurements are much more reliable than X ⊗X +measurements. This is because errors introduced in the +cluster states due to faulty X⊗X measurements, referred +to as biased fusion failures, give rise to a two-dimensional +system symmetry [24] which considerably simplifies the +decoding problem, leading to a substantial improvement +in the threshold to biased fusion noise. +Our construction is motivated by dual-rail qubits in +linear optics [25–28], which is the most widely studied +platform in the framework of FBEC [3, 19, 29]. Linear- +optic fusions on dual rail qubits are inherently proba- +bilistic. +The simplest fusion circuit fails with proba- +bility 1/2 [19]. The failure probability can be reduced +to 1/2n using an ancillary (2n − 2)-photon entangled +state [30], although for the special case of 1/4 failure +probability, 4 unentangled photons are also sufficient [31]. +Notably, when a fusion attempt fails, the X ⊗ X infor- +mation is completely erased but Z ⊗ Z can still be re- +covered [3, 19]. The architecture proposed here leverages +this biased noise structure to achieve record-high thresh- +olds to fusion failures. +With numerical simulations of +the fusion-based XZZX cluster state with photonic dual- +rail qubits and entangled ancillae, we find a threshold +to fusion failures exceeding 25% in the experimentally +relevant regime of non-zero loss rate per photon. This +is the highest known threshold to fusion failures in lin- +ear optics without additional encodings on the photonic +state [3, 23, 32, 33], and for the first time allows scalable +FBEC using an ancilla of only two entangled photons or +four unentangled photons. +The XZZX cluster state– We start with a review of +the XZZX cluster state. +It is a specific instance of a +generalized cluster state [16], a stabilizer state defined on +a decorated graph G = (V, E) with two types of vertices +arXiv:2301.00019v1 [quant-ph] 30 Dec 2022 + +2 +V = X ⊔Z. Each vertex represents a qubit of the cluster +state; we refer to v ∈ X as X-type qubits and denote +them by �, and we refer to v ∈ Z as Z-type qubits and +denote them by ⃝. The N-qubit generalized cluster state +is the +1 eigenstate of N mutually commuting stabilizers, +one centered at each qubit v ∈ V , given by +� +� +� +� +� +� +� +� +� +� +� +� +� +Xv +� +(v,w)∈E +w∈Z +Xw +� +(v,u)∈E +u∈X +Zu, +v ∈ X +Zv +� +(v,w)∈E +w∈Z +Xw +� +(v,u)∈E +u∈X +Zu, +v ∈ Z +. +(1) +Essentially, for each v ∈ X we have a stabilizer given by +the product of Xv on that qubit, and some combination of +X and Z operators on the neighboring qubits. Similarly, +for each v ∈ Z we have a stabilizer given by the product +of Zv on that qubit, and some combination of X and Z +operators on the neighbors. These neighboring X and Z +operators are chosen to ensure all stabilizers commute. +The XZZX cluster state is defined on a periodic 3D +graph, a unit cell of which is shown in Fig. 1(a), along +with stabilizers centered at an X-type qubit and Z-type +qubit. +Note that there are no edges between Z-type +qubits. +The XZZX cluster state has the same geome- +try as the RHG cluster state [9, 14], and can be obtained +from it by local Clifford operations. Taking the product +of the stabilizers centered on the faces of a unit cell gives +the cell stabilizer shown in Fig. 1(b), which is used to +correct errors. Performing a measurement-based compu- +tation using the XZZX cluster state involves measuring +all Z-type qubits in the Z basis and all X-type qubits in +the X basis. Once we have measured all qubits in their +respective bases, we use the cell stabilizers to check for +errors in our measurement outcomes. A Z error on an +X-type qubit or an X error on a Z-type qubit causes the +qubit’s two neighboring cell stabilizers to flip to (−1), as +shown in Fig. 1(c). Importantly, we note that Z errors +on X-type qubits only cause pairs of defects that are re- +stricted to 2D planes. This 2D system symmetry simpli- +fies the decoding problem—for example, a matching de- +coder only needs to match defects in 2D—and ultimately +leads to higher thresholds for Z-biased noise [16, 24]. +The XZZX cluster state may be prepared using +controlled-not and controlled-phase gates, and the ro- +bustness of this cluster state to gate noise has been stud- +ied previously [16]. In this work we consider the alterna- +tive approach of preparing this state by fusing copies of +few-body entangled resource states, which is the standard +approach for realizing cluster states in photonic dual-rail +platforms [3, 19, 25, 34]. We will consider two schemes +for the XZZX cluster state, which expand on schemes +introduced in Ref. [3] for the RHG cluster state. The +important distinction is that in both of our schemes bi- +ased fusion failures create pairs of defects restricted to +2D planes, leading to improved thresholds. +FIG. 1. (a) A unit cell of the XZZX cluster state, with two +examples of stabilizers centered on the faces of the cell as +given by Eq. (1). (b) If we multiply the stabilizers centered +at all faces of a unit cell, we get the cell stabilizer shown. (c) +The value of the cell stabilizer allows us to detect errors. X +errors on Z-type qubits or Z errors on X-type qubits cause the +value of the neighboring cell stabilizers to flip. Importantly, +Z errors cause pairs of defects restricted to 2D planes, which +allows for more effective decoding of Z errors. +Construction from 4-star resource states— In the fol- +lowing discussion we introduce the principle underlying +our construction. Consider a cluster state defined on a +graph G = (V, E). Let G have a Z-type qubit at a degree- +1 vertex vi ∈ Z with an edge to a qubit at vi′, and an +X-type qubit at degree-1 vertex vj ∈ X with an edge to a +qubit at vj′ ̸= vi′. We will refer to the qubits on degree-1 +vertices as dangling qubits. As shown in Fig. 2(a), per- +forming Xi ⊗ Xj and Zi ⊗ Zj measurements on the pair +of dangling qubits disentangles them from the rest of the +system while entangling their neighbors, removing ver- +tices vi, vj and edges (vi, vi′), (vj, vj′) and adding a new +edge (vi′, vj′). Consequently, the stabilizers centered at +vi and vj are removed and we obtain two new stabiliz- +ers centered at vi′ and vj′ defined according to Eq. (1). +To ensure that the new cluster state is the +1 eigenstate +of these new stabilizers, a Pauli correction is applied to +the qubits at vi′ and vj′ according to the outcomes of +the Xi ⊗ Xj measurement (mXX = 0 or 1) and Zi ⊗ Zj +measurement (mZZ = 0 or 1). If vi′ ∈ X and vj′ ∈ Z, +the correction is ZmXX +i′ +⊗ XmZZ +j′ +, while if vi′, vj′ ∈ X, +the correction is ZmXX +i′ +⊗ ZmZZ +j′ +. It is not necessary to +physically apply these Pauli corrections and instead they +may just be tracked in software. Observe that in case + +3 +FIG. 2. +The 4-star construction. +(a) Performing a fusion +measurement of Xi ⊗Xj and Zi ⊗Zj on a dangling pair of X- +and Z-type qubits removes them from the cluster but forms an +edge between their neighbors. (b) The two five-qubit cluster +states we use in our construction. (c) The arrangement of five- +qubit cluster states we use to build the XZZX cluster state. +(d) Because the qubits of the resulting cluster state will be +measured in the X/Z basis, and because these measurements +commute with the fusion measurements, we can measure the +center qubits before performing the fusions. Equivalently, we +can instead start with the simpler four-qubit resource states +that result from measuring the center qubit as (+1) in the +X/Z basis. In this case, the central qubits making up the +cluster state are never physically realized, but exist as virtual +qubits whose measurement outcomes are tracked in software. +of unreliable Xi ⊗ Xj (Zi ⊗ Zj) measurements, we can- +not correctly determine the proper Pauli correction on +vi′ (vj′) which results in an effective error on that qubit. +In fact, a complete erasure of Xi ⊗ Xj measurement out- +come that arises due to fusion failure in linear-optics is +equivalent to applying I or Z to the X-type qubit at vi′ +with 50% probability [35]. This type of Z-biased error at +a known location in the cluster state (vi′), marked by the +location of the failed (Xi⊗Xj) measurement, is classified +as Z-biased fusion failure. +With the above discussion in mind we introduce the +two five-qubit cluster states shown in Fig. 2(b) with sta- +bilizers defined according to Eq. (1). One has a Z-type +qubit at the center and the other has a X-type qubit. +The center qubits are marked in red to indicate that these +will eventually form the desired XZZX cluster state. The +Z-centered and X-centered states are placed at the loca- +tion of Z- and X-type qubits respectively in the desired +cluster state, as shown in Fig. 2(c). The arrangement +of the 5-body states ensures that neighboring dangling +qubits are always opposite types; we can thus fuse the +neighboring dangling qubits according to Fig. 2(a). The +fused qubits are removed from the cluster and new bonds +appear between the red qubits, resulting in the desired +XZZX cluster state. Finally, the cluster state qubits can +be measured in the appropriate basis described in the +previous section for error correction. Note that each cen- +ter qubit is entangled into the final cluster state after four +fusion measurements on its neighboring dangling qubits; +consequently, four Pauli corrections need to be accounted +for on this qubit. This accounting may be done in soft- +ware by simply re-interpreting the outcome of the final +measurement of cluster state qubits. This is because a X +(Z) measurement of the X- (Z-) type qubit in the cluster +state after a Pauli Z (X) is applied to it is equivalent to +a X (Z) measurement followed by a classical flip of the +measurement outcome 0 → 1, 1 → 0. +It is possible to simplify the initial 5-qubit cluster +states to 4-qubit resource states by noting that measuring +the qubits comprising the final cluster state commutes +with fusion measurements, as these measurements are +performed on different qubits and Pauli corrections due +to fusions can simply be accounted for in software. That +is, we can equally well measure the center qubits which +would form the XZZX cluster state before performing the +fusion measurements. Once the fusions are realized, the +outcomes of the prior center-qubit measurements can be +flipped conditional on the fusion outcomes. Measuring +the center X- and Z-type qubits of the 5-qubit states in +the X and Z basis respectively leaves us with the simpler +4-star resource states shown in Fig. 2(d). Thus we can +directly start with these resource states assuming that +the center qubits have been measured in the X/Z basis +with measurement outcome 0. In this approach, the cen- +tral qubit acts like a virtual qubit. It is never physically +realized and never physically measured, and its effective +measurement outcome is entirely determined by the Pauli +corrections that are tracked in software [3]. +Construction from 6-ring resource states— While the +previous construction relied on fusing an X-type qubit +with a Z-type qubit, this second construction relies on +fusing two qubits of the same type. While our construc- +tion is reminiscent of the approach in [3], we provide an +alternate intuitive interpretation of why it works. +Consider a cluster state defined on a graph G = (V, E) +with Z (X)-type qubits at vertices vi, vj ∈ V such that +vi and vj are not neighbors and share no neighbors in +common. +Measuring Xi ⊗ Xj (Zi ⊗ Zj) on these two +qubits projects them into an effective two-dimensional +subspace with the Pauli operators ¯X = Xi (Xi ⊗ Xj) +and ¯Z = Zi ⊗ Zj (Zi). As shown in Fig. 3(a), a new +cluster state is obtained with vertices vi, vj replaced by a +single vertex vij and an effective Z (X)-type qubit placed +at this vertex. All the edges incident at vi and vj are in- +cident at vij in the new graph. To ensure that the new +cluster state is the +1 eigenstate of all the stabilizers, a + +4 +Pauli correction determined by the Xi ⊗ Xj (Zi ⊗ Zj) +measurement outcome, mXX = 0 or 1 (mZZ = 0 or 1), +must be applied to the qubits that were originally ad- +jacent to vj. +Specifically ZmXX (ZmZZ) is applied to +adjacent X-type qubits and XmXX (XmZZ) is applied to +adjacent Z-type qubits. +With the above merging principle in mind, we intro- +duce the 6-ring resource cluster state in Fig. 3(b). A copy +of this state is placed at two opposite corners of each unit +cell of the XZZX cluster state as shown in Fig. 3(c). Two +qubits of the same type share a face centre or edge centre. +If we measure X ⊗ X (Z ⊗ Z) for each pair of Z (X)- +type qubits sharing a face/edge, and apply the required +Pauli corrections, we obtain the desired XZZX cluster +state comprised of the effective qubits. Note that an un- +reliable X ⊗X measurement on Z-type qubits only leads +to an incorrect Pauli Z correction to adjacent X-type +qubits. Finally, we need to measure the effective Pauli +¯X = X ⊗ X of the effective X-type qubits and effective +Pauli ¯Z = Z ⊗ Z of the effective Z-type qubits for error +correction. Note that an unreliable X ⊗ X measurement +in the second set of measurements is like a ¯Z error on the +effective X-type qubits. As before, Pauli corrections from +the first set of measurements that create the cluster state +may be simply tracked in software by re-interpreting the +outcomes of the second set of measurements that are used +for error correction. Overall then, error correction is im- +plemented in our construction by measuring commuting +observables X ⊗X and Z ⊗Z, that is performing fusions, +on pairs of qubits that share an edge or a face centre. The +above discussion implies that biased fusion failure, as is +the case in linear optics, effectively leads to biased Pauli +Z noise on the X-type qubits of the XZZX cluster state. +Application +to +dual-rail +qubits +with +linear-optic +fusions—A photonic dual-rail qubit is encoded in the +presence of a single photon in one of two orthogonal +modes, |¯0⟩ = |01⟩ , |¯1⟩ = |10⟩ and is a leading candidate +for linear-optical quantum computing [19, 25, 28, 36]. In +this platform, all single qubit gates, like the Hadamard +gate, can be performed deterministically using passive +linear optical elements [37]. Multi-qubit operations, like +the fusion measurements, are non-deterministic. Fault- +tolerant FBEC in this platform is based on heralded gen- +eration of few-body entangled resource states, followed +by non-deterministic fusion measurements [3, 29]. +Re- +markably, the resource states used in our proposal differ +from the ones considered in previous works [3] only by +Hadamard transformations and these states can be eas- +ily generated using well-known linear-optics circuits. For +clarity, in [35] we also give circuits for generation of our +four-star and six-ring states. The required fusion mea- +surements can be realized using a so-called type-II fusion +circuit comprised of beamsplitters and photon number re- +solving detectors [19, 27]. The circuit consumes the two +dual-rail qubits which need to be fused and outputs the +photon number/clicks observed at each detector. Con- +FIG. 3. Building the XZZX cluster state using 6-ring resource +states. (a) Measuring Xi ⊗ Xj (Zi ⊗ Zj) on two Z (X)-type +qubits projects those qubits onto a two-dimensional subspace +with effective Pauli operators ¯ +Xij = Xi; (Xi ⊗Xj) and ¯Zij = +Zi ⊗ Zj (Zi). In terms of the effective Pauli operators, the +two Z (X)-type qubits are replaced by a single Z (X)-type +qubit vij, with all edges to neighbors intact. (b) The 6-ring +resource state. (c) The fusion pattern used to join the 6-ring +states to make the XZZX cluster state. +ditional on the observed clicks, one of two operations is +implemented by the circuit on the input qubits. Either a +successful X⊗X and Z⊗Z measurement is implemented, +in which case we say the fusion has succeeded and the +measurement outcomes are inferred from the detector +clicks. Or, each input qubit is independently measured +in the Z-basis, in which case we say that the fusion has +failed and the independent Z measurement outcomes are +again inferred from detectors clicks [3, 35]. Consequently, +in a failed-fusion event, Z ⊗Z can be recovered by multi- +plying the independent Z measurement outcomes, while +the X ⊗ X measurement is completely erased. Without +any ancilla photons, the probability of fusion failure is +pfail = 1/2. With additional (2n − 2)-photon entangled +ancilla, for a total of 2n photons in the fusion circuit, the +failure probability may be reduced to pfail = 1/2n [30]. +For the particular case of n = 2, the failure probability +may also be reduced to 1/4 with a 4-photon unentangled +ancilla [31]. However, n > 2 requires entangled states +that get progressively more complicated to realize. +So far we have ignored hardware imperfections that +can introduce photon loss. Fortunately, the ideal fusion +circuit preserves the number of photons, that is, the to- +tal detector clicks must be equal to the number of input +photons. A loss of any photon in the fusion circuit will +be heralded by observing fewer than expected clicks at +the detector and result in an erasure of both X ⊗ X and +Z ⊗ Z measurement outcomes. If the probability of loss +per photon is ploss, then the probability of such an erasure +in the boosted fusion circuit with a total of 1/pfail pho- +tons is pfull erase = 1 − (1 − ploss)1/pfail [3]. This equation +highlights the tradeoff between the rate at which only the + +5 +X ⊗ X outcome is erased due to fusion-failure and the +rate at which both X ⊗X and Z ⊗Z outcomes are erased +due to photon loss. As we decrease pfail by adding more +photons to the fusion circuit, we increase the probability +of photon loss from the fusion circuit pfull erase. +We evaluate the performance of our fusion architec- +tures under the linear optical error model including fusion +failures and photon loss as described above. We perform +Monte-Carlo simulations of errors when the four-star and +six-ring states are used for FBEC with d × d × d XZZX +cluster states. The syndrome data generated by the er- +rors is arranged on a graph and decoded using the linear- +time erasure-decoder [38]. We evaluate the logical error +rates for different cluster state sizes d, and error param- +eters pfail, and ploss and estimate the threshold which +is plotted in Fig. 4. The solid red and blue curves are +the thresholds for the 4-star and 6-ring constructions re- +spectively. +If ploss and pfail lie under the curves, then +these errors are correctable, otherwise they are not. We +also compare our results with the thresholds obtained +with (a) the 4-star and 6-ring construction from a previ- +ous work [3] to construct the XZZX cluster state and (b) +our 4-star and 6-ring constructions based on the adaptive +error-correction strategy introduced in [33]. The thresh- +olds with both (a) and (b), shown in Fig. 4 using dashed +lines, are identical to each other as they have identical +error syndrome graphs and are consistent with known +results [3]. +We observe, first, that the thresholds for the six-ring +construction are higher than the four-star construction +as it has fewer fusions, and hence a lower probability of +error, per cluster state qubit. +In the absence of pho- +ton loss, the numerically obtained threshold for biased +fusion-failure using our scheme is 34.7% for the six-ring +construction and 20.6% for the four-star construction. +Without photon loss, these thresholds can also be derived +analytically (see Supplement [35]) and are significantly +higher than the threshold for fusion-failure obtained with +previous proposals, which are correspondingly ∼ 24% for +the six-ring construction [3] and ∼ 14.5% for the four- +star construction [33], that fail to leverage the noise bias +in fusion failures. This is because, as argued earlier, in +our approach fusion failure leads to two-dimensional sys- +tem symmetry and hence a two-dimensional syndrome +graph which is easier to decode. In contrast, in previ- +ous strategies the syndrome graph resulting from fusion +failures is three-dimensional, which is harder to decode. +When ploss ̸= 0, we must deal with both full fusion +erasure along with fusion failure. +Increase in pfull erase +means we must decrease pfail by using additional pho- +tons. However, this also increases pfull erase due to the +possibility of these additional photons being lost. This +relationship between pfull erase, pfail, and ploss gives the +overall “inverted-u” shape of the threshold curve. Im- +portantly, from Fig. 4 we see that our scheme with the +six-ring resource states can tolerate up to 0.37% of pho- +FIG. 4. +Threshold with four-star (red) and six-ring (blue) +construction protocols as proposed here under the linear op- +tical error model. For comparison, thresholds based on previ- +ous approaches which do not leverage the noise bias in fusion +failures is also shown [3, 33]. Excluding XZZX FBEC points +on the x-axis, all points are simulated thresholds for sets of +cluster states of size d × d × d, with d up to 17. XZZX FBEC +points on the x-axis are thresholds for sets of d×d×4 cluster +states, with d ≤ 71. From left to right, the black dotted ver- +tical lines represent fusion success boosting by 30, 14, 6, and +2 additional entangled ancillae photons. +ton loss with 25% fusion failure. +That is, it is possi- +ble to achieve fault tolerance with only 2 additional en- +tangled ancilla photons for boosted fusions [30]. More +importantly, this means that we could even reach fault +tolerance with boosted fusions using only 4 additional +unentangled ancilla photons [31] and ploss ≤ 0.25%. On +the other hand, the maximum fusion failure that previ- +ous schemes can tolerate is < 25%, meaning they could +not achieve fault-tolerance with only 2 additional entan- +gled ancilla photons or 4 additional unentangled ancilla +photons. +Conclusion By taking advantage of the biased struc- +ture of fusion failures, we have introduced new resource +states and fusion strategies for FBEC that allow for more +efficient error correction of biased fusion failures. This +FBEC strategy is particularly relevant to linear-optical +quantum computers based on dual-rail photonic qubits, +where biased fusion failures are the dominant source of +error. Our resource states and fusion strategies require +no additional overhead to realize compared to the previ- +ous approach of Ref. [3], but result in higher thresholds +to fusion failures for both the 4-star and 6-ring resource +states. In the 6-ring construction in particular, our strat- +egy has a threshold over 25% to fusion failures, which +can be reached using only a 2-photon entangled ancilla +or a 4-photon unentangled ancilla; thus, our construction +overcomes a key barrier for photonic quantum comput- +ing. +Our construction achieves higher thresholds because: +(1) the fusion operations in linear optics have a natural +bias towards Z errors, (2) we construct our cluster state +in a bias-preserving way so that the final cluster state also + +6 +has a bias towards Z errors, and (3) XZZX cluster state +is designed to correct Z errors [16]. A similar strategy +should apply to any other hardware with a similar bias in +the entangling operations. 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A scheme for efficient quantum computation with +linear optics. nature, 409(6816):46–52, 2001. +[37] Alberto Politi, Jonathan CF Matthews, Mark G Thomp- +son, and Jeremy L O’Brien. Integrated quantum photon- +ics. IEEE Journal of Selected Topics in Quantum Elec- +tronics, 15(6):1673–1684, 2009. +[38] Nicolas Delfosse and Gilles Z´emor. Linear-time maximum +likelihood decoding of surface codes over the quantum era- +sure channel. Phys. Rev. Research, 2:033042, 2020. + +Supplemental Material: Tailoring fusion-based error correction for high thresholds to +biased fusion failures +Kaavya Sahay, Jahan Claes, and Shruti Puri +Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA +Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA +(Dated: January 3, 2023) +4-STAR CONSTRUCTION +Here we show the evolution of stabilizers when a fusion operation is conducted between two dangling qubits in the +XZZX cluster state. Continuing with the notation in the main text, we use a cluster state defined on a graph G. +Let G have a dangling qubit at vertex vi ∈ Z and vj ∈ X, each with an edge to a qubit at vertex vi′ and vj′ ̸= vi′ +respectively. Qubit vi′ (vj′) has further edges to a set of qubits {vi′′ } ({vj′′ }). In Fig. S1, we show the evolution of +stabilizers and associated Pauli frame updates when vi′ ∈ X and vj′ ∈ Z. In Fig. S2 we show the same for vi′, vj′ ∈ X. +The set of peripheral qubits {vi′′ } and {vj′′ } are not shown for simplicity, and indeed we shall see that they are not +affected by the fusion. Note that after the fusion, the qubits at vertices vi, vj are disentangled from the cluster state. +(a) Pre- +fusion +stabilizers += +′ , +′ = +′ +∏ +′ ′ +′′′ , +{ +′′ } = { +′′ +′ }, += +′ , +′ = +′ +∏ +′ ′ +′′′ , +{ +′′ } = { +′′ +′ } +(b) Fusion +success +(i) Checks measured +[outcome] += +, += +(ii) Fused system now +eigenstate of [eigenvalue] +′ ⋅ +⋅ += +′ +′ ∏ +′ ′ +′′′ +(-1) +, +′′ = +′′ +′ +1 , +⋅ +⋅ +′ = +′ +′ ∏ +′ ′ +′′′ +(-1) +, +′′ = +′′ +′ +1 +(iii) Pauli frame update to +impose +1 eigenvalue +′ +, +′ +(c) Fusion +failure +(i) Checks measured +[outcome] += +, += +(ii) System now in +eigenstate of [eigenvalue] +⋅ +⋅ +⋅ +′ = +′ +′ ∏ +′ ′ +′′′ +(-1) ++ +, +′′ = +′′ +′ +1 , +′′ = +′′ +′ +1 +(iii) Pauli frame update to +impose +1 eigenvalue +′ ++ +(iv) Projector on +′ +(1 + (-1) +′ /2 +1 + −1 +2 +{ +′′ } +{ +′′ } +(a) +(b) +(c) ++ +) +FIG. S1. Tracking the stabilizers and deriving the required Pauli frame updates for a fusion conducted on qubits at vertices +vi, vj when vi′ ∈ X and vj′ ∈ Z. We look at the state of the system (a) pre-fusion, (b) post-successful fusion, and (c) on fusion +failure. Note that Pi′′ (Pj′′ ) denotes a Pauli operator on vi′′ (vj′′ ). +Pre-fusion, the stabilizer generators {Sr} are defined according to Eq. (1) in the main text. If the fusion succeeds +(fails), we measure checks Xi ⊗ Xj, Zi ⊗ Zj (Zi, Zj) with measurement outcomes mXX, mZZ (mi, mj). Post-fusion, +the system is constrained by the surviving stabilizer set generated by (i) the original stabilizers that commute with +the measurements and (ii) the commuting products of those stabilizers that do not. The system will now be in the +eigenstate of this new generating set, given in row b(ii) of the table in Fig. S1,S2, with eigenvalues defined by the +measurement outcomes. Thus, in order to restore the system to the +1 eigenstate of a given stabilizer generator Sp, +we apply a Pauli frame update that is identified by locating the correction that anticommutes with Sp but no other +stabilizer generator. These corrections are shown in row b(iii). In practice, this Pauli correction is tracked in software. +In the case when a fusion failure occurs, the stabilizer of the combined system centered at vj′ is recovered by adding +the fusion measurement outcomes. In particular, when vj′ ∈ Z (vj′ ∈ X), the conditional Pauli frame update applied + +9 +(a) Pre- +fusion +stabilizers += +′ , +′ = +′ +∏ +′ ′ +′′′ , +{ +′′ } = { +′′ +′ }, += +′ , +′ = +′ +∏ +′ ′ +′′′ , +{ +′′ } = { +′′ +′ } +(b) Fusion +success +(i) Checks measured +[outcome] += +, += +(ii) Fused system now +eigenstate of [eigenvalue] +′ ⋅ +⋅ += +′ +′ ∏ +′ ′ +′′′ +(−1) +, +′′ = +′′ +′ +′ +1 , +⋅ +⋅ +′ = +′ +′ ∏ +′ ′ +′′′ +(−1) +, +′′ = +′′ +′ +′ [+1] +(iii) Pauli frame update to +impose +1 eigenvalue +′ +, +′ +(c) Fusion +failure +(i) Checks measured +[outcome] += +, += +(ii) System now in eigenstate +of [eigenvalue] +⋅ +⋅ +⋅ +′ = +′ +′ ∏ +′ ′ +′′′ +(−1) ++ +, +′′ = +′′ +′ [+1], +′′ = +′′ +′ [+1] +(iii) Pauli frame update to +impose +1 eigenvalue +′ ++ +(iv) Projector on +′ +(1 + (-1) +′ /2 +1 + −1 +′ +2 ++ +{ +′′ } +{ +′′ } +(a) +(b) +(c) +) +FIG. S2. Tracking the stabilizers and deriving the required Pauli frame updates for a fusion conducted on vi, vj when vi′, vj′ ∈ X. +We look at the state of the system (a) pre-fusion, (b) post-successful fusion, and (c) on fusion failure. +to it is an X (Z) operator. However, a projector (1 + (−1)miZ)/2 is applied to vertex vi′, which destroys its X +measurement outcome information. This is equivalent to a successful fusion followed by a Z measurement on vi′, a +process described by the error channel ρ �→(ρ + Zi′ρZi′)/2. +Additionally, we note that the sets of stabilizers {Si′′ }, {Sj′′ } centered at the neighbouring qubits remain undis- +turbed in all cases. We shall use this fact to simplify our analysis in the next section. +6-RING CONSTRUCTION +(a) Pre-fusion +stabilizers += +1 +2, +1 = +1 +, +2 = +2 +, += +1 +2, +1 = +1 +, +2 = +2 +(b) Fusion +success +(i) Checks measured to create +virtual qubit [outcome] += +[ +] +(ii) Measured system now +eigenstate of [eigenvalue] +Note += +⋅ += +1 +2 +1 +2 [+1], +1 = +1 ++1 , +2 = +2 ++1 , +Sj1 ⋅ +2 = +1 +2 +1 , +1 ⋅ += +1 +[ −1 +], +2 ⋅ += +2 +[ −1 +] +(iii) Pauli frame update to +impose +1 eigenvalue +1 +2 +(c) Fusion +failure +(i) Checks measured +[outcome] += +, += +(ii) Measured system now in +eigenstate of [eigenvalue] +⋅ += +1 +2 +−1 +, +⋅ += +1 +2 +−1 +, +1 ⋅ +2 = +1 +2 +1 , +1 ⋅ +2 = +1 +2 +1 , +(iii) Effec�ve correlated +projector on +(1 +(−1) ++ +1 +2)/2 +FIG. S3. Tracking the stabilizers and deriving the required Pauli frame updates for the six-ring construction. Measurements +are conducted on vi, vj with vi′, vj′ ∈ Z. We look at the system (a) pre-fusion, (b) post-successful fusion, and (c) on fusion +failure. +In this section we show how measuring X ⊗X (Z ⊗Z) for Z-type (X-type) qubits in six-ring cluster states produces +virtual qubits with Pauli corrections on neighboring qubits and derive the noise introduced in the cluster state due to + +10 +erasure of X ⊗ X measurement outcome. Let the cluster state graph G have a degree-2 vertex vi (vj ) with an edge +to two qubits at vertices {vik} ({vjk}) with {vjk}∩{vjk} = ∅. An effective cluster state qubit vij is formed when such +a measurement is performed between vi and vj. In Fig. S3, we show the evolution of stabilizers and associated Pauli +frame updates when vi, vj ∈ Z. Note that we have disregarded further qubits connected to {vik}, and {vjk} since +their stabilizers remain unchanged during this process, just like in the 4-star construction discussed in the previous +section. +Similar to the previous section, we update the stabilizers according to the checks measured. Clearly we see that +measuring Xi⊗Xj creates a virtual qubit with effective Pauli ¯Z = Zi⊗Zj. Note that on fusion failure, due to the form +of the system stabilizers in this setup, two X-type qubits initially connected to vj experience a correlated projection +into an eigenstate of Zj1Zj2. Individual errors on either of the two qubits vj1, vj2 connect unit cell syndromes in +mutually perpendicular directions in the cluster state. As a result, this correlated error effectively connects a unit cell +syndrome with its diagonally displaced neighbour. +4-STAR RESOURCE STATE GENERATION +Four-star resources states can probabilistically be generated from single photons by a sequence of beam splitters +and photon detectors, conditioned on a certain detector output. A potential generation circuit is shown in Fig. S4, +based on prior constructions for three-body GHZ states [1, 2]. This circuit succeeds if a single photon is output at +each detector pair, where a detector pair is defined at the two output ports of a single beam splitter. +������� +������� +������� +������� +��� ++++ + + ��� � +�2 +000 + + 111 � +�2 +��� +��� +�������������������� +������������������� +��������������� +� +� +� +� +� +� +� +� +FIG. S4. Construction of the four-star resource state for the XZZX cluster state. (a) Representations of the two resource states +(b) Probabilistic photonic fusion circuit to construct the state (| + + + +⟩ + | − − − −⟩) / +√ +2. (c) Beam splitters are applied +to three qubits in the state output from (b) to obtain the second resource state (|000+⟩ + |111−⟩) / +√ +2. +6-RING RESOURCE STATE GENERATION +As described in Fig. S5(a), a six-ring resource state can be constructed by performing Type-I fusions [3] on a set +of GHZ states. We present a potential photonic circuit to probabilistically construct six-rings from single photons +in Fig. S5(b,c). The base GHZ state (| + ++⟩ + | − −−⟩) / +√ +2 (upto heralded Pauli corrections) is generated by the +circuit shown in Fig. S5(b) when one photon is detected at each pair of detectors. This occurs with probability +1/32, which can be effectively increased to near-unity via multiplexing. Two beam splitters performing Hadamard +operations to specific qubits can be applied to this GHZ state in order to get the complete input set of GHZ states. +These input states are fed into the circuit in Fig. S5(c), which performs Type-I fusions between the peripheral qubits +of the individual GHZ states. Note that this secondary circuit layer has a success probability (1/2)3 which can be +increased by further multiplexing. Derivation of a more efficient circuit is left to future work. + +11 +������� +������� +������� +��� +��� +���� +������ +������� +��� +������� +� +� +� +� +� +� +������� +������ +������� +���� +������ +������� +������� +������ +������� +������� +������ +������� +������� +������� +������� +������� +������� +FIG. S5. Construction of the 6-ring resource state for the XZZX cluster state. (a) (top) Type-I fusions are conducted between +input GHZ states to form the resource state. (bottom) Representations of the input GHZ states. (b) Probabilistic photonic +circuit to construct the state (| + ++⟩ + | − −−⟩) / +√ +2 [2]. Hadamards can be applied to specific qubits to obtain the full input +state set. (c) Photonic circuit to conduct Type-I fusions between output GHZ states to construct a six-ring state. +BIASED FUSION FAILURES IN BOOSTED TYPE-II FUSION +The aim of the fusion measurements is to measure Xi ⊗ Xj and Zi ⊗ Zj between two qubits labelled by i, j. That +is, in terms of the dual rail encoding |¯0⟩ = |10⟩, |¯1⟩ = |01⟩, the aim is to discriminate between the four states +|ψ±⟩ = (|¯0i¯1j⟩ ± |¯1i¯0j⟩)/ +√ +2 and |φ±⟩ = (|¯0i¯0j⟩ ± |¯1i¯1j⟩)/ +√ +2 which are the simultaneous eigenstates of XiXj and +ZiZj: +ZiZj |ψ±⟩ = − |ψ±⟩ , XiXj |ψ±⟩ = ± |ψ±⟩ , ZiZj |φ±⟩ = |φ±⟩ , XiXj |φ±⟩ = ± |φ±⟩ +(S1) +In a direct type-II fusion [3, 4], the four modes of the two dual-rail qubits are interfered using two 50 : 50 beam-splitters +������� +������� +FIG. S6. Photonic circuit for type-II fusion between two dual rail qubits using beam splitters and photon detectors. +as shown in Fig. S6. The transformed states at the ouput of the network are: +|ψ+⟩ → +i +√ +2(|1100⟩ + |0011⟩) +|ψ−⟩ → +i +√ +2(|1001⟩ − |0110⟩) +|φ+⟩ → i +2(|2000⟩ + |0200⟩ + |0020⟩ + |0002⟩) +|φ−⟩ → i +2(|2000⟩ − |0200⟩ + |0020⟩ − |0002⟩) +(S2) +As can be seen, the photon number distribution n across the four output modes, which can be measured using PNRDs, +carries information about the input states. The output photon number distribution generated if |ψ+⟩ is input into + +12 +the interference network, N|ψ+⟩ = {1100} or {0011}, discriminates |ψ+⟩ from other states. Similarly, the output +photon number distribution generated if |ψ−⟩ is input into the interference network, N|ψ−⟩ = {1001} or {0110}, +discriminates |ψ−⟩ from other states. Unfortunately, however, it is not possible to discriminate between |φ+⟩ and +|φ−⟩ using PNRDs as they produce the same output photon number distribution: N = {2000} or N = {0200} or +N = {0020} or N = {0002}. In this case we say that the fusion fails. Given an equal distribution of Bell states at the +input (which is the case in 4-star and 6-ring fusions), the probability of failure is 50%. However, note that measuring +N = {2000} or N = {0020} discriminates (|φ+⟩ + |φ−⟩)/ +√ +2 = |0i0j⟩ and the output photon number distribution +N = {0200} or N = {0002} discriminates (|φ+⟩ − |φ−⟩)/ +√ +2 = |1i1j⟩. Thus, even when fusion fails, Zi and Zj of each +dual-rail qubit is still measured. +Refs. [5, 6] introduced protocols for fusions with success probability boosted by lifting the degeneracy in the output +photon-number distribution when |φ±⟩ are input into a interference network. Here, we give a broad overview of the +two protocols without repeating the details in Refs. [5, 6]. The broad overview will be sufficient to see that even when +boosted fusions fail, Zi and Zj of each dual-rail qubit is still measured. +The protocols in [5, 6] use auxillary entangled-photons, |Γ⟩. The exact form of this entangled state is not important +for our purpose here, but |Γ⟩ required for the protocol in [5] is different from that in [6]. |Γ⟩, together with the state +of the two duil-rail qubits, transform through the interference network as follows, +|ψ+⟩ |Γ⟩ → | |N ′ +ψ+⟩ +|ψ−⟩ |Γ⟩ → | |N ′ +ψ−⟩ +|φ+⟩ |Γ⟩ → x| |N ′ +φ+⟩ + y(|A⟩ + |B⟩) +|φ−⟩ |Γ⟩ → x| |N ′ +φ−⟩ + y(|A⟩ − |B⟩) +(S3) +Here, x, y are c-numbers for normalization and | |N ′ +ψ±⟩, | |N ′ +φ±⟩, |A⟩, |B⟩ are states with distinct photon-number distri- +butions. | |N ′ +ψ±⟩, | |N ′ +φ±⟩ may be a superposition of many Fock states (indicated by | | ⟩). Clearly, the protocol uniquely +identifies |ψ±, φ±⟩ if PNRDs measure photon number distributions consistent with | |N ′ +ψ±⟩ , | |N ′ +φ±⟩. However, the +protocol fails if photon number distribution corresponding to the states |A⟩ or |B⟩ is measured. Measuring photon +number distribution consistent with |A⟩ or |B⟩ discriminates (|φ+⟩ + |φ−⟩)/ +√ +2 or (|φ+⟩ − |φ−⟩)/ +√ +2 respectively and, +like the direct type-II measurement, this is equivalent to measuring Zi, Zj. This feature of fusion measurements has +also been identified in [7]. The key difference in the protocols of [5] and [6] is the nature of |Γ⟩ and the success +probability. In [5] a success probability of (1 − 2−n) is obtained with (2n − 2)-photon entangled state, while in [6], +twice as many photons are required to reach the same success probability. Importantly, with the protocol of [6], |Γ⟩ +for 75% successful fusion is an unentangled state of 4 photons. +ANALYTIC THRESHOLD IN CASE OF ONLY BIASED FUSION FAILURES +Consider the four-star construction from Fig. 2 in the main text. Suppose the probability of erasing only X ⊗ X +measurement outcome or the probability of a biased fusion failure is p. An incorrect X ⊗X measurement implies that +an incorrect Pauli Z correction may be applied to a X type qubit that eventually makes the cluster state. Each such +X type qubit gets a Pauli Z correction from the fusions of three neighboring dangling qubits. Thus, effectively the +probability of a Z error on each X type qubit forming the cluster state is 3p(1 − p)2 + 3p2(1 − p) + p3. The syndrome +graph in 2D is the 44 tiling of the plane with a bond percolation threshold of 50% [8]. Thus, the threshold pth may +be found by requiring that 3pth(1 − pth)2 + 3p2 +th(1 − pth) + p3 +th = 0.5 or pth = 20.63%. +In order to justify the threshold for biased fusion failures with the six-ring construction, we examine the syndrome +graph formed by fusion failures between resource states. A fusion failure between two X-type qubits creates a an +erasure on this qubit, leading to (−1) stabilizer outcomes on adjoining unit cells within a plane. When a fusion +between two Z-type qubits fails, the two adjacent X-type qubits in the resource state are projected into an eigenstate +of Z ⊗ Z. This leads to a pair of (−1) stabilizer outcomes on unit cells diagonally displaced from one another in the +same plane. Note that all such diagonals have the same orientation. As a result, the syndrome graph in 2D results +in a 36 tiling of the plane. This has a percolation threshold of 2 sin(π/18) ≈ 34.73% [9], which agrees with numerical +simulations of this construction at large lattice sizes. + +13 +[1] Michael Varnava, Daniel E Browne, and Terry Rudolph. How good must single photon sources and detectors be for efficient +linear optical quantum computation? Physical Review Letters, 100(6):060502, 2008. +[2] Ying Li, Peter C Humphreys, Gabriel J Mendoza, and Simon C Benjamin. Resource costs for fault-tolerant linear optical +quantum computing. Physical Review X, 5(4):041007, 2015. +[3] Daniel E Browne and Terry Rudolph. Resource-efficient linear optical quantum computation. Physical Review Letters, +95(1):010501, 2005. +[4] Alexei Gilchrist, AJF Hayes, and TC Ralph. Efficient parity-encoded optical quantum computing. Physical Review A, +75(5):052328, 2007. +[5] Warren P Grice. +Arbitrarily complete bell-state measurement using only linear optical elements. +Physical Review A, +84(4):042331, 2011. +[6] Fabian Ewert and Peter van Loock. 3/4-efficient bell measurement with passive linear optics and unentangled ancillae. +Physical review letters, 113(14):140403, 2014. +[7] Sara Bartolucci, Patrick Birchall, Hector Bombin, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, +Konrad Kieling, Naomi Nickerson, Mihir Pant, et al. Fusion-based quantum computation. arXiv preprint arXiv:2101.09310, +2021. +[8] Thomas M Stace, Sean D Barrett, and Andrew C Doherty. Thresholds for topological codes in the presence of loss. Physical +review letters, 102(20):200501, 2009. +[9] M. F. Sykes and J. W. Essam. Exact critical percolation probabilities for site and bond problems in two dimensions. Journal +of Mathematical Physics, 5(8):1117–1127, 1964. + diff --git a/ztAyT4oBgHgl3EQfPPYT/content/tmp_files/load_file.txt b/ztAyT4oBgHgl3EQfPPYT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ff6dbc533b2007b27e6ef384e9bdf2f72d0a9537 --- /dev/null +++ b/ztAyT4oBgHgl3EQfPPYT/content/tmp_files/load_file.txt @@ -0,0 +1,545 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf,len=544 +page_content='Tailoring fusion-based error correction for high thresholds to biased fusion failures Kaavya Sahay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Jahan Claes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' and Shruti Puri Department of Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Yale University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' New Haven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Connecticut 06511,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' USA Yale Quantum Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Yale University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' New Haven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Connecticut 06511,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' USA (Dated: January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 2023) We introduce fault-tolerant (FT) architectures for error correction with the XZZX cluster state based on performing measurements of two-qubit Pauli operators Z ⊗ Z and X ⊗ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' or fusions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' on a collection of few-body entangled resource states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Our construction is tailored to be effective against noise that predominantly causes faulty X ⊗ X measurements during fusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This feature offers practical advantage in linear optical quantum computing with dual-rail photonic qubits, where failed fusions only erase X ⊗ X measurement outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' By applying our construction to this platform, we find a record high FT threshold to fusion failures exceeding 25% in the experimentally relevant regime of non-zero loss rate per photon, considerably simplifying hardware requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Introduction— Fault-tolerant (FT) error correction en- ables arbitrary suppression of errors as long as the error rate is below a constant threshold, making scalable quan- tum computation possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' It is important to take into consideration the underlying physical operations avail- able when designing a FT architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' For example, if the entangling operations are inherently probabilistic or if the noise in these operations destroys the qubits, then the framework of FT measurement-based error cor- rection (MBEC) is more natural [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' MBEC is imple- mented using a cluster state [4–8], which is a many-body entangled state and may be obtained from a stabilizer error correcting code using a process called foliation [9– 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Outcomes of single-qubit measurements performed on the cluster state are used to reconstruct the underly- ing stabilizers and correct errors [9, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The measured qubits are removed from the entangled state, allowing considerable flexibility in how the measurements are real- ized in hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' For example, it is possible for the mea- surement to be destructive and erase the measured qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The most well known cluster state is the Raussendorf- Harrington-Goyal (RHG) cluster state [9, 13, 14] which is a foliation of the standard surface code [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Recently, the XZZX cluster state was introduced [16], which is a foliation of the XZZX surface code [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In the most common MBEC framework, the cluster state is generated using a set of commuting two-qubit entangling gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Alternatively, one could start with a collection of few-body entangled states and then stitch them together into a many-body entangled cluster state using measurements of two-qubit Pauli operators X ⊗ X and Z ⊗ Z, also called fusions or Bell measurements, which may be implemented destructively [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This ap- proach has been referred to as fusion-based error cor- rection (FBEC) [3] and is a more natural choice for ar- chitectures where high-fidelity fusions are native to the hardware like discrete variable photonic qubits [19], con- tinuous variable qubits [20, 21], and Majorana-based qubits [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The FBEC framework has been studied for error correction with the RHG cluster state [3] and, more recently, with the foliated floquet color code [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In this paper, we introduce fusion-based architecture for error correction with the XZZX cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' We present two constructions, one based on using a collection of 4-body entangled resource states and the other based on using a set of 6-body entangled resource states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Im- portantly, both the constructions offer practical advan- tage when noise in the fusion circuit is biased so that the Z ⊗Z measurements are much more reliable than X ⊗X measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This is because errors introduced in the cluster states due to faulty X⊗X measurements, referred to as biased fusion failures, give rise to a two-dimensional system symmetry [24] which considerably simplifies the decoding problem, leading to a substantial improvement in the threshold to biased fusion noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Our construction is motivated by dual-rail qubits in linear optics [25–28], which is the most widely studied platform in the framework of FBEC [3, 19, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Linear- optic fusions on dual rail qubits are inherently proba- bilistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The simplest fusion circuit fails with proba- bility 1/2 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The failure probability can be reduced to 1/2n using an ancillary (2n − 2)-photon entangled state [30], although for the special case of 1/4 failure probability, 4 unentangled photons are also sufficient [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Notably, when a fusion attempt fails, the X ⊗ X infor- mation is completely erased but Z ⊗ Z can still be re- covered [3, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The architecture proposed here leverages this biased noise structure to achieve record-high thresh- olds to fusion failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' With numerical simulations of the fusion-based XZZX cluster state with photonic dual- rail qubits and entangled ancillae, we find a threshold to fusion failures exceeding 25% in the experimentally relevant regime of non-zero loss rate per photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This is the highest known threshold to fusion failures in lin- ear optics without additional encodings on the photonic state [3, 23, 32, 33], and for the first time allows scalable FBEC using an ancilla of only two entangled photons or four unentangled photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The XZZX cluster state– We start with a review of the XZZX cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' It is a specific instance of a generalized cluster state [16], a stabilizer state defined on a decorated graph G = (V, E) with two types of vertices arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content='00019v1 [quant-ph] 30 Dec 2022 2 V = X ⊔Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Each vertex represents a qubit of the cluster state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' we refer to v ∈ X as X-type qubits and denote them by �, and we refer to v ∈ Z as Z-type qubits and denote them by ⃝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The N-qubit generalized cluster state is the +1 eigenstate of N mutually commuting stabilizers, one centered at each qubit v ∈ V , given by � � � � � � � � � � � � � Xv � (v,w)∈E w∈Z Xw � (v,u)∈E u∈X Zu, v ∈ X Zv � (v,w)∈E w∈Z Xw � (v,u)∈E u∈X Zu, v ∈ Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (1) Essentially, for each v ∈ X we have a stabilizer given by the product of Xv on that qubit, and some combination of X and Z operators on the neighboring qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Similarly, for each v ∈ Z we have a stabilizer given by the product of Zv on that qubit, and some combination of X and Z operators on the neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' These neighboring X and Z operators are chosen to ensure all stabilizers commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The XZZX cluster state is defined on a periodic 3D graph, a unit cell of which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 1(a), along with stabilizers centered at an X-type qubit and Z-type qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Note that there are no edges between Z-type qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The XZZX cluster state has the same geome- try as the RHG cluster state [9, 14], and can be obtained from it by local Clifford operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Taking the product of the stabilizers centered on the faces of a unit cell gives the cell stabilizer shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 1(b), which is used to correct errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Performing a measurement-based compu- tation using the XZZX cluster state involves measuring all Z-type qubits in the Z basis and all X-type qubits in the X basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Once we have measured all qubits in their respective bases, we use the cell stabilizers to check for errors in our measurement outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' A Z error on an X-type qubit or an X error on a Z-type qubit causes the qubit’s two neighboring cell stabilizers to flip to (−1), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Importantly, we note that Z errors on X-type qubits only cause pairs of defects that are re- stricted to 2D planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This 2D system symmetry simpli- fies the decoding problem—for example, a matching de- coder only needs to match defects in 2D—and ultimately leads to higher thresholds for Z-biased noise [16, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The XZZX cluster state may be prepared using controlled-not and controlled-phase gates, and the ro- bustness of this cluster state to gate noise has been stud- ied previously [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In this work we consider the alterna- tive approach of preparing this state by fusing copies of few-body entangled resource states, which is the standard approach for realizing cluster states in photonic dual-rail platforms [3, 19, 25, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' We will consider two schemes for the XZZX cluster state, which expand on schemes introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' [3] for the RHG cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The important distinction is that in both of our schemes bi- ased fusion failures create pairs of defects restricted to 2D planes, leading to improved thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (a) A unit cell of the XZZX cluster state, with two examples of stabilizers centered on the faces of the cell as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (b) If we multiply the stabilizers centered at all faces of a unit cell, we get the cell stabilizer shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (c) The value of the cell stabilizer allows us to detect errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' X errors on Z-type qubits or Z errors on X-type qubits cause the value of the neighboring cell stabilizers to flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Importantly, Z errors cause pairs of defects restricted to 2D planes, which allows for more effective decoding of Z errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Construction from 4-star resource states— In the fol- lowing discussion we introduce the principle underlying our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Consider a cluster state defined on a graph G = (V, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Let G have a Z-type qubit at a degree- 1 vertex vi ∈ Z with an edge to a qubit at vi′, and an X-type qubit at degree-1 vertex vj ∈ X with an edge to a qubit at vj′ ̸= vi′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' We will refer to the qubits on degree-1 vertices as dangling qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 2(a), per- forming Xi ⊗ Xj and Zi ⊗ Zj measurements on the pair of dangling qubits disentangles them from the rest of the system while entangling their neighbors, removing ver- tices vi, vj and edges (vi, vi′), (vj, vj′) and adding a new edge (vi′, vj′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Consequently, the stabilizers centered at vi and vj are removed and we obtain two new stabiliz- ers centered at vi′ and vj′ defined according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' To ensure that the new cluster state is the +1 eigenstate of these new stabilizers, a Pauli correction is applied to the qubits at vi′ and vj′ according to the outcomes of the Xi ⊗ Xj measurement (mXX = 0 or 1) and Zi ⊗ Zj measurement (mZZ = 0 or 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' If vi′ ∈ X and vj′ ∈ Z, the correction is ZmXX i′ ⊗ XmZZ j′ , while if vi′, vj′ ∈ X, the correction is ZmXX i′ ⊗ ZmZZ j′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' It is not necessary to physically apply these Pauli corrections and instead they may just be tracked in software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Observe that in case 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The 4-star construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (a) Performing a fusion measurement of Xi ⊗Xj and Zi ⊗Zj on a dangling pair of X- and Z-type qubits removes them from the cluster but forms an edge between their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (b) The two five-qubit cluster states we use in our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (c) The arrangement of five- qubit cluster states we use to build the XZZX cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (d) Because the qubits of the resulting cluster state will be measured in the X/Z basis, and because these measurements commute with the fusion measurements, we can measure the center qubits before performing the fusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Equivalently, we can instead start with the simpler four-qubit resource states that result from measuring the center qubit as (+1) in the X/Z basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In this case, the central qubits making up the cluster state are never physically realized, but exist as virtual qubits whose measurement outcomes are tracked in software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' of unreliable Xi ⊗ Xj (Zi ⊗ Zj) measurements, we can- not correctly determine the proper Pauli correction on vi′ (vj′) which results in an effective error on that qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In fact, a complete erasure of Xi ⊗ Xj measurement out- come that arises due to fusion failure in linear-optics is equivalent to applying I or Z to the X-type qubit at vi′ with 50% probability [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This type of Z-biased error at a known location in the cluster state (vi′), marked by the location of the failed (Xi⊗Xj) measurement, is classified as Z-biased fusion failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' With the above discussion in mind we introduce the two five-qubit cluster states shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 2(b) with sta- bilizers defined according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' One has a Z-type qubit at the center and the other has a X-type qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The center qubits are marked in red to indicate that these will eventually form the desired XZZX cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The Z-centered and X-centered states are placed at the loca- tion of Z- and X-type qubits respectively in the desired cluster state, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The arrangement of the 5-body states ensures that neighboring dangling qubits are always opposite types;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' we can thus fuse the neighboring dangling qubits according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The fused qubits are removed from the cluster and new bonds appear between the red qubits, resulting in the desired XZZX cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Finally, the cluster state qubits can be measured in the appropriate basis described in the previous section for error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Note that each cen- ter qubit is entangled into the final cluster state after four fusion measurements on its neighboring dangling qubits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' consequently, four Pauli corrections need to be accounted for on this qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This accounting may be done in soft- ware by simply re-interpreting the outcome of the final measurement of cluster state qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This is because a X (Z) measurement of the X- (Z-) type qubit in the cluster state after a Pauli Z (X) is applied to it is equivalent to a X (Z) measurement followed by a classical flip of the measurement outcome 0 → 1, 1 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' It is possible to simplify the initial 5-qubit cluster states to 4-qubit resource states by noting that measuring the qubits comprising the final cluster state commutes with fusion measurements, as these measurements are performed on different qubits and Pauli corrections due to fusions can simply be accounted for in software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' That is, we can equally well measure the center qubits which would form the XZZX cluster state before performing the fusion measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Once the fusions are realized, the outcomes of the prior center-qubit measurements can be flipped conditional on the fusion outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Measuring the center X- and Z-type qubits of the 5-qubit states in the X and Z basis respectively leaves us with the simpler 4-star resource states shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Thus we can directly start with these resource states assuming that the center qubits have been measured in the X/Z basis with measurement outcome 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In this approach, the cen- tral qubit acts like a virtual qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' It is never physically realized and never physically measured, and its effective measurement outcome is entirely determined by the Pauli corrections that are tracked in software [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Construction from 6-ring resource states— While the previous construction relied on fusing an X-type qubit with a Z-type qubit, this second construction relies on fusing two qubits of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' While our construc- tion is reminiscent of the approach in [3], we provide an alternate intuitive interpretation of why it works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Consider a cluster state defined on a graph G = (V, E) with Z (X)-type qubits at vertices vi, vj ∈ V such that vi and vj are not neighbors and share no neighbors in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Measuring Xi ⊗ Xj (Zi ⊗ Zj) on these two qubits projects them into an effective two-dimensional subspace with the Pauli operators ¯X = Xi (Xi ⊗ Xj) and ¯Z = Zi ⊗ Zj (Zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 3(a), a new cluster state is obtained with vertices vi, vj replaced by a single vertex vij and an effective Z (X)-type qubit placed at this vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' All the edges incident at vi and vj are in- cident at vij in the new graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' To ensure that the new cluster state is the +1 eigenstate of all the stabilizers, a 4 Pauli correction determined by the Xi ⊗ Xj (Zi ⊗ Zj) measurement outcome, mXX = 0 or 1 (mZZ = 0 or 1), must be applied to the qubits that were originally ad- jacent to vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Specifically ZmXX (ZmZZ) is applied to adjacent X-type qubits and XmXX (XmZZ) is applied to adjacent Z-type qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' With the above merging principle in mind, we intro- duce the 6-ring resource cluster state in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' A copy of this state is placed at two opposite corners of each unit cell of the XZZX cluster state as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Two qubits of the same type share a face centre or edge centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' If we measure X ⊗ X (Z ⊗ Z) for each pair of Z (X)- type qubits sharing a face/edge, and apply the required Pauli corrections, we obtain the desired XZZX cluster state comprised of the effective qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Note that an un- reliable X ⊗X measurement on Z-type qubits only leads to an incorrect Pauli Z correction to adjacent X-type qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Finally, we need to measure the effective Pauli ¯X = X ⊗ X of the effective X-type qubits and effective Pauli ¯Z = Z ⊗ Z of the effective Z-type qubits for error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Note that an unreliable X ⊗ X measurement in the second set of measurements is like a ¯Z error on the effective X-type qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' As before, Pauli corrections from the first set of measurements that create the cluster state may be simply tracked in software by re-interpreting the outcomes of the second set of measurements that are used for error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Overall then, error correction is im- plemented in our construction by measuring commuting observables X ⊗X and Z ⊗Z, that is performing fusions, on pairs of qubits that share an edge or a face centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The above discussion implies that biased fusion failure, as is the case in linear optics, effectively leads to biased Pauli Z noise on the X-type qubits of the XZZX cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Application to dual-rail qubits with linear-optic fusions—A photonic dual-rail qubit is encoded in the presence of a single photon in one of two orthogonal modes, |¯0⟩ = |01⟩ , |¯1⟩ = |10⟩ and is a leading candidate for linear-optical quantum computing [19, 25, 28, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In this platform, all single qubit gates, like the Hadamard gate, can be performed deterministically using passive linear optical elements [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Multi-qubit operations, like the fusion measurements, are non-deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Fault- tolerant FBEC in this platform is based on heralded gen- eration of few-body entangled resource states, followed by non-deterministic fusion measurements [3, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Re- markably, the resource states used in our proposal differ from the ones considered in previous works [3] only by Hadamard transformations and these states can be eas- ily generated using well-known linear-optics circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' For clarity, in [35] we also give circuits for generation of our four-star and six-ring states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The required fusion mea- surements can be realized using a so-called type-II fusion circuit comprised of beamsplitters and photon number re- solving detectors [19, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The circuit consumes the two dual-rail qubits which need to be fused and outputs the photon number/clicks observed at each detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Con- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Building the XZZX cluster state using 6-ring resource states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (a) Measuring Xi ⊗ Xj (Zi ⊗ Zj) on two Z (X)-type qubits projects those qubits onto a two-dimensional subspace with effective Pauli operators ¯ Xij = Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (Xi ⊗Xj) and ¯Zij = Zi ⊗ Zj (Zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In terms of the effective Pauli operators, the two Z (X)-type qubits are replaced by a single Z (X)-type qubit vij, with all edges to neighbors intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (b) The 6-ring resource state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (c) The fusion pattern used to join the 6-ring states to make the XZZX cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ditional on the observed clicks, one of two operations is implemented by the circuit on the input qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Either a successful X⊗X and Z⊗Z measurement is implemented, in which case we say the fusion has succeeded and the measurement outcomes are inferred from the detector clicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Or, each input qubit is independently measured in the Z-basis, in which case we say that the fusion has failed and the independent Z measurement outcomes are again inferred from detectors clicks [3, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Consequently, in a failed-fusion event, Z ⊗Z can be recovered by multi- plying the independent Z measurement outcomes, while the X ⊗ X measurement is completely erased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Without any ancilla photons, the probability of fusion failure is pfail = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' With additional (2n − 2)-photon entangled ancilla, for a total of 2n photons in the fusion circuit, the failure probability may be reduced to pfail = 1/2n [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' For the particular case of n = 2, the failure probability may also be reduced to 1/4 with a 4-photon unentangled ancilla [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' However, n > 2 requires entangled states that get progressively more complicated to realize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' So far we have ignored hardware imperfections that can introduce photon loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Fortunately, the ideal fusion circuit preserves the number of photons, that is, the to- tal detector clicks must be equal to the number of input photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' A loss of any photon in the fusion circuit will be heralded by observing fewer than expected clicks at the detector and result in an erasure of both X ⊗ X and Z ⊗ Z measurement outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' If the probability of loss per photon is ploss, then the probability of such an erasure in the boosted fusion circuit with a total of 1/pfail pho- tons is pfull erase = 1 − (1 − ploss)1/pfail [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This equation highlights the tradeoff between the rate at which only the 5 X ⊗ X outcome is erased due to fusion-failure and the rate at which both X ⊗X and Z ⊗Z outcomes are erased due to photon loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' As we decrease pfail by adding more photons to the fusion circuit, we increase the probability of photon loss from the fusion circuit pfull erase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' We evaluate the performance of our fusion architec- tures under the linear optical error model including fusion failures and photon loss as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' We perform Monte-Carlo simulations of errors when the four-star and six-ring states are used for FBEC with d × d × d XZZX cluster states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The syndrome data generated by the er- rors is arranged on a graph and decoded using the linear- time erasure-decoder [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' We evaluate the logical error rates for different cluster state sizes d, and error param- eters pfail, and ploss and estimate the threshold which is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The solid red and blue curves are the thresholds for the 4-star and 6-ring constructions re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' If ploss and pfail lie under the curves, then these errors are correctable, otherwise they are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' We also compare our results with the thresholds obtained with (a) the 4-star and 6-ring construction from a previ- ous work [3] to construct the XZZX cluster state and (b) our 4-star and 6-ring constructions based on the adaptive error-correction strategy introduced in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The thresh- olds with both (a) and (b), shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 4 using dashed lines, are identical to each other as they have identical error syndrome graphs and are consistent with known results [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' We observe, first, that the thresholds for the six-ring construction are higher than the four-star construction as it has fewer fusions, and hence a lower probability of error, per cluster state qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In the absence of pho- ton loss, the numerically obtained threshold for biased fusion-failure using our scheme is 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content='7% for the six-ring construction and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content='6% for the four-star construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Without photon loss, these thresholds can also be derived analytically (see Supplement [35]) and are significantly higher than the threshold for fusion-failure obtained with previous proposals, which are correspondingly ∼ 24% for the six-ring construction [3] and ∼ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content='5% for the four- star construction [33], that fail to leverage the noise bias in fusion failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This is because, as argued earlier, in our approach fusion failure leads to two-dimensional sys- tem symmetry and hence a two-dimensional syndrome graph which is easier to decode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In contrast, in previ- ous strategies the syndrome graph resulting from fusion failures is three-dimensional, which is harder to decode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' When ploss ̸= 0, we must deal with both full fusion erasure along with fusion failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Increase in pfull erase means we must decrease pfail by using additional pho- tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' However, this also increases pfull erase due to the possibility of these additional photons being lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This relationship between pfull erase, pfail, and ploss gives the overall “inverted-u” shape of the threshold curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Im- portantly, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 4 we see that our scheme with the six-ring resource states can tolerate up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content='37% of pho- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Threshold with four-star (red) and six-ring (blue) construction protocols as proposed here under the linear op- tical error model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' For comparison, thresholds based on previ- ous approaches which do not leverage the noise bias in fusion failures is also shown [3, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Excluding XZZX FBEC points on the x-axis, all points are simulated thresholds for sets of cluster states of size d × d × d, with d up to 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' XZZX FBEC points on the x-axis are thresholds for sets of d×d×4 cluster states, with d ≤ 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' From left to right, the black dotted ver- tical lines represent fusion success boosting by 30, 14, 6, and 2 additional entangled ancillae photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ton loss with 25% fusion failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' That is, it is possi- ble to achieve fault tolerance with only 2 additional en- tangled ancilla photons for boosted fusions [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' More importantly, this means that we could even reach fault tolerance with boosted fusions using only 4 additional unentangled ancilla photons [31] and ploss ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content='25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' On the other hand, the maximum fusion failure that previ- ous schemes can tolerate is < 25%, meaning they could not achieve fault-tolerance with only 2 additional entan- gled ancilla photons or 4 additional unentangled ancilla photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Conclusion By taking advantage of the biased struc- ture of fusion failures, we have introduced new resource states and fusion strategies for FBEC that allow for more efficient error correction of biased fusion failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This FBEC strategy is particularly relevant to linear-optical quantum computers based on dual-rail photonic qubits, where biased fusion failures are the dominant source of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Our resource states and fusion strategies require no additional overhead to realize compared to the previ- ous approach of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' [3], but result in higher thresholds to fusion failures for both the 4-star and 6-ring resource states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In the 6-ring construction in particular, our strat- egy has a threshold over 25% to fusion failures, which can be reached using only a 2-photon entangled ancilla or a 4-photon unentangled ancilla;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' thus, our construction overcomes a key barrier for photonic quantum comput- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Our construction achieves higher thresholds because: (1) the fusion operations in linear optics have a natural bias towards Z errors, (2) we construct our cluster state in a bias-preserving way so that the final cluster state also 6 has a bias towards Z errors, and (3) XZZX cluster state is designed to correct Z errors 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Integrated quantum photon- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' IEEE Journal of Selected Topics in Quantum Elec- tronics, 15(6):1673–1684, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' [38] Nicolas Delfosse and Gilles Z´emor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Linear-time maximum likelihood decoding of surface codes over the quantum era- sure channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Research, 2:033042, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Supplemental Material: Tailoring fusion-based error correction for high thresholds to biased fusion failures Kaavya Sahay, Jahan Claes, and Shruti Puri Department of Applied Physics, Yale University, New Haven, Connecticut 06511, USA Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA (Dated: January 3, 2023) 4-STAR CONSTRUCTION Here we show the evolution of stabilizers when a fusion operation is conducted between two dangling qubits in the XZZX cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Continuing with the notation in the main text, we use a cluster state defined on a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Let G have a dangling qubit at vertex vi ∈ Z and vj ∈ X, each with an edge to a qubit at vertex vi′ and vj′ ̸= vi′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Qubit vi′ (vj′) has further edges to a set of qubits {vi′′ } ({vj′′ }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S1, we show the evolution of stabilizers and associated Pauli frame updates when vi′ ∈ X and vj′ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S2 we show the same for vi′, vj′ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The set of peripheral qubits {vi′′ } and {vj′′ } are not shown for simplicity, and indeed we shall see that they are not affected by the fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Note that after the fusion, the qubits at vertices vi, vj are disentangled from the cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (a) Pre- fusion stabilizers = ′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ′ = ′ ∏ ′ ′ ′′′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' { ′′ } = { ′′ ′ },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' = ′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ′ = ′ ∏ ′ ′ ′′′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' { ′′ } = { ′′ ′ } (b) Fusion success (i) Checks measured [outcome] = ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' = (ii) Fused system now eigenstate of [eigenvalue] ′ ⋅ ⋅ = ′ ′ ∏ ′ ′ ′′′ (-1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ′′ = ′′ ′ +1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ⋅ ⋅ ′ = ′ ′ ∏ ′ ′ ′′′ (-1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ′′ = ′′ ′ +1 (iii) Pauli frame update to impose +1 eigenvalue ′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ′ (c) Fusion failure (i) Checks measured [outcome] = ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' = (ii) System now in eigenstate of [eigenvalue] ⋅ ⋅ ⋅ ′ = ′ ′ ∏ ′ ′ ′′′ (-1) + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ′′ = ′′ ′ +1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ′′ = ′′ ′ +1 (iii) Pauli frame update to impose +1 eigenvalue ′ + (iv) Projector on ′ (1 + (-1) ′ /2 1 + −1 2 { ′′ } { ′′ } (a) (b) (c) + ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Tracking the stabilizers and deriving the required Pauli frame updates for a fusion conducted on qubits at vertices vi, vj when vi′ ∈ X and vj′ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' We look at the state of the system (a) pre-fusion, (b) post-successful fusion, and (c) on fusion failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Note that Pi′′ (Pj′′ ) denotes a Pauli operator on vi′′ (vj′′ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Pre-fusion, the stabilizer generators {Sr} are defined according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (1) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' If the fusion succeeds (fails), we measure checks Xi ⊗ Xj, Zi ⊗ Zj (Zi, Zj) with measurement outcomes mXX, mZZ (mi, mj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Post-fusion, the system is constrained by the surviving stabilizer set generated by (i) the original stabilizers that commute with the measurements and (ii) the commuting products of those stabilizers that do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The system will now be in the eigenstate of this new generating set, given in row b(ii) of the table in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S1,S2, with eigenvalues defined by the measurement outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Thus, in order to restore the system to the +1 eigenstate of a given stabilizer generator Sp, we apply a Pauli frame update that is identified by locating the correction that anticommutes with Sp but no other stabilizer generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' These corrections are shown in row b(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In practice, this Pauli correction is tracked in software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In the case when a fusion failure occurs, the stabilizer of the combined system centered at vj′ is recovered by adding the fusion measurement outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' when vj′ ∈ Z (vj′ ∈ X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' the conditional Pauli frame update applied 9 (a) Pre- fusion stabilizers = ′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ′ = ′ ∏ ′ ′ ′′′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' { ′′ } = { ′′ ′ },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' = ′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ′ = ′ ∏ ′ ′ ′′′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' { ′′ } = { ′′ ′ } (b) Fusion success (i) Checks measured [outcome] = ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' = (ii) Fused system now eigenstate of [eigenvalue] ′ ⋅ ⋅ = ′ ′ ∏ ′ ′ ′′′ (−1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ′′ = ′′ ′ ′ +1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ⋅ ⋅ ′ = ′ ′ ∏ ′ ′ ′′′ (−1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ′′ = ′′ ′ ′ [+1] (iii) Pauli frame update to impose +1 eigenvalue ′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ′ (c) Fusion failure (i) Checks measured [outcome] = ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' = (ii) System now in eigenstate of [eigenvalue] ⋅ ⋅ ⋅ ′ = ′ ′ ∏ ′ ′ ′′′ (−1) + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ′′ = ′′ ′ [+1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ′′ = ′′ ′ [+1] (iii) Pauli frame update to impose +1 eigenvalue ′ + (iv) Projector on ′ (1 + (-1) ′ /2 1 + −1 ′ 2 + { ′′ } { ′′ } (a) (b) (c) ) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Tracking the stabilizers and deriving the required Pauli frame updates for a fusion conducted on vi, vj when vi′, vj′ ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' We look at the state of the system (a) pre-fusion, (b) post-successful fusion, and (c) on fusion failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' to it is an X (Z) operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' However, a projector (1 + (−1)miZ)/2 is applied to vertex vi′, which destroys its X measurement outcome information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This is equivalent to a successful fusion followed by a Z measurement on vi′, a process described by the error channel ρ �→(ρ + Zi′ρZi′)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Additionally, we note that the sets of stabilizers {Si′′ }, {Sj′′ } centered at the neighbouring qubits remain undis- turbed in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' We shall use this fact to simplify our analysis in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 6-RING CONSTRUCTION (a) Pre-fusion stabilizers = 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 1 = 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 2 = 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' = 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 1 = 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 2 = 2 (b) Fusion success (i) Checks measured to create virtual qubit [outcome] = [ ] (ii) Measured system now eigenstate of [eigenvalue] Note = ⋅ = 1 2 1 2 [+1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 1 = 1 +1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 2 = 2 +1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Sj1 ⋅ 2 = 1 2 +1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 1 ⋅ = 1 [ −1 ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 2 ⋅ = 2 [ −1 ] (iii) Pauli frame update to impose +1 eigenvalue 1 2 (c) Fusion failure (i) Checks measured [outcome] = ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' = (ii) Measured system now in eigenstate of [eigenvalue] ⋅ = 1 2 −1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ⋅ = 1 2 −1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 1 ⋅ 2 = 1 2 +1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 1 ⋅ 2 = 1 2 +1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (iii) Effec�ve correlated projector on (1 +(−1) + 1 2)/2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Tracking the stabilizers and deriving the required Pauli frame updates for the six-ring construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Measurements are conducted on vi, vj with vi′, vj′ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' We look at the system (a) pre-fusion, (b) post-successful fusion, and (c) on fusion failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In this section we show how measuring X ⊗X (Z ⊗Z) for Z-type (X-type) qubits in six-ring cluster states produces virtual qubits with Pauli corrections on neighboring qubits and derive the noise introduced in the cluster state due to 10 erasure of X ⊗ X measurement outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Let the cluster state graph G have a degree-2 vertex vi (vj ) with an edge to two qubits at vertices {vik} ({vjk}) with {vjk}∩{vjk} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' An effective cluster state qubit vij is formed when such a measurement is performed between vi and vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S3, we show the evolution of stabilizers and associated Pauli frame updates when vi, vj ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Note that we have disregarded further qubits connected to {vik}, and {vjk} since their stabilizers remain unchanged during this process, just like in the 4-star construction discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Similar to the previous section, we update the stabilizers according to the checks measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Clearly we see that measuring Xi⊗Xj creates a virtual qubit with effective Pauli ¯Z = Zi⊗Zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Note that on fusion failure, due to the form of the system stabilizers in this setup, two X-type qubits initially connected to vj experience a correlated projection into an eigenstate of Zj1Zj2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Individual errors on either of the two qubits vj1, vj2 connect unit cell syndromes in mutually perpendicular directions in the cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' As a result, this correlated error effectively connects a unit cell syndrome with its diagonally displaced neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 4-STAR RESOURCE STATE GENERATION Four-star resources states can probabilistically be generated from single photons by a sequence of beam splitters and photon detectors, conditioned on a certain detector output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' A potential generation circuit is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S4, based on prior constructions for three-body GHZ states [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This circuit succeeds if a single photon is output at each detector pair, where a detector pair is defined at the two output ports of a single beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ������� ������� ������� ������� ��� +++ + + ��� � �2 000 + + 111 � �2 ��� ��� �������������������� ������������������� ��������������� � � � � � � � � FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Construction of the four-star resource state for the XZZX cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (a) Representations of the two resource states (b) Probabilistic photonic fusion circuit to construct the state (| + + + +⟩ + | − − − −⟩) / √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (c) Beam splitters are applied to three qubits in the state output from (b) to obtain the second resource state (|000+⟩ + |111−⟩) / √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 6-RING RESOURCE STATE GENERATION As described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S5(a), a six-ring resource state can be constructed by performing Type-I fusions [3] on a set of GHZ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' We present a potential photonic circuit to probabilistically construct six-rings from single photons in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S5(b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The base GHZ state (| + ++⟩ + | − −−⟩) / √ 2 (upto heralded Pauli corrections) is generated by the circuit shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S5(b) when one photon is detected at each pair of detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This occurs with probability 1/32, which can be effectively increased to near-unity via multiplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Two beam splitters performing Hadamard operations to specific qubits can be applied to this GHZ state in order to get the complete input set of GHZ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' These input states are fed into the circuit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S5(c), which performs Type-I fusions between the peripheral qubits of the individual GHZ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Note that this secondary circuit layer has a success probability (1/2)3 which can be increased by further multiplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Derivation of a more efficient circuit is left to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 11 ������� ������� ������� ��� ��� ���� ������ ������� ��� ������� � � � � � � ������� ������ ������� ���� ������ ������� ������� ������ ������� ������� ������ ������� ������� ������� ������� ������� ������� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Construction of the 6-ring resource state for the XZZX cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (a) (top) Type-I fusions are conducted between input GHZ states to form the resource state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (bottom) Representations of the input GHZ states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (b) Probabilistic photonic circuit to construct the state (| + ++⟩ + | − −−⟩) / √ 2 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Hadamards can be applied to specific qubits to obtain the full input state set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' (c) Photonic circuit to conduct Type-I fusions between output GHZ states to construct a six-ring state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' BIASED FUSION FAILURES IN BOOSTED TYPE-II FUSION The aim of the fusion measurements is to measure Xi ⊗ Xj and Zi ⊗ Zj between two qubits labelled by i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' That is, in terms of the dual rail encoding |¯0⟩ = |10⟩, |¯1⟩ = |01⟩, the aim is to discriminate between the four states |ψ±⟩ = (|¯0i¯1j⟩ ± |¯1i¯0j⟩)/ √ 2 and |φ±⟩ = (|¯0i¯0j⟩ ± |¯1i¯1j⟩)/ √ 2 which are the simultaneous eigenstates of XiXj and ZiZj: ZiZj |ψ±⟩ = − |ψ±⟩ , XiXj |ψ±⟩ = ± |ψ±⟩ , ZiZj |φ±⟩ = |φ±⟩ , XiXj |φ±⟩ = ± |φ±⟩ (S1) In a direct type-II fusion [3, 4], the four modes of the two dual-rail qubits are interfered using two 50 : 50 beam-splitters ������� ������� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Photonic circuit for type-II fusion between two dual rail qubits using beam splitters and photon detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The transformed states at the ouput of the network are: |ψ+⟩ → i √ 2(|1100⟩ + |0011⟩) |ψ−⟩ → i √ 2(|1001⟩ − |0110⟩) |φ+⟩ → i 2(|2000⟩ + |0200⟩ + |0020⟩ + |0002⟩) |φ−⟩ → i 2(|2000⟩ − |0200⟩ + |0020⟩ − |0002⟩) (S2) As can be seen, the photon number distribution n across the four output modes, which can be measured using PNRDs, carries information about the input states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The output photon number distribution generated if |ψ+⟩ is input into 12 the interference network, N|ψ+⟩ = {1100} or {0011}, discriminates |ψ+⟩ from other states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Similarly, the output photon number distribution generated if |ψ−⟩ is input into the interference network, N|ψ−⟩ = {1001} or {0110}, discriminates |ψ−⟩ from other states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Unfortunately, however, it is not possible to discriminate between |φ+⟩ and |φ−⟩ using PNRDs as they produce the same output photon number distribution: N = {2000} or N = {0200} or N = {0020} or N = {0002}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In this case we say that the fusion fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Given an equal distribution of Bell states at the input (which is the case in 4-star and 6-ring fusions), the probability of failure is 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' However, note that measuring N = {2000} or N = {0020} discriminates (|φ+⟩ + |φ−⟩)/ √ 2 = |0i0j⟩ and the output photon number distribution N = {0200} or N = {0002} discriminates (|φ+⟩ − |φ−⟩)/ √ 2 = |1i1j⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Thus, even when fusion fails, Zi and Zj of each dual-rail qubit is still measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' [5, 6] introduced protocols for fusions with success probability boosted by lifting the degeneracy in the output photon-number distribution when |φ±⟩ are input into a interference network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Here, we give a broad overview of the two protocols without repeating the details in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The broad overview will be sufficient to see that even when boosted fusions fail, Zi and Zj of each dual-rail qubit is still measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The protocols in [5, 6] use auxillary entangled-photons, |Γ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The exact form of this entangled state is not important for our purpose here, but |Γ⟩ required for the protocol in [5] is different from that in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' |Γ⟩, together with the state of the two duil-rail qubits, transform through the interference network as follows, |ψ+⟩ |Γ⟩ → | |N ′ ψ+⟩ |ψ−⟩ |Γ⟩ → | |N ′ ψ−⟩ |φ+⟩ |Γ⟩ → x| |N ′ φ+⟩ + y(|A⟩ + |B⟩) |φ−⟩ |Γ⟩ → x| |N ′ φ−⟩ + y(|A⟩ − |B⟩) (S3) Here, x, y are c-numbers for normalization and | |N ′ ψ±⟩, | |N ′ φ±⟩, |A⟩, |B⟩ are states with distinct photon-number distri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' | |N ′ ψ±⟩, | |N ′ φ±⟩ may be a superposition of many Fock states (indicated by | | ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Clearly, the protocol uniquely identifies |ψ±, φ±⟩ if PNRDs measure photon number distributions consistent with | |N ′ ψ±⟩ , | |N ′ φ±⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' However, the protocol fails if photon number distribution corresponding to the states |A⟩ or |B⟩ is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Measuring photon number distribution consistent with |A⟩ or |B⟩ discriminates (|φ+⟩ + |φ−⟩)/ √ 2 or (|φ+⟩ − |φ−⟩)/ √ 2 respectively and, like the direct type-II measurement, this is equivalent to measuring Zi, Zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This feature of fusion measurements has also been identified in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The key difference in the protocols of [5] and [6] is the nature of |Γ⟩ and the success probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In [5] a success probability of (1 − 2−n) is obtained with (2n − 2)-photon entangled state, while in [6], twice as many photons are required to reach the same success probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Importantly, with the protocol of [6], |Γ⟩ for 75% successful fusion is an unentangled state of 4 photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' ANALYTIC THRESHOLD IN CASE OF ONLY BIASED FUSION FAILURES Consider the four-star construction from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 2 in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Suppose the probability of erasing only X ⊗ X measurement outcome or the probability of a biased fusion failure is p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' An incorrect X ⊗X measurement implies that an incorrect Pauli Z correction may be applied to a X type qubit that eventually makes the cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Each such X type qubit gets a Pauli Z correction from the fusions of three neighboring dangling qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Thus, effectively the probability of a Z error on each X type qubit forming the cluster state is 3p(1 − p)2 + 3p2(1 − p) + p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' The syndrome graph in 2D is the 44 tiling of the plane with a bond percolation threshold of 50% [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Thus, the threshold pth may be found by requiring that 3pth(1 − pth)2 + 3p2 th(1 − pth) + p3 th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content='5 or pth = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content='63%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' In order to justify the threshold for biased fusion failures with the six-ring construction, we examine the syndrome graph formed by fusion failures between resource states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' A fusion failure between two X-type qubits creates a an erasure on this qubit, leading to (−1) stabilizer outcomes on adjoining unit cells within a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' When a fusion between two Z-type qubits fails, the two adjacent X-type qubits in the resource state are projected into an eigenstate of Z ⊗ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This leads to a pair of (−1) stabilizer outcomes on unit cells diagonally displaced from one another in the same plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Note that all such diagonals have the same orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' As a result, the syndrome graph in 2D results in a 36 tiling of the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' This has a percolation threshold of 2 sin(π/18) ≈ 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content='73% [9], which agrees with numerical simulations of this construction at large lattice sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 13 [1] Michael Varnava, Daniel E Browne, and Terry Rudolph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' How good must single photon sources and detectors be for efficient linear optical quantum computation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Physical Review Letters, 100(6):060502, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' [2] Ying Li, Peter C Humphreys, Gabriel J Mendoza, and Simon C Benjamin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Resource costs for fault-tolerant linear optical quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Physical Review X, 5(4):041007, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' [3] Daniel E Browne and Terry Rudolph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Resource-efficient linear optical quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Physical Review Letters, 95(1):010501, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' [4] Alexei Gilchrist, AJF Hayes, and TC Ralph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Efficient parity-encoded optical quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Physical Review A, 75(5):052328, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' [5] Warren P Grice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Arbitrarily complete bell-state measurement using only linear optical elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Physical Review A, 84(4):042331, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' [6] Fabian Ewert and Peter van Loock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' 3/4-efficient bell measurement with passive linear optics and unentangled ancillae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Physical review letters, 113(14):140403, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' [7] Sara Bartolucci, Patrick Birchall, Hector Bombin, Hugo Cable, Chris Dawson, Mercedes Gimeno-Segovia, Eric Johnston, Konrad Kieling, Naomi Nickerson, Mihir Pant, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' Fusion-based quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztAyT4oBgHgl3EQfPPYT/content/2301.00019v1.pdf'} +page_content=' arXiv preprint arXiv:2101.' metadata={'source': 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